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statistika socialur

mecnierebebSi

mesame gamocema

__________________

harold kisi

Tbilisi

2007

socialur mecnierebaTa seria

Authorized translation from the English language edition, entitled STATISTICAL CONCEPTS FOR THE BEHAVIORAL SCIENCES, 3rd Edition, ISBN 0205332919, by KIESS, HAROLD O., published by Pearson Education, Inc, publishing as Allyn & Bacon, Copyright © 2002, 1996, 1989 by Allyn & Bacon, A Pearson Education Company. 75 Arlington St. Boston, MA 02116, Internet : www.ablongman.com. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage retrieval system, without permission from Pearson Education, Inc. GEORGIAN language edition published by the UNION – CENTER FOR SOCIAL SCIENCES, Copyright Georgian Translation © 2007 Center for Social Sciences 10, Chovelidze St. Tbilisi 0108 Georgia, E-mail: [email protected] Internet : www.ucss.ge

harold kisis wignis, «statistika socialur mecnierebebSi», inglisuri gamocemis avtorizebuli

Targmani (mesame gamocema, ISBN 0205332919), gamomcemeli Pearson Education, Inc, Allyn & Bacon. saavtoro ufleba inglisur gamocemaze © 2002, 1996, 1989 by Allyn & Bacon, A Pearson Education Company. bostoni, arlingtonis q. 75. indeqsi MA 02116, internet gverdi : www.ablongman.com. yvela saavtoro ufleba daculia. Pearson Education, Inc.–is Tanxmobis gareSe ar SeiZleba wignis romelime nawilis aslis gadaReba, gamravleba an gavrceleba raime formiT, eleqtronuli an meqanikuri saSualebiT, fotokopirebiT an informaciuli Senaxvis raime sxva saSualebiT. wignis qarTuli Targmani gamocemulia kavSir «socialur mecnierebaTa centris» mier. qarTuli Targmani © socialur mecnierebaTa centri, 2007

q. Tbilisi, 0108, T. WoveliZis q. # 10 el. fosta: [email protected] internet gverdi: www.ucss.ge mTargmnelTa jgufi: Teona aslaniSvili maia robaqiZe lili xeCuaSvili nino wereTeli mTavari samecniero redaqtori: elizbar nadaraia samecniero redaqtorebi: nino javaxiSvili qeTevan manjgalaZe saredaqcio jgufis wevrebi: zaza xeCinaSvili TinaTin zurabiSvili enobrivi redaqtori: lia kaWarava dakabadoneba, ydis dizaini: giorgi bagrationi

ISBN: 978-99940-873-1-0 EAN : 9789994087310 ISSN 1512-4576

wignis qarTuli Targmani momzadda da gamoica “socialur

mecnierebaTa centris” (Center for Social Sciences) mier, ×ÏÍÃÉÓ OSI – Zug da budapeStis Ria sazogadoebis institutis umaRlesi ganaTlebis mxardaWeris programis (HESP) finansuri xelSewyobiT.

Georgian translation of the book has been published by the Center for

Social Sciences, sponsored by the OSI-Zug Foundation and the Higher Education Support Program (HESP) of the Open Society Institute Budapest (OSI).

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winasityvaoba saqarTveloSi socialuri mecnierebebis ganviTarebis bunebrivi gza meoce saukuneSi sabWoeTis oficialuri ideologiuri wnexis arsebobis gamo mniSvnelovnad Seferxda, ramac seriozuli ziani miayena Tanamedrove standartebis kvaldakval socialur mecnierebebSi kvlevis ganviTarebas. saqarTvelos axali kanoni umaRlesi ganaTlebis Sesaxeb aqtualurad miiCnevs umaRles saswavleblebSi swavlebisa da kvlevis reintegracias, xolo samagistro da sadoqtoro programebis dafinansebis axali modeli mimarTulia kvlevaze dafuZnebuli swavlebis danergvisaken. umaRlesi ganaTlebis axali kanonis Sesabamisad umaRlesi saswavlebelebi da, gansakuTrebiT, universitetebi cdiloben fexi auwyon reformebis process da Secvalon saswavlo programebi axali moTxovnebis Sesabamisad. cvlilebebi TvalsaCinos xdis saswavlo masalebis deficits da gansakuTrebiT ki kvlevis meTodebis mimarTulebiT. harold kisis wigni “statistika socialur mecnierebebSi” erTiandeba saswavlo literaturis im aucilebel CamonaTvalSi, romlis gareSec sruliad SeuZlebelia romelime mkvlevars hqondes seriozuli pretenzia socialur mecnierebebSi mopovebuli monacemebi warmoadginos, rogorc kvlevis sando Sedegi. ufro metic, dResdReobiT tradiciulad ara raodenobriv meTodebze orientirebuli disciplinebic ki, rogoricaa politikuri mecniereba da saerTaSoriso urTierTobebi sadoqtoro programebis kvleviTi seminarebis aucilebel komponentad miiCneven raodenobriv meTodebSi seriozul trenings (amis magaliTad, mxolod eseqsis universitetis gamocdilebac ki sakmarisia, romelic 1967 wlidan yovelwliurad marTavs zafxulis skolas socialur mecnierebaTa monacemebis Segrovebasa da analizSi). saqarTvelos ganaTlebis ministris 2005 wlis #633 brZanebis safuZvelze ganisazRvra bakalavriatis specialobebis CamonaTvali, romelSic socialuri menierebebis grifiT gaerTianebulia Semdegi specialobebebi – “ekonomika, geografia, fsiqologia, sociologia, politikuri mecnierebani, saerTaSoriso urTierTobebi, sazogadoebrivi urTierTobebi, Jurnalistika, sabiblioTeko, saarqivo da sainformacio saqme da sxva” (http://www.mes.gov.ge). TiTqmis yvela es dargi iswavleba saqarTvelos sxvadasxva umaRles saswavlebelsa da universitetSi da harold kisis qarTul enaze Targmnili wigni udavod SenaZenia socialur mecnierebebSi dakavebuli pedagogebisa da studentebisaTvis. socialur mecnierebaTa centri 2002 wlidan gamoscems socialur mecnierebaTa serias, romlis mizania socialur mecnierebebSi qarTul enaze saswavlo da samecniero literaturis danaklisis Sevseba da samecniero enisa da terminologiis ganviTareba. 2004 wels pirvelad qarTul enaze gamoica politikur da socialur mecnierebaTa terminTa

http://www.mes.gov.ge/

leqsikoni-cnobari. socialur mecnierebaTa seriiT ukve gamocemulia 16 damxmare saxelmZRvanelo da 23 gafarToebuli saswavlo saleqcio kursi. statistikis es saxelmZRvaneloc am seriis nawilia. harold kisis wignis es Targmani mravali adamianis Zalisxmevis Sedegia. TiToeuli maTganis gansakuTrebuli wvlilis gareSe Zneli iqneboda samuSaos bolomde miyvana. socialur mecnierebaTa centri gansakuTrebul madlobas moaxsenebs mTavar samecniero redaqtors baton elizbar nadaraias; samecniero redaqtorebs qalbatonebs nino javaxiSvilsa da qeTevan manjgalaZes, baton zaza xeCinaSvils; enobriv redaqtors qalbaton lia kaWaravas; mTargmnelebs qalbatonebs Teona aslaniSvils, maia robaqiZes, nino wereTelsa da lili xeCuaSvils; socialur mecnierebaTa centris mTargmnelobiTi programis xelmZRvanels qalbaton manana miqaberiZes. Targmnis procesSi TiToeuli terminis Sesatyvisis moZebnasa da SeTanxmebas didi dro daeTmo. Tu socialuri mecnierebebisaTvis terminologia aqilevsis quslia statistikis terminologiuri aparati nadvilad arsebobs, magram rig SemTxvevebSi is an rusuli kliSea. magaliTad, stiudentis kriteriumi; an erTi da igive termini sxvadasxva konotaciiT gvxvdeba (erTi mxriv, uwyveti da wyvetili cvladi, da meore mxriv, uwyveti da diskretuli cvladi), an sxvadasxva sityviT aRiniSneba erTi da igive Sinaarsi (magaliTad sandooba da mniSvnelovneba). samecniero redaqtorebis mier mizanSewonilad iqna miCneuli, rom teqstSi moyvanili yofiliyo terminis meore variantic gaugebrobebisa da daumsaxurebeli SeniSvnebis Tavidan asacileblad. vimedovnebT, rom proeqtis ganxorcielebaSi Cadebuli dro da energia, samecniero redaqtorebis, mTargmnelebisa da organizatorebis mondomeba saTanado nayofs gamoiRebs da Sedegad qarTvel studentebs statistikis Sesaswavlad SesaniSnavi saxelmZRvanelo eqnebaT da uaxloes momavalSi socialur mecnierebebSi axali Taobis mkvlevarTa mier ganxorcilebuli kvlevebi maqsimalurad iqneba dazRveuli zedapiruli spekulaciebisa da artefaqtuli daskvnebisagan. marine CitaSvili, Tsu profesori socialur mecnierebaTa centris direqtori Tbilisi, 2006

1 w i n a s i t y v a o b a

winamdebare saxelmZRvanelos pirveli da meore gamocemis mizani ZiriTadi

statistikuri meTodebis konceptualuri ganviTarebis aRwera da bihevioristul

mecnierebebSi maTi gamoyeneba iyo. am miznis misaRwevad mkveTri aqcenti davsviT

Tanamedrove kvlevaSi arsebuli problemebis bihevioristul mecnierebebSi

gamoyenebaze, radgan migvaCnia, rom realurad arsebuli problemebi ufro

adekvaturad axdenen mecnierul kvlevaSi statistikuri analizis aucileblobis

ilustrirebas. saxelmZRvanelos winamdebare, mesame gamocema wina gamocemebis

gadasinjvaa, Tumca, SevecadeT, rom saxelmZRvanelos mizani ucvleli darCeniliyo.

teqstis organizacia

winamdebare gamocemis struqtura warmodgenili cnebebis Tanmimdevrulad

mowodebis principiTaa agebuli. statistikis gamoyeneba warmodgenilia

konceptualurad, logikuri Tanmimdevrobis dacviT. pirvel xuT TavSi aRwerilia

statistikis, qcevis gazomvis, qulebis sixSireTa ganawilebis saxiT, grafikulad

warmodgenis saWiroeba da centraluri tendenciisa da cvalebadobis sazomebi. me-6

da me-7 TavebSi warmodgenilia normaluri ganawileba, albaToba, SerCeviTi

ganawilebebi da statistikis gamoyeneba parametrebis SefasebisTvis. statistikuri

hipoTezebis Semowmebis safuZvlebi warmodgenilia me-8 TavSi .z kriteriumiTa da

erTi SerCevis t kriteriumiT. me-9-12 Tavebi eZRvneba saSualoebs Soris gansxvavebis

dadgenis mizniT hipoTezebis Semowmebas, parametruli kriteriumis gamoyenebiT.

detalurad aris warmodgenili dispersieuli analizi, vinaidan dReisaTvis is

yvelaze farTod gamoyenebadi statistikuri kriteriumia bihevioristul

mecnierebebSi da studentebma aucileblad unda icodnen misi gamoyeneba. srulad

aris aRwerili cdis pirTa Soris erTfaqtoriani, cdis pirebs SigniT

erTfaqtoriani da cdis pirTa Soris faqtoruli dispersiuli analizi; aseve,

aRwerilia mravaljeradi Sedarebis kriteriumebis gamoyeneba da gazomvis efeqtis

siZliere. garda amisa, mniSvnelovani yuradReba eTmoba urTierTqmedebis

interpretacias faqtorul analizSi. me-13 da me-14 Tavebi korelaciasa da regresias

eZRvneba. me-13 TavSi ganxilulia pirsonisa da spirmenis korelaciis koeficientebi

da TiToeul maTganTan dakavSirebuli statistikuri hipoTezebis Semowmeba. me-14

TavSi mocemulia martivi wrfivi regresiis safuZvlebis mimoxilva. am ori Tavis

saxelmZRvanelos bolosken moTavseba SeiZleba ucnaurad mogeCvenoT, vinaidan

mraval sxva saxelmZRvaneloSi am TavebSi mocemuli sakiTxebis ganxilva aRweriTi

statistikis Semdeg aris xolme mocemuli; Tumca, migvaCnia, rom es Cveuli

Tanmimdevroba studentebs garkveul sirTuleebs uqmnis. iseTi aRweriTi

statistikebis ganxilvas, rogoricaa saSualo da standartuli gadaxra, bunebrivad

2 mivyavarT SerCevis saSualoebs Soris arsebuli gansxvavebis statistikuri

Semowmebis ganxilvisken; aRweriT statistikebsa da statistikuri hipoTezebis

Semowmebas Soris korelaciisa da regresiis sakiTxebis Casma logikur jaWvs wyvets.

amrigad, vinaidan saxelmZRvanelo cnebebis Tanmimdevrulad mowodebis principiTaa

agebuli, korelaciisa da regresiis sakiTxebi parametruli statistikur

kriteriumebis ganxilvis Semdeg movaTavseT. saxelmZRvanelo me-15 TaviT sruldeba,

sadac warmodgenilia sami araparametruli statistikuri kriteriumi: _ xi-kvadratis

kriteriumi, man-uitnis U kriteriumi da uilkoqsonis niSniani rangebis kriteriumi.

saxelmZRvanelos maxasiaTeblebi

studentebi statistikis rTuli Teoriuli safuZvlebis Seswavlisas mraval

problemas awydebian. winamdebare saxelmZRvanelos aqvs ramdenime meToduri midgoma,

romelic am problemebis daZlevaSi dagexmarebaT da saSualebas mogcemT, rom

warmodgenili masala ukeT gaigoT da realur kvlevebs daukavSiroT.

realuri problemebi

yvela statistikuri meTodi bihevioristuli kvlevis konteqstSia warmodgenili.

mravali problema Tanamedrove publikaciebidan aris amoRebuli; amasTan, wyaro

imdenad srulad aris dasaxelebuli, rom dainteresebul students Tavisuflad

SeuZlia misi moZieba.

statistikebis konceptualuri ganviTareba

mravali statistika cnebiT doneze, ganmartebiTi formulis gamoyenebiT aris

warmodgenili. mniSvnelovania, rom studentebma TvalsaCinod dainaxon, Tu rogor

moqmedebs mocemuli statistika monacemebze. amis gaTvaliswinebiT, SevecadeT, rom

ama Tu im statistikis aRwerisas studentebi ar gadagvetvirTa maTematikuri

formulebiT. saxelmZRvanelos pirvel TavebSi aRwerili statistikebi mxolod

martiv maTematikas saWiroeben, Tumca, studentebs mainc aSinebT xolme pirvel TavSi

warmodgenili maTematikis sawyisebi. amitom winamdebare saxelmZRvaneloSi

maTematikuri cnebebi mxolod saWiroebis SemTxvevaSi aris mocemuli da axsnili;

xolo maTTvis, vinc ufro metad dainteresdeba, maTematikuri operaciebi srulad

aris mocemuli b danarTSi.

„SeamowmeT Tqveni codna“ da „kiTxvebi“

TiToeul TavSi gafantulia qveTavebi _ „SeamowmeT Tqveni codna“, Tavis bolos ki

warmodgenilia qveTavi _ „kiTxvebi“. am qveTavebSi mocemuli kiTxvebi studentebs

saSualebas aZlevs, gaimeoron masala da Seamowmon, Tu rogor gaiges da aiTvises.

bevr kiTxvaSi realuri kvlevebidan aRebuli problemebia warmodgenili. amrigad,

3 maT ara mxolod mimoxilvis funqcia aqvT, aramed sxvadasxva statistikuri meTodis

gamoyenebis magaliTebsac iZlevian. studentebs vurCevT, rom axal masalaze

gadasvlamde srulad upasuxon kiTxvebs da Seasrulon mocemuli savarjiSoebi.

prezentacia samecniero statiebSi

bevri statistikuri kriteriumis ganxilvisas mocemulia amonaridebi samecniero

Jurnalebidan da axsnilia maTSi warmodgenili informacia; amas Tan axlavs

samecniero JurnalSi gamoqveynebuli kvlevis Sedegebis wakiTxvis savarjiSoebi.

saxelmZRvanelos winamdebare gamocemaSi es maxasiaTebeli ufro farTod aris

warmodgenili da Seesabameba amerikis fsiqologiuri asociaciis publikaciebis

saxelmZRvanelos me-4 gamocemaSi (Publication Manual of the American Psychological Association.

American Psychological Association, 1994) mocemul saredaqcio stils.

teqstis Tanmxlebi ganmartebebi da terminebis leqsikoni

winamdebare gamocemaSi Sevida mniSvnelovani terminebis ganmartebebi, romlebic

Tan axlavs teqsts, da am terminTa leqsikoni. mniSvnelovani terminebis Srifti

gamuqebulia da teqstSia ganmartebuli. saswavlo masalis gansamtkiceblad es

ganmartebebi kidev erTxel aris mocemuli teqstis Semdeg. saxelmZRvaneloSi

gamoyenebuli yvela termini ganmartebulia g danarTSi.

statistikuri simboloebi

statistikaSi gamoyenebuli simboloebis mravalferovnebam SeiZleba iolad

dagabnioT. statistikis enis aTvisebisas daxmarebas gagiwevT d danarTi, romelSic

Tavmoyrilia saxelmZRvaneloSi warmodgenili yvela simbolo.

Tanmimdevroba

teqsti Tanmimdevrulad ikiTxeba, mas ar wyvets CarCoSi Casmuli CanarTebi.

TiToeuli qveTavi wina masalidan gamomdinareobs. am uwyvetobis gaumjobesebis

mizniT, dispersiuli analizis nedli qulebis gamosaTvleli formulebi Tavis

boloSi, statistikur damatebaSia mocemuli. amrigad, maTTvis, vinc yuradRebis

miRma tovebs gamosaTvlel formulebs (kompiuteruli statistikuri programebis

gamoyenebis SemTxvevebSi), ar irRveva teqstis Tanmimdevruloba.

vimedovnebT, rom am saxelmZRvaneloTi statistikis Seswavla saintereso da

gamosadegi iqneba TqvenTvis.

madloba1

am saxelmZRvalos SeqmnaSi mraval adamians miuZRvis wvlili. pirveli gamocemis

idea alen uorkmens (Allen Workmen, gamomcemlobidan „Allyn and Bacon“) ekuTvnis,

1 ibeWdeba mcireodeni SemoklebiT (mTargmnelis SeniSvna).

4 romelmac gamoTqva mosazreba, rom fsiqologiuri kvlevis meTodebis

saxelmZRvanelodan _ „fsiqologiuri kvlevis meTodebi, konceptualuri midgoma“ _

aRebul midgomas viyenebdi; es saxelmZRvanelo Cemi da doqtor duglas u.

blumquisTis (Dr. Douglas W. Bloomquist) TanaavtorobiT gamoica. aqedan gamomdinare,

winamdebare saxelmZRvaneloSi gamoyenebuli mravali konceptualuri mosazreba

xsenebul saxelmZRvaneloSi warmodgenilis msgavsia. me madlobas movaxseneb duglas

blumquisTs am mosazrebaTa CamoyalibebaSi Setanili wvlilisTvis da masalisTvis;

aseve, imisTvis, rom man am masalis winamdebare saxelmZRvaneloSi gamoyenebis neba

damrTo. gamomcemlobis _ „Allyn and Bacon“ vice-prezidentma da gamomcemelma suzan

bedgerma (Susan Badger) daamtkica es gamocema da amiT proeqtis dasrulebis impulsi

momca. „Allyn and Bacon“-is aRmasrulebelma redaqtorma fsiqologiis dargSi laura

pirsonma (Laura Pearson) mniSvnelovani roli Seasrula wignis saboloo saxiT

CamoyalibebaSi. saxelmZRvanelos gadasinjvisas misi ideebi da winadadebebi yvelaze

sasargeblo aRmoCnda. redaqtoris asistenti jenifer normandini (Jennifer Normandin)

xalisiT da moxerxebulad ukmlavdeboda xelnaweris momzadebasTan dakavSirebul

Cem mier dasmul mraval sakiTxs. madlobas movaxseneb Allyn and Bacon“-is prezidents

bil barks (Bill Barke), romelmac 12-ze meti wlis win damarwmuna da wamaxalisa, rom

kolejis saxelmZRvanelos dawera mecada.

warmoebis ZiriTad process moxerxebulad gaumklavdnen sintia niubai (Cynthia

Newby) Chestnut Hill Enterprises, Inc. da raia maisi (Raeia Maes) Maes Associates. uiliam o.

tomasi (William O. Thomas) yvelaze aqtiurad CarTuli da gulmodgine redaqtori iyo,

romelmac daxvewa Cemi nawerebi, madlierebiT minda movixsenio.

garda amisa, winamdebare saxelmZRvanelos SeqmnaSi Cemma mravalma profesionalma

kolegam miiRo monawileoba. zogierTma maTganma pasuxebi gasca detalur kiTxvars,

romelic Sinaarssa da midgomebis sxvadasxva variants Seexeboda. sxvebma naxes

xelnaweris sruli varianti an misi fragmentebi. maT mier nawerebis guldasmiT

gadamowmebam fasdaudebeli samsaxuri gamiwia azris naTlad gadmocemaSi da Tavidan

amarida konceptualuri da maTematikuri Secdomebi. cxadia, saxelmZRvaneloSi

darCenil nebismier uzustobaze mxolod me vageb pasuxs. gaweuli daxmarebisTvis

madlobas vuxdi: pol amrheins niu meqsikos universitetidan (Paul Amrhein, University of

New Mexico), denis kogans texasis teqnikuri universitetidan (Dennis Cogan, Texas Tech

University), mark fainmens, samxreT koneqtikutis saxelmwifo universiteti (Mark Fineman,

Southern Connecticut State University), maik greles, centraluri misuris saxelmwifo

universiteti (Mike Grelle, Central Missouri State University), rik hoils, kentukis

universitetidan (Rick Hoyle, University of Kentucky), artur d. kemfs centraluri misuris

saxelmwifo universiteti (Arthur D. Kemp, Central Missouri State University), melani m. kerCers,

sam hiustonis saxelmwifo universiteti (Melanie M. Kercher, Sam Houston State University),

robert langs, centraluri floridis universiteti (Robert Lange, University of Central

5 Florida), j. u. mur umcross, marSalis universiteti (J.W.Moore, Jr., Marshall University),

jon riCs, fulertonis kaliforniis universiteti (Jon Rich, California State University at

Fullerton), riCard serks, tulsas koleji (Richard Serkes, Tusla Junior College), n. s. silvers,

centraluri floridis universiteti (N. C. Silver, University of Central Florida), david d.

simfsons, karolis koleji (David D. Simpson, Carroll College), maikl terels, stounhilis

koleji (Michael Tirrell, Stonehill College) da pol iarnolds, Crdilo-dasavleTis

universitetis samedicino skola (Paul Yarnold, Northwestern University Medical School).

aseve, madloba minda gadavuxado CRC Press, Inc.-s danarTSi mravali statistikuri

cxrilis gamoyenebis nebarTvis mocemisTvis.

da bolos, madlobas vuxdi maT, viszec yvelaze metad imoqmeda winamdebare

saxelmZRvaneloze muSaobam _ Cems Svilebs, qimberlisa da jefris, da Cems meuRles

sandras. maTi mxardaWeris, moTminebis, waxalisebisa da siyvarulis gareSe es wigni

verasodes dasruldeboda da ver ixilavda dRis sinaTles.

h.o.k.

Tavi I. statistikis Sesavali 6

Tavi I. statistikis Sesavali

statistikis Seswavla: rogor gamoviyenoT winamdebare

saxelmZRvanelo

mniSvnelovani terminebi

SeamowmeT Tqveni codna

kiTxvebi

maTematikuri cnebebi

ra aris statistika

statistikis gamoyeneba: sami magaliTi

aRwera da daskvna

eqsperimenti da statistikuri hipoTezis Semowmeba

korelacia da regresia

ratom aris saWiro statistika


„o, es saSineli maRviZara“, _ buzRuniT aCumebT saaTs da cdilobT, kidev cota

xans daiZinoT. ra mokle Rame iyo! rogorRac dgebiT da acnobierebT, rom Tu

xanmokle SxapiTa da swrafi sauzmiT SemoifarglebiT, ar daigvianebT leqciaze,

romelic 8 saaTzea daniSnuli. manqanaSi Cajdomamde amCnevT, rom adreuli

gazafxulis sasiamovno, Tbili dila gaTenebula; gacilebiT ukeTesi, vidre

guSindeli civi, wvimiani dRe iyo. amasobaSi gzatkecilze gadixarT _ moZraoba

Cveulebrivze metadaa gadatvirTuli da ise swrafad ver moZraobT, rogorc

moelodiT. daigvianebT mecadineobaze? sabednierod, ar dagagviandaT. dRes erT-erTi

saukeTeso leqcia gqondaT mTeli semestris ganmavlobaSi, magram gaSinebT

laboratoriuli samuSao, romelic 8 saaTzea daniSnuli; is xSirad imdenad didxans

grZeldeba, rom waxemsebisTvis dro aRar grCebaT xolme; Tumca, dRes yvelaferi

sxvagvarad, SesaniSnavad aris da droulad gamoxvediT mecadineobidan, amitom

sakmarisi dro dagrCaT da or saukeTeso megobarTan erTad auCqareblad da

mxiarulad waixemseT. Semdeg samuSaod midixarT did savaWro centrSi.

guSindelisgan gansxvavebiT, momxmareblebi dRes Zalze Tavazianebi arian, albaT,

kargi amindis gamo. sasiamovno momxmareblebTan urTierTobaSi SuadRec male modis,

Tqven isev gzatkecilze xarT da saxlisken miiCqariT. dRes Cveulebrivze gacilebiT

naklebad datvirTulia moZraoba, albaT, yvela adre wavida samsaxuridan. samecadino

bevri gaqvT, amitom sadilis Semdeg im imediT sxdebiT magidasTan, rom nayofierad

imecadinebT mTeli saRamo. vinaidan (yovel SemTxvevaSi, sadilis Semdeg mecadineobis

sxva mraval SemTxvevasTan SedarebiT) asec xdeba, grZnobT, rom namdvilad mzad

xarT kviris bolos daniSnuli statistikis gamocdisTvis. ajildovebT sakuTar Tavs

_ erT saaTs televizorTan atarebT da dasaZineblad midixarT. cota adre Tu

daiZinebT, iqneb Rame ufro grZeli geCvenoT.

es mokle epizodi, albaT, Tqveni cxovrebis erT Cveulebriv dRes asaxavs an ar

asaxavs. nebismier SemTxvevaSi, albaT, dagainteresebT, ra kavSirSia es epizodi

statistikasTan. is gviCvenebs, rom Tqven cxovrebaSi amCnevT garkveul

kanonzmierebas: _ drois Cveuli odenoba, romelsac uTmobT Sxapsa Tu sauzmes,

adreuli gazafxulisTvis Cveuli amindi, kolejisken mimavali gzisTvis

damaxasiaTebeli garkveulwilad gadatvirTuli moZraoba da dro, romelic

gWirdebaT kolejamde misvlisTvis, mecadineobis sainteresoobis zogadi done,

laboratoriuli samuSaos saSualo xangrZlivoba, mxiarulebis Cveuli zoma

waxemsebisas, savaWro centris momxmarebelTa tipuri qceva da saRamos

mecadineobisas Cveuli nayofiereba. amCnevT, rom Tqvens cxovrebaSi aris

kanonzomiereba da sistematuroba da es faqti imazec miuTiTebs, rom

cvalebadobasac ganicdiT. zogierTi Rame, danarCenTan SedarebiT, ufro grZeli

geCvenebaT, dRidan dRemde varirebs im drois xangrZlivoba, romelsac Sxapis

miRebasa da sauzmes uTmobT, weliwadis am dros iSviaTad aris kargi amindi,


gzatkecilze moZraoba met-naklebad gadatvirTulia xolme, sxvadasxva dRes (da

sxvadasxva kursze) mecadineoba met-naklebad saintereso SeiZleba iyos,

laboratoriuli samuSao zogjer saSualoze didxans grZeldeba, zogjer ki _ ufro

mcire xans, myidvelTa qceva sakmaod cvalebadia da zogjer ufro nayofierad

mecadineobT, vidre sxva dros.

movlenebis es cvalebadoba garemoSi cvladebis arsebobas gulisxmobs. cvladi

aris garemos nebismieri mdgomareoba an movlena, nebismieri stimuli, pirovnuli

maxasiaTebeli an qceva, romelmac SeiZleba miiRos sxvadasxva mniSvneloba

sxvadasxva dros an sxvadasxva

adamianTan. cvladia im drois xangrZlivoba,

ramden xansac geZinaT, Sxapis miRebis

xangrZlivoba, gzatkecilze moZraobis

intensivoba, mecadineobis sainteresoobis done,

laboratoriuli samuSaos xangrZlivoba,

myidvelTa guneba-ganwyobileba da mecadineobis

nayofiereba. statistika aris mecniereba, romelic poulobs raRac mudmivs da am

mudmivis mimarT cvladebadobas zomavs. miuxedavad imisa, rom statistikis aseTi

gansazRvreba araformaluri da arasrulia, am etapze sruliad sakmarisia.

cxadia, Tqven ar xarT erTaderTi adamiani, romelic amCnevs movlenaTa

kanonzomierebebs. farTo interesebis mqone belgiel profesor lambert adolf Jak

ketles (Lambert Adomphe Jacques Quetelet, 1796-1874) miaCnda, rom cvalebadobis aseT

kanonzomierebebs Tavisi mizezebi aqvs da Tu is daadgenda am kanonzmierebebs, Semdeg

maTi mizezebis povnasac SeZlebda. amrigad, man ganizraxa am kanonzomierebebis

gamovlena da muSaoba daiwyo imis daxasiaTebiT, rasac Tavad „saSualo adamians“

uwodebda. amisaTvis man didi raodenobis frangi jariskacebisgan Sekriba iseTi

cvladebis anTropologiuri monacemebi (anu sxeulebrivi gazomvebis Sedegad

miRebuli monacemebi), rogoricaa wona da simaRle (Stigler, 1986). mas imedi hqonda, rom

saSualo individis daxasiaTebiT is SeZlebda adamianis maxasiaTeblebis

cvladi (variable): _ garemos

nebismieri mdgomareoba, nebismieri

stimuli, pirovnuli maxasiaTebeli

an qceva, romelmac SeiZleba miiRos

sxvadasxva mniSvneloba sxvadasxva

dros an sxvadasxva adamianTan.

cxrili N1.1. gasamarTlebaTa sixSire safrangeTis sasamarTloSi, rogorc wlisa da

braldebulis wera-kiTxvis unaris funqcia

braldebulis wera-kiTxvis unari weli

saSualo 1828 1829 1830

ar SeuZlia wera an kiTxva

SeuZlia wera-kiTxva, magram arasrulyofilad

kargad wers da kiTxulobs

aqvs umaRlesi ganaTleba

0.63

0.62

0.56

0.35

0.63

0.60

0.55

0.48

0.62

0.58

0.52

0.37

0.627

0.600

0.543

0.400

ketles monacemebis mixedviT. adaptirebulia s. m. stigleris (S.M.Stigler, 1986) Sromidan statistikis istoria: ganusazRvrelobis gazomva 1900 wlamde, cxrili N5.2.


kanonzomierebebis povnas da Seadarebda mamakacebisa da qalebis saSualoebs, an

moaxdenda Sedarebas sacxovrebeli adgilis, asakisa da rasis mixedviT. ketles

saSualo adamianiT daintereseba vrceldeboda imazec, rasac igi „moralis

statistikas“ uwodebda anu kanonzomierebebs iseT cvladebSi, rogoricaa suicidis

an danaSaulis sixSire da gasamarTlebis sixSire. magaliTad, is wlebis manZilze

Seiswavlida safrangeTis sasamarTloSi gasamarTlebaTa sixSires da moaxdina maTi

kategorizacia imis mixedviT, Tu ramdenad SeeZlo braldebuls wera-kiTxva. Sedegebi

mocemulia cxrilSi N1.1, romelSic ricxviTi mniSvnelobebi albaTobebs gamoxatavs.

magaliTad, Tu Tqven safrangeTis sasamarTloSi 1829 wels gqondaT saqme da mxolod

arasrulyofilad SegeZloT wera-kiTxva, albaToba imisa, rom Tqven damnaSaved

gcnobdnen, iqneboda 0.60 (anu 6 Sansi 10-dan). yuradReba miaqcieT, rom aq Cans aSkara

kavSiri (anu Sesabamisoba) braldebulis wera-kiTxvis unarebsa da damnaSaved cnobis

albaTobas Soris. am statistikuri kvlevebis dawyebisTvis ketles xSirad

socialuri mecnierebis mamas uwodeben (Dudycha & Dudycha, 1972).

statistikis Seswavla: rogor gamoviyenoT winamdebare

saxelmZRvanelo

rogorc SesavalSi vnaxeT, statistika Seexeba movlenaTa kanonzmierebebs da

cvalebadobas, romelsac yvela Cvengani amCnevs cxovrebaSi. amrigad, garkveuli

azriT, yvelani damwyebi statistikosebi varT: miuxedavad amisa, bevr adamians

afrTxobs statistikis srul kursze fiqri, Tumca, es ase ar unda iyos. statistikis

Seswavla axali enis aTvisebas hgavs. statistikis ena sityvebisgan, simboloebisgan

da formulebisgan Sedgeba. nebismieri axali enis msgavsad, is seriozul

mecadineobas moiTxovs am komponentebis dasaZlevad da asaTviseblad, magram

praqtikasTan da gamocdilebasTan erTad is iseTive bunebrivi unda gaxdes TqvenTvis,

rogorc mSobliuri ena. winamdebare saxelmZRvanelos zogierTi Tavisebureba

daxmarebas gagiwevT statistikis Seswavlisas.

mniSvnelovani terminebi da simboloebi

teqstSi warmodgenili mniSvnelovani terminebi gamuqebulia da Semdeg calkea

gamotanili maTi ganmartebebi. rodesac teqstSi SegxvdebaT gamuqebuli termini,

unda mixvdeT, rom is mniSvnelovania statistikis gagebisaTvis. nu gaagrZelebT

teqstis kiTxvas, sanam ar SeiswavliT termins. amasTanave, terminTa leqsikoni

mocemulia g danarTSi. Tu teqstSi SegxvdebaT termini, romlis mniSvneloba

dagaviwydaT, CaixedeT am leqsikonSi. statistikuri simboloebis leqsikoni

mocemulia d danarTSi.


qveTavebi „SeamowmeT Tqveni codna“

TiToeul TavSi mocemulia qveTavebi _ „SeamowmeT Tqveni codna“, romlebic

SesaZleblobas gaZlevT, rom Semdeg qveTavze gadasvlamde gaimeoroT masala da

naxoT, Tu ramdenad gaigeT is. am kiTxvaTa didi nawili mimarTulia kvlevis realur

problemebze. Sesabamisad, maTSi mocemulia Tqvens mier ukve naswavli statistikuri

meTodebis gamoyenebis magaliTebi. am qveTavebis kiTxvebze pasuxebis didi nawili

mocemulia saxelmZRvanelos bolo e danarTSi _ amosaxsneli amocanebis pasuxebi.

kiTxvebi

TiToeuli Tavis bolos mocemulia ganvlili masalis gasameorebeli kiTxvebi.

rogorc qveTavebis „SeamowmeT Tqveni codna“ SemTxvevaSi, aqac SegiZliaT upasuxoT

ganvlili masalis gasameorebel kiTxvebs da SeadaroT Tqveni pasuxebi e danarTSi

mocemul pasuxebs.

maTematikuri cnebebi

winamdebare saxelmZRvaneloSi gamoyenebulia elementaruli maTematika, romelic

mxolod Sekrebis, gamoklebis, gamravlebisa da gayofis operaciebs saWiroebs.

maTematikuri simboloebi da operaciebi imdenad aris axsnili, ramdenadac saWiroa

isini. Tu raime problema SegeqmnebaT romelime maTematikur operaciasTan

dakavSirebiT, naxeT danarTi b _ maTematikuri cnebebi, sadac ganxilulia ZiriTadi

maTematikuri simboloebi da operaciebi.

ra aris statistika

ketles Semdeg

bihevioristuli mecnierebis

warmomadgenlebi cdiloben

adamianisa da cxovelis

qcevis gagebasa da axsnas.

isini amas monacemebis

Segrovebis saSualebiT

akeTeben. monacemebi aris is

qulebi an qcevis Tu

maxasiaTeblebis gazomvis

Sedegebi, romlebic miiReba

adamianebze an cxovelebze

dakvirvebis Sedegad.

procedurebs, romlebic gamoiyeneba monacemebis mimarT maTi sistematurobis da

cvalebadobis dasadgenad, statistika ewodeba. amrigad, statistika niSnavs im

meTodebsa Tu procedurebs, romlebic gamoiyeneba monacemebis mowesrigebis,

analizisa da maTgan daskvnebis gamosatanad. statistika (statistikuri sidide) aris

monacemebi (Data _ mravlobiTi, datum _ mxolobiTi): _ is

qulebi an qcevis Tu maxasiaTeblebis gazomvis Sedegebi,

romelic miiReba adamianebze an cxovelebze dakvirvebis

Sedegad.

statistika (mecniereba) (Statistics): _ is meTodebi Tu

procedurebi, romlebic gamoiyeneba monacemebis

sistematizaciisTvis, gasaanalizeblad da maTgan daskvnebis

gamosatanad.

statistika (statistikuri sidide) (Statistic): _ monacemebze

dayrdnobiT gamoTvlili calkeuli ricxvi, romelic

SeiZleba gamoyenebul iqnes SerCevis monacemTa

erTobliobis aRsawerad, an am monacemTa ufro sruli

gaanalizebisTvis.


monacemebze dayrdnobiT gamoTvlili calkeuli ricxvi, romelic SeiZleba

gamoyenebul iqnes SerCevis monacemTa erTobliobis aRsawerad an am monacemTa ufro

sruli gaanalizebisaTvis.

statistikis gamoyeneba: sami magaliTi

bihevioristul mecnierebebSi statistikur procedurebs sxvadasxvagvarad iyeneben.

qvemoT mocemulia bihevioristul mecnierebaSi statistikuri meTodebis gamoyenebis

sami magaliTi. aseve, mocemulia mTeli rigi terminebi, romlebic Semdeg SegxvdebaT

teqstSi; amitom unda iswavloT isini. ar aReldeT, Tu ver gaigebT warmodgenili

magaliTebis yvela detals. TiToeuli midgomis Sesaxeb ufro dawvrilebiT da

srulad Semdeg TavebSi visaubrebT.

aRwera da daskvna

yoveldRiurad presidan didi raodenobiT monacemTa statistikur aRwerebs

vecnobiT. magaliTad, gazeTi USA Today gvTavazobs 1980-iani wlebis tipuri

individebis saqmianobaTa aRweras („The Way“, 1985). erT-erTi SemoTavazebuli

daxasiaTebis mixedviT, saSualo ojaxSi yoveldRiurad televizori CarTuli iyo 7-

12 saaTis ganmavlobaSi. es ricxobrivi maCvenebeli gviCvenebs statistikuri

meTodebis gamoyenebis or mniSvnelovan gzas: _ aRwerasa da daskvnas. miuxedavad

imisa, rom ricxobrivi maCvenebeli _ 7-12 saaTi mowodebulia mTlianad aSS-is

ojaxebis populaciis dasaxasiaTeblad im drois xangrZlivobis mixedviT, romlis

manZilzec yoveldRiurad CarTulia televizori, is ar aris miRebuli qveyanaSi

mcxovrebi yvela ojaxze dakvirvebisa da gazomvis Sedegad. metic, yvela ojaxis

mxolod nawils, anu SerCevas daakvirdnen da gazomes. monacemTa arsebuli

erToblioba miiRes im saaTebis raodenobis Caweris Sedegad, ramden xansac hqondaT

yoveldRiurad CarTuli televizori SerCevaSi Sesul ojaxebs. amis Semdeg SerCevaSi

Semavali sxvadasxva ojaxidan miRebuli monacemebi Seajames anu gamoTvales erTi

ricxvi, aRweriTi statistika _ 7-12 saaTi. es statistika mowodebulia im saaTebis

yovedRiuri raodenobis aRsawerad, romlis ganmavlobaSic SerCevaSi Semaval tipur

ojaxs CarTuli aqvs televizori. Semdeg am statistikis safuZvelze askvnian, anu

zomaven im saaTebis raodenobas, romlis ganmavlobaSic yoveldRiurad CarTuli aqvs

televizori aSS-Si mcxovrebi yvela ojaxis populaciaSi Semaval tipur ojaxs.

Cven gamoviyeneT terminebi populacia da SerCeva. populacia aris saerTo

maxasiaTeblis mqone adamianTa, cxovelTa an

movlenaTa sruli erToblioba. magaliTad,

aSS-is kolejis yvela studenti Seadgens

populacias. am populaciis saerTo

maxasiaTebelia is, rom masSi Sesuli yvela individi swavlobs kolejSi. aseve,

populaciis magaliTia aSS-is yvela ojaxi, CrdiloeT karolinis yvela mozrdili

populacia (Population): _ aris saerTo

maxasiaTeblis mqone adamianTa, cxovelTa

an movlenaTa sruli erToblioba.


qali, nebraskis fermerebi an yvela adamiani, visac erTi an meti sabanko sakredito

baraTi aqvs; TiToeuli dasaxelebuli jgufi Seadgens populacias. individTa

nebismieri jgufi SeiZleba ganvixiloT rogorc populacia mas Semdeg, rac

mocemuli saerTo maxasiaTeblis mqone yvela individi Sesuli iqneba jgufSi.

bihevioristuli mecnierebis warmomadgenlebs xSirad populaciis daxasiaTeba

iseTi debulebebiT surT, rogoricaa, magaliTad, „tipur amerikul ojaxs

yoveldRiurad 7-12 saaTis ganmavlobaSi aqvs CarTuli televizori“, an „aSS-Si

mcxovrebi saSualo mamakacis Semosavali weliwadSi $19,438-s Seadgens maSin, roca

qalis _ mxolod $9,584-s“ („The Way“, 1985); Tumca, naTelia, rom SemTxvevaTa

umravlesobaSi SeuZlebelia aseTi maxasiaTeblebis misaRebad mTeli populaciis

gazomva. Sesabamisad, mecnierebi, rogorc wesi, mxolod populaciis SerCevas iyeneben.

SerCeva aris populaciidan mTeli

rigi wesebis dacviT (raTa SerCeva

populaciis reprezentaciuli iyos)

arCeuli qvesimravle anu qvejgufi.

SerCevis arCevis erT-erTi

gavrcelebuli formaa SemTxveviTi

SerCeva. martiv SemTxveviT SerCevaSi

individebi SerCeulni arian ise, rom

populaciis TiToeul wevrs SerCevaSi

moxvedris toli Sansi aqvs da erTi

wevris arCeva ar aris damokidebuli

populaciis sxva wevrebis arCevaze.

Semdeg SerCevis wevrebisgan miiReba

monacemebi da maT aRsawerad

gamoiTvleba aRweriTi statistika,

romelic gamoiyeneba populaciis

maxasiaTeblis Sesaxeb daskvnis

gasakeTeblad. sityva „daskvna“

gulisxmobs raRac cnobilze

dayrdnobiT raRac ucnobis Sesaxeb msjelobas. amrigad, statistikuri daskvna aris

SerCevidan miRebul statistikebze dayrdnobiT populaciis ucnobi maxasiaTeblis

(magaliTad, im saaTebis raodenoba, romlis ganmavlobaSic yoveldRiurad rTavs

televizors tipuri amerikuli ojaxi) anu parametrebis Sesaxeb daskvnebis gamotanis

procesi.

aRwerisa da daskvnisaTvis statistikis gamoyenebis mravali magaliTia cnobili.

magaliTad, aSS-is mosaxleobis aRweris biuro agrovebs sxvadasxvagvar informacias

amerikaSi cxovrebis Sesaxeb da zogierT maTgans yovelwliurad aqveynebs

SerCeva (Sample): _ populaciidan mTeli rigi

wesebis dacviT (raTa SerCeva populaciis

reprezentaciuli iyos) arCeuli qvesimravle anu

qvejgufi.

SemTxveveiTi SerCeva (Random sample): _ SerCeva,

romelSic individebi SerCeulni arian ise, rom

populaciis TiToeuls wevrs SerCevaSi

moxvedris toli Sansi aqvs da erTi wevris

arCeva ar aris damokidebuli populaciis sxva

wevrebis arCevaze.

aRweriTi statistika (Descriptive statistics): _

statistikuri procedurebi, romlebic

gamoiyeneba SerCevidan miRebuli monacemebis

Sesajameblad da aRsawerad.

statistikuri daskvna (Statistical inference): _

populaciis maxasiaTeblis Sefaseba SerCevidan

miRebul statistikebze dayrdnobiT.

parametri (Parameter): _ ricxvi, romelic aRwers

populaciis maxasiaTebels.


SeerTebuli Statebis statistikuri cnobaris saxiT. 1999 wlis aseTi gamocema

gvamcnobs, rom

• aSS-Si mcxovreb ojaxTa 98%-s aqvs televizori, maTgan 84.6%-s ki _ video

magnitofoni;

• 1997 wels saSualo statistikurma moqalaqem tansacmelSi $353.30, xolo

$957.30 samedicino momsaxurebaSi daxarja;

• mamakacebis 58% da qalebis 54.9% daojaxebulni arian.

TiToeuli es mniSvneloba aRweriTi statistikaa _ ricxvi, romelic aRwers

SerCevis wevrebis tipur pasuxs. cxadia, praqtikulad SeuZlebelia aSS-Si arsebuli

TiToeuli televizorisa Tu video magnitofonis daTvla, anda TiToeuli

moqalaqisTvis tansacmelSi daxarjuli Tanxebis Sesaxeb gamokiTxva. amrigad,

TiToeuli am mniSvnelobaTagani statistikaa, romelic populaciaSi amave

maCveneblis mniSvnelobas afasebs. is, Tu ramdenad kargad afasebs calkeuli

aRweriTi statistika populaciis mniSvnelobas, damokidebulia rogorc SerCevis

Seqmnis meTodze, ise _ mis reprezentaciulobaze. SerCevis Sedgenisa da aRweris da

populaciaze daskvnebis gamotanis procedurebs ufro dawvrilebiT me-3-7 TavebSi

ganvixilavT.

SeamowmeT Tqveni codna I.1

1. ganmarteT: _ monacemebi, parametri, populacia, SemTxveviTi SerCeva,

SerCeva, statistika (mecniereba), statistikuri daskvna, statistika

(statistikuri sidide), cvladi.

2. bolo dros Catarebul arCevnebSi winaswar gamokiTxes saarCevno olqis

600 potenciuri amomrCeveli. gamokiTxvis mizani iyo imis garkveva, Tu

visTvis apirebdnen amomrCevlebi xmis micemas. Sedegebma aCvena, rom 312

amomrCeveli, anu SerCevis 52% A kandidatisTvis apirebda xmis micemas, 240

amomrCeveli anu 40% _ B kandidatisTvis, 48 amomrCevelma, anu 8%-ma

miuTiTa, rom ar apirebda arCevnebSi monawileobis miRebas an ar aniWebda

upiratesobas arc erT kandidats. gamokiTxvis am Sedegebze dayrdnobiT

mkvlevrebma daadgines, rom kandidati A miiRebda arCevnebSi Semosuli

yvela xmis 52%-s.

a) axseniT, ratom aris aucilebeli aRweriTi statistikebi am kvlevis

Sedegebis SejamebisTvis.

b) miRebul statistikebze dayrdnobiT, ra daskvnebi gakeTda populaciis

Sesaxeb?


eqsperimenti da statistikuri hipoTezis Semowmeba

bihevioristuli mecnierebis warmomadgenlebi dainteresebulni arian ara marto

SerCevis aRweriTa da daxasiaTebiT, aramed calkeuli individebisa Tu jgufebis

qcevis mizezebis povniTac. am sakiTxisadmi erT-erTi midgomaa eqsperimentis Catareba.

eqsperimentis dros mkvlevari gamoyofs cvlads, romelsac damoukidebeli cvladi

ewodeba da romelic, misi azriT, gavlenas axdens pirovnebis qcevaze. qcevas,

romelzec, mkvlevris molodiniT, gavlenas moaxdens damoukidebeli cvladi,

damokidebuli cvladi ewodeba.

magaliTad, SeiZleba movelodeT, rom alkoholis moxmareba gavlenas axdens

pirovnebisagan Zaladobis reaqciaze: _ mosalodnelia, rom alkoholis zemoqmedebis

qveS myofi adamianebisTvis ufro misaRebi iyos Zaladoba, vidre im adamianebisTvis,

romlebic ar arian alkoholis zemoqmedebis qveS (Gustafson, 1987a). am SemTxvevaSi

alkoholis moxmareba aris damoukidebeli cvladi. eqsperimentatori akontrolebs

_ miiRebs individi alkohols Tu ara da moelis, rom is (alkoholi) gavlenas

damoukidebeli cvladi (Independent variable): _ cvladi, romliTac manipulireben eqsperimentSi,

damokidebul cvladze misi gavlenis gansazRvris mizniT.

damokidebuli cvladi (Dependent variable): _ cvladi eqsperimentSi, romelic damokidebulia

damoukidebel cvladze. SemTxvevaTa umravlesobaSi damokidebuli cvladi qcevis garkveuli

sazomia.

kvlevis hipoTeza (Research hypothesis): _ debuleba or an met cvlads Soris mosalodneli an

nawinaswarmetyvelebi kavSiris Sesaxeb. eqsperimentSi kvlevis hipoTeza aris

nawinaswarmetyvelebi kavSiri damoukidebel da damokidebul cvladebs Soris.

cdis piri (Subject): _ pirovneba, romelic monawileobs eqsperimentSi.

ekvivalenturi jgufebi (Equivalent groups): _ cdis pirTa jgufebi, romlebic, savaraudod, ar

gansxvavdebian erTmaneTisgan raime mudmivi maxasiaTebliT eqsperimentis damoukidebeli

cvladis miwodebamde.

cdis pirTa Soris sqema (Between-subject design): _ kvlevis sqema, romelSic monawileobs ori an

meti jgufi.

qula (Score): _ cdis piris mier davalebis Sesrulebis gazomviT miRebuli ricxviTi

mniSvneloba.

nedli monacemebi (Raw data): _ yvela cdis pirisgan miRebuli qula, romelTac jer ar

gauvliaT statistikuri analizi.

statistikuri hipoTezis Semowmeba (Statistical hypothesis testing) _ procedurebi, romlebic

jgufebs Soris SemTxveviTi gansxvavebis zomis winaswarmetyvelebisTvis gamoiyeneba.

SemTxveviTi gansxvaveba (Chance difference): _ gansxvaveba ekvivalentur jgufebs Soris,

romelic miiReba SemTxveviT an gauTvaliswinebeli mizeziT.


moaxdens Zaladobis mimReblobaze. individis mier Zaladobis mimReblobis sazomi

aris damokidebuli cvladi. varaudoben, rom Zaladobis mimRebloba damokidebulia

imaze, aqvs Tu ara individs miRebuli alkoholi. debulebas, romelic asaxavs

mkvlevris molodins imis Taobaze, rom alkoholis moxmareba gavlenas moaxdens

Zaladobis mimReblobaze, kvlevis hipoTeza ewodeba. kvlevis hipoTeza aris debuleba

or an met cvlads Soris mosalodneli an nawinaswarmetyvelebi kavSiris Sesaxeb.

eqsperimentSi kvlevis hipoTeza aris nawinaswarmetyvelebi kavSiri damoukidebel da

damokidebul cvladebs Soris.

kvlevis im hipoTezis Sesamowmebeli umartivesi eqsperimenti, romelic alkoholis

moxmarebas akavSirebs Zaladobis mimReblobasTan, adamianTa ori ekvivalenturi

jgufis SedgeniT iwyeba. im adamianebs, romlebic monawileoben samecniero kvlevebSi,

mecnierebi cdis pirebs an monawileebs uwodeben. ekvivalenturi jgufebi, romlebic,

savaraudod, ar gansxvavdebian erTmaneTisgan raime mudmivi maxasiaTebliT

eqsperimentis damoukidebeli cvladis miwodebamde. eqsperiments, romelSic

monawileobs ori an meti jgufi, cdis pirTa Soris sqemis mqone eqsperimenti

ewodeba. ekvivalenturi jgufebis Seqmnis Semdeg mkvlevari manipulirebs

damoukidebeli cvladiT anu cvlis (axdens varirebas) mas da zomavs damokidebul

cvlads; amasTan, akontrolebs yvela im sxva cvlads, romelmac SeiZleba gavlena

moaxdinos damokidebul cvladze. Cvens magaliTSi eqsperimentatorma SeiZleba cdis

pirebis pirvel jgufs misces 0.8 mililitri alkoholi amave odenobis forToxlis

wvenTan erTad sxeulis 1 kilogramze, meore jgufs ki _ 1.6 mililitri sufTa,

ualkoholo forToxlis wveni. erTaderTi, riTac es ori jgufi SeiZleba

gansxvavdebodes erTmaneTisgan, aris miRebuli sasmelis raodenoba. damokidebuli

cvladi SeiZleba gaizomos imiT, rom cdis pirs ayurebinon Zaladobis Semcveli

filmi da Semdeg Seafasebinon is nanaxi Zaladobis mimReblobis mixedviT. Sefasebis

Sedegad TiToeuli cdis pirisgan miiReba qula. qula aris cdis piris mier

davalebis Sesrulebis gazomviT miRebuli ricxviTi mniSvneloba. qula aris

damokidebuli cvladi, romelic, Cveni molodinis Tanaxmad, damokidebulia imaze,

miiRo cdis pirma filmis naxvamde alkoholi Tu ara. yvela cdis pirisgan miRebuli

qula Seadgens eqsperimentis nedl monacemebs. am nedli monacemebis mixedviT unda

gadawyvitos mkvlevarma, axdens Tu ara damoukidebeli cvladi _ alkoholis miReba

gavlenas cdis piris mier filmSi nanaxi Zaladobis Sefasebaze.

procesi, romlis saSualebiTac askvnis eqsperimentatori, axdens Tu ara

damoukidebeli cvladi gavlenas damokidebul cvladze, ori etapisgan Sedgeba: _

pirveli etapia TiToeuli jgufis mier mocemuli Sefasebis aRwera; amisaTvis

iTvleba aRweriTi statistikebi, romlebic warmoadgenen TiToeul jgufSi Semavali

cdis pirebis mier mocemul tipur Sefasebebs. es etapi monacemTa aRweris zemoT


ganxiluli procesis identuria. Tqven, albaT, ukve icnobT iseT zogierT cnobil

aRweriT statistikas, rogoricaa moda, mediana an saSualo.

meore etapze irkveva: _ qulaTa ori jgufis aRweriT statistikebs Soris

miRebuli gansxvaveba aris Tu ara sakmarisad didi imisaTvis, rom is damoukidebeli

cvladis gavlenis Sedegad iqnes miCneuli. am etapze saWiroa statistikuri

hipoTezis Semowmebis gamoyeneba. statistikuri hipoTezis Semowmebis mizania,

gansazRvros or jgufs Soris SemTxveviTi gansxvavebis mosalodneli zoma.

eqsperimentSi ekvivalenturi jgufebi yovelTvis gansxvavdebian erTmaneTisgan

raime gauTvaliswinebeli niSniT, rasac SemTxveviT gansxvavebas vuwodebT.

SemTxveviTi gansxvaveba eqsperimentSi maSinac ki miiReba, rodesac damoukidebeli

cvladi ar axdens gavlenas damokidebul cvladze. statistikuri hipoTezis

Semowmebisas or jgufs Soris eqsperimentSi realurad miRebuli gansxvaveba

Sedardeba maT Soris mosalodnel SemTxveviT gansxvavebasTan. Tu iseTive didi

SemTxveviTi gansxvavebis miRebis albaToba sakmarisad mcirea, rogoricaa realurad

miRebuli gansxvaveba, maSin mkvlevari gadawyvets, rom miRebuli gansxvaveba ar aris

SemTxveviTi. amasTan, miRebuli gansxvaveba damoukidebeli cvladis gavlenas

miewereba.

eqsperimentirebisa da statistikuri hipoTezebis Semowmebis magaliTad Semdeg

kiTxvaze pasuxis gacemis procesi ganvixiloT: _ gqoniaT SemTxvevebi, rom xandaxan

dRis manZilze gasakeTebeli zogierTi saqme dagviwyebiaT? cxadia, TiToeul Cvengans

gvqonia aseTi SemTxveva, Tumca, sainteresoa, Tu ramdenad aumjobesebs dRis ganrigis

damaxsovrebas mnemonikuri teqnikebis gamoyeneba. harisma da blaiserma (Harris and

Blaiser, 1997) eqsperimentulad ikvlies es sakiTxi, risTvisac isini adamianebs dRis

manZilze Sesasrulebel saqmeTa damaxsovrebas sTxovdnen. kvlevis monawileebi

SemTxveviTad gayves or ekvivalentul jgufad da TiToeul maTgans 10 iseTi saqmis

damaxsovreba daavales, rogoricaa gaqiravebis punqtSi videokasetis dabruneba,

telefonze darekva da sursaTis maRaziaSi Sesvla. damaxsovrebidan 24 saaTis Semdeg

Seamowmes, Tu risi damaxsovreba SeZles maT. damoukidebel cvlads monawilisTvis

micemuli instruqciis forma, damokidebul cvlads ki _ TiToeul monawilis mier

gaxsenebuli saqmeebis raodenoba warmoadgenda. sakontrolo jgufSi Semaval

adamianebs ar eZleodaT instruqcia imis Taobaze, Tu rogor daemaxsovrebinaT

masala, eqsperimentul jgufs ki sityvebis erTmaneTTan dakavSirebis teqnikas

aswavlidnen, rac niSnavs imas, rom sityvas ricxvi eriTmeba, magaliTad, „erTi aris

merxi, ori aris doli, sami aris skami, oTxi _ noxi, xuTi _ yuTi, eqvsi _ nemsi,

Svidi _ nivTi, rva _ Tma, cxra _ ca, aTi _ Tafli“. riTmis daswavlis Semdeg

individi dasamaxsovrebeli saqmis dakavSirebas cdilobs vizualuri warmodgenis

xatTan; magaliTad, Tu I saqme gaqiravebis punqtSi video kasetis dabrunebaa, man


SeiZleba warmoidginos, rom kaseta merxze devs, xolo Tu II saqme telefonze

darekvaa, warmoidgonos, rom telefoni dolze dgas.

harisma da blaiserma ivaraudes, rom eqsperimentul jgufSi Semavali adamianebi

am teqnikis gamoyenebiT ufro met saqmes gaixsenebdnen, vidre sakonrolo jgufi.

eqsperimentuli jgufis mier gaxsenebul saqmeTa saSualo raodenoba 7.1, xolo

sakontrolo jgufis mier gaxsnebul saqmeTa saSualo raodenoba _ 4.1 iyo. imis

gasarkvevad, eqsperimentuli da sakontrolo jgufebis mier gaxsenebul saqmeTa

raodenobebs Soris sxvaoba ubralo SemTxveviTobaze metia Tu ara, harisma da

blaiserma dispersiuli analizi gamoiyenes, romelic aCvenebs ori jgufis mier

gaxsenebuli saqmeebis raodenobebis gansxvaveba mosalodnel SemTxveviT

gansxvavebaze metia Tu ara. gairkva, rom eqsperimentul jgufSi Semavali individebis

mier gaxsenebul saqmeTa raodenoba sandod aRemateboda sakontrolo jgufisas. es

Sedegi adasturebda kvlevis hipoTezas: _ Cven ukeT vimaxsovrebT gasakeTebel

saqmeebs sityvebis mibmis mnemonikuri teqnikis gamoyenebis SemTxvevaSi.

statistika, rogorc mecniereba, farTod gamoiyeneba bihevioristul mecnierebaSi,

Tumca, masSi Semavali cnebebis gasagebad saWiroa aRweriTi statistikis,

statistikuri daskvnisa da Sefasebis Teoriis codna, romelsac me-3-7 TavebSi

ganvixilavT. statistikur kriteriumebze me-8-13 TavebSi visaubrebT.


1. ganmarteT: _ cdis pirTa Soris sqema, SemTxveviTi gansxvaveba, damokidebuli

cvladi, ekvivalenturi jgufebi, damoukidebeli cvladi, nedli monacemebi,

kvlevis hipoTeza, qula, cdis piri.

2. baletSi ZiriTadi poziciebis swavlisas studentebs xSirad uWirT fexebis

swori poziciis miReba. aq erT-erTi problema isaa, rom fexebi TavisTavad

ixreba muxlebSi, rodesac quslebi mityupebulia da cerebi gareT iyureba.

am problemis dasaZlevad klarksonma, jemsma, uotkinsma da foleim (Clarkson,

James, Watkins, Foley, 1986) Caatares eqsperimenti, romelSic monawileobda

qalTa ori ekvivalenturi jgufi, romelTac ar hqondaT baletSi

mecadineobis gamocdileba. orive jgufs aswavles baletis ZiriTadi

poziciebi da daTvales fexebis moxris raodenoba. erT jgufs, cdis pirebis

gasakvirad, misces fexebis muxlebSi moxris ufleba, meore jgufs ki

araferi uTxres amis Taobaze. mkvlevrebma aRmoaCines, rom sacdeli

periodis manZilze pirvel jgufSi cdis pirebi mxolod 3,1 wamis manZilze

xridnen muxlebSi fexebs maSin, roca meore jgufSi (romelsac araferi

uTxres) es dro 21,8 wams Seadgenda. statistikurma Semowmebam daadastura,

rom muxlebSi fexebis moxris droSi gansxvaveba jgufebs Soris realuri

iyo da ara _ SemTxveviTi.


a) daasaxeleT damoukidebeli cvladi, romlis manipulirebasac

axdendnen klarksoni da sxvebi.

b) daasaxeleT damokidebuli cvladi, romelsac zomavdnen.

g) axseniT, ratom aris es kvleva eqsperimenti.

d) axseniT, ratom aris am eqsperimentSi statistikuri hipoTezis

Semowmeba aucilebeli.

korelacia da regresia

kvlevisas eqsperimentators xSirad surs

ara damoukidebeli cvladiT manipulireba,

aramed mxolod is ainteresebs,

ukavSirdeba Tu ara erTmaneTs anu

korelirebs Tu ara erTmaneTTan

individebis ori an meti maxasiaTebeli. amboben, rom ori cvladi korelirebs, Tu

erTi cvladis cvlileba ukavSirdeba meore cvladis SeTanxmebul sistematur

cvlilebas. magaliTad, Tu pirovnebis SfoTvis tipuri done dakavSirebulia mis mier

alkoholis miRebasTan ise, rom SfoTvis maRali done ukavSirdeba didi raodenobiT

alkoholis miRebas, xolo SfoTvis dabali done _ naklebi raodenobis alkoholis

miRebas, SegviZlia vTqvaT, rom SfoTva da alkoholis miReba korelaciaSia

erTmaneTTan.

imis dasadgenad, ori cvladi

korelaciaSia Tu ara, vzomavT adamianTa

SerCevas da TiToeuli individisgan

viRebT or qulas: _ SfoTvis qulas da

miRebuli alkoholis qulas. Semdeg

viTvliT korelaciis koeficients. korelaciis koeficienti aris statistika,

romelic ricxobrivad gviCvenebs maCvenebelTa ori jgufis dakavSirebulobis

xarisxs da am kavSiris mimarTulebas. am koeficientis mniSvneloba SeiZleba

icvlebodes -1.00-dan +1.00-mde.

statistikuri hipoTezis Semowmeba korelaciis koeficientTan mimarTebaSic

gamoiyeneba. yovelTvis arsebobs ori gansxvavebuli cvladis mniSvnelobebs Soris

SemTxveviTi kavSiris gamovlenis albaToba. amrigad, ibadeba kiTxva: _ aris Tu ara

korelaciis koeficientiT gamoxatuli, realurad miRebuli kavSiri mosalodnel

SemTxveviT kavSirze meti? am kiTxvaze pasuxs korelaciis koeficientis

statistikuri Semowmeba iZleva.

Tu davadgineT, rom

maCvenebelTa or jgufs

Soris kavSiri

korelacia (Covary): _ ori cvladi

korelaciaSi imyofeba erTmaneTTan,

rodesac erTi cvladis cvlileba

ukavSirdeba meore cvladis SeTanxmebul

sistematur cvlilebas.

korelaciis koeficienti (Correlation coefficient):

_ statistika, romelic ricxobrivad

gviCvenebs maCvenebelTa ori jgufis

dakavSirebulobis xarisxs da am kavSiris

mimarTulebas.

regresiuli analizi (Regression analysis): _ statistikuri

meTodebis gamoyeneba qulaTa erTi erTobliobis mixedviT

qulaTa meore erTobliobis Sesaxeb winaswarmetyvelebisTvis.


SemTxveviT kavSirze metia, SeiZleba SevecadoT erTi maCvenebelis mixedviT meoris

Sesaxeb winaswarmetyvelebas. Tu viciT pirovnebis SfoTvis maCvenebeli, SevZlebT

miRebuli alkoholis maCveneblis winaswarmetyvelebas? da Tu SegviZlia miRebuli

alkoholis maCveneblis winaswarmetyveleba, ramdenad zusti iqneba es

winaswarmetyveleba? erTi cvladis mniSvnelobis mixedviT meore cvladis

mniSvnelobis winaswarmetyveleba gulisxmobs regresiuli analizis gamoyenebas.

korelaciuri da regresiuli analizis teqnikebi farTod gamoiyeneba

bihevioristuli mecnierebis mraval sferoSi. sailustraciod ganvixiloT uoerisa

da qavelis (Woehr and Cavel, 1993) mier Catarebuli kvleva, romelic im faqtorebis

gamovlenas Seexeboda, rac, SesaZloa, fsiqologiis safuZvlebSi pirvel Sualedur

testSi naCveneb Sedegs ukavSirdebodes. mkvlevrebma 325 studentis iseTi

maCveneblebi Seiswavles, rogoricaa skolaSi miRebuli niSnebis saSualo, sagamocdo

masalis wakiTxvis raodenoba da kviris manZilze televizoris yurebaze daxarjuli

dro. Semdeg daiTvales TiToeuli dasaxelebuli maCveneblisa da pirvel Sualedur

testSi miRebul qulas Soris korelacia. skolaSi miRebuli niSnebis saSualosa da

Sualedur gamocdaSi miRebul qulas Soris korelaciis koeficienti +.31 gamovida

(anu rac ufro maRali iyo skolaSi miRebuli niSnebis saSualo, miT ufro maRali

iyo Sualedur SefasebaSi miRebuli qula); sagamocdo masalis wakiTxvaTa

raodenobasa da Sualedur gamocdaSi miRebul qulas Soris korelaciis

koeficienti _ +.22 (anu rac ufro metjer waikiTxa studentma masala, miT ufro

maRali qula miiRo), kviris manZilze televizoris yurebaze daxarjuli drosa da

Sualedur gamocdaSi miRebul qulas Soris korelaciis koeficienti ki _ _.13 (anu

rac ufro met dros xarjavda studenti kviris manZilze televizoris yurebaSi, miT

ufro naklebi qula miiRo). statistikuri Semowmeba aCvenebs, rom korelacia araa

ubralo SemTxveviTi kavSiri. avtorebi Tvlian, rom es Sedegebi, sxva monacemebTan

erTad, SesaZloa, sasargeblo aRmoCndes studentebisTvis sakuTari Sedegebis

gasaumjobeseblad. korelaciasa da regresias me-13 da me-14 TavebSi ganvixilavT.


1. ganmarteT: _ korelaciis koeficienti, korelacia, regresiuli analizi.

2. bevr adamians sjera iseTi paranormaluri fenomenebis arsebobis, rogoricaa

naTelxilva, eqstrasensoruli aRqma da fsiqokinezi. ukavSirdeba Tu ara aseTi

fenomenebis rwmenis xarisxi imas, Tu ra tipis ganaTleba miiRo individma

skolaSi? tobasikma, milerma da jonsma (Tobacyk, Miller & Jones) 193 me-11

klasels gauzomes paranormaluri fenomenebis arsebobis rwmena. maT, aseve,

aRnusxes TiToeuli moswavlis mier gavlili kursebis saerTo raodenoba.

rodesac gavlili samecniero kursebis raodenoba paranormaluri fenomenebis

arsebobis rwmenas daukavSires, maT Soris Sebrunebuli kavSiri aRmoCnda. rac


ufro meti samecniero kursi hqonda moswavles gavlili, miTYufro susti iyo

misgan paranormaluri movlenebis arsebobis rwmena.

a) axseniT, ratom aris saWiro am kvlevaSi korelaciuri statistika.

b) romeli ori sazomi gamoiyenes am kvlevaSi?

ratom aris saWiro statistika

zemoT warmogidgineT bihevioristul mecnierul kvlevaSi statistikuri analizis

gamoyenebis sami ZiriTadi meTodi, Tumca, Tqven SeiZleba ibuzRunoT da SegawuxoT

kiTxvam imis Taobaze, Tu risTvis aris statistikuri meTodebi ase aucilebeli da

saWiro bihevioristul mecnierebaSi. pasuxi sakmaod pirdapiria da ucvleli

TiToeuli warmodgenili statistikuri meTodisTvis: _ cxovrebis sasiamovno da

aRmafrTovanebeli faqtia, rom adamianebi mravali maxasiaTeblis xarisxiT

gansxvavdebian erTmaneTisgan. samyaro mosabezrebeli iqneboda TiToeuli adamiani

meoris identuri rom yofiliyo, Tumca, aseTi garemoebis upiratesoba statistikuri

meTodebis saWiroebis ararseboba iqneboda. ratom? ganvixiloT statistikis zemoT

warmodgenili sxvadasxvagvari gamoyeneba.

pirveli magaliTi individebisgan miRebuli raodenobrivi Sefasebis Sedegebis

aRweras Seexeba. yvela adamiani rom erTnairi yofiliyo, nebismieri erTi pirovnebis

SefasebiT miRebuli monacemebi aRwerda yvela danarCen adamians. magaliTad, yvela

maTganis Semosavali iqneboda $19.438 weliwadSi. populaciis Sesaxeb daskvnis

gasakeTeblad sakmarisi iqneboda mxolod erTi adamianis Sefaseba. Sesabamisad, ar

iqneboda aRweriTi statistikis saWiroeba, romelic warmoadgenda SerCevis wevrTa

tipur maCvenebels.

meore magaliTi Seexeboda statistikuri hipoTezis Semowmebas da, Sesabamisad,

pasuxis gacemas kiTxvaze: _ axdens Tu ara damoukidebeli cvladi gavlenas qcevaze.

warmoidgineT, ra advili iqneboda amis gansazRvra, TiToeuli individis qceva

nebismieri sxva individis qcevis identuri rom yofiliyo. davuSvaT, gvinda

gavarkvioT, amcirebs Tu ara konkretuli medikamenti guliscemis sixSires.

medikamentis gareSe yvela adamianis guliscemis sixSirea 60 dartyma wuTSi,

medikamentis miRebis SemTxvevaSi ki _ 55 dartyma wuTSi. kidev eWvqveSaa, amcirebs Tu

ara mocemuli medikamentis miReba guliscemis sixSires? vinaidan ar arsebobs

individidan individze cvalebadobis Sansi, ueWveli iqneba, rom medikamenti amcirebs

guliscemis sixSires, statistikuri hipoTezis Semowmeba ki aRar dagvWirdeba.

mesame magaliTi korelaciisa da regresiuli analizis problemas Seexeba. aris Tu

ara ori qceva erTmaneTTan kavSirSi da Tu aris, ramdenad SegviZlia erTi qcevis

ganxorcielebis prognozireba meore qcevis ganxorcielebis mixedviT? es kiTxva

mxolod imitom Cndeba, rom adamianebi gansxvavdebian erTmaneTisgan da

sxvadasxvagvarad iqcevian erTsa da imave situaciaSi. TiToeuli individi yvela sxva


individis identuri rom yofiliyo, yvela qceva srulad prognozirebadi iqneboda. am

SemTxvevaSi ar iqneboda cvalebadoba individidan individze da srulyofilad

SevZlebdiT yvela qcevis winaswarmetyvelebas, korelaciuri da regresiuli

analizis gamoyenebis gareSe.

erTi individidan meoreze cvalebadobis ararsebobis pirobebSi sruliad

moxsnili iqneboda statistikuri analizis saWiroeba, magram ramdenad mosawyeni

iqneboda es samyaro. sabednierod, Cveni samyaro ar aris aseTi; Tumca,

kanonzimierebebiTa da cvalebadobiT savse cxovreba statistikur meTodebsac

saWiroebs, raTa aRiweros da aixsnas qceva.

Sejameba

• statistika efuZneba meTodebsa da procedurebs, romlebic gamoiyeneba

monacemebis Sejamebis, analizisa da maTze dayrdnobiT daskvnebis

gamosatanad.

• statistika (statistikuri sidide) aris monacemebze dayrdnobiT gamoTvlili

ricxvi, romelic SeiZleba gamoyenebul iqnes SerCevidan miRebuli qulebis

aRsawerad.

• statistika, aseve, SeiZleba gamoyenebul iqnes populaciis parametris Sesaxeb

daskvnis gasakeTeblad.

• statistikuri hipoTezis Semowmeba gamoiyeneba imis gansasazRvrad,

eqsperimentSi or jgufs Soris miRebuli gansxvaveba aris Tu ara sakmarisad

didi imisaTvis, rom miCneul iqnes damoukidebeli cvladis moqmedebis

Sedegad da ara _ SemTxveviTad miRebulad.

• korelaciuri statistika gamoiyeneba imis gansasazRvrad, arsebobs Tu ara

kavSiri qulaTa or jgufs Soris.

• regresiuli analizi gamoiyeneba qulaTa erTi jgufis mixedviT qulaTa

meore jgufis Sesaxeb prognozirebisTvis.

ZiriTadi terminebi da simboloebi

aRweriTi statistika

damokidebuli cvladi

damoukidebeli cvladi

ekvivalenturi jgufebi

kvlevis hipoTeza

korelacia

korelaciis koeficienti

monacemebi

nedli monacemebi

parametri

populacia

regresiuli analizi

statistika (mecniereba)

statistika (statistikuri sidide)

statistikuri daskvna

statistikuri hipoTezis Semowmeba

qula

SemTxveviTi gansxvaveba

SerCeva

cdis pirebi

cdis pirTa Soris sqema

cvladi

kiTxvebi


1. daasaxeleT oTxi movlena an aqtivoba Tqvens cxovrebaSi, romelic

kanonzmierebiT xasiaTdeba (magaliTad, drois Cveuli raodenoba, romelsac

uTmobT Zils an yoveldRiuri samuSao saaTebis raodenoba).

2. moifiqreT da daasaxeleT sami sxvadasxva populaciis magaliTi. ra saerTo

maxasiaTebliT erTiandebian TiToeul maTganSi wevrebi?

3. qriusma, Sirefsma, tomasma, krahenbalma da helfriCma (Crews, Shirreffs, Thomas,

Krahenbuhl & Helfrich, 1986) gazomes 23 profesionali golfis moTamaSe qalis

sxeulSi cximis Semcveloba. miRebuli Sedegebi maT daukavSires golfis

moTamaSeTa mier erTi wlis manZilze SejibrebSi miRebul saSualo qulebs

da aRmoaCines, rom maT SeeZloT golfSi miRebuli qulebis prognozireba

sxeulSi cximis Semcvelobis odenobis mixedviT. sxeulis cximis nakleb

procentul Semcvelobas ukavSirdeboda golfSi dabali saSualo qula

maSin, roca sxeulis cximis Semcvelobis ufro maRali maCvenebeli

dakavSirebuli iyo golfSi maRal saSualo qulasTan.

a) am TavSi ganxiluli romeli statistikuri meTodi gamoiyenes am

kvlevaSi?

b) axseniT, risTvis iyo saWiro am kvlevaSi korelaciuri statistika.

g) romeli ori sazomi iyo gamoyenebuli am kvlevaSi?

4. kavasakis sindromi bavSvebis davadebaa, romlis simptomebia maRali sicxe,

damskdari tuCebi, dawiTlebuli ena, xelisgulebisa da fexisgulebis

siwiTle da gamonayari. rogorc wesi, am daavadebas aspiriniT mkurnaloben,

Tumca, ukanasknelma kvlevebma aCvena, rom aspirinze gemaglobulinis

damateba SeiZleba ufro efeqturi iyos, vidre mxolod aspirini. kvlevis am

hipoTezis Sesamowmeblad niuburgerma da sxvebma (Newburger et al, 1986) 168

bavSvi SemTxveviTad gaanawiles or jgufSi: _ erTi jgufi iRebda mxolod

aspirins, meore ki _ aspirinze damatebul gemaglobulins. maT aRmoaCines,

rom mkurnalobis pirveli dRis Sedegad im bavSvebis temperatura,

romlebic aspirinze damatebul gemaglobulins iRebdnen, saSualod 2.3°F-iT

(0.26°C) Semcirda maSin, roca meore jgufSi (anu im bavSvebTan, vinc mxolod

aspirins iRebda) es cvlileba mxolod 0.8°F (0.09°C) iyo. statistikurma

Semowmebam aCvena, rom am or jgufs Soris temperaturis Semcirebis

mixedviT gansxvaveba ar iyo SemTxveviTi; aspirinze damatebuli

gemaglobulinis miRebis Sedegad ufro metad Semcirda temperatura, vidre

mxolod aspirinis miRebisas.


kvlevaSi?

b) daasaxeleT, romeli damoukidebeli cvladis manipulirebas axdendnen

niubergeri da sxvebi?


g) daasaxeT damokidebuli cvladi.

d) axseniT, ratom aris es kvleva eqsperimenti.

e) axseniT, risTvis aris saWiro statistikuri hipoTezis Semowmeba am

eqsperimentSi.

5. jakobi (Jacoby, 2000) aRniSnavs, rom amerikelTa erovnuli satelefono

gamokiTxvis Sedegebis mixedviT, mamakacebis 72%-s da qalebis 58%-s surs,

materialurad uzrunvelyofili iyos. respondentTa 19% miuTiTebs, rom

fuliT SesaZlebelia bednierebis yidva, 56%-ki sjera, rom rac ufro meti

fuli gaqvs, miT naklebad stresulia cxovreba.


kvlevaSi?

b) axseniT, risTvis aris saWiro aRweriTi statistikis gamoyeneba am

kvlevis Sedegebis Sesajameblad.

g) romeli aRweriTi statistika gamoiyenes am kvlevaSi?

6. miuTiTeT, ras aRwers qvemoT mocemuli TiToeuli debuleba: _ nedl

monacemebs, statistikas Tu daskvnas populaciis parametris Sesaxeb.

a) linda samsaxurSi misvlas yovel dRe 36 wuTs andomebs.

b) saSualo amerikeli mamakaci mgzavri yoveldRiurad 42 wuTs

transportSi atarebs.

g) 75 mgzavrisgan Semdgari SerCeva yoveldRiurad saSualod 46 wuTs

mgzavrobs.

d) benJamin da anita hiustonebs sabanko angariSze $5126 aqvT Senaxuli.

e) tipur amerikel mozrdils sabanko angariSze $6024 aqvs Senaxuli.

v) samxreTis patara qalaqis SemTxveviTad SerCeuli 50 moqalaqes sabanko

angariSze saSualod $5692 aqvs Senaxuli.

7. moifiqreT magaliTi, romelSic erTi cvladis quliT SeZlebT meore

cvladis qulis winaswarmetyvelebas (magaliTad, qolesterinis done,

romliTac SesaZlebelia gulis daavadebis Sansis prognozireba).

Tavi II. mecnieruli kvleva

ra aris mecniereba

mecnieruli kvlevis kiTxvebi

kvlevis meTodebi

operaciuli gansazRvrebebi

gazomva

gazomvis skalebi

Tvisebrivi da raodenobrivi monacemebi

diskretuli (wyvetili) da uwyveti cvladebi

თავი II. მეცნიერული კვლევა 2

statistikuri meTodebi aTaswleulebis manZilze viTardeboda praqtikuli

problemebis gadasaWrelad. sityva „statistika“ laTinuri status-idan anu

sazogadoebrivi mdgomareobidan (State) momdinareobs; statistikis adreuli gamoyeneba

saxelmwifos Tu imperiis Sesaxeb codnis miRebas isaxavda miznad. romaeli

imperatorebi yovelTvis awarmoebdnen axali dapyrobili teritoriebis aRweras.

aRweriT miRebul informaciaze dayrdnobiT, romelic warmoadgenda mosaxleobisa

da maT mflobelobaSi arsebuli sakuTrebis statistikas, wesdeboda gadasaxadebi.

miuxedavad imisa, rom statistika am miznisTvis dResac gamoiyeneba, misi farglebi

gafarTovda da moicavs meTodebs, romlebic mowodebulia adamianebisa da

cxovelebis qcevis SeswavlisTvis.

ra aris mecniereba

mecniereba codnis SeZenis saSualebaa, romelic kiTxvis dasmas, am kiTxvaze

savaraudo pasuxis gacemas kvlevis hipoTezis formiT, hipoTezis empiriulad

SemowmebisTvis kvlevis sqemis Camoyalibebas, monacemebis Segrovebas, monacemTa

analizs da monacemTa analizze dayrdnobiT hipoTezis Sesaxeb daskvnis gamotanas

moicavs.


mecnieruli kiTxva empiriuli monacemebis

mogrovebis gziT pasuxis miRebis saSualebas

iZleva. sityva „empiriuli“ sensorul

gamocdilebas an dakvirvebas gulisxmobs.

amrigad, empiruli monacemebi aris sistematur dakvirvebasa da sensorul

gamocdilebaze dafuZnebuli gazomvis Sedegad miRebuli qulebi. I TavSi ganxiluli

TiToeuli magaliTi mecnierul kiTxvebs efuZneboda da empiriuli monacemebis

mogrovebis gziT pasuxis miRebis SesaZleblobas iZleoda.

dasmul kiTxvaze eqsperimenti xSirad savaraudo pasuxs kvlevis hipoTezis saxiT

gvTavazobs. I TavSi kvlevis hipoTeza ganvsazRvreT, rogorc debuleba or an met

cvlads Soris mosalodneli an savaraudo kavSiris Sesaxeb. uoletma da ioSim

(Ouellet and Joshi, 1986) gamoTqves hipoTeza, rom martooba da depresia

urTierTdakavSirebuli maxasiaTeblebia. vinaidan kvlevis hipoTezaSi mocemulia

cvladebi da miTiTebulia maT Soris mosalodneli kavSiris ragvaroba, is

mimarTulebas aZlevs mkvlevars kvlevis sqemis SerCevisas da monacemTa

mogrovebisas.


kvlevis hipoTezis Sesamowmeblad empiriuli

monacmebis mogroveba saWiroebs kvlevis meTods.

empiriuli monacemebi (Empirical data): _

dakvirvebasa da sensorul

gamocdilebaze dafuZnebuli gazomvis

Sedegad miRebuli qulebi.

kvlevis meTodi (Research method): _

midgoma monacemebis mogrovebisadmi.


kvlevis meTodi aris midgoma, romelsac mecnieri iyenebs monacemTa mosagroveblad.

bihevioristul mecnierebebSi mravali meTodi gamoiyeneba. aq ZiriTadi meTodebidan

mokled mimovixilavT mxolod oTxs. TiToeulis SemTxvevaSi mogrovili monacemebis

sistematizaciisa da analizisTvis gamoiyeneba statistika.

bunebrivi dakvirveba

bunebrivi dakvirveba gulisxmobs bunebriv

pirobebSi ganxorcielebul qcevebze, situaciaSi

Carevis gareSe, dakvirvebas. bunebrivi dakvirvebis

magaliTad SeiZleba movitanoT trinkosis (Trinkaus,

1982, 1983, 1988, 1993) kvleva, romelic 10 wlis

manZilze sistematurad akvirdeboda mZRolebis qcevas erT gzajvaredinze. erT-erTi

misi aRmoCenaa, rom dakvirvebis periodSi Semcirda SuqniSnis wiTel Suqze

damorCileba. am dakvirvebisas eqsperimentatori praqtikulad ar ereoda procesSi;

mZRolebma arc ki icodnen, rom vinme akvirdeboda wiTeli Suqisadmi maT

damorCilebas. maT rom scodnodaT amis Sesaxeb, mosalodneli iqneboda maTi qcevis

Secvla.

saarqivo Canawerebi

saarqivo Canawerebi gulisxmobs kiTxvebze

pasuxis gacemas arsebuli Canawerebis gamoyenebiT.

sazogadoebebSi, rogorc wesi, grovdeba

mravalferovani informacia maTi wevrebis Sesaxeb:

_ dabadebis, qorwinebisa da gardacvalebis Canawerebi; kriminaluri angariSebi,

mosaxleobis aRweris Canawerebi da a.S. es arsebuli Canawerebi SeiZleba mecnierul

kiTxvaze pasuxisTvis saWiro monacemebs iZleodnen. magaliTad, londonelma

maRaziis mflobelma jon grantma (John Grant, 1620-1674) mamakacebisa da qalebis

sikvdili mizezebSi gansxvavebebis dasadgenad dabadebisa da gardacvalebis

Canawerebi gamoiyena. uaxloes warsulSi zusne (Zusne, 1986-87) dainteresebuli iyo

Semdegi sakiTxiT: _ SeuZliaT Tu ara sikvdilis piras myof individebs sikvdilis

gadavadeba. man Seiswavla 3000-ze meti adamianis dabadebisa da gardacvalebis

Canawerebi, romelic aCvenebda, rom es SesaZlebelia, Tumca, is gansxvavdeba qalebTan

da mamakacebTan.

meore magaliTi: tomsonma (Thomson, 1993) gaaanaliza aSS-is janmrTelobis

statistikis nacionaluri centris monacemebi mkvlelobaTa sixSiris Sesaxeb. am

kvlevam aCvena, rom mkvlelobaTa ricxvi yvelaze metia kolumbiis olqSi; aq

gardacvalebis yoveli 1000 SemTxvevidan daaxloebiT 39 SemTxveva mkvlelobaa. es

ricxvi yvelaze naklebia aiovaSi, sadac 1000 SemTxvevidan mxolod 2 mkvlelobaa.

bunebrivi dakvirveba (Naturalistic

observation): _ kvleva, romelic

gulisxmobs bunebriv pirobebSi

ganxorcielebul qcevebze

dakvirvebas.

saarqivo Canawerebi (Archival recodrs):

_ kvleva, romelSic gamoiyeneba

arsebuli Canawerebi.


gamokiTxva

gamokiTxva gulisxmobs monacemebis

miRebas adamianebis zepiri an werilobiTi

interviuebiT miiReba. gamokiTxva farTod

aris gavrcelebuli sazogadoebaSi. Tu Tqven erTxel mainc SegivsiaT restoranSi

baraTi, romelSic mocemuli iyo kiTxvebi sakvebisa Tu momsaxurebis xarisxis

Sesaxeb an Tu erTxel mainc SegivsiaT produqtis sagarantio furceli, sadac

mocemuli iyo kiTxvebi Tqveni Semosavlis, ojaxuri mdgomareobisa da hobis Sesaxeb,

Tqven monawileoba migiRiaT gamokiTxvaSi. gamokiTxvis kidev erTi cnobili

magaliTia arCevnebze xmis micemis procedura.

eqsperimenti

eqsperimenti aris zedmiwevniT

gakontrolebuli situacia, sadac mecnieri

axdens erTi an meti damoukidebeli cvladis

manipulirebas da akvirdeba, Tu ra gavlenas

axdens es damokidebul cvladze. magaliTad,

harisma da blaiserma (1997, ix. gv. XX)

operacionalurad gansazRvres mnemonikuri teqnika, rogorc cdis pirebisTvis

sityvis da misi Sesatyvisi riTmis mqone ricxvebis Sewyvilebis swavlebaa. Amis

mixedviT, „erTi aris merxi, ori aris doli, sami aris skami, oTxi _ noxi, xuTi _

yuTi, eqvsi _ nemsi, Svidi _ nivTi, rva _ Tma, cxra _ ca, aTi _ Tafli“. amrigad, Tu

sxva mecnieri moisurvebs am kvlevis gagrZelebas, mas ukve ecodineba, Tu rogor

gansazRvres harisma da blaiserma mnemonikuri teqnika.

operaciuli gansazRvrebebi

sxvadasxva kvlevis sqemebis gamoyenebiT

miRebuli mecnieruli codna ganmeorebadi

unda iyos da mis Sesaxeb unda

Tanxmdebodnen sxvadasxva damkvirveblebi.

mecnieruli codna qcevis Sesaxeb maSin iZens

Zalas da xdeba yvelasTvis misaRebi,

rodesac sxva mecnierebs SeuZliaT am qcevis gameoreba sxvadasxva monawileebTan

sxvadasxva garemoSi da sxvadasxva adgilze. codna rom mecnieruli gaxdes, srulad

unda iyos cnobili masTan dakavSirebuli mosazrebebi da Sedegebi ise, rom sxvas

SeeZlos Sedegebis gameoreba, gakritikeba da gavrcoba. kvlevis Sesaxeb informaciis

gadacemis procesSi gadamwyveti nabijia im operaciebis zustad gansazRvra, romelic

gamoyenebul iqna dakvirvebis, damoukidebeli cvladebiT manipulirebisa Tu

damokidebuli cvladis gazomvis dros. am process mecnierebi kvlevaSi gamoyenebuli

procedurebis operaciulad gansazRvras uwodeben. erT-erT kvlevaSi uoletma da

gamokiTxva (Survey research): _ kvleva,

romelSic monacemebi adamianebis zepiri

an werilobiTi interviuebiT miiReba.

eqsperimenti (Experiment): _ zedmiwevniT

gakontrolebuli situacia, sadac

mecnieri axdens erTi an meti

damoukidebeli cvladis manipulirebas

da akvirdeba, Tu ra gavlenas axdens es

damokidebul cvladze.

operaciuli gansazRvreba (Operational

definition): _ dakvirvebisas, damoukidebeli

cvladiT manipulirebisas an

damokidebuli cvladis gazomvisas

gamoyenebuli operaciebis detaluri

aRwera.


ioSim operaciulad gansazRvres martooba, rogorc martoobis skalaze cdis piris

mier miRebuli qula (Russell, Peplau & Cutrona, 1980). amrigad, Tu romelime mecniers

moesurveba am kvlevis gagrZeleba, man unda icodes, Tu rogor gazomes martooba

uoletma da ioSim.

SeamowmeT Tqveni codna II.1

1. ganmarteT: _ saarqivo Canawerebi, empiriuli monacemebi, eqsperimenti,

bunebrivi dakvirveba, operaciuli gansazRvreba, kvlevis meTodi, kvlevis

hipoTeza, gamokiTxva.

2. qvemoT mocemuli TiToeuli kiTxvisTvis daasaxeleT kvlevis meTodi,

romelic, Tqveni azriT, unda gamoiyenon masze pasuxis gasacemad.

a) ras fiqrobs mocemuli dasaxlebuli punqtis mosaxleoba dasaxlebis

teritoriaze narCenebis gadamamuSavebeli qarxnis funqcionirebis

Sesaxeb, romelmac SeiZleba safrTxe Seuqmnas maT janmrTelobas?

b) rogor Seicvleba cdis pirebis damokidebuleba imis mixedviT, Tu ra

tipis _ SiSis gamomwvev Tu informacis matarebel _ Setyobinebas

gadavcemT?

g) saSualod ra asakSi uCndebaT arizonel mamakacebs pirveli Svili?

d) ramdenad xSirad TamaSoben 4 wlis bavSvebi sami bavSvisgan Semdgar

jgufebSi sabavSvo baRebSi?

gazomva

mecnierul kiTxvaze pasuxis

gasacemad saWiroa gazomva. gazomva

aris garkveuli wesebis mixedviT

cvladebisTvis ricxviTi mniSvnelobebis miniWebis procesi. rogorc wesi, gazomvis

Sedegi miekuTvneba oTxidan erT-erT skalas. es skalebia: _ saxeldebis, rigis,

intervalebisa da Sefardebis skalebi.

gazomvis skalebi

saxeldebis skala

saxeldebis skala aris gazomili

cvladis sxvadasxva kategoriebSi

gaerTianeba. magaliTad, iseTi cvladi

rogoricaa individis eTnikuri warmomavloba, saxeldebis skalaze izomeba; am

cvladis mixedviT aSS-Si mcxovrebi adamianebi SeiZleba gavaerTianoT kategoriebSi:

_ aborigeni, azieli amerikeli, afroamerikeli, TeTrkaniani an espanuri warmoSobis

amerikeli da Semdeg TiToeul kategorias mivaniWoT pirobiTi nomeri (aqedan

momdinareobs dasaxeleba saxeldebis, romelic niSnavs „saxelis darqmevas“),

gazomva (Measurement): _ garkveuli wesebis

mixedviT cvladebisTvis ricxviTi

mniSvnelobebis miniWeba.

saxeldebis skala (Nominal measurement): _

gazomili cvladis sxvadasxva kategoriebSi

gaerTianeba.


magaliTad, 1 = aborigeni, 2 = azieli amerikeli, 3 = afroamerikeli, 4 = TeTrkaniani,

5 = espaneli warmoSobis amerikeli. erTsa da imave kategoriaSi gaerTianebul yovel

individs mieniWeba erTi da igive nomeri. cxadia, es nomeri ar aris ricxobrivi

mniSvnelobis matarebeli; igive warmatebiT kategoriebis aRsaniSnad cifrebis

nacvlad SesaZlebelia asoebis gamoyeneba. bihevioristuli mecnierebis mravali

warmomadgeneli saxeldebis skalas ar miiCnevs realur sazomad, radgan

kategoriebisTvis miniWebuli ricxvebi mxolod identifikaciis funqcias asruleben;

isini ar iZlevian raodenobriv informacias gazomili cvladis Sesaxeb.

rigis skala

rodesac cvladis sidide

mniSvnelobebis mixedviT aris

dalagebuli skalaze, es niSnavs, rom

gamoyenebulia rigis skala. magaliTad, adamianebs SeiZleba vTxovoT

multiplikaciuri filmis seriebis dalageba skalaze ufro sasacilodan naklebad

sasacilomde. xSirad ricxviTi mniSvneloba 1 eniWeba stimuls an individs,

romelsac gasazomi maxasiaTebeli yvelaze didi odenobiT aqvs, mniSvneloba 2 _ im

stimuls an individs, romelsac aReniSneba mocemuli maxasiaTeblis Semdegi yvelaze

didi odenoba da a.S. magaliTad, Tu SfoTva rigis skalaze izomeba, individs,

romelic yvelaze met SfoTvas avlens, mieniWeba 1; individs, romelic avlens Semdeg

yvelaze met SfoTvas, eniWeba 2, Semdegs _ 3 da a.S. ase dalagdebian individebi ufro

mSfoTvaredan naklebad mSfoTvaremde.

rigis skala, faqtobrivad, ranJirebas axdens, sadac viRaca an raRaca, romelsac

mieniWa rangi 1, ufro metad xasiaTdeba mocemuli cvladiT, vidre is, visac mieniWa

rangi 2; Tumca, rigis skala ar iZleva imis gansazRvris saSualebas, Tu ramdenad

gansxvavebulia gazomili cvladi rangebs Soris. gazomili cvladis mixedviT

SeiZleba mciredi gansxvaveba iyos 1 da 2 rangebs Soris, magram es gansxvaveba

SeiZleba gacilebiT didi iyos 2 da 3 rangebs Soris. magaliTad, maraTonSi

monawile pirveli sami morbenalis dro SeiZleba iyos 3 sT da 19 wT, 3 sT da 20 wT

da 3 sT da 36 sT, Sesabamisad. finiSze am morbenalTa rangebi iqneba 1, 2 da 3;

Tumca, droSi gansxvaveba pirvel da meore morbenals Soris iqneba 1 wT maSin, roca

meore da mesame morbenals Soris gansxvaveba 16 wT-s Seadgens.

rigis skala ar aCvenebs gazomili cvladis sidides; is iZleva informacias

mxolod am cvladis mixedviT individebis Tu sagnebis Tanmimdevrobis (rigis)

Sesaxeb. ver vityviT, rom rangi 1 aucileblad ukavSirdeba gazomili cvladis met

odenobas; verc imas vityviT, rom umciresi rangis mqone individs an sagans yvelaze

naklebad axasiaTebs gazomili maxasiaTebeli. individma, romelsac moeTxoveboda

multiplikaciuri filmis seriebis dalageba ufro sasacilodan naklebad

sasacilomde, SeiZleba verc erTi maTgani ver aRiqvas sasacilod; amasTan, am

rigis skala (Ordinal measurement): _ skalaze

cvladis sidide dalagebulia mniSvnelobebis

mixedviT.


individis mier multiplikaciuri filmisTvis miniWebuli rangi 1 imave informaciis

matarebelia, romelic gaaCnia rangs 1, romelic mianiWa individma, romelmac yvela

seria Zalze sasacilod aRiqva. miuxedavad damaxasiaTebeli naklovanebebisa, rigis

skala pirveli nabijia ufro zusti sazomebis ganviTarebisken. magaliTad, autizmi

ganviTarebis darRvevaa, romelic gavlenas axdens bavSvis metyvelebis ganviTarebaze,

azrovnebis procesze da socialur da fizikur gamRizianeblebze reagirebis unarze.

naklebad aris nakvlevi, Tu rogor SeiZleba gansxvavdebodnen erTmaneTisgan

autizmis diagnozis mqone bavSvebi. am mimarTulebiT pirveli nabiji iyo bavSvis mier

ganxorcielebeuli autisturi qcevis xarisxis sazomi rigis skala, romelic

dihofma, heznekerma brosviCma, karpenterma da hofmanma (Dihoff, Hetznecker, Brosvic,

Carpenter & Hoffman, 1993) Seqmnes. am skalaze izomeba qcevis eqvsi komponenti: _

metyvelebis unari, garemoze reagirebis unari, socialur gamRizianeblebze

reagirebis unari, fizikur gamRizianeblebze reagirebis unari, masalebis

gamoyenebisadmi adaptaciis unari da kognituri funqcionireba. TiToeuli

komponentis farglebSi gamoiyo funqcionirebis xuTi done. es doneebi rigis

skalaze gansxvavdeba erTmaneTisgan da dalagebulia funqcionirebis umdablesi

donidan umaRles donemde. magaliTad, metyvelebis ganviTarebis komponentis

farglebSi es doneebia:

• araferi

• Jestebi

• komunikaciis survili

• komunikaciis survili da sityvebi

• sxva mimarTuli komunikacia

Jestebi miuTiTeben ufro metad ganviTarebul metyvelebaze, vidre metyvelebis

saerTod ganuviTarebloba, komunikaciis survili da sityvebi ki _ ufro metad

ganviTarebul metyvelebaze, vidre mxolod Jestebi, magram Cven ar viciT, ramdenad

didia gansxvaveba kategoriebs Soris; amitom es rigis skalaa.


1. ganmarteT: _ skala, saxeldebis skala, rigis skala.

2. CamoTvaleT gazomvis oTxi skala.

3. ra nakli aqvs saxeldebis skalas?

4. ra informacias gvawvdis individisTvis miniWebuli rangi 2 miRwevis

motivaciis skalaze?

5. depresiisken midrekilebis sklaze (Zlierad midrekilidan naklebad

midrekilisken) 13 individisgan Semdgar jgufSi jenifers mieniWa rangi 1,

debis _ 2, anas _ 12 da martis _ 13.


a) axseniT, ratom ver daaskvniT, rom jeniferi midrekilia depresiisken

da rom marti _ saerTod ara.

b) SeiZleba davaskvnaT, rom depresiisken midrekilebiT jeniferi da debi

iseve gansxvavdebian erTmaneTisgan, rogorc ana da marti? axseniT

Tqveni pasuxi.

intervalebis skala

Tu dakmayofilebulia rigis skalis

moTxovnebi da, amasTan miniWebul

ricxvebs Soris gansxvaveba asaxavs

cvladis mniSnelobebs Soris igive

sididis sxvaobas, maSin saqme gvaqvs

intervalebis skalasTan, Tumca,

intervalTa skalas ar aqvs absoluturi nuli, sadac saerTod ar iqneba mocemuli

gasazomi cvladi. intervalTa skalaze nulovani mniSvneloba pirobiTad aris

arCeuli sawyis wertilad. is SeiZleba Canacvldes nebismieri sxva mniSvnelobiT da

am mniSvnelobidan daviwyoT aTvla.

intervalebis skalis magaliTebia farenhaitisa da celsiusis temperaturis

skalebi. nulovani maCvenebeli arc erTze ar niSnavs temperaturis saerTod arqonas.

arc erT am skalaTagans ar aqvs absoluturi nuli, radgan temperatura SeiZleba

nul gradusze qvemoTac daeces. swori iqneba, Tu vityviT, rom temperaturis

mixedviT iseTive sxvaobaa 20°C-sa da 40°C-s Soris, rogoric 40°C-sa da 60°C-s Soris,

magram ar iqneba swori, Tu vityviT, rom 40°C orjer ufro maRalia, vidre _ 20°C.

naTeli gaxdeba, Tu ratom ar SegviZlia amis Tqma, rodesac SevadarebT celsiusisa

da farengeitis skalebze gazomil Tanabar temperaturebs:

Tanabari temperaturebi

°C °F

20

40

60

68

104

140

yuradReba miaqcieT, rom 20°C-iT sxvaoba yovelTvis tolia 36°F-iT sxvaobisa,

magram vinaidan TiToeuli skala pirobiTi nulovani wertilidan iwyeba, erT

skalaze erTmaneTis gverdigverd mdebare qulebis fardoba ar udris meore skalis

erTmaneTis gverdigverd mdebare qulebis fardobas. magaliTad, Sefardeba

40°C/20°C=2.0, xolo Sesabamisi temperaturebis fardoba 104°F/68°F=1.5.

bihevioristul mecnierebSi skalebi intervalebis dones zustad iSviaTad

Seesabameba, Tumca, Sefasebis skalebi, romelTac xSirad iyeneben adamianebis iseTi

intervalebis skala (Interval measurement): _

cvladis mniSvnelobebi dalagebulia skalaze

da miniWebul ricxvebs Soris gansxvaveba

asaxavs cvladis mniSnelobebs Soris erTi da

igive sididis sxvaobas. intervalTa skalaze

nuli pirobiTad sawyisi wertilia.


maxasiaTeblebis Sesafaseblad, rogoricaa pirovnuli maxasiaTeblebi,

damokidebulebebi, ojaxuri bedniereba, sagnebis SiSi, miRwevis motivacia an

lideroba, xSirad gamoiyeneba, rogorc intervalebis skala. rogorc wesi, am

skalebSi warmodgenilia debuleba da pasuxebis skala ori ukiduresi polusiTa da

maT Soris moTavsebuli ramdenime wertiliT. magaliTad, damokidebulebebis skalaze

individs SeiZleba moeTxovebodes, rom upasuxos debulebas: _ „iaraRis tarebis

makontrolebeli Zlieri kanoni amcirebs danaSauls“ da aRniSnos Tavisi pasuxi aseT

skalaze:

sruliad

veTanxmebi

veTanxmebi ver gadamiwyvetia ar veTanxmebi sruliad ar

veTanxmebi

sxva formis Sefasebis skalaze SeiZleba mogeTxovebodeT pirovnebis Sefaseba

bipolaruli zedsarTavi saxelebiT:

aragulwrfeli _____ : ______ : ______ : ______ : ______ : ______ : ______ gulwrfeli

mimzidveli _____ : ______ : ______ : ______ : ______ : ______ : ______ aramimzidveli

civi ______ : ______ : ______ : ______ : ______ : ______ : ______ cxeli

Sefasebis skalebis or ukidures poluss Soris SeiZleba nebismieri raodenobis

kategoria moTavsdes; Sesabamisad, am skalebs ewodebaT 5-quliani, 7-quliani, 9-

quliani da a.S. imisda mixedviT, Tu ramdeni kategoriaa masze mocemuli. xSirad

skalis polusebze moTavsebuli zedsarTavi saxelebis prioritetuloba inversiulia

debulebaTa naxevrisTvis. amrigad, prioritetul zedsarTav saxelTa naxevari

skalis marcxena nawilSi Tavsdeba, xolo meore naxevari _ marjvena nawilSi, rac

Tavidan gvaridebs pasuxebis gacemisas wamoWril iseT problemebs, rogoricaa

mxolod skalis marjvena mxares moTavsebul debulebaTa gadasinjva.

mas Semdeg, rac pirovneba gascems pasuxebs, skalis TiToeul kategorias eniWeba

ricxobrivi mniSvneloba:

aragulwrfeli _____ : ______ : ______ : ______ : ______ : ______ : ______ gulwrfeli

1 2 3 4 5 6 7

mimzidveli _____ : ______ : ______ : ______ : ______ : ______ : ______ aramimzidveli

1 2 3 4 5 6 7

civi ______ : ______ : ______ : ______ : ______ : ______ : ______ cxeli

1 2 3 4 5 6 7

xSirad mocemulia ramdenime skala da cdis piris saboloo, erTi qulis

misaRebad saWiroa calkeul skalebze miRebul qulaTa dajameba. aseT skalebs


jamuri Sefasebis skalebs uwodeben. magaliTad, rotonma da kelim (Rotton & Kelly, 1985)

Seqmnes e.w. mTvaris efeqtis rwmenis skala, romliTac fasdeboda qcevaze savse

mTvaris gavlenis rwmena. skala cxra debulebisgan Sedgeba. respondenti irCevs

ricxvs 1-dan (sruliad ar veTanxmebi) 9-mde (sruliad veTanxmebi) da amiT miuTiTebs,

Tu ramdenad eTanxmeba TiToeul debulebas. magaliTad, skalis erT-erTi debuleba1

aseTia:

kargi azria savse mTvaris dros saxlSi darCena.

1 2 3 4 5 6 7 8 9

sruliad ar

veTanxmebi

arc

veTanxmebi,

arc ar

veTanxmebi

sruliad

veTanxmebi

Sefasebis skalisTvis saWiro saboloo qula miiReba cxrave debulebaze miRebuli

qulebis dajamebiT. qulaTa SesaZlebeli diapazonia 9-dan (Tu yoveli debuleba

Sefasebulia 1 quliT) 81 qulamde (Tu yoveli debuleba Sefasebulia 9 quliT).

yuradReba miaqcieT, rom am skalaze qulaTa mocemuli rigi pirobiTia; Tu

TiToeuli debulebis Sesafasebeli sayrdeni punqtebi (romelTac raodenoba am

skalaze aris 9) dalagebuli iqneboda 10-dan 90-mde (da ara 1-dan 9-mde), maSin jamuri

Sefasebis skalis diapazoni iqneboda 90-dan 810 qulamde.

gazomvis romel skalas miekuTvneba mTvaris efeqtis rwmenis skalaze da Sefasebis

sxva msgavs skalebze miRebuli qulebi? cxadia, rom TiToeuli debulebis Sefaseba

moTavsebulia, sul mcire, rigis skalaze. qula 4 miuTiTebs debulebaSi aRwerili

konkretuli maxasiaTeblis ufro metad mocemulobaze, vidre _ qula 3, xolo qula

3 _ ufro metad mocemulobaze, vidre qula 2; Tumca, ver vityviT gadaWriT, rom

gansxvaveba 4-sa da 3-s Soris maxasiaTebelSi imdenive gansxvavebas gamoxatavs,

ramdensac gansxvaveba 3-sa da 2-s Soris. Tumca, aseTi Sefasebebi ufro mets unda

iZleodes, vidre mxolod rigis skalis informaciaa. Cven SeiZleba movelodeT, rom

respondenti subieqturad dayofs kategoriebs daaxloebiT tol intervalebad

skalis polusebs Soris. Tu davajamebT respondentisgan ramdenime calkeul

skalaze miRebul Sefasebebs, samarTliani iqneba, Tu davuSvebT, rom saboloo qula

ufro mets gvaZlevs, vidre mxolod rigis Sesaxeb informacias. gardneris (Gardner,

1975) mixedviT, jamur skalebze miRebuli qulebi rigisa da intervalebis skalebs

Soris xvdeba. amrigad, ar aris aucilebeli, rom romelime konkertuli skala

zustad ergebodes gazomvis oTxi skalidan erT-erTs; Tumca, rodesac statistikuri

1 debuleba ibeWdeba avtorebis _ j. rotonisa da i. u. kelis da gamomcemlis nebarTviT, „A

Scale for assessing belief in lunar effects: reliability and concurrent validity“, Psychological Reports, 1985, 57, 239-245,

© Psychological Reports, 1985.


analizis saWiroeba dgeba, Sefasebis skalebs mkvlevrebi xSirad intervalebis

skalebad ganixilaven.

Sefardebis skala

intervalebis skalis pirobiTi

nulovani wertili Sefardebis

skalaze Canacvlebulia

absoluturi nuliT, romelic

Seesabameba gasazomi cvladis

srul ararsebobas. Sefardebis

skalaze SesaZlebelia vTqvaT, rom erTi obieqti (magaliTad, gamRizianebeli,

movlena an individi) orjer, naxevarjer an samjer metad xasiaTdeba gasazomi

cvladiT, vidre _ meore. Sefardebis skala maSin gamoiyeneba bihevioristul

mecnierebebSi, rodesac izomeba iseTi sidideebi, rogoricaa reaqciis dro an

amocanis Sesrulebaze daxarjuli dro. am sidideebs aqvT absoluturi nuli, skalis

erTeulebi ki Tanabari intervalebiT arian erTmaneTisgan daSorebuli. magaliTad,

Tu vzomavT dros, romelsac individi Tavsatexis amoxsnas andomebs, 15 wamsa da 20

wams Soris (anu 10 wami) sxvaoba tolia 50 wamsa da 60 wams Soris (isev 10 wami)

sxvaobisa; amasTanave, is, vinc 60 wamSi xsnis Tavsatexs, orjer met dros andomebs

amas, vidre is, vinc Tavsatexis amoxsnas 30 wamSi axerxebs.

perceptuli iluziis sididis gazomva xSirad Sefardebis skalazea

SesaZlebeli. iluzia Cndeba maSin, rodesac gamRizineblis Cveneuli aRqma ar

Seesabameba mis fizikur maxasiaTeblebs; Cven arasworad aRviqvamT imas, rac

arsebobs. amis cnobili magaliTia miuler-laieris iluzia, romelic N2.1 suraTzea

mocemulia. aq Tqven xedavT (an metad albaTuria, rom dainaxoT), rom (a) xazi (b)

xazze grZelia, Tumca, fizikurad es xazebi tolia. Tqveni aRqma mcdaria. am iluziis

sidide SeiZleba gavzomoT imis gansazRvriT, Tu ramdenad gansxvavdeba stimulis

aRqmuli sigrZe misi realuri sigrZisgan. milimetrebiT (mm) gazomili perceptuli

Secdoma ganekuTvneba Sefardebis skalas. sruliad SesaZlebelia, rom cdis piris

Secdoma iyos 0 mm (anu ar iyos Secdoma), rodesac stimulis aRqmuli sigrZe ar

gansxvavdeba misi realuri sigrZisgan. 5 mm-iT Secdomasa da 8 mm-iT Secdomas Soris

gansxvaveba igivea, rac 10 mm-iT Secdomasa da 13 mm-iT Secdomas Soris. garda amisa,

10 mm-iT Secdoma gamoxatavs sididis iluzias, romelic orjer aRemateba 5 mm-iT

Secdomas.

Sefardebis skala (Ratio measurement): _ cvladis sidide

dalagebulia skalaze da miniWebul ricxvebs Soris

gansxvaveba asaxavs cvladis mniSnelobebs Soris igive

sididis gansxvavebas. Sefardebis skalaze arsebobs

WeSmaritad nulovani wertili, romelic Seesabameba

gasazomi maxasiaTeblis srul ararsebobas.


Tvisebrivi da raodenobrivi monacemebi

adamianebisa Tu sagnebis sxvadasxva

kategoriebSi gasaerTianeblad

saxeldebis skalaze gamoiyeneba

ricxvebi, romlebic ar iZlevian

raodenobriv informacias im cvladis

Sesaxeb, romlis mixedviTac moxda

maTi kategoriebSi gaerTianeba.

amrigad, saxeldebis skala gvaZlevs

Tvisebriv monacemebs. kategoriebad gaerTianebuli adamianebi an sagnebi raRac

TvisebiT arian gansxvavebulni. rigis, intervalebisa da Sefardebis skalebi

gvaZleven raodenobriv monacemebs. cvladisTvis miniWebuli ricxvebi gamoxataven am

cvladis sidides anu raodenobas. am sami skalidan bihevioristuli mecnierebebis

wamomadgenlebi upiratesobas aniWeben intervalebisa da Sefardebis skalebs, radgan

isini gasazomi cladis ZiriTadi Tvisebebisa Tu maxasiaTeblebis raodenobis ufro

zustad gansazRvris saSualebas iZlevian, vidre rigis skala. garda amisa, vinaidan

intervalebisa da Sefardebis skalebi iseTi maTematikuri operaciebis gamoyenebis

saSualebas iZlevian, rogoricaa Sekreba, gamokleba, gamravleba da gayofa, am

skalebze SesaZlebelia metad mravalferovani statistikuri teqnikis gamoyenebac.

aqedan gamomdinare, winamdebare saxelmZRvaneloSi warmodgenili mravali

statistikuri meTodi mowodebulia intervalebis an Sefardebis skalebze miRebuli

monacemebisTvis.


1. ganmarteT: _ intervalebis skala, Tvisebrivi monacemebi, raodenobrivi

monacemebi, Sefardebis skala.

2. liderobis unaris Sesafasebel rigis skalaze cdis pirebma miiRes

Semdegi qulebi: rozana _ 84, eva _ 78, pirsi _ 71, viqtoria _ 65, nensi _ 42

da uiliami _ 0.

a) marTebuli iqneba, Tu davaskvniT, rom rozanasa da evas liderobis

unarebs Soris iseTive gansxvavebaa, rogoric pirsisa da viqtorias

unarebs Soris? axseniT Tqveni pasuxi.

b) marTebuli iqneba, Tu davaskvniT, rom rozanas liderobis orjer meti

unari aqvs, vidre nensis? axseniTYTqveni pasuxi.

g) marTebuli iqneba, Tu davaskvniT, rom radgan uiliamma 0 qula miiRo,

mas saerTod ar aqvs liderobis unari? axseniTYTqveni pasuxi.

3. kolejis ramdenime studentma monawiloba miiRo Zilis kvlevaSi,

romelSic Tvalebis swrafi moZraobis fazaSi (REM Zili; faza, rodesac

Tvisebrivi monacemebi (Qualitative data): _

saxeldebis skalaze miRebuli monacemebi,

romlebic miuTiTeben, rom cvladebi

Tvisebrivad gansxvavdebian erTmaneTisgan.

raodenobrivi monacemebi (Quantitative data): _

rigis, intervalebisa an Sefardebis skalaze

miRebuli monacemebi, romlebic miuTiTeben, Tu

ra sididis aris cvladi.


adamiani sizmars xedavs) gatarebuli dro (wuTebiT)

eleqtroencefalogramis gamoyenebiT izomeboda. erT Rames REM Zilis

xangrZlivoba iyo: dominikTan _ 204 wT, gretasTan _ 194 wT, andresTan _

175 wT, judisTan _ 165 da karmenTan _ 102 wT.

a) marTebuli iqneba, Tu davaskvniT, rom sxvaoba dominikisa da gretas

REM Zilis xangrZlivobas Soris imdenivea, ramdenic andresa da judis

REM Zilis xangrZlivobas Soris? axseniTYTqveni pasuxi.

b) marTebuli iqneba, Tu davaskvniT, rom dominikma orjer meti dro

gaatara REM Zilis fazaSi, vidre karmenma? axseniTYTqveni pasuxi.

g) davuSvaT, erT-erTma studentma, martinma REM Zilis fazaSi gaatara 0

wT. marTebuli iqneba, Tu davaskvniT, rom im Rames mas ar hqonia REM

Zilis faza? axseniTYTqveni pasuxi.

diskretuli (wyvetili) da uwyveti cvladebi

cvladebi SeiZleba iyos diskretuli (wyvetili) da uwyveti

diskretuli (wyvetili) cvladebi

diskretulia (wyvetilia) cvladi,

romelmac SeiZleba miiRos

mniSvnelobebi mxolod sasruli an

potenciurad Tvladi simravlidan. sityva „sasruli“ niSnavs SesaZlo mniSvnelobebis

gansazRvrul simravles. mocemuli kolejis studentebis raodenoba diskretuli

cvladia: am kolejSi SeiZleba swavlobdes 5652 studenti an 5653 studenti, magram

ara _ 5652½ studenti. studentebis raodenoba SeiZleba gaizardos an Semcirdes

erTi mTliani, magram ara _ ½ an ⅓ studentis mimatebiT an gamoklebiT. amrigad,

studentebis raodenoba iRebs mxolod gansazRvrul Tvlad mniSvnelobebs.

Cveulebisamebr, cvladebs, romelTa SesaZlo mniSvnelobebi SeiZleba daviTvaloT,

diskretul (wyvetil) cvladebs uwodeben: qalaqSi mcxovreb adamianTa raodenoba,

ojaxSi bavSvebis raodenoba, qurdobebis raodenoba erTi wlis manZilze, StatSi

regisgrirebuli avtomanqanebis raodenoba da a.S.

uwyveti cvladebi

uwyvetia cvladi, romelmac SeiZleba miiRos

mniSvnelobebi cvladis nebismier or dones

Soris moTavsebuli usasrulo simravlidan.

amrigad, uwyvet cvlads aqvs mniSvnelobaTa

SemousazRvreli simravle. aseTi cvladis magaliTia Tqveni wonis gazomva: is

SeiZleba iyos 55.6 kg an 55.64 kg. Tqveni wona ar SemoisazRvreba iseTi diskretuli

mniSvnelobebiT, rogoricaa 55 kg an 56 kg. Tumca, Tqveni wona mainc ar iqneboda

diskretuli (wyvetili) cvladi (Discrete variable):

_ cvladi, romelmac SeiZleba miiRos

sasruli an Tvladi mniSvneloba.

uwyveti cvladi (Continuous variable): _

cvladi, romelmac SeiZleba miiRos

usasrulo mniSvneloba cvladis

nebismier or dones Soris.


SemosazRvruli am konkretuli mniSvnelobebiT; man SeiZleba miiRos nebismieri

mniSvneloba 55 kg-sa da 56 kg-s Soris. analogiurad, mravali fsiqologiuri cvladi,

rogoricaa, magaliTad, SfoTva an inteleqti, arsebobs uwyveti cvladis saxiT,

Tumca, maTi gazomva SesaZlebelia skalebiT, romlebzec cdis pirs diskretuli

qulebi eniWeba.

gazomvis realuri sazRvrebi. rodesac vzomavT uwyvet cvlads, cvladis arsebul

sidideebs eniWeba konkretuli ricxviTi mniSvnelobebi, magram es aris cvladis

realur sidideebTan mxolod miaxloebuli mniSvnelobebi. aqedan gamomdinare,

ganurCevlad imisa, gazomvis romel skalas avirCevT, unda vicodeT, ra sizustiT

vzomavT. uwyveti cvladis gazomvis sizuste moTavsebulia miniWebuli ricxvis

realur sazRvrebSi.

gazomvis realuri sazRvrebis cnebis

gageba gaioldeba, Tu ganvixilavT drois

cvladis gazomvis magaliTs. davuSvaT,

gvainteresebs, ramden xans SeuZlia

adamians 40°F (5°C) wyalSi xelis gaCereba.

am davalebas xSirad iyeneben tkivilis

usafrTxod gamosawvevad eqsperimentebSi,

sadac ikvleven tkivilisadmi

tolerantobis gazrdaSi relaqsaciis

strategiebis rols. davuSvaT, erTma cdis

pirma, Cveni gazomvis mixedviT

(vakvirdebodiT mas da vuyurebdiT wamzoms), wyalSi xelis gaCereba SeZlo 37 wm.

ramdenad zustia gazomvis es Sedegi? vinaidan Cven vzomavdiT uaxloesi mTeli wamis

sizustiT, dafiqsirebuli dro ver iqneboda 36 an 38 wm, magram SesaZlebelia, rom

realuri dro 37,2 wm an 36,7 wm yofiliyo? pasuxia: _ „diax“; Cven SegveZlo 37 wm-mde

dagvemrgvalebina 36,7 wamic da 37,2 wamic. am gazomvis (37 wm) sizuste mocemulia

ricxvis realuri sazRvrebiT. ricxvis realuri sazRvrebi aris ricxvisa da misi

wina ricxvis Suawertili an ricxvisa da misi momdevno ricxvis Suawertili im

skalaze, romelsac viyenebT mocemuli cvladis gasazomad. miRebuli

mniSvnelobisTvis _ 37 wm-isTvis, Sesabamisad, realuri qveda sazRvari iqneba 36,5 wm,

xolo realuri zeda sazRvari _ 37,5 wm. 36,5 wamisTvis realuri qveda zRvari aris

Sualedi 36 wamsa (skalaze momdevno gazomvad mniSvnelobasa) da 37 wams Soris. 37,5

wamisTvis realuri zeda zRvari aris Sualedi 37 wamsa da 38 wams (skalze momdevno

gazomvad mniSvnelobas) Soris.

axla davuSvaT, rom vzomavT eleqtronuli wamzomiT, romelsac SeuZlia gazomos

uaxloesi 100

1 wamis sizustiT da miviReT, magaliTad, 37,21 wm. ra iqneba am gazomvis

ricxvis realuri sazRvrebi (Real limits of a

number): _ ricxvisa da misi wina ricxvis

Suawertili an ricxvisa da misi momdevno

ricxvis Suawertili im skalaze, romelsac

viyenebT mocemuli cvladis gasazomad.

ricxvis realuri qveda sazRvari (Lower real

limit of a number) _ ricxvisa da misi wina

ricxvis Suawertili.

ricxvis realuri zeda sazRvari(Upper real

limit of a number) _ ricxvisa da misi momdevno

ricxvis Suawertili.


realuri zeda da qveda zRvari? am SemTxvevaSic viyenebT igive wess: _ vpoulobT

momdevno umdables da umaRles mniSvnelobebs Soris Sua wertils im skalaze,

romelzec vzomavT cvlads. amrigad, realuri qveda zRvari iqneba Suawertili 37,20

wamsa (momdevno umdables mniSvnelobas) da 37,21 wams Soris anu 37,205 wm, xolo

realuri zeda zRvari _ Suawertili 37,21 wamsa da 37,22 wams (momevno umaRles

mniSvnelobas) Soris anu 37,215 wm.

uwyveti cvladis yovel ricxviT mniSvnelobas aqvs masTan dakavSirebuli

realuri zeda da qveda zRvrebi. davuSvaT, viyenebT Sefasebis 7-qulian skalas,

romelic mocemulia N2-2 suraTze. vTqvaT, cdis pirma am skalaze aRniSna pasuxi 3.

zemoT aRwerili wesebis mixedviT, am qulis realuri qveda da zeda zRvari,

Sesabamisad, aris 2,5 da 3,3-ia.

gazomvis realuri sazRvrebi mniSvnelovania qulebis sixSireTa ganawilebis

agebisas da statistikur gamoTvlebSi mniSvnelobebis damrgvalebisas. sixSireTa

ganawilebis Sesaxeb me-3 TavSi visaubrebT.

suraTi N2.2. 7-qulian skalaze gacemuli pasuxis (aRniSnulia -iT) realuri zeda da

qveda zRvrebi.

1 2 3 4 5 6 7

arasodes

Zalian

xSirad

2,5

realuri

qveda

zRvari

3,5

realuri

zeda

zRvari

damrgvaleba: gamoTvlis saboloo pasuxi. davuSvaT, sam adamians vTxoveT N2-2

suraTze mocemuli skalis Sevseba da miviReT Semdegi qulebi: _ 4, 4 da 3. tipuri

pasuxis aRsawerad unda SevkriboT es ricxvebi (miviRebT 11-s) da miRebuli jami

gavyoT 3-ze, Sedegad gveqneba 6666,33

11= . aRniSvna (‾) bolo 6-ianis Tavze niSnavs, rom

6-ianebi usasrulod meordeba. mZimis Semdeg ramdeni TanrigiT unda warmoadginoT

miRebuli pasuxi?

erTi SeTanxmebaa, rom gamoTvlis saboloo mniSvneloba unda damrgvaldes mZimis

Semdeg or Tanrigamde realurad miRebuli mniSvnelobis Semdeg. warmodgenil

magaliTSi qulebi mocemulia erTeulebSi: cdis pirma unda airCios 3 an 4 da ara _

3,2 an 4,6. am wesis mixedviT, gamoTvlis saboloo Sedegi unda damrgvaldes arCeuli

pasuxis Semdeg or Tanrigamde anu 3,67-mde.

meore SeTanxmebiT, gamoTvlis saboloo Sedegi mrgvaldeba mZimis Semdeg erT

Tanrigamde realurad miRebuli qulis Semdeg. am midgomis mixedviT, mocemul


magaliTSi gamoTvlis saboloo Sedegi damrgvaldeba miRebuli qulis mZimis Semdeg

erT Tanrigamde anu iqneba 3,7. qvemoT ganxilulia orive midgomis sailustracio

magaliTi.

damrgvaleba: gamoTvlis Sualeduri nabijebi. Tu saboloo pasuxis gamoTvlas

ramdenime nabiji sWirdeba, Sualeduri ricxviTi mniSvnelobebi, sul mcire, ori

TanrigiT metiT unda gadavitanoT saboloo pasuxis gamoTvlis etapamde. magaliTad,

Tu 3

11 -is mniSvneloba Semdgom gamoTvlebSic unda gamoviyenoT, is ar unda

damrgvaldes 3,67-mde; metic, Semdgomi gamoTvlebisTvis is ori damatebiTi TanrigiT

unda gadavitanoT _ 3,6667. am midgomas minimumamde dayavs gamoTvlebSi damrgvalebis

Secdoma.

damrgvalebis wesebi. davuSvaT, gamoTvlis Sedegad miviReT pasuxi 3,6AB, sadac A

da B SesaZlo ricxviTi mniSvnelobebia. magaliTad, Tu A=2 da B=9, maSin 3,6AB=3.629.

vTqvaT, gvsurs am ricxvis mZimis Semdeg or Tanrigamde damrgvaleba. ricxvebis

damrgvaleba, Cvelebriv, ramdenime wesis dacviT xdeba.

• Tu B-Ti aRniSnuli ricxvi 5-ze metia, A izrdeba 1-iT da ugulebeliyofa B

sidide. am wesis mixedviT, 3,629 (aq A=2 da B=9) mrgvaldeba 3,63-mde.

• Tu B 5-ze naklebia, A-s mniSvneloba ucvleli rCeba da ugulebeliyofa B

sidide. am wesis mixedviT, 3,623 (aq A=2 da B=3) mrgvaldeba 3,62-mde.

• Tu B-Ti aRniSnuli ricxvi 5-is tolia, A izrdeba 1-iT, Tu A kentia; xolo A-

s mniSvneloba ucvleli rCeba, Tu A luwia, B sidide ki ugulebeliyofa. am

wesis mixedviT, 3,635 (aq A=3 da B=5) mrgvaldeba 3,64-mde, xolo 3,625 (aq A=2 da

B=5) _ 3,62-mde da ara 3,63-mde.

SeamowmeT Tqveni codna II-4

1. ganmarteT: _ uwyveti cvladi, diskretuli (wyvetili) cvladi, ricxvis

realuri sazRvrebi, realuri qveda zRvari, realuri zeda zRvari.

2. gaarkvieT, qvemoT CamoTvlili cvladebidan romelia diskretuli da

romeli _ uwyveti.

a) individis xelis sigrZe, gazomili santimetrebiT.

b) sasamarTlos darbazSi bralmdebelisa da braldebulis TvalebiT

kontaqtis xangrZlioba, gazomili wamebiT.

g) tomsonis gazelis naxtomebis (vertikaluri naxtomi, rodesac oTxive

fexi erTdroulad scildeba miwas) raodenoba 1 saaTis ganmavlobaSi.

d) mgalobeli Citis simReris xangrZlivoba (wamebSi) Sewyvilebis

periodSi.


e) multiplikaciur filmebze sicili, romelic izomeba im drois

xangrZlivobis mixedviT (wamebSi), ramden xansac icinis maTze cdis

piri.

v) sabavSvo baRis bavSvebis altruistul qcevaTa raodenoba erTi dRis

ganmavlobaSi.

z) bavSvis megobrebis raodenoba.

T) statistikis gamocdaze sworad gacemuli pasuxebis raodenoba.

3. ipoveT TiToeuli qulis realuri zeda da qveda zRvrebi.

a) 152,4 sm simaRle.

b) TvalebiT kontaqtis xangrZlioba 17,3 wm.

g) 7-qulian skalaze micemuli Sefaseba 4.

d) reaqciis dro 0,437 wm.

e) 6,9 mm Secdoma miuler-laieris iluziis aRqmisas.

v) xelis sigrZe 67,81 sm.

4. daamrgvaleT Semdegi mniSvnelobebi frCxilebSi miTiTebul Tanrigamde.

a) 27,436 (or Tanrigamde).

b) 119,0278 (sam Tanrigamde).

g) 1,445 (or Tanrigamde).

d) 1263.75915 (oTx Tanrigamde).

e) 0,352 (or Tanrigamde).

v) 13,2659 (or Tanrigamde).

Sejameba

• statistika qcevis mecnieruli Seswavlis saSualebaa.

• mecnieruli kvleva iwyeba kiTxvis dasmiT, romelzec pasuxis gacema

SesaZlebelia empiriuli monacemebis mopovebis saSualebiT.

• kvlevis meTodi aris monacemebis mogrovebisas gamoyenebuli midgoma. aq

ganxilulia kvlevis oTxi meTodi: _ bunebrivi dakvirveba, saarqivo

Canawerebi, gamokiTxva da eqsperimenti.

• bunebrivi dakvirveba aris kvleva, romelic gulisxmobs bunebriv pirobebSi

ganxorcielebul qcevebze dakvirvebas.

• saarqivo CanawerebiT kvlevisas gamoiyeneba arsebuli Canawerebi.

• gamokiTxva aris kvleva, romelSic monacemebi adamianebis zepiri an

werilobiTi interviuebiT miiReba.

• eqsperimenti aris zedmiwevniT gakontrolebuli situacia, sadac mecnieri

axdens erTi an meti damoukidebeli cvladis manipulirebas da akvirdeba, Tu

ra gavlenas axdens es damokidebul cvladze.


• operaciuli gansazRvreba aRwers dakvirvebisas an damoukidebeli cvladiT

manipulirebisas gamoyenebul operaciebs Tu etapebs.

• gazomva aris garkveuli wesebis mixedviT cvladebisTvis ricxviTi

mniSvnelobebis miniWeba.

• saxeldebis skala aris gazomili cvladis sxvadasxva kategoriebSi

gaerTianeba. saxeldebis skala iZleva Tvisebriv monacemebs.

• rigis skala mocemuli parametris mixedviT TanmimdevrobiT ganalagebs

cvladis sidideebs.

• intervalebis skalaze Tanabrad izrdeba gasazomi cvladis sidide, Tumca, aq

mas ara aqvs absoluturi nuli.

• Sefardebis skala xasiaTdeba intervalebis skalis TvisebebiT da damatebiT

aqvs absoluturi nulovani wertili.

• rigis, intervalebisa da Sefardebis skalebi raodenobriv monacemebs iZleva.

• diskretulma cvladma SeiZleba miiRos mniSvnelobebi mxolod sasruli an

potenciurad Tvladi simravlidan.

• uwyvetma cvladma SeiZleba miiRos mniSvnelobebi cvladis nebismier or dones

Soris moTavsebuli usasrulo simravlidan.

• ricxvis realuri sazRvrebi aris ricxvsa da mis wina da momdevno umdables

da umaRles ricxvebs Soris Sua wertilebi im skalaze, romelsac viyenebT

mocemuli cvladis gasazomad.


bunebrivi dakvirveba

gamokiTxva

diskretuli cvladi

empiriuli monacemi

eqsperimenti

Tvisebrivi monacemebi

intervalebis skala

kvlevis sqema

kvlevis hipoTeza

operaciuli gansazRvreba

raodenobrivi monacemebi

rigis skala

ricxvis realuri zeda

zRvari

ricxvis realuri zRvari

ricxvis realuri qveda

zRvari

saarqivo Canawerebi

saxeldebis skala

skala

uwyveti cvladi

Sefardebebis skala

cvladi

kiTxvebi

1. davuSvaT, gTxoves qvemoT CamoTvlil kiTxvebze pasuxis gasacemad

monacemebis Segroveba. kvlevis romel sqemas airCevdiT TiToeul SemTxvevaSi

da ratom?

a) statistikis kursze Tqveni jgufelebi, Cveulebriv, ramden dros uTmoben

saSinao davalebis Sesrulebas?


b) ramdenad xSirad isvris kalaTburTeli burTs kalaTSi maSin, rodesac

msaji uSualod srolis win drois amowurvis niSans iZleva?

g) zrdis Tu ara gamyidvelTa daxelovnebis programa im adamianTa navaWris

raodenobas, romlebmac gaiares es programa?

d) ra tipur qcevebs axorcielebs patara bavSvi maRaziidan gamosvlisas?

2. davuSvaT, gTxoves TanamSromelTa jgufSi samsaxureobrivi stresis gazomva.

rogor gansazRvravT operaciulad samsaxureobrivi stresis zomas?

3. gazomes policiis oficerTa orientacia profesiul miRwevaze. amisaTvis

gamoiyenes oTxi teqsti. TiToeul maTganSi aRwerili iyo profesiul

miRwevaze sxvadasxvagvari oriantacia, romlis Sinaarsi icvleboda 1-dan

(invidividis profesiul miRwevasTan yvelaze miaxloebulidan) 4-mde

(individis profesiul miRwevisgan yvelaze daSorebulamde).

a) ra ZiriTadi cvladi gazomes am kvlevaSi?

b) gazomvis romel skalas miekuTvneboda gamoyenebuli sazomi?

4. profesionali golfis moTamaSe qalebi awones, raTa daedginaT,

ukavSirdeboda Tu ara maTi sxeulis wona maTi TamaSis xarisxs.



g) golfis erTi moTamaSis wona iyo 54,6 kg. ra iqneba am gazomvis realuri

zeda da qveda zRvari.

d) 6 golfis moTamaSe awones kilogramis sizustiT. maTi saSualo wona

iyo 57,648 kg. daamrgvaleT es ricxvi.

5. pirvelklaseli bavSvebis mSoblebi Seafases emociuri stabilobis

pirovnuli maxasiaTeblis mixedviT. amisaTvis gamoiyenes Sefasebis skala.



g) vTqvaT, cdis pirma daTanxmeba _ ardaTanxmebis 9-qulian skalaze

debulebas mianiWa 8. ra iqneba am gazomvis realuri qveda da zeda

zRvari?

d) vTqvaT, am skalis mixedviT SeafaseT eqvsi adamiani da mocemuli

debulebis maTi tipuri Sefaseba iyo 5,833. ramdenad sworad aris

mocemuli es Sedegi mZimis Semdeg warmodgenili Tanrigebis raodenobis

TvalsazrisiT? axseniT Tqveni pasuxi.

6. Tqven ikvlevT, Tu ramden dros uTmoben studentebi yoveldRiurad saSinao

davalebis Sesrulebas. erTma studentma aRniSna, rom qimias 97 wuTs uTmobs.

a) gazomvis romel skalas miekuTvneboda gamoyenebuli sazomi?

b) ra iqneba studentis mier dasaxelebuli 97 wuTis realuri zeda da

qveda zRvari?


g) samma studentma aRniSna, rom saSinao davalebis Sesrulebas uTmobs 97,

68 da 83 wuTs, Sesabamisad. maTi qulebis saSualoa 82,666666.

daamrgvaleT es ricxvi.

7. pirovnebaTSorisi Rirebulebebis Sesafaseblad cdis pirs moeTxoveboda

Rirebulebebis amsaxveli 12 debuleba daelagebina yvelaze

mniSvnelovanidan yvelaze naklebad mniSvnelovanamde. gazomvis romel

skalas miekuTvneboda gamoyenebuli sazomi?

8. sabavSo baRSi bavSvebis urTierTqmedeba gaarTianes kategoriebSi:

agonisturi (anu kamaTi an Cxubi), neitraluri (ar iyo interaqcia) da

altruistuli (anu daxmarebis gamwevi). gazomvis romel skalas

miekuTvneboda gamoyenebuli sazomi?

9. gansazRvreT, qvemoT CamoTvlili cvladebi diskretulia Tu uwyveti.

a) Tqveni jgufelebis raodenoba statistikis kursze.

b) im adamianebis mier daklebuli wona kilogramebSi, romlebic dietas

icaven da varjiSoben.

g) caciebis ricxvi 100 kacian jgufSi.

d) individis sisxlis wneva, romelic izomeba vercxlis wylis svetis

simaRliT.

10. gansazRvreT qvemoT CamoTvlili maCveneblebis realuri zeda da qveda

zRvari.

a) wona 61,3 kg.

b) 34 wuTi, romelic eTmoba video TamaSs.

g) manqaniT gavlili 175.6 km.

d) sxeuli temperatura 36,6°C.

e) sistoluri sisxlis wneva vercxlis wylis simaRlis 109 mm.

11. daamrgvaleT Semdegi ricxvebi mZimis Semdeg or Tanrigamde.

a) 42,255

b) 1,165

g) 163,651

d) 27,426

e) 13,895

v) 24,6250

z) 19,2751

12. sadilis dasrulebis Semdeg restornis mepatroneebi avseben baraTebs. mas

sTxoven, rom TiToeul baraTSi 5-qulian skalze (cudidad saukeTesomde)

Seafasos sakvebis xarisxi, momsaxurebisas keTilganwyobis xarisxi,

momsaxurebis siswrafe da sisufTave. TiToeuli baraTidan miRebuli

saboloo qulaa 4-dan 20-mde. gazomvis romeli skalaa gamoyenebuli aq?

axseniT Tqveni pasuxi.

13. fsiqologi ikvlevda, Tu ramdenad SeeZloT adamainebs tkivilis atana;

amisaTvis cdis pirebi xels yofdnen 5°C-ian wyalSi; izomeboda drois


xangrZlivoba, ramden xansac SeeZloT maT xelis gaCereba wyalSi. man

Caatara ori kvleva.

a) pirvel kvlevaSi cdis pirebi dalagda imis mixedviT, Tu ramden xans

SeeZloT xelis gaCereba wyalSi da miiRes Semdegi Tanmimdevroba: liza

_ 1, stacia _ 2, marita _ 3, robini _ 4 da judi _ 5. gazomvis romeli

skala gamoiyena mkvlevarma?

b) meore kvlevaSi tkivilis amtanobis xangrZlivoba wamebiT gaizoma da

miiRes Semdegi Sedegebi: liza _ 111,3 wm, stacia _ 97,8 wm, marita _ 71,3

wm, robini _ 18,7 wm da judi _ 13,6 wm. gazomvis romeli skala gamoiyena

mkvlevarma?

14. jandacvis msoflio organizaciis mier janmrTelobis dacvis sistemebis

ranJirebisas (Mishra, 2000) aSS-is janmrTelobis dacvis sistemam 37-e rangi

miiRo. pirveli da meore adgilebi, Sesabamisad, safrangeTisa da italiis

janmrTelobis dacvis sistemebma daikaves. iaponias mieniWa me-10, xolo

Sveicarias _ me-20 rangi.

a) ramdenad ukeTesia safrangeTis janmrTelobis dacvis sistema

italiasTan SedarebiT?

b) aris Tu ara aSS-is janmrTelobis dacvis sistema orjer ukeTesi,

vidre Sveicariis? axseniT Tqveni pasuxi.

g) ras geubnebaT aSS-is janmrTelobis dacvis sistemis xarisxis Sesaxeb

is faqti, rom mas 37-e rangi mieniWa?

Tavi III. monacemTa aRwera: sixSireTa ganawileba da grafikebi

sixSireTa ganawileba daujgufebeli monacemebisTvis

martiv sixSireTa ganawileba

fardobiT sixSireTa ganawileba

sixSireTa ganawileba procentebSi

informaciis miReba sixSireTa ganawilebidan

dajgufebul sixSireTa ganawileba

dajgufebul sixSireTa ganawilebis ageba

procentili da procentuli rangebi

monacemis procentuli rangis povna

ganawilebis procentilis povna

sixSireTa ganawilebis grafikulad gamosaxva

histograma

sixSireTa poligoni

fardobiTi sixSiris histograma da poligoni

informaciis miReba histogramidan an sixSireTa

poligonidan

foTlebiani Reroebis diagrama

Tvisebrivi monacemebis grafikulad gamosaxva

sixSireTa ganawilebis formebi

sixSireTa simetriuli ganawileba

sixSireTa asimetriuli ganawileba

modaloba

თავი III. მონაცემთა აღწერა: სიხშირეთა განაწილება და გრაფიკები 2

gifiqriaT rodesme: _ „SfoTvis kontroli rom SemZleboda, ukeTesad

Cavabarebdi gamocdebs“? Tu ki, getyviT, rom marto Tqven ar gawuxebT es

problema. ukanesknel wlebSi konsultantebma da fsiqologebma daadgines, rom

gamocdebTan dakavSirebuli SfoTva zemoqmedebas axdens studentTa akademiur

warmatebaze. gamocdebTan dakavSirebuli SfoTvis gazomvis mizniT spilbergerma

(Spielberger, 1980) SeimuSava gamocdebTan dakavSirebuli SfoTvis kiTxvari (TAI). es

kiTxvari Sedgeba 20 iseTi debulebisgan, rogoricaa, magaliTad, „aforiaqebuli da

SeSinebuli var didi gamocdis win.“ yovel debulebas aqvs oTxi savaraudo pasuxi:

_ TiTqmis arasdros, xandaxan, xSirad da TiTqmis yovelTvis. kiTxvaris Sevsebisas

studentma TiToeuli debulebisTvis unda airCios erT-erTi pasuxi, romelic ukeT

miesadageba mas. mas Semdeg, rac kiTxvari Sevsebulia, TiToeul variants eniWeba

ricxviTi mniSvneloba: _ TiTqmis arasdros = 1; xandaxan = 2; xSirad = 3; TiTqmis

yovelTvis = 4. saboloo qula miiReba arCeuli variantebisTvis miniWebuli

ricxviTi mniSvnelobebis SekrebiT. amrgad, es kiTxvari jamuri Sefasebis skalaa,

romlis qulebi 20-dan 80-mde meryeobs. 20 qula gviCvenebs SfoTvis minimalur

dones (am kiTxvaris mixedviT), 80 qula ki _ maqsimalurs. rogorc me-2 TavSi

vnaxeT, jamuri Sefasebis skala, savaraudod, skalaTa Soris Tavsdeba; Tumca,

statistikuri miznebisTvis Sefasebis skalebze miRebuli qulebi intervalTa

skalis monacemebad ganixileba. qvemoT ganxilul magaliTSic am tradicias

mivsdevT.

davuSvaT, 20 studentisgan

(romelsac gamocdebTan dakavSirebuli

SfoTvis kontrolSi daxmareba

esaWiroeboda) Semdgar SerCevas

kolejis konsultantma gamocdebTan

dakavSirebuli kiTxvari misca.

miRebuli Sedegebi mocemulia cxrilSi

N3.1. aseT qulebs nedl monacemebs an

nedl qulebs uwodeben, radgan isini

uSualod kvlevis dros arian Segrovili

da ar arian statistikurad

damuSavebulni. laTinuri aso N

gamoiyeneba miRebuli qulebis saerTo

raodenobis aRsaniSnad; amrigad, am

magaliTSi N aris 20.

ganvixiloT studentis SemTxveva,

romelmac am kiTxvaris mixedviT 60 qula

miiRo. ras gveubneba es qula am studentis gamocdasTan dakavSirebuli SfoTvis

N: _ jgufSi miRebuli qulebis saerTo

raodenoba.

nedli monacemebi an nedli qulebi (raw data or

raw scores) _ kvlevis monawileebisgan

miRebuli qulebi, romelTac jer ar

gauvliaT statistikuri analizi.

cxrili N3.1. gamocdebTan

dakavSirebuli SfoTvis kiTxvarSi 20

studentis mier miRebuli hipoTeturi

qula.

61 67 63 60 62 67 59 60 68 64

57 55 59 59 59 64 65 62 59 60


donis Sesaxeb? is sxva studentebis mier miRebul qulebze metia Tu naklebi?

SegviZlia ki am kiTxvaze pasuxis gacema mxolod am erTi qulis codnis

pirobebSi? cxadia, ara da aseTive uaryofiTia pasuxi bihevioristul mecnierebaSi

gamoyenebuli mravali sazomis SemTxvevaSi, radgan erTi calkeuli qula mxolod

sxva qulebTan Sedarebis Semdeg iZens mniSvnelobas. amrigad, miRebuli qulebis

analizis da maTgan informaciis miRebis pirveli nabijia monacemebis warmodgena

sixSireTa ganawilebis saxiT. dasawyisisTvis ganvixilavT intervalebisa da

Sefardebis skalebze miRebuli raodenobrivi monacemebis warmodgenas.

sixSireTa ganawileba daujgufebeli monacemebisTvis

martiv sixSireTa ganawileba

sixSireTa ganawilebas uwodeben

cxrils, romelSic mocemulia

TiToeul qula da is sixSire, ra

sixSiriTac gvxvdeba es qula. cxrilSi N3.1 mocemuli gamocdebTan dakavSirebul

kiTxvarSi miRebuli qulebis daujgufebel sixSireTa ganawileba warmodgenilia

cxrilSi N3.2.

cxrili N3.2. daujgufebel sixSireTa ganawileba N3-1 cxrilSi

mocemuli (gamocdebTan dakavSirebuli SfoTvis kiTxvarSi

miRebuli) qulebisTvis.

(a) (b) (g) (d) (e)

qula Tvlis

Sedegi

sixSire

(ƒ)

fardobiTi

sixSire

(rƒ)

sixSire

procentebSi

(%ƒ)

68 67 66 65 64 63 62 61 60 59 58 57 56 55

/ // / // / // /

/// /////

/ /

1 2 0 1 2 1 2 1 3 5 0 1 0 1

.05

.10 .00 .05 .10 .05 .10 .05 .15 .25 .00 .05 .00 .05

5 10 0 5 10 5 10 5 15 25 0 5 0 5

sixSireTa ganawileba (Frequency distribution): _

cxrili, romelSic mocemulia TiToeuli qula da

is sixSire, ra sixSiriTac gvxvdeba es qula.

daujgufebel sixSireTa ganawileba (Ungroup frequency distribution): _ sixSireTa ganawileba,

romelic agebulia miRebul minimalur da maqsimalur qulaTa Soris yvela SesaZlo

mniSvnelobis dalagebiT da Semdeg qulis gaswvriv Tvlis Sedegis aRniSvnis CxiriT (/)

imdenjer moTavsebiT, ramdenjerac iqna miRebuli es qula.


es daujgufebel sixSireTa ganawileba agebulia miRebul umdables da umaRles

qulaTa Soris yvela SesaZlo mniSvnelobis (cxrilis N3.2 (a) sveti) dalagebiT da

Semdeg qulis gaswvriv Tvlis Sedegis aRniSvnis Cxiris (/) imdenjer moTavsebiT,

ramdenjerac iqna miRebuli es qula (amave cxrilis (b) sveti). yuradReba miaqcieT,

rom (a) svetSi mocemulia yvela SesaZlo qula ganurCevlad imisa, miRebul iqna

is realurad Tu ara. im qulebis gaswvriv, romlebic ar iqna miRebuli qulaTa

mocemul simravleSi, ar iwereba Tvlis Sedegis aRmniSvneli Cxiri. TiToeuli

qulis gaswvriv moTavsebuli Cxirebis raodenoba daiTvleba da Seva (g) svetSi,

rogorc TiToeuli qulis Sexvedris martivi sixSire (aRniSneba ƒ-iT). yuradReba

miaqcieT, rom qula 60 Cvens mier ganxiluli ganawilebis daaxloebiT Sua nawils

warmoadgens da qulaTa daaxloebiT naxevari masze naklebi, meore naxevari ki _

masze metia.

fardobiT sixSireTa ganawileba

daujgufebel sixSireTa ganawilebaSi

qulebis Sexvedris sixSire SeiZleba

fardobiTi sixSireebis saxiTac

warmovadginoT. fardobiT sixSireTa

ganawileba miiReba TiToeuli qulis sixSiris gayofiT ganawilebaSi qulaTa

saerTo raodenobaze. amrigad, fardobiTi sixSire (aRiniSneba rƒ-iT) ase

gamoisaxeba:

qulis rƒ = qulis ƒ/N

magaliTad, cxrilidan N3.2 59-is Sexvedris fardobiTi sixSiris

gamosaTvlelad misi sixSire (anu ƒ=5) unda gavyoT qulaTa saerTo raodenobaze

(N=20) da miviRebT .25-s1. gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli

qulebis fardobiTi sixSireebi mocemulia cxrilis N3.2 (d) svetSi.

sixSireTa ganawileba procentebSi

sixSireTa ganawileba procentebSi gamoxatavs

qulis Sexvedris sixSires miRebuli qulebis

saerTo raodenobis procentis saxiT. qulis

procentebSi gamoxatuli sixSiris (aRiniSneba %ƒ-

1 teqstSi meaTedi wiladebi mocemulia amerikis fsiqologTa asociaciis poblikaciebis

saxelmZRvanelos (Publication Manual of the American Psychological Association, 1994) mixedviT, rac

gulisxmobs, rom rodesac wiladi 1-ze naklebia, mZimis win nuli ar iwereba. amitom

fardobiTi sixSireebi mocemulia mZimis win nulis gareSe.

ƒ: _ sixSireTa ganawilebaSi TiToeuli qulis Sexvedris martivi sixSire.

fardobiTi sixSire (rƒ) (Relative frequency):

_ qulis sixSire gayofili miRebul

qulaTa saerTo raodenobaze.

sixSiris procenti (%ƒ) (Relative

frequency): _ qulis fardobiTi

sixSire gamravlebuli 100-ze.


iT) gamosaTvlelad qulis fardobiTi sixSire mravldeba 100-ze anu

qulis %ƒ = (qulis rƒ)x100

cxrilSi N3.2 qulis sixSiris procenti mocemulia (e) svetSi. yuradReba

miaqcieT, rom am cxrilis (g), (d) da (e) svetebSi mocemulia erTi da igive

informacia cxrilSi N3.1 warmodgenili qulebis Sesaxeb. fardobiTi sixSireebis

an procentebSi gamoxatuli sixSiris saxiT sixSireebis gamosaxvis upiratesoba

isaa, rom maTi mniSvnelobebi ar aris damokidebuli miRebuli qulebis saerTo

raodenobaze. amrigad, Tu gvaqvs studentTa sxva jgufisgan miRebuli 35 qulisgan

Semdgari SerCeva (N=35), ufro martivi iqneba sxvadasxva zomis ori SerCevis

sixSireTa ganawilebis urTierTSedareba, rodesac sixSireTa ganawileba

warmodgenilia fardobiTi sixSireebiT an sixSireTa procentebis saxiT, vidre

mxolod martivi sixSireebis saxiT warmodgenis SemTxvevaSi.

informaciis miReba sixSireTa ganawilebidan

cxrilSi N3.2 mocemuli martivi sixSireebi, fardobiTi sixSireebi an sixSireebi

procentebSi garkveul warmodgenas gviqmnis gamocdebTan dakavSirebuli SfoTvis

kiTxvarSi miRebul qulebze, rac arc ise naTlad Cans cxrilSi N3.1 mocemuli

nedli monacemebidan. magaliTad, TiToeuli ganawilebis SemTxvevaSi SeiZleba

SeamCnioT, rom:

• yvelaze dabali qulaa 55;

• yvelaze maRali qulaa 68;

• yvelaze xSirad gvxvdeba qula 59;

• 59-sa da 65-s Soris moTavsebulia TxuTmeti qula.

naTelia, rom nedli monacemebis warmodgena sixSireTa ganawilebis saxiT

aiolebs am monacemebis mniSvnelovani Tvisebebis danaxvas.

magaliTi N3.1

sixSireTa ganawilebis ageba

amocana

fsiqologebi dainteresdnen orazrovani figurebis aRqmiT. maT miaCndaT, rom

aseTi figurebis aRqmis Taviseburebebis SeswavliT samyaros Cveulebrivi aRqmis

Sesaxeb moipovebdnen codnas. suraTze N3.1 mocemuli naxati, romelic meCis

piramidis (The Mach Pyramid) saxeliT aris cnobili, orazrovani figuris magaliTia,

vinaidan sxvadasxvanairad aris SesaZlebeli misi siRrmeSi aRqma. is SeiZleba

aRviqvaT rogorc piramida, romlis mcire kvadrati Cvens mxares aris moqceuli,

fuZe ki furcelzea ganlagebuli; an rogorc oTaxi, romelSic igive mcire

kvadrati Cvens pirdapir mdebare, daSorebul kedlad aRiqmeba. vTqvaT, gyavdaT 25


cdis piri, romelTagan TiToeuli 1 wuTis ganmavlobaSi uyurebda am figuras da

afiqsirebdiT, ramden xans SeeZlo TiToeul maTgans piramidis danaxva. Sedegad

miiReT Semdegi droebi (1 wamis sizustiT):

2, 7, 10, 5, 5, 3, 9, 11, 5, 8, 2, 4, 7, 6, 10, 7, 3, 9, 7, 11, 4, 6, 3, 1 da 8.

es qulebi gamosaxeT martiv sixSireTa ganawilebis, fardobiT sixSireTa

ganawilebisa da sixSireTa procentebSi ganawilebis saxiT. ra daskvnebamde

SeiZleba mixvideT am ganawilebebze dayrdnobiT?

amoxsna

qvemoT mocemul cxrilSi warmodgenilia

sixSireTa dasaxelebuli ganawilebebi.

daskvnebi

qulebi icvleba 1-dan 11 wamamde da yvelaze xSirad gvxvdeba 7 wami. yvelaze

xSirad dafiqsirebuli 7 wamis garda, yvela danarCeni qula daaxloebiT Tanabari

sixSiriT gvxvdeba.

SeamowmeT Tqveni codna III.1

1. ganmarteT: _ sixSireTa ganawileba, sixSireTa ganawileba procentebSi,

%ƒ, nedli monacemebi, nedli qulebi, fardobiT sixSireTa ganawileba, rƒ,

sixSireTa ganawileba daujgufebeli monacemebisTvis.

2. gudma da gudma (Good & Good, 1973) Seqmnes warmatebis SiSis Sefasebis

skala. iTvleba, rom am skalaze maRali maCveneblis mqone individebi Tavs

arideben dawyebul saqmeebSi didi warmatebis miRwevas sxvebis

arakeTilganwyobis gamowvevis SiSis gamo. warmatebis SiSis skalaze

miRebuli qulebi SeiZleba icvlebodes 0-dan (rac miuTiTebs, rom ar

aRiniSneba warmatebis SiSi am skalaze) 29-mde (rac miuTiTebs warmatebis

maqsimalur SiSze am skalaze). davuSvaT, Sefasebis es skala mieciT 20

cdis pirs da miiReT Semdegi qulebi: _ 6, 13, 4, 19, 3, 12, 9, 8, 5, 11, 6, 8, 15,

5, 8, 10, 9, 14, 4 da 8.

qula Tvlis Sedegi

ƒ rƒ %ƒ

11 10 9 8 7 6 5 4 3 2 1

// // // // //// // /// // /// // /

2 2 2 2 4 2 3 2 3 2 1

.08

.08

.08

.08

.16

.08

.12

.08

.12

.08

.04

8 8 8 8 16 8 12 8 12 8 4

suraTi N3.1. meCis piramida


a) es qulebi gamosaxeT martiv sixSireTa ganawilebis, fardobiT

sixSireTa ganawilebisa da procentebSi gamoxatuli sixSiris saxiT.

b) miRebul qulebs Soris romelia maqsimaluri da minimaluri qula?

g) romeli qula gvxvdeba yvelaze xSirad?

sixSireTa ganawileba dajgufebuli monacemebisTvis

Tu miRebulia didi raodenobiT qulebi, an

Tu monacemebSi qulaTa mniSvnelobebis farTo

diapazonia, cxrilSi N3.2 mocemuli

daujgufebel sixSireTa ganawileba SeiZleba

Zalze gafantuli gamovides da amiT

garTuldes monacemTa naTeli suraTis danaxva.

am SemTxvevaSi sasargebloa dajgufebul

sixSireTa ganawilebis gamoyeneba, sadac qulebi dajgufebulia intervalebSi.

intervalebi lagdeba yvelaze maRlidan yvelaze dablisken, iTvleba TiToeuli

intervalis qulebis sixSireebi da gamoisaxeba cxrilis saxiT. dajgufebul

sixSireTa ganawilebis sailustraciod, davuSvaT, rom konsultantma gadawyvita,

gaezoma gamocdebTan dakavSirebuli SfoTva 100 studentisgan Semdgari

SerCevisTvis, romelTac ar miumarTavT daxmarebisTvis am problemis gamo.

miRebuli qulebi mocemulia cxrilSi N3.3.

cxrili N3.3. gamocdebTan dakavSirebuli SfoTvis

kiTxvarSi 100 studentisgan Semdgari SerCevis mier

miRebuli qulebi.

42 28 37 45 58 41 39 52 40 34 25 43 49 45 48 42 44

44 38 41 41 38 37 50 40 31 50 41 46 37 40 38 26 41

41 44 22 32 35 41 38 47 52 48 55 41 28 26 41 44 55

36 34 48 43 46 43 42 28 41 38 41 65 42 46 34 38 40

39 40 39 36 25 31 40 39 38 45 70 36 44 39 42 44

50 46 37 60 60 47 36 73 43 46 37 46 34 29 40 37

dajgufebul sixSireTa ganawilebis ageba


(Grouped frequency distribution): _

ganawileba, sadac qulebi

dajgufebulia intervalebSi da

TiToeuli maTganSi moxvedrili

qulebis sixSireebi gamoisaxeba

cxrilis saxiT.

intervali (Class interval): _ intervalis sidide, romelic gamoiyeneba dajgufebul sixSireTa

ganawilebaSi nedli qulebis dasajgufeblad. intervalis sidide aRiniSneba i-iT.


dajgufebul sixSireTa ganawilebaSi nedli qulebi nawildeba intervalebSi.

intervali aris qulaTa diapazoni, romelSic jgufdeba nedli qulebi. cxrilSi

N3.3 mocemuli qulebis dajgufebul sixSireTa ganawileba warmodgenilia

cxrilSi N3.4. am SemTxvevaSi intervalis zoma aris 5. dajgufebul sixSireTa

iseTi ganawilebis agebas, rogoric mocemulia cxrilSi N3.4, sWirdeba ori ram: _

(1) gamoyenebul intervalTa raodenoba da (2) intervalis zoma.

cxrili N3.4. dajgufebul sixSireTa ganawileba cxrilSi N3.3 mocemuli gamocdebTan

dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebisTvis.

(a) (b) (g) (d) (e) (v) (z) (T) (i) (k)

intervali

realuri zeda da qveda

sazRvari

intervalis Suawertili

Tvlis Sedegi ƒ rƒ %ƒ cƒ crƒ c%ƒ

70-74 65-69 60-64 55-59 50-54 45-49 40-44 35-39 30-34 25-29 20-24

69.5-75.5 64.5-69.5 59.5-64.5 54.5-59.5 49.5-54.5 44.5-49.5 39.5-44.5 34.5-39.5 29.5-34.5 24.5-29.5 19.5-24.5

72 67 62 57 52 47 42 37 32 27 22

// / // /// ///// //////////////// ///////////////////////////////// //////////////////////// /////// /////// /

2 1 2 3 5 15 33 24 7 7 1

.02 .01 .02 .03 .05 .15 .33 .24 .07 .07 .01

2 1 2 3 5 15 33 24 7 7 1

100 98 97 95 92 87 72 39 15 8 1

1.00 .98 .97 .95 .92 .87 .72 .39 .15 .08 .01

100 98 97 95 92 87 72 39 15 8 1

intervalTa raodenoba

imas, Tu ramden intervals gamoviyenebT, ganapirobebs qulaTa diapazoni;

Cveulebisamebr, gamoiyeneba 10-dan 20-mde intervali. rogorc wesi, intervalebis

raodenobas irCeven ise, rom intervalTa sidide iyos 1, 2, 3, 5 an 5-is jeradi.

cxrilSi N3.3 mocemul qulaTa diapazonia 51 [22-dan (yvelaze dabali qula) 73-mde

(yvelaze maRali qula)]. qulaTa aseTi diapazonis SemTxvevaSi monacemebis

dasajgufeblad sakmarisi iqneba 10 intervali.

intervalis zoma

intervalTa raodenobis gansazRvris Semdeg gamoiTvleba intervalis zoma.

amisaTvis maqsimaluri da minimaluri qulebis sxvaoba iyofa intervalTa

raodenobaze.

statistikaSi simbolo X xSirad gamoiyeneba qulis aRsaniSnad. aqedan

gamomdinare, intervalis sidide, romelic aRiniSneba i-iT, ase gamoiTvleba:

raodenoba intervalTa

minimaluriimaqsimalur XXi

−=

X _ simbolo, romelic calkeuli, individualuri qulis aRsaniSnad gamoiyeneba.

i _ intervalis zoma dajgufebul sixSireTa ganawilebaSi.


sadac Xmaqsimaluri aris yvelaze maRali qula, xolo Xminimaluri _ yvelaze dabali

qula ganawilebaSi. cxrilSi N3.3 mocemuli monacemebisTvis, sadac gvaqvs 10

intervali, xolo Xmaqsimaluri = 73 da Xminimaluri = 22, intervalis sidide tolia:

10.51051

102273

==−

=i

Tu am formulaSi mniSvnelobebis Setanis Semdeg miviRebT aTwilads (rogorc

ganxilul magaliTSi _ 5.10), is mrgvaldeba intervalTa klasis uaxloes

rekomendebul sigrZemde (anu 1, 2, 3, 5 an 5-is jeradamde). magaliTad, am

SemTxvevaSi i mrgvaldeba 5-mde. Tu qulebi gamosaxulia 1-ze naklebi aTwiladiT,

intervalis sidide ar mrgvaldeba mTel ricxvamde; am dros gamoiyeneba iseTi

intervalebi, rogoricaa 0.3, 0.5, 0.03 an 0.05.

intervalis ageba

intervalis asagebad unda sruldebodes ori ZiriTadi piroba:

• qveda intervali unda moicavdes yvelaze dabal qulas;

• pirveli intervalis qveda sazRvari (anu pirveli intervalis yvelaze

dabali SesaZlo qula) unaSTod unda iyofodes intervalis sigrZeze.

cxrilSi N3.3 mocemul qulaTa Soris yvelaze dabalia 22. Sesabamisad,

cxrilis N3.4 (a) svetSi yvelaze dabali intervali iwyeba qveda sazRvriT 20,

radgan 20 aris pirveli qula, romelic naklebia 22-ze da romelic unaSTod iyofa

intervalis sidideze anu 5-ze. Semdegi intervalia 25-29. damatebiTi intervali

gamoiTvleba im wertilis zeviT, romelic saWiroa bolo intervali aigeba ise,

rom masSi CarTuli iyos yvelaze maRali qula, mocemul magaliTSi, es aris

intervali 70-74.

intervalis dadgenili da realuri

sazRvrebi. cxrilis N3.4 (a) svetSi mocemul

TiToeul intervals aqvs zeda da qveda

dadgenili sazRvari (romelsac nominalur

sazRvars an qulis sazRvarsac uwodeben).

intervalis zeda da qveda sazRvari aris maqsimaluri da minimaluri qulebi,

romelic SeiZleba moxvdes am intervalSi. magaliTad, 40-44 intervalisTvis qveda

dadgenili sazRvaria 40, xolo zeda dadgenili sazRvari _ 44. 40 aris minimaluri

qula, xolo 44 _ maqsimaluri qula, romelic SeiZleba Sevides intervalSi.

suraTze N3-2 warmodgenilia 40-44 intervalis dadgenili zeda da qveda sazRvari.

ricxvis realuri sazRvrebi

me-2 TavSi ganmartebulia,

rogorc ricxvisa da mis wina

ricxvis da ricxvisa da mis

intervalis dadgenili sazRvari

(Stated limits of a class interval): _

maqsimaluri da minimaluri qulebi,

romelic SeiZleba moxvdes am

intervalSi.

intervalis realuri sazRvari (Real limits of a class

interval): _ intervalis dadgenil sazRvrebsa da am

intervalis wina an momdevno intervalebis dadgenil

sazRvrebs Soris Suawertili.


momdevno ricxvis Suawertili im skalaze, romelsac viyenebT mocemuli cvladis

gasazomad. realuri sazRvrebi dajgufebuli sixSireebis ganawilebis

intervalebsac aqvs da isini moiZebneba iseve, rogorc ricxvis realuri sazRvrebi.

intervalis qveda realuri sazRvari aris intervalis qveda dadgenil sazRvarsa

da winamdebare intervalis zeda dadgenil sazRvars Soris Suawertili.

magaliTad, vipovoT intervalis 40-44 (cxrilidan N3.4) qveda realuri sazRvari. am

intervalis qveda miRebuli sazRvaria 40. Semdegi dabali intervalis (anu

intervalis 35-39) zeda miRebuli sazRvaria 39. Sesabamisad, intervalis 40-44 qveda

realuri sazRvaria 39-sa da 40-s Soris Suawertili anu 39.5. es qveda realuri

sazRvaric mocemulia suraTze N3.2.

intervalis zeda realuri sazRvari aris intervalis zeda dadgenil sazRvarsa

da Semdegi maRali intervalis qveda dadgenil sazRvars Soris Suawertili.

intervalis 40-44 zeda realuri sazRvari miiReba Semdegi intervalis 40-44 zeda

dadgenil sazRvarsa (anu 44) da Semdegi maRali intervalis 45-49 qveda dadgenil

sazRvars Soris Suawertilis povniT. am intervalis qveda dadgenili sazRvaria

45. amrigad, intervalis 40-44 zeda realuri sazRvaria Suawertili 44-sa da 45-s

Soris anu 44.5. es zeda realuri sazRvaric naCvenebia suraTze N3.2. yuradReba

miaqcieT, rom intervalis 40-44 zeda realuri sazRvari Semdegi 45-49 intervalis

intervalis Suawertili (Midpoint of a class interval): _ intervalis realur sazRvrebs Soris

Suawertili.

suraTi N3.2. qveda da zeda dadgenili sazRvrebis, qveda da zeda realuri sazRvrebisa da

dajgufebuli sixSireebis ganawilebis intervalis Suawertilis ilustracia.

intervali

35-39

intervali

40-44

intervali

45-49

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

zeda dadgenili

zRvari

zeda dadgenili

zRvari

qveda dadgenili

zRvari

qveda dadgenili

zRvari


39.5 intervali 40-44 qveda realuri sazRvari

intervali 35-39 zeda realuri sazRvari

44.5 intervali 40-44 zeda realuri sazRvari

intervali 45-49 qveda realuri sazRvari


qveda realuri sazRvaricaa. intervalis zeda da qveda realuri sazRvrebi

warmodgenilia cxrilis N3.4 (b) svetSi.

intervalis Suawertili. TiToeul intervals Suawertilic aqvs. intervalis

Suawertili aris intervalis realur sazRvrebs Soris Suawertili. 40-44

intervalis Suawertili aris 42, radgan 42 aris 39.5-sa da 44.5-s Soris

Suawertili. intervalis Suawertilis povna SesaZlebelia intervalis realuri

sazRvrebis mniSvnelobaTa jamis 2-ze gayofiT. 40-44 intervalis Suawertili

gamoiTvleba Semdegnairad:

422

842

5.445.39==

+

TiToeuli intervalis Suawertili warmodgenilia N3.4 cxrilis (g) svetSi.

miuxedavad imisa, rom am cxrilSi erTad aris mocemuli intervalis zeda da qveda

realuri sazRvrebi da Suawertili, dajgufebul sixSireTa ganawileba xSirad

mxolod (a) svetSi mocemuli intervalebiT aris xolme warmodgenili.

dajgufebul martiv sixSireTa ganawileba

dajgufebul martiv sixSireTa ganawileba miiReba im intervalSi qulebis

daTvlis Sedegad, romelSic es qulebi xvdeba (ix. cxrili N3.4, (d) sveti).

magaliTad, qula 35 daTvlilia intervalSi 35-39, qula 52 _ intervalSi 50-54.

mocemuli intervalis martivi sixSiris (ƒ) misaRebad jamdeba daTvlis Sedegebi

intervalSi.

dajgufebul fardobiT sixSireTa ganawileba da sixSireTa ganawileba

procentebSi

dajgufebul sixSireTa ganawilebisTvis SesaZlebelia fardobiT sixSireTa

ganawilebisa da procentebSi sixSireTa ganawilebis miRebac, rac iseve

xorcieldeba, rogorc daujgufebel sixSireTa ganawilebis SemTxvevaSi.

dajgufebul fardobiT sixSireTa

ganawileba. dajgufebul fardobiT

sixSireTa ganawilebis gamosaTvlelad

intervalSi qulaTa raodenoba iyofa qulaTa

saerTo raodenobaze ganawilebaSi. amgvarad,

Nf

rf qulaTa intervalSi

qulaTa intervalSi =

gamocdebTan dakavSirebul SfoTvis kiTxvarSi miRebuli qulebis dajgufebul

fardobiT sixSireTa ganawileba mocemulia cxrilis N3.4 (v) svetSi.

dajgufebul fardobiT sixSireTa

ganawileba (Grouped relative frequancy

distribution): _ dajgufebul sixSireTa

ganawileba miiReba qulaTa

fardobiTi sixSiris 100-ze

gamravlebiT.

dajgufebul sixSireTa ganawileba procentebSi (Grouped


dajgufebul sixSireTa

ganawileba procentebSi.


ganawileba procentebSi

miiReba formuliT

100% ×=N

ff

qulaTa i intervalS qulaTa intervalSi

gamocdebTan dakavSirebul SfoTvis kiTxvarSi miRebuli qulebis dajguebul

sixSireTa ganawileba procentebSi mocemulia cxrilis N3.4 (z) svetSi.

dagrovil sixSireTa ganawileba

dagrovili intervaluri

sixSire. yvela ganxiluli

ganawileba: _ martivi sixSire,

fardobiTi sixSire da sixSire

procentebSi _ dagrovil

sixSireTa saxiTac SeiZleba

warmovadginoT. qulis

dagrovili sixSire (aRiniSneba

cƒ-iT an cum ƒ-iT) aris am

qulis Sexvedris sixSires

damatebuli yvela ufro

naklebi qulis sixSireTa jami.

dagrovili intervaluri

sixSire aris am intervalSi

qulebis Sexvedris raodenobas

damatebuli yvela ufro dabal

intervalSi qulebis sixSireTa

jami. magaliTad, 30-34 intervalis (ix. cxrili N3.4) dagrovili sixSire aris 15,

radgan mTlianobaSi 15 qula gvxvdeba intervalSi 20-24 (1 qula), intervalSi 25-29

(7 qula) da intervalSi 30-34 (7 qula). TiToeuli dagrovili intervaluri sixSire

mocemulia cxrilis N3.4 (T) svetSi.

dagrovili intervaluri fardobiTi sixSiris ganawileba da sixSiris

ganawileba procentebSi. fardobiTi sixSireebi da sixSireebi procentebSi

SeiZleba warmodgenil iqnes dagrovili sixSireebis saxiTac. intervalis

dagrovili intervaluri fardobiTi sixSire (aRiniSneba crƒ-iT an cum rƒ-iT) aris

intervalSi qulebis fardobiT sixSires damatebuli yvela ufro dabal

intervalSi qulaTa fardobiT sixSireTa jami.

percentage frecuancy distribution): _ dajgufebul sixSireTa

ganawileba, sadac intervalSi qulaTa sixSire

Sefardebulia ganawilebaSi qulaTa saerTo

raodenobasTan.

qulis dagrovili sixSire (Cumulative frecuency of a score):

_ qulis Sexvedris sixSires damatebuli yvela

ufro naklebi qulis sixSireTa jami.

dagrovili intervaluri sixSire (Cumulative frecuency of

a class interval): _ intervalSi qulebis Sexvedris

raodenobas damatebuli yvela ufro dabal

intervalSi qulebis sixSireTa jami.

dagrovili intervaluri fardobiTi sixSire

(Cumulative grouped relative frecuency of a class interval): _

intervalSi qulebis fardobiT sixSires damatebuli

yvela ufro dabal intervalSi qulaTa fardobiT

sixSireTa jami.

dagrovili intervaluri sixSiris ganawileba

procentebSi ( Cumulative grouped percentage frecuency of a

class interval): _ intervalSi qulebis sixSires

procentebSi damatebuli yvela ufro dabal

intervalSi qulaTa sixSireTa jami procentebSi.


procentebSi gamoxatuli dagrovili intervaluri sixSiris ganawileba

(aRiniSneba c%ƒ-iT an cum %ƒ-iT) aris intervalSi qulebis procentebSi gamoxatul

sixSires damatebuli yvela ufro dabal intervalSi procentebSi gamoxatul

qulaTa sixSireTa jami. dagrovili intervaluri fardobiTi sixSireebi da

sixSireebi procentebSi TiToeuli intervalisTvisA mocemulia cxrilis N3.4 (i)

da (k) svetebSi, Sesabamisad.

informaciis miReba dajgufebul sixSireTa ganawilebidan

cxrilSi N3.4 mocemuli dajgufebul sixSireTa sxvadasxva ganawileba

gviCvenebs, rom qulaTa umravlesoba moTavsebulia 30-sa da 54-s Soris. garda

amisa, ganawilebebi gviCvenebs, rom SfoTvis qulaTa daaxloebiT erTi mesamedi

(100-dan 33) moTavsebulia intervalSi 40-44, Tumca, am cxrilSi ar Cans qulaTa

zusti ricxviTi mniSvnelobebi. Cven ver davasaxelebT konkretul maqsimalur an

minimalur, anda yvelaze xSir qulas, magram informaciis es danakargi im

simartivis sazRauria, romliTac SegviZlia ganawilebis mTliani, zogadi formis

aRqma.

procentili da procentuli rangebi

dagrovil sixSireTa ganawilebebi

xSirad gamoiyeneba procentilebisa da

procentiluri rangebis gamosaTvlelad.

procentili aris qula, romelzec qulaTa

garkveuli procenti ganawilebaSi tolia

an naklebia am qulaze. magaliTad, Tu 35-e

procentili gamocdaze aris 70, gamocdaze

miRebul qulaTa 35% tolia an naklebia

70-ze. qulis procentuli rangi gviCvenebs ganawilebaSi im qulaTa procents,

romlebic tolia an naklebia am qulaze. amrigad, ganxilul magaliTSi testSi

miRebuli 70 qulis procentuli rangi aris 35. qulaTa 35% tolia an naklebia 70-

ze.

qulis procentilis an procentuli rangis gamoTvliT viRebT informacias

ganawilebaSi sxva qulebis mimarT am qulis adgilmdebareobis Sesaxeb. davuSvaT,

Tqven im 100 studentTa Soris xarT, vinc Seavso gamocdebTan dakavSirebuli

SfoTvis kiTxvari da Tqveni qulaa 41. es 41 qula maRalia Tu dabali sxva

studentebis qulebTan SedarebiT? Tu Tqven getyvian, rom 41 aris 49-e procentili,

an 41 qulis procentuli rangia 49, Tqven gecodinebaT, rom qulaTa 49% tolia an

naklebia 41-ze.

qulis procentuli rangis povna

procentili (Percentile): _ qula, romelzec

qulaTa garkveuli procenti

ganawilebaSi tolia an naklebia am

qulaze.

qulis procentuli rangi (Percentile rank of

a score): _ im qulaTa procenti

ganawilebaSi, romlebic tolia an

naklebia am qulaze.


davuSvaT, gvsurs gamovTvaloT cxrilSi N3.4 mocemuli ganawilebidan 41 qulis

procentuli rangi. es qula Sedis intervalSi 40-44. am intervalis qveda realur

sazRvramde yvela qulis dagrovili sixSire procentebSi aris 39 (ix. sveti (k) anu

c%ƒ), xolo am intervalis zeda realur sazRvramde qulaTa dagrovili sixSire

procentebSi _ 72 (ix. (k) sveti). amrigad, 41-is procentuli rangi iqneba 39-sa da 72-

s Soris. zusti procentuli rangis gamosaTvlelad unda movaxdinoT 41-is

interpolireba 40-44 intervalSi. qvemoT mocemulia formula, romliTac xdeba

interpolireba da gamoiTvleba qulis procentiluri rangi:

( )[ ] 100/×

−+==

NfiXXcfP iLL

Xrangi riprocentilu

sadac PX = X qulis procentuli rangi.

cƒL = X-is Semcveli intervalis qveda realur sazRvramde qulebis

dagrovili sixSire.

X = qula, romlisTvisac gamoiTvleba procentuli rangi.

XL = X-is Semcveli intervalis qveda realuri sazRvari.

i = intervalis sidide.

ƒ i = X-is Semcveli intervalis qulaTa sixSire.

N = qulaTa saerTo raodenoba ganawilebaSi.

cxrilSi N3.4 mocemuli 41 qulisTvis es mniSvnelobebi iqneba:

PX = P41.

cƒL = 39 (ix. cxrili N3.4, (T) sveti).

X = 41.

XL = 39.5 (ix. cxrili N3.4, (b) sveti).

i = 5.

ƒ i = 33 (ix. cxrili N3.4, (e) sveti).

N = 100.

Tu am mniSvnelobebs CavsvamT formulaSi, miviRebT:

( )[ ]

( )( )[ ]

( )( )[ ]

9.48

100100

9.48

100100

9.939

100100

333.39

100100

335/5.139

100100

335/5.39413941

=

×=

×+

=

×+

=

×+

=

×−+

=P


procentuli rangi, rogorc wesi, mrgvaldeba mTel ricxvamde, amitom

P41 = 49

41 qulis procentuli rangi aris 49 anu cxrilSi N3.4 qulaTa 49% tolia an

naklebia 41-ze. es formula daujgufebel sixSireTa ganawilebisTvisac SeiZleba

gamoviyenoT; am SemTxvevaSi i = 1.

ganawilebis procentilis povna

Semdegi amocanaa mocemuli procentis Sesabamisi qulis povna. magaliTad,

davuSvaT, gvsurs cxrilSi N3.4 warmodgenili ganawilebis 49-e procentilis povna.

yuradReba miaqcieT, rom es procentili aris qula, romelic Sedis intervalSi

40-44, qulaTa 39% xvdeba am intervalis qveda realur sazRvars qvemoT, xolo

qulaTa 72% _ zeda realur sazRvars qvemoT. im konkretuli qulis sapovnelad,

romelic aris 49-e procentili, unda movaxdinoT interpolireba. qvemoT

mocemulia formula, romliTac xdeba interpolireba da procentilis gamoTvla:

if

cfPNXXi

LLP

−+==procentili

sadac XP = qula konkretuli procentilis toli (anu qula, romlis povnac

gvsurs).

XL = konkretuli procentilis Semcveli intervalis qveda realuri

sazRvari.

P = mocemuli procenti, mocemuli 0-dan 1.00-mde wiladis saxiT.

N = qulaTa saerTo raodenoba ganawilebaSi.

cƒL = XP-is Semcveli intervalis qveda realur sazRvramde qulebis

dagrovili sixSire.

ƒ i = XP-is Semcveli intervalSi qulaTa sixSire.

i = intervalis sidide.

49-e procentilisTvis es mniSvnelobebi iqneba:

XP = X49.

XL = 39.5 (ix. cxrili N3.4, (b) sveti).

P = .49

N = 100.

cƒL = 39 (ix. cxrili N3.4, (T) sveti).

ƒ i = 33 (ix. cxrili N3.4, (e) sveti).

i = 5.

Tu am mniSvnelobebs CavsvamT formulaSi, miviRebT:


02.4152.15.39

533105.39

533

39495.39

533

39)100(49.5.3949

=+=

+=

−

+=

−

+=X

rodesac am mniSvnelobas mTel ricxvamde davamrgvalebT, miviRebT: X49 = 41. 49-e

procentili aris 41 qula. ganawilebaSi qulaTa 49% tolia an naklebia 41-ze. es

formula daujgufebel sixSireTa ganawilebisTvisac SeiZleba gamoviyenoT; am

SemTxvevaSi i = 1.

magaliTi N3-2


amocana

davuSvaT, vikvlieT N3-1 magaliTSi ganxiluli orazrovani naxatebis aRqmis

problema da gavzomeT drois xangrZlioba, romlis manZilzec cdis pirebs

SeeZloT piramidis danaxva meCis piramidaSi. gvyavda 50 cdis piri da miviReT

Semdegi monacemebi (wamis sizustiT): _ 18, 2, 7, 23, 16, 10, 5, 30, 5, 15, 17, 3, 31, 9, 11, 13,

5, 13, 8, 19, 16, 2, 14, 4, 7, 7, 6, 2, 10, 6, 7, 24, 3, 8, 9, 7, 19, 7, 11, 9, 4, 7, 6, 1, 3, 10, 1, 26, 8

da 5.

a) am qulebisTvis aageT dajgufebul sixSireTa Semdegi ganawilebebi: _ martiv

sixSireTa ganawileba, fardobiT sixSireTa ganawileba, sixSireTa

ganawileba procentebSi, dagrovili intervaluri sixSire, dagrovili

fardobiTi sixSire da dagrovil sixSireTa ganawileba procentebSi.

gansazRvreT intervalebis qveda da zeda realuri sazRvrebi da TiToeuli

intervalis Suawertilebi.

b) ra daskvnebis gakeTeba SegiZliaT am monacemebis Sesaxeb am ganawilebebze

dayrdnobiT?

g) ipoveT 16 wamis procentiluri rangi.

d) am ganawilebaSi romelia 25-e procentili?

amoxsna

a) saWiroa intervalTa raodenobis gansazRvra, intervalTa sididis gamoTvla da

Semdeg dajgufebul sixSireTa ganawilebis ageba am intervalebis gamoyenebiT.

intervalTa raodenoba: qulaTa diapazonia 1-dan 31 wamamde anu 30 wami.

monacemebis dasajgufeblad sakmarisi iqneba 10 intervali.


intervalis sidide: i= (31-1)/10=3.0.

ganawilebis ageba: miRebul qulaTa Soris yvelaze dabali qulaa 1 wami,

intervalis sidide _ 3 wami. amitom pirveli intervali iqneba 0-2. es intervali

Seicavs yvelaze dabal qulas da misi qveda dadgenili sazRvari (anu 0)

unaSTod iyofa intervalis sidideze (anu 3-ze). dajgufebul sixSireTa

intervalebi warmodgenilia qvemoT mocemuli cxrilis (a) svetSi. TiToeuli

intervalis zeda da qveda sazRvrebi miRebulia momijnave intervalebs Soris

Suawertilebis povnis Sedegad. magaliTad, zeda da qveda realuri sazRvrebi

intervalisTvis 24-26 aris Suawertili 23-sa da 24-s (qveda realuri sazRvari)

Soris da Suawertili 26-sa da 27-s (zeda realuri sazRvari) Soris,

Sesabamisad. cxrilis (b) svetSi mocemulia TiToeuli intervalis realuri

sazRvrebi, (g) svetSi _ TiToeuli intervalis Suawertilebi, (d) svetSi _

qulebis Tvlis Sedegi, xolo (e), (v), (z), (T), (i) da (k) svetebSi _ dajgufebul

sixSireTa ganawilebebi.

(a) (b) (g) (d) (e) (v) (z) (T) (i) (k)

intervali

realuri zeda da qveda

sazRvari


Tvlis Sedegi ƒ rƒ %ƒ cƒ crƒ c%ƒ

30-32 27-29 24-26 21-23 18-20 15-17 12-14 9-11 6-8 3-5 0-2

29.5-32.5 26.5-29.5 23.5-26.5 20.5-23.5 17.5-20.5 14.5-17.5 11.5-14.5 8.5-11.5 5.5-8.5 2.5-5.5 -0.5-2.5

31 28 25 22 19 16 13 10 7 4 1

// // / /// //// /// //////// ///////////// ///////// /////

2 0 2 1 3 4 3 8 13 9 5

.04

.00

.04

.02

.06

.08

.06 .16 .26 .18 .10

4 0 4 2 6 8 6 16 26 18 10

50 48 48 46 45 42 38 35 27 14 5

1.00 .96 .96 .92 .90 .84 .76 .70 .54 .28 .10

100 96 9 92 90 84 76 70 54 28 10

b) daskvnebi: martivi sixSireebidan (sveti (e)), fardobiTi sixSireebidan (sveti (v))

anda sixSireebidan procentebSi (sveti (z)) vxedavT, rom qulebi yvelaze

xSirad gvxvdeba intervalSi 6-8. dagrovil sixSireTa ganawileba gviCvenebs,

rom 0.70 anu qulaTa 70% aris 11 wami an masze naklebi.

g) qulis procentuli rangi gamoiTvleba formuliT

( )[ ] 100/×

−+=

NfiXXcfP iLL

X

16 wamis procentuli rangis gamosaTvlelad saWiro mniSvnelobebia:

PX = P16

cƒL = 38

X = 16

XL = 14.5


i = 3

ƒ i = 4

N = 50

CavsvaT es mniSvnelobebi formulaSi. miviRebT:

( )[ ]

[ ]

8010080.

1005040

10050

238

10050

4)5(.38

10050

43/5.138

10050

43/5.14163816

=×=

×=

×+

=

×+

=

×+

=

×−+

=P

16 wamis procentuli rangi aris 80. es niSnavs, rom drois miRebuli

xangrZliobebis 80% aris 16 wami an 16 wamze naklebi.

d) 25-e procentili gamoiTvleba formuliT

if

cfNPXXi

LLP

−+=

)(

25-e procentilisTvis es mniSvnelobebi iqneba:

XP = X25.

XL = 2.5

P = .25

N = 50

cƒL = 5

ƒ i = 9

i = 3

CavsvaT es mniSvnelobebi formulaSi. miviRebT:

0.55.25.2

395.75.2

39

55.125.2

39

5)50(25.5.216

=+=

+=

−

+=

−

+=X

25-e procentilia qula 5.



1. ganmarteT: _ intervali, intervalis dagrovili sixSire, qulis

dagrovili sixSire, intervalis dagrovili sixSire procentebSi,

intervalis dagrovili fardobiTi sixSire, cƒ, c%ƒ, crƒ, cum ƒ, cum %ƒ, cum

rƒ, dagrovil sixSireTa ganawileba, intervalis qveda realuri sazRvari,

intervalis qveda dadgenili sazRvari, intervalis Suawertili,

procentili, procentuli rangi, intervalis zeda realuri sazRvri,

intervalis zeda dadgenili sazRvari.

2. dajgufebul sixSireTa ganawilebis agebisas ra raodenobis

intervalebis gamoyenebaa rekomendebuli?

3. ganawilebaSi maqsimaluri qulaa 8 da minimaluri _ 14. Tqven apirebT

dajguebul sixSireTa ganawilebis agebas 15 intervaliT. ra sigrZis

iqneba Tqveni intervali?

4. ganawilebaSi yvelaze didi qulaa 95 ada yvelaze mcire _ 62. Tqven

gsurT, rom gqondeT intervali sididiT 3. ramdeni intervali geqnebaT?

5. ra ori ZiriTadi piroba unda sruldebodes dagjufebul sixSireTa

ganawilebis agebisas?

6. dajgufebul sixSireTa ganawilebis intervalebia 120-122, 123-125, 126-128,

129-131 da 132-134.

a) ipoveT 123-125 intervalis qveda da zeda dadgenili sazRvrebi;

b) ipoveT 129-131 intervalis qveda da zeda dadgenili sazRvrebi;

g) ipoveT 123-125 intervalis qveda da zeda realuri sazRvrebi;

d) ipoveT 129-131 intervalis qveda da zeda realuri sazRvrebi;

e) ipoveT 120-122 intervalis Suawertili;

v) ras udris i am ganawilebisTvis?

7. mtaceblis miaxloebisas tompsonis qurcikebi xtian ise, rom oTxive fexs

erTdroulad aSoreben miwas. es qceva, romelsac maRali naxtomi

ewodeba, rogorc Cans, mniSvnelovania qurcikisTvis, radgan is

saSualebas aZlevs mas, Tavi aaridos mtacebelTan Setakebas. am naxtomis

Seswavlis mizniT karom (Caro, 1986) bunebrivi dakvirveba gamoiyena. is

ikvlevda qurcikebs serengetis erovnuli parkSi, tanzaniaSi. erT-erTi

maCvenebeli iyo qurciksa da mtacebels (magaliTad, gepards) Soris

manZili, rodesac qurciki pirvel maRali naxtoms axorcielebda. vTqvaT,

aRinusxa gepardis qurcikTan miaxloebis 50 SemTxveva da dafiqsirda

Semdegi manZilebi (metrebSi) qurciksa da gepards Soris pirveli maRali

naxtomis ganxorcielebamde: 47, 61, 72, 63, 56, 54, 67, 75, 91, 72, 83, 140, 31, 37,

25, 38, 57, 62, 49, 205, 66, 39, 43, 81, 66, 72, 42, 29, 190, 64, 76, 54, 38, 67, 61, 33,

48, 55, 59, 68, 18, 49, 37, 83, 70, 154, 91, 50, 58 da 66.


a) am qulebisTvis aageT Semdegi dajgufebuli ganawilebebi: _ martiv

sixSireTa ganawileba, fardobiT sixSireTa ganawileba, sixSireTa

ganawileba procentebSi, dagrovil martiv sixSireTa ganawileba,

dagrovil sixSireTa fardobiTi ganawileba da dagrovil sixSireTa

ganawileba procentebSi. ipoveT intervalTa qveda da zeda realuri

sazRvrebi da TiToeuli intervalis Suawertili. gamoiyeneT 20

intervali.

b) ra daskvnebis gakeTeba SegiZliaT qurcikis maRali naxtomis sigrZis

Sesaxeb dajgufebul sixSireTa ganawilebebSi mocemul

informaciaze dayrdnobiT?

g) ipoveT 68 metriani manZilis procentuli rangi.

d) am ganawilebaSi romelia 50-e procentili?

sixSireTa ganawilebis grafikulad gamosaxva

sixSireTa ganawileba iZleva monacemebis sistematuri gziT aRqmis saSualebas.

monacemebis ukeT aRsaqmelad sixSireTa ganawilebiT mocemuli informacia xSirad

gamoisaxeba iseTi grafikebis saxiT, rogoricaa histograma, sixSireTa poligoni

an foTlebiani Reroebis diagrama. Tavdapirvelad gavecnoT grafikebis agebis

zogad wesebs.

• grafikis ori RerZi erTmaneTis mimarT marTi (90°-iani) kuTxiT aris

ganlagebuli. horizontalur RerZs ewodeba x RerZi anu abscisa,

vertikalurs ki _ y RerZi anu ordinata.

• SesaZlo qulebi an dajgufebul sixSireTa SesaZlo qulebis intervalebi

moTavsebulia x RerZis gaswvriv. sixSiris sazomi (magaliTad, martivi

sixSire, fardobiTi sixSire, sixSire procentebSi, dagrovili sixSire da a.S.)

moTavsebulia y RerZze.

• TiToeul RerZs erqmeva iseTi saxeli, rom mkiTxveli mixvdes, Tu ra

informaciaa mocemuli grafikze.

• TiToeuli RerZis sawyisi wertili, Cveulebisamebr, aris nuli. Tu skalis

nawili sawyis nulovan wertilsa da pirvel mocemul qulas an qulis

sixSires Soris carielia RerZis marcxena mxares, RerZze mocemulia amis

maniSnebeli niSani -/ /-.

• y RerZis skala ise unda iqnes SerCeuli, rom masze gadazomili sixSiris

maqsimaluri simaRle x RerZis daaxloebiT mde dan −−43

32

siganes Seadgendes.

histograma


histogramaSi vertikaluri

svetebis simaRle gamoiyeneba

intervalSi qulis

Sexvedris sixSiris

saCveneblad. TiToeuli intervalis sidide warmodgenilia abscisaze svetis

siganiT. histogramebs, Cveulebriv, mxolod dajgufebuli sixSireebisTvis ageben,

sixSiris sazomad ki iyeneben martiv intervalur sixSires, an fardobiT

intervalur sixSires (ix. cxrili N3.4, (e) da (v) svetebi, Sesabamisad). gamocdebTan

dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebis dajgufebul martiv

sixSireTa histograma mocemulia suraTze N3.3. rogorc wesi, abscisaze qulebi

mieTiTeba TiToeuli intervalis Suawertilebis ricxviTi mniSvnelobebiT,

rogorc es naCvenebia suraTebze N3.3 da N3.5. qulaTa moxvedris sixSire

gamosaxulia ordinataze da TiToeuli svetis simaRle Seesabameba im intervalTa

sixSires, romelsac is warmoadgens.

suraTi N3.3. martiv sixSireTa histograma gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebisTvis

(cxrilidan N3.3)

0

10

20

30

40

0 22 27 32 37 42 47 52 57 62 67 72

gamocdebTan dakavSirebuli SfoTvis kiTxvaris qula

six

Sir

e

sixSireTa poligoni

sixSireTa poligonis asagebad saWiroa,

rom abscisaTa RerZze ganlagdes

intervalis Suawertilebi, intervalis

sixSire ki aRiniSneba Suawertilebis

Tavze wertilebiT. es wertilebi Semdeg erTdeba swori xazebiT. suraTze N3.4

mocemulia gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli

dajgufebuli qulebis (ix. cxrili N3.4) sixSireTa poligoni. aq gamoyenebulia

histograma (Histogram): _ svetebiani grafikis nairsaxeoba,

sadac intervalebSi qulaTa Sexvedris sixSire

gamosaxulia svetis simaRliT, TiToeuli intervalis

sigrZe ki _ abscisaze svetis siganiT.

sixSireTa poligoni (Frequency polygon): _

grafiki, romlis abscisebia intervalTa

Suawertilebi, ordinatebi ki _ am

intervalTa Sesabamisi sixSireebi.


martiv sixSireTa ganawileba, romelic mocemulia am cxrilis (e) svetSi.

yuradReba miaqcieT, rom suraTze N3.4 ganawilebis orive boloSi mocemulia

TiTo-TiTo damatebiTi intervali, romelic ar Seicavs arc erT qulas da rom

sixSireTa poligoni grafikis orive boloSi miyvanilia nulamde

(Suawertilebamde 17 da 77). sixSireTa poligonis agebisas es yovelTvis ase xdeba.

suraTi N 3.4. fardobiT sixSireTa poligoni gamocdebTan dakavSirebuli SfoTvis

kiTxvaris qulebisTvis (cxrilidan N3.3)

0

10

20

30

40

17 22 27 32 37 42 47 52 57 62 67 72 77

gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebi

far

dobiT

i six

Sir

e

fardobiTi sixSiris histograma da poligoni

histogramisa da sixSireTa poligonis asagebad, martivi sixSireebis (suraTebi

N3.3 da N3.4) nacvlad, fardobiTi sixSireebic SeiZleba gamoviyenoT. gamocdebTan

dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebis fardobiT sixSireTa (ix.

cxrilis N3.4 (v) sveti) histograma da sixSireTa poligoni warmodgenilia

suraTebze N3.5 da N3.6, Sesabamisad.

histogramis an sixSireTa poligonisTvis fardobiTi sixSireebis gamoyenebis

upiratesoba isaa, rom fardobiTi sixSireebis ricxviTi mniSvnelobebi gviCveneben

qulaTa wils, romelic sxvadasxva intervalSi xvdeba. amrigad, Tu SevadarebT

qulaTa sxvadasxva raodenobaze agebul sixSireTa or gansxvavebul ganawilebas,

fardobiTi sixSireebis SemTxvevaSi pirdapiri Sedareba ufro iolia, vidre

martivi sixSireebisas.


suraTi N3.5. fardobiT sixSireTa histograma gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebisTvis

(cxrilidan N3.3)

0

10

20

30

40

0 22 27 32 37 42 47 52 57 62 67 72

gamocdebTan dakavSirebuli SfoTvis kiTxvaris qula

far

dobiT

i six

Sir

e

informaciis miReba histogramidan an sixSireTa poligonidan

histograma da sixSireTa poligoni, sixSireTa ganawilebis msgavsad, erTsa da

imave informacias gvawvdis. histogramisa da sixSireTa poligonis upiratesoba

isaa, rom xSirad gacilebiT iolia grafikulad warmodgenili informaciis aRqma.

nebismier suraTze, iqneba es suraTi N3.3, N3.4, N3.5 Tu N3.6, SegviZlia davinaxoT,

rom SfoTvis qulebis ganawilebis piki mdebareobs qula 40-is midamoebSi da rom

qulaTa umravlesoba SfoTvis skalis qulaTa qveda sazRvarTan (anu 20-Tan) ufro

axlosaa ganlagebuli, vidre _ zeda sazRvarTan (anu 80-Tan), Tumca, qulaTa

raodenoba mcirdeba maRali SfoTvis qulebis mimarTulebiT.

suraTi N3.6. fardobiT sixSireTa poligoni gamocdebTan dakavSirebuli SfoTvis

kiTxvaris qulebisTvis (cxrilidan N3.3)

0

10

20

30

40

17 22 27 32 37 42 47 52 57 62 67 72 77

gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebi

far

dobiT

i six

Sir

e

.

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foTlebiani Reroebis diagrama

foTlebiani Reros diagrama, romelic

tukis (Tukey, 1977) ekuTvnis, xasiaTdeba

sixSireTa ganawilebisa da monacemTa

grafikulad gamosaxvis Tvisebebis

kombinaciiT. gamocdebTan dakavSirebuli SfoTvis

kiTxvarSi miRebuli qulebis (ix. cxrili N3.3)

foTlebiani Reroebis diagarama warmodgenilia

cxrilSi N3.5. am grafikze marcxena svetSi mocemulia

Reroebi, romelic qulis pirvel cifrs aRniSnavs.

magaliTad, qulisTvis 3.5 Reroa 3. Reroebi,

Cveulebisamebr, dalagebulia ise, rom yvelaze dabali

Rero (am magaliTSi, 2-dan) moTavsebulia svetis bolos,

yvelaze didi (am magaliTSi, 7) ki _ svetis TavSi.

marjvena svetSi mocemuli ricxvebi warmoadgenen

foTlebs da Seesabamebian qulis meore cifrs; 35-is

SemTxvevaSi, es iqneba 5. magaliTad, pirvel rigSi, sadac

mocemulia Rero 7 da foTlebi 0 da 3, warmodgenilia

qulebi 70 da 73, Sesabamisad. Tu gvqvs samniSna qula,

rogoricaa, magaliTad, 321 wami, pirveli ori cifri (anu

32) gamoiyeneba Reros formirebisTvis, xolo bolo

cifri (anu 1) iqneba foToli.

miuxedavad amisa, mainc rTulia cxrilSi N3.5

mocemuli informaciis wakiTxva, gansakuTrebiT me-3 da

me-4 grZel Reroebze. am sirTulis dasaZlevad martivi

foTlebiani Reros diagrama SeiZleba ise

modificirdes, rom TiTo striqonze mxolod ramdenime

foToli darCes. cxrilSi N3.6 swored aseTi

modificirebuli variantia mocemuli, sadac TiToeuli

Rero xuTi striqoniTaa warmodgenili. yuradReba

miaqcieT ·, t, ƒ, s da * svetebs, romelic ricxviT aRniSnul svetebs mosdevs. aseT

ganlagebaSi ·-is gaswvriv mocemulia 0-isa da 1-is toli mniSvnelobebi, t-is

gaswvriv _ 2 da 3, ƒ-is gaswvriv _ 4 da 5, s-is gaswvriv _ 6 da 7, xolo *-is

foTlebiani Reroebis diagrama (Stem-and-leaf

display): _ monacemTa gamosaxvis forma, sadac

qulis pirveli (an pirveli da meore) cifri

aris Rero, meore (an bolo) ki _ foToli.

cxrili N3.5. gamocdebTan dakavSirebuli

SfoTvis kiTxvarSi miRebuli qulebis (ix.

cxrili N3.3) foTlebiani Reroebis diagrama.

Rero foTlebi

7 6 5 4 3 2

03 005 00022558 000000011111111111222223333444444555666666778889 1124444566666777777888888899999 25568889

cxrili N3.6. gamocdebTan

dakavSirebuli SfoTvis

kiTxvarSi miRebuli

qulebis (ix. cxrili 3.3)

foTlebiani Reroebis

modificirebuli diagrama.

Rero foTlebi

t 3

7 · 0

6 *

s

ƒ 5

t

6 · 00

5 * 8

s

ƒ 55

t 22

5 · 000

4 * 8889

s 66666677

ƒ 444444555

t 222223333

4 · 000000011111111111

3 * 888888899999

s 66666777777

ƒ 44445

t 2

3 · 11

2 * 8889

s 6

ƒ 55

t 2

2 ·


gaswvriv _ 8 da 9. aseTi modifikacia cxrils ufro advilad wasakiTxs xdis da

ganawilebis formis ukeT da iolad danaxvis saSualebas iZleva.

foTlebiani Reros diagrama martivad aigeba nedl monacemebze dayrdnobiT; am

dros ar ikargeba informacia, rogorc es xdeba dajgufebuli sixSireebis

ganawilebisas. am diagramidan yvela nedli qulis aRdgenaa SesaZlebeli. cxrilis

N3-6 ganxilviTa da qulebis ubralod daTvliT swrafad SegviZlia ganvsazRvroT,

rom qulebi icvleba 22-dan 73-mde da rom qula 41 yvelaze xSirad gvxvdeba.

foTlebiani Reroebis kidev erTi magaliTis saxiT ganvixiloT mTvaris efeqtis

rwmenis skala, romelic rotonsa da kelis (Rotton & Kelly, 1985) ekuTvniT da qcevaze

savse mTvaris gavlenis rwmenas afasebs. SegaxsenebT, rom am skalaze minimaluri

qulaa 9, rac mTvaris efeqtis minimalur rwmenaze an am rwmenis arqonaze

miuTiTebs, xolo maqsimaluri qulaa 81, rac mTvaris efeqtis maqsimaluri rwmenis

arsebobas gulisxmobs. rotonma da kelim es skala kolejis 157 students

miawodes. miRebuli qulebi mocemulia foTlebiani Reroebis diagramis saxiT2.

Rero warmoadgens qulis aTeuls, xolo

foToli _ erTeuls. amrigad, Rero 3 da

foToli 7 aris qula 37. rotonma da kelim

daxvewes Reroebiani foTlebis TavianTi

diagrama. yuradReba miaqcieT wertilebiTa da

varskvlavebiT Sedgenil svets. wertili

foTlebis 0-dan 4-is CaTvliT mniSvnelobis

aRsaniSnad gamoiyeneba, varskvlavi ki foTlis 5-

dan 9-is CaTvliT mniSvnelobebze miuTiTebs.

magaliTad, 1 Reros mqone qulebi cxrilSi

Semdegnairad aris mocemuli:

1 * .

556666778888899 122334

foTlis im mniSvnelobaTa diapazoni, romelTac win uZRvis wertili, 1-dan 4-is

CaTvliTaa. isini gamosaxaven qulebs 11, 12, 12, 13, 13 da, Sesabamisad. foTlis im

mniSvnelobebsac, romelTac win uZRvis varskvlavi (*), aqvT Rero 1, Tumca, maTi

diapazoni 5-dan 9-is CaTvliTaa. Rero 1-is SemTxvevaSi es mniSvnelobebi qmnis

qulebs 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 19 da 19, Sesabamisad.

foTlebiani Reroebis am diagramidan SegviZlia davinaxoT, rom miRebul

qulaTa diapazonia 11-dan 72-mde; yvelaze xSirad _ 9-jer gvxvdeba qula 45. am

2 ibeWdeba avtorebis _ j. rotonisa da i. u. kelis da gamomcemlis nebarTviT, „A Scale for assessing belief in lunar effects: reliability and concurrent validity“, Psychological Reports, 1985, 57, 239-245, © Psychological Reports, 1985.

Reroebi foTlebi

7 . 12

6 * .

59 0011114

5 * .

5666668999 000111111222334

4 * .

555555555666678888899 001111223333344444

3 * .

566666677788899999 0000112223344444

2 * .

55667777899999 00001122234444

1 * .

556666778888899 122334


diagramaze ar ikargeba informacia, SesaZlebelia 157-dan TiToeuli qulis

aRdgena.

Tvisebrivi monacemebis grafikulad gamosaxva

zemoT ganxiluli sakiTxebi Seexeboda raodenobrivi monacemebis grafikulad

gamosaxvas. Tvisebrivi monacemebi mxolod imas gviCvenebs, rom gasazomi cvladi

Tvisebrivad gansxvavebulia. magaliTad, trinkusma (Trinkaus, 1982, 1983, 1988, 1993)

bunebrivi dakvirvebis Sedegad mZRolebis qceva _ gzatkecilze Sesabamis niSanze

gaCereba _ miakuTvna samidan erT-erT kategorias: _ (1) srulad gaCereba, (2) svlis

Seneleba, (3) ar gaCereba. es kategoriebi Tvisebrivad gansxvavdeba erTmaneTisgan.

isini aRniSnaven qcevas, magram arafers gveubnebian imis Sesaxeb, Tu ramdenad

gansxvavdeba es qcevebi erTmaneTisgan raodenobrivad. trinkusma daiTvala

TiToeuli qcevis Sexvedris raodenoba da TiToeuli kategoriisTvis gamoTvala

SexvedraTa sixSiris procenti. procentebiT gamosaxuli es sixSireebi SeiZleba

warmodgenil iqnes grafikulad iseTi svetebiani diagramis saxiT, rogoric

mocemulia suraTze N3.7.

svetebiani diagrama (Bar graph) _ raodenobrivi monacemebis sixSireTa ganawilebis

grafikulad gamoxatvis saSualeba. mocemul kategoriaSi qulaTa gamoCenis sixSires

svetis simaRle Seesabameba.

svetebiani diagrama histogramas

hgavs. gansxvaveba isaa, rom

svetebi gancalkevebuladaa

ganlagebuli X RerZze, rac imaze

miuTiTebs, rom kategoriebi ar

warmoadgenen uwyvet cvlads.

svetebs Soris manZili SeiZleba

iyos nebismieri, magram

erTmaneTis toli. TiToeuli

kategoriis Sexvedris sixSiris

gamosaxatad SeiZleba gamoviyenoT

martivi sixSireebi, fardobiTi

sixSireebi anda sixSireebi

procentebSi (rogorc mocemulia

suraTze N3.7). am diagramidan advilad davinaxavT, rom gzajvaredinebze yvelaze

xSirad dafiqsirebuli qceva ekuTvnis kategoria 3-s da es niSnavs saerTod ar

gaCerebas.


suraTi N3.7. gaCerebis qcevis sami kategoriis Sexvedris sixSire

procentebSi (Trinkaus, 1982, 1983, 1988, 1993).

0

10

20

30

40

50

60

70

80

90

100

srulad

gaCereba

svlis

Seneleba

ar gaCereba

six

Sir

e pr

ocenteb

Si


1. „SeamowmeT Tqveni codna III.2“-is 7a punqtSi mogeTxovebodaT qurcikis

mier pirvelad Sesrulebuli maRali naxtomis dros mtacebelsa da

qurciks Soris arsebuli manZilis sixSireTa ganawilebis ageba.

gamoiyeneT sixSireTa es ganawileba da upasuxeT (a) da (b) kiTxvebs.

a) dajgufebuli ganawilebis martivi sixSireebis gamoyenebiT

(intervalTa klasis sogrZe 10 metri) miRebuli manZilebisTvis aageT

histograma da sixSireTa poligoni.

b) aageT fardobiT sixSireTa histograma da fardobiT sixSireTa

poligoni 1a kiTxvaSi aRwerili dajgufebuli ganawilebisTvis.

g) ra daskvnebis gakeTeba SegiZliaT qurcikis maRali naxtomis sigrZis

Sesaxeb Tqvens mier agebul grafikebze dayrdnobiT?

2. aageT foTlebiani Reroebis diagrama „SeamowmeT Tqveni codna III.2-is“ me-

7 kiTxvaSi mocemuli 50 manZilisTvis.

a) romelia minimaluri dafiqsirebuli manZili?

b) romelia maqsimaluri dafiqsirebuli manZili?

g) romelia yvelaze xSirad dafiqsirebuli manZili?

sixSireTa ganawilebis formebi

qulaTa ganawilebam SeiZleba miiRos nebismieri forma arsebuli mTeli rigi

sxvadasxvagvari formebidan. amrigad, rogorc wesi, ar arsebobs sixSireTa

ganawilebis formis ramdenime sityviT zustad aRweris martivi gza; Tumca,

arsebobs ramdenime zogadi aRweriTi termini, romelic gamoiyeneba ganawilebis

formis mimarT da aiolebs ganawilebis sityvebiT aRweras. suraTze N3.8

warmodgenilia sxvadasxva formis sixSireTa poligonis magaliTebi.


sixSireTa simetriuli ganawileba

Tu sixSireTa simetriul

ganawilebas orad

gadavkecavT Suawertilis

gaswvriv, miviRebT erTnairi

formis or naxevars. ganawilebis erTi naxevari warmoadgens meoris sarkisebur

asaxvas. suraTze N3.8 ganawilebebi (a), (b) da (g) simertiulia, (d) da (e) ki _ ara;

isini asimetriuli ganawilebebia. simetriuli zaris formis anu normalur

ganawilebas, romelic naCvenebia suraTze N3.8a, gansakuTrebuli mniSvneloba aqvs

bihevioristul mecnierebaSi. amaze ufro dawvrilebiT me-6 TavSi visaubrebT.

asimetriuli ganawileba

nedli monacemebis srulyofilad

simetriuli ganawileba iSviaTad miiReba.

ufro mosalodnelia, rom sixSireTa

ganawileba iyos asimetriuli anu

gadaxrili. asimetriul ganawilebaSi

qulebi Tavmoyrilia ganawilebis erT

boloSi; ganawilebis meore boloSi (anu kudSi) ki ganlagebulia iSviaTad

dafiqsirebuli qulebi. ganawilebas ewodeba dadebiTad asimetriuli, Tu maRali

qulebis Semcveli bolo (anu kudi) ganawilebis marjvena mxares mdebareobs

(suraTi N3.8d), xolo ganawilebas ewodeba uaryofiTad asimetriuli, Tu dabali

sixSireTa simetriuli ganawileba (Symmetrical frecuency

distribution): _ sixSireTa ganawileba, romlis

Suawertilze orad gadakecvis SemTxvevaSi miiReba

erTnairi formis ori naxevari.

asimetriuli ganawileba (Skewed

distribution): _ sixSireTa ganawileba, sadac

qulebi Tavmoyrilia ganawilebis erT

boloSi; ganawilebis meore boloSi (anu

kudSi) ki ganlagebulia iSviaTad

dafiqsirebuli qulebi.

suraTi N3.8.sixSireTa hipoTeturi ganawilebebis formebi.

(g)

marTkuTxedi

simetriuli

(d)

asimetriuli

dadebiTad asimetriuli uaryofiTad asimetriuli

(e)

(a) (b)

bimodaluri


qulebis Semcveli bolo (anu kudi) mdebareobs ganawilebis marcxena mxares

(suraTi N3.8e).

modaloba

ganawileba aRiwereba imiTac, Tu

ramdeni modiT xasiaTdeba is. moda

qulaTa ganawilebaSi yvelaze xSirad dafiqsirebuli qulaa. unimodalur

ganawilebas mxolod erTi yvelaze xSirad dafiqsirebuli qula aqvs (ix. suraTi

N3.8(a), (d) da (e)). ganawilebas, romelsac ori moda aqvs, bimodaluri ewodeba (ix.

suraTi N3.8b). bihevioraluri mecnierebis mravali warmomadgeneli ganawilebas

bimodalurad miiCnevs, Tu is ori gancalkevebuli pikiT xasiaTdeba; im

SemTxvevaSic ki, rodesac pikebi ar gamoxataven toli sixSiriT dafiqsirebul

qulebs. multimodalur ganawilebaSi warmodgenilia sami an meti qula, romelic

yvelaze xSirad fiqsirdeba. ganawileba maSinac ki SeiZleba CaiTvalos

multimodalurad, rodesac masSi mocemuli sami an meti pikis sixSire ar aris

toli. ar gaaCnia moda marTkuTxedis formis iseT ganawilebas, rogoric suraTze

N3.8g-zea naCvenebi. aseT ganawilebaSi yvela qula Tanabrad an TiTqmis Tanabrad

fiqsirdeba.


1. ganmarteT: _ asimetriuli ganawileba, bimodaluri ganawileba, moda,

multimodaluri ganawileba, uaryofiTad asimetriuli ganawileba,

dadebiTad asimetriuli ganawileba, simetriuli ganawileba,

unimodaluri ganawileba.

2. aRwereT suraTze N3.9 mocemuli sixSireTa ganawilebebi simetriis,

asimetriulobisa da modalobis mixedviT.

moda (Mode): _ qulaTa ganawilebaSi

yvelaze xSirad dafiqsirebuli qula.


Sejameba

• sixSireTa ganawileba aris cxrili, romelic gviCvenebs individTa jgufis

mier miRebul TiToeul qulas da am qulis miRebis sixSires.

• sixSireTa ganawileba gamoiyeneba monacemebis mowesrigebisTvis.

• daujgufebel sixSireTa ganawileba Seileba gamoviyenoT mcire raodenobis

monacemebis SemTxvevaSi.

• dajgufebul sixSireTa ganawileba gamoiyeneba didi raodenobiT qulebis,

an qulaTa farTo diapazonis SemTxvevaSi.

• daujgufebel an dajgufebul sixSireTa ganawileba SeiZleba gamoisaxos

martivi, fardobiTi an procentebSi sixSireebis saxiT.

• procentili aris qula, romlis qvemoTac Tavsdeba ganawilebaSi qulaTa

garkveuli procenti.

• qulis procentuli rangi gviCvenebs ganawilebaSi im qulaTa procents,

romelic tolia an naklebia am qulaze.

• sixSireTa ganawileba SeiZleba gamoisaxos grafikulad histogramis,

sixSireTa poligoniT an foTlebiani Reroebis diagramis saxiT.

• Tvisebrivi monacemebi SeiZleba gamoisaxos grafikulad svetebiani

diagramis saxiT.

(a) (b)

(d) (g)

suraTi N3-.9. sixSireTa ganawileba me-2 kiTxvisTvis.


• sixSireTa ganawilebis forma aRiwereba simetriulobis, asimetriulobisa

da modalobis terminebiT.


asimetriuli ganawileba

bimodaluri ganawileba

dagrovili sixSire

procentebSi

dadebiTad asimetriuli

ganawileba

daujgufebel sixSireTa

ganawileba


ganawileba

intervali

intervalis dagrovili

sixSire

intervalis zeda

dadgenili sazRvari

intervalis zeda

realuri sazRvari

intervalis dadgenili

sazRvrebi

intervalis realuri

sazRvrebi

intervalis qveda realuri

sazRvari

intervalis qveda dadgenili

sazRvari


moda

multimodaluri ganawileba

nedli monacemebi

nedli qulebi

procentili

procentuli rangi

svetebiani diagrama

simetriuli ganawileba

sixSireTa ganawileba

sixSireTa ganawileba

procentebSi

sixSireTa poligoni

uaryofiTad asimetriuli

ganawileba

unimodaluri ganawileba

fardobiT sixSireTa

ganawileba

foTlebiani Reroebis

diagrama

qulis dagrovili sixSire

histograma

cƒ anu cum ƒ

crƒ anu cum rƒ

c%ƒ anu cum %ƒ

%ƒ

rƒ

kiTxvebi

1. hiksi, misTri, luCero, meriqali da pelegrini (Hicks, Mistry, Lucero, Marical and

pellegrini, 1990) aRniSnaven, rom kolejis studentebis gamokiTxvis Sedegebis

mixedviT, maT dRe-RameSi saSualod 6 saaTi da 87 wuTi (anu 412 wuTi)

ZinavT. Tqven dainteresdiT, Tu ramdenad SeiZleba am monacemis

gavrceleba kolejis mamrobiTi sqesis studentebze. amitom sTxoveT

kolejis 50 mamrobiTi sqesis students, rom CaeweraT, Tu ramden xans

eZinaT erTi Ramis ganmavlobaSi. Tqven miiReT wuTebSi Cawerili Semdegi

droebi: _ 440, 223, 309, 427, 463, 2371, 75, 315, 290, 356, 407, 399, 263, 417, 328, 377,

406, 275, 316, 391, 350, 422, 431, 382, 350, 371, 410, 295, 305, 327, 400, 227, 275, 340,

364, 279, 300, 412, 388, 296, 320, 342, 371, 258, 470, 496, 384, 439, 284, 351, 417.

a) am monacemebiT aageT dajgufebul sixSireTa ganawileba

procentebSi da dajgufebul dagrovil sixSireTa ganawileba

procentebSi. gamoiyenebT 14 intervali.


b) romeli inervali Seicavs yvelaze met qulas? Tanxvdeba Tu ara am

intervalis Suawertili kvlevaSi dasaxelebul 412 wuTs?

g) dajgufebul dagrovil sixSireTa ganawilebaze dayrdnobiT

SeafaseT drois is raodenoba, romelic ganawilebas or naxevrad

yofs anu romlis qvemoTac da zemoTac moTavsebulia 25-25 cdis piri.

d) ipoveT am ganawilebis 50-e procentili. Tanxvdeba Tu ara is (g)

punqtSi miRebul pasuxs?

e) ra iqneba 431 wuTis procentuli rangi?

v) aageT histograma da sixSireTa poligoni (a) punqtSi agebuli

ganawilebis martivi sixSireebis gamoyenebiT.

z) aRwereT ganawilebis forma asimetriulobisa da modalobis

terminebiT.

2. mamri beRura mondomebiT aSenebs budes da mas naWuWebiT, feradi

foTlebiTa da TuTiyuSis cisferi bumbulebiT rTavs, radgan

SewyvilebaSi warmateba budis morTulobis odenobasTan aris

dakavSirebuli. bevrjer iqna SemCneuli, rom mamri beRurebi sxva

frinvelebis budeebidan iparaven morTulobebs da maT TavianTi

budeebisTvis iyeneben. borgia da gori (Borgia & Gore, 1986) am qcevas axal

samxreT uelsSi, avstraliaSi akvirdebodnen. maTi dakvirvebis obieqti iyo

59 beRura. maT daniSnes budeSi arsebuli bumbulebi da iniSnavdnen, Tu

romeli bumbulebi iyo moparuli sxva budeebidan. davuSvaT, dakvirvebis

periodis ganmavlobaSi maT CainiSnes TiToeuli frinvelis mier moparuli

bumbulebis Semdegi raodenobebi: _ 2, 6, 14, 1, 10, 3, 0, 26, 37, 47, 4, 0, 6, 21, 3,

6, 6, 51, 8, 11, 0, 0, 4, 2, 5, 9, 4, 6, 24, 4, 1, 18, 63, 1, 5, 6, 4, 7, 12, 5, 3, 4, 3, 15, 46,

0, 5, 0, 26, 5, 17, 1, 17, 1, 58, 9.

a) am monacemebiT aageT dajgufebul fardobiT sixSireTa ganawileba

da dagrovil dajgufebul fardobiT sixSireTa ganawileba.

gamoiyeneT 13 intervali.

b) romeli intervali Seicavs yvelaze bevr qulas?

g) ipoveT am ganawilebis 50-e procentili.

d) ra iqneba 24 qulis procentuli rangi?

e) am ganawilebisTvis aageT foTlebiani Reroebis grafiki. romelia

minimaluri da maqsimaluri qula am ganawilebaSi? romelia yvelaze

xSirad dafiqsirebuli qula?

v) aageT histograma da sixSireTa poligoni 2a kiTxvaSi agebuli

ganawilebis martivi sixSireebis gamoyenebiT.

v) aRwereT ganawilebis forma asimetriulobisa da modalobis

terminebiT.


3. davuSvaT, daakvirdiT mamrobiTi da mdedrobiTi sqesis mZRolebs

avtosadgomze da daiTvaleT, Tu ra sixSiriT aCerebdnen isini

avtomanqanebs iseT adgilebSi, romlebic ar ekuTvnoda avtosadgoms.

samdRiani dakvirvebis manZilze Tqven SeamCnieT, rom mamrobiTi sqesis 100

mZRolidan 12 da mdedrobiTi sqesis 100 mZRolidan 7 arasaTanado

adgilze aCerebda avtomanqanas. aageT arasaTanado adgilze avtomanqanis

gaCerebis fardobiTi sixSireebis svetebiani diagrama mamrobiTi da

mdedrobiTi sqesis mZRolebisTvis.

4. stargenma da birma (Sturgeon & Beer, 1990) aCvenes, rom im studentebis

waxaliseba, romlebic regularulad dadian leqciebze, semestris bolos

gamocdidan ganTavisuflebiT, zrdis leqciebze maT daswrebas. davuSvaT,

erTmaneTs SeadareT ori skolis daswrebis monacemebi; aqedan erT

skolaSi xorcieldeboda aseTi waxaliseba, meoreSi ki _ ara. Tqven miiReT

gacdenebis martiv sixSireTa Semdegi saxis ganawileba:

a) TiToeuli skolisTvis aageT

fardobiTi sixSireebis ganawileba.

b) ra iqneba 50-e procentili TiToeuli

skolisTvis?

g) ra iqneba TiToeul skolaSi oTxi

gacdenis procentuli rangi?

d) gansxvavdebian Tu ara skolebi

erTmaneTisgan gacdenebis sixSiris

mixedviT? aRwereT ganawilebebi.

5. vinaidan cxovrebis tempi Zalze

Cqaria, gviWirs erTi calkeuli

dRis gaxseneba. bunebriv

pirobebSi mexsierebis kvlevis farglebSi fsiqologma kolejis 90

students sTxova, rom CamoeweraT maTTvis Cveuli saqmianobebi dasvenebis

dReebSi. davuSvaT, miRebuli Sedegebi fsiqologma qvemoT mocemuli

foTlebiani Reros diagramis saxiT warmoadgina.

Rero foToli

2 * s ƒ t 2 · 1 * s ƒ t 1 · 0 *

888899 66667 44555 2222333 0000000111111 888999999 6666677777 44455555 2233333 0001111

gacdenili dReebis

raodenoba

waxalisebis mqone skola

waxalisebis ar mqone skola

25-ze meti 24-25 22-23 20-21 18-19 16-17 14-15 12-13 10-11 8-9 6-7 4-5 2-3 0-1

2 0 2 0 1 0 2 0 3 8 4 13 23 42

6 6 1 8 0 4 22 10 18 20 3 14 34 54

jami 100 200


s ƒ t 0 ·

888999 6677 555

a) aageT nedli qulebis cxrili im studentebisTvis, romelTac 14-dan

19 saqmianoba gaixsenes.

b) romelia dasaxelebuli yvelaze maRali da dabali qulebi?

g) romelia yvelaze xSirad dasaxelebuli qula?

d) gamosaxeT qulebi histogramiT. gamoiyeneT intervali, romlis

sididea i=2.

e) foTlebiani Reros diagramasTan SedarebiT ra informacia ikargeba

qulebis histogramis saxiT gamosaxvis SemTxvevaSi?

6. pirvel TavSi iTqva, rom 25-54 wlis amerikeli qalebi, rogorc wesi, dReSi

4.4 saaTs (264 wT) televizorTan atareben. socialuri fsiqologiis

studentebis saklaso proeqtis farglebSi am asakis 100 qali gamoikiTxa,

romelTac sTxoves maT mier dRis manZilze televizorTan gatarebuli

saaTebis raodenobis Sefaseba. dasaxelebuli dro (wuTebSi) dajgufebul

sixSireTa Semdegi ganawilebis saxiTaa mocemuli:

a) ra sididis (i) intervalia gamoyenebuli am

cxrilSi?

b) ra iqneba 210-239 intervalis qveda da zeda

realuri zRvari?

g) gansazRvreT 210-239 intervalis Suawertili.

d) am qulebisTvis aageT martiv sixSireTa

ganawileba da dagrovil sixSireTa

ganawileba.

e) ipoveT am ganawilebis 75-e procentili.

v) ipoveT procentiluri rangi televizorTan

gatarebuli 264 wuTisTvis

7. gamajansaRebeli klubis 25 mepatronem

Seafasa, Tu dRis ganmavlobaSi ra dros

(wuTebiT) uTmobda TiToeuli maTgani

varjiSs. Sedegad dasaxelda Semdegi ricxvebi: _ 67, 73, 79, 78, 50, 80, 98, 88,

79, 71, 57, 81, 41, 65, 50, 71, 91, 75, 60, 79, 79, 71, 65, 85 da 64.

a) am qulebisTvis i=5 intervalis gamoyenebiT aageT fardobiTi

sixSireebisa da dagrovili fardobiTi sixSireebis ganawilebebi.

ra iqneba pirveli intervalis qveda dadgenili sazRvari? ratom?

b) ra formisaa es ganawileba?

g) ipoveT 78 wuTis procentiluri rangi.

intervali Tvlis Sedegi

510-539 480-509 450-479 420-449 390-419 360-389 330-359 300-329 270-299 240-269 210-239 180-209 150-179 120-149 90-119 60-89 30-59 0-29

/ // ///// ////// ///// ///////// //////// ///////// ////////// ///////// /// /////// // //////// ///// /////// ////


d) davuSvaT, pirovnebas surs, rom sakmarisi raodenobiT ivarjiSos

ise, rom am ganawilebaSi pirvel 25%-Si Soris moxvdes. sul mcire

ramdeni wuTi unda ivarjiSos man yoveldRiurad, rom man es

SeZlos?

Tavi IV. monacemTa aRwera: centraluri tendenciis

sazomebi

moda

mediana

medianis gamoTvla kenti raodenobis monacemebiT:

monacemebi ar aris dajgufebuli

medianis gamoTvla luwi raodenobis monacemebiT:

monacemebi ar aris dajgufebuli

medianis gamoTvla procentilis formuliT

medianis Tvisebebi

SerCevis saSualo

SerCevis saSualos gamoTvla

SerCevis saSualos Tvisebebi

SerCevis saSualos warmodgena samecniero statiebSi

populaciis saSualo

centraluri tendenciis sazomebis Sedareba

centraluri tendenciis sazomebis aRweriTi gamoyeneba

centraluri tendenciis sazomebis daskvniTi gamoyeneba

თავი IV. მონაცემთა აღწერა: ცენტრალური ტენდენციის საზომები 2

sixSireTa ganawileba da grafikebi did informacias gvaZlevs qulaTa

ganawilebis formisa da miRebuli qulebis diapazonis Sesaxeb; Tumca, xSirad

mecnierebs sWirdebaT, rom qulaTa ganawileba mxolod erTi ricxviT aRiweros. am

ricxvebs aRweriTi statistikebi ewodebaT, radgan isini nedl monacemebs erTi

ricxviT aRweren. Cveulebisamebr, gamoiyeneba ori tipis aRweriTi statistikebi: _

erTni, romelTac centraluri tendenciis sazomebs uwodeben, aRweren miRebul

tipur qulas; meoreni ki, romelTac gafantvis sazomebs uwodeben, miuTiTeben

tipuri qulis irgvliv cvalebadobisa da gafantvis zomaze.

aRweriT statistikebs ori funqcia aqvT: _ (1) SerCevaSi miRebuli qulebis

aRwera; (2) SerCevis monacemebze dayrdnobiT populaciis maxasiaTeblebis Sefaseba

anu maT Sesaxeb daskvnebis gakeTeba. magaliTad, me-3 TavSi xsenebuli mkvlevari

SeiZleba dainteresebuli iyos Tavisi kolejis studentTa populaciaSi

gamocdebTan dakavSirebuli SfoTvis tipuri donis SefasebiT. misi Sefaseba

dafuZnebuli iqneba 100 gogonasgan Semdgari SerCevis qulebisgan miRebul

aRweriT statistikaze.

winamdebare Tavi Seexeba centraluri tendenciis sazomebis aRweriT

gamoyenebas. cvalebadobis sazomebi da aRweriTi statistikebis gamoyeneba

populaciis maxasiaTeblebis Sesaxeb daskvnebis gakeTebisas ganxilulia me-5 da me-

7 TavebSi.

centraluri tendenciis sazomebi aris ricxvebi, romlebic warmoadgenen

SerCevidan miRebul saSualo an tipur qulas. Cveulebisamebr, gamoiyeneba

centraluri tendenciis sami sazomi: _ moda, mediana da saSualo ariTmetikuli.

moda

moda aris qula, romelic yvelaze xSirad gvxvdeba qulaTa ganawilebaSi. moda

moiZebneba martivi, daujgufebeli sixSireebis ganawilebis gadasinjviT da imis

dafiqsirebiT, Tu romeli qula gvxvdeba yvelaze xSirad. magaliTad, cxrilSi N3-

2 yvelaze xSirad (xuTjer) gvxvdeba qula 59. Sesabamisad, am ganawilebis modaa 59.

centraluri tendenciis sazomebi (Measures of central tendency): _ ricxvebi, romlebic

warmoadgenen SerCevidan miRebul saSualo an tipur qulas.

cvalebadobis sazomi (measures of central tendency) _ ricxvi, romelic miuTiTebs, Tu ramdenad

gansxvavdebian qulebi erTmaneTisgan da ramdenad gansxvavdeba maTgan scentraluri

tendenciis sazomi.

moda (Mode): _ qula, romelic yvelaze xSirad gvxvdeba ganawilebaSi.

unimodaluri (Unimodal): _ ganawileba, romelsac aqvs erTi moda.

bimodaluri (Bimodal): _ ganawileba, romelsac aqvs ori moda.

multimodaluri (Multimodal): _ ganawileba, romelsac aqvs orze meti moda.


vinaidan qulaTa am erTobliobaSi mxolod erTi modaa, am ganawilebas

unimodaluri ewodeba. qulaTa ganawileba SeiZleba iyos bimodaluri (anu hqondes

ori moda), an multimodaluri (anu hqondes orze meti moda). SegaxsenebT, rom

sixSireTa ganawilebis formis aRweris erT-erTi gza modalobas efuZneba.

unimodaluri da bimodaluri ganawilebebi mocemulia suraTze N3.8.

miuxedavad imisa, rom moda advilad gamoiTvleba martivi, daujgufebeli

sixSireebis ganawilebidan, bihevioristul mecnierul kvlevaSi is iSviaTad

gamoiyeneba centraluri tendenciis sazomad. Tu ganawilebaSi erTze meti modaa,

romeli maTgani iqneba yvelaze tipuri qula? amrigad, modam SeiZleba ver

uzrunvelyos ganawilebis aRsawerad sakmarisi erTi tipuri qula.

modis iSviaTad gamoyenebis meore mizezi isaa, rom erTi qulis cvlilebamac ki

SeiZleba mniSvnelovnad Secvalos ganawilebis moda. magaliTad, davuSvaT, rom

cdis pirebs mieciT xanmokle mexsierebis testi da sTxoveT 10-cifriani

mwkrividan SeZlebisdagvarad meti cifris gameoreba. vTqvaT, 6 cdis pirisgan

miiReT cifrebis Semdegi raodenoba:

cdis

pirebi

cifrebis

raodenoba

1 2 3 4 5 6

5 6 7 7 8 9

am ganawilebis modaa 7, radgan orma cdis pirma gaimeora 7 cifri. davuSvaT,

meoTxe cdis pirma 7-is nacvlad gaimeora 5 cifri. axla ganawilebis moda 7-is

nacvlad aris 5. mxolod erTi qulis cvlilebam mniSvnelovnad Secvala modac.

axla davuSvaT, rom mesame cdis pirma 7-is nacvlad gaixsena 10 cifri. am

SemTxvevaSi ganawilebas ar eqneba moda, TiToeuli qula gveqneba toli

raodenobiT. vinaidan moda ase Zalian aris damokidebuli mxolod ramdenime

qulaze, is ar aris centraluri tendenciis stabiluri sazomi.

mediana

mediana (SemoklebiT Mdn) ukavSirdeba ganawilebaSi

im qulas, romlis zemoT da qvemoT ganlagebulia

qulaTa toli raodenoba. amrigad, mediana 50-e

procentilia ganawilebaSi. qulaTa ganawilebis

umravlesobisTvis martivad SeiZleba medianis povna.

mediana (Median, Mdn): _ qula

ganawilebaSi, romlis zemoT

da qvemoT ganlagebulia

qulaTa toli raodenoba,

mediana aris 50-e procentili

ganawilebaSi.


medianis gamoTvla kenti raodenobis monacemebiT: monacemebi ar aris

dajgufebuli

ganawilebaSi Semavali kenti raodenobis monacemebisTvis, rodesac monacemebi

ar aris dajgufebuli, rogorc wesi, mediana sixSireTa ganawilebaSi SuaTana

qulis povniT ganisazRvreba. magaliTad, gamocdebTan dakavSirebuli SfoTvis

kiTxvaris qulebis _ 22, 34, 36, 37, 38, 44 da 72 _ medianaa 37. esaa meoTxe qula da

Svidquliani ganawilebis SuaSi xvdeba. sami qula naklebia 37-ze da sami _ metia

masze. yuradReba miaqcieT, rom medianis gansasazRvrad daTvlis procedura

gamoiyeneba, xolo Tavad qulaTa ricxviTi mniSvnelobebi mxedvelobaSi ar miiReba.

medianis gamoTvla luwi raodenobis monacemebiT: monacemebi ar aris

dajgufebuli

ganawilebaSi Semavali luwi raodenobis monacemebisTvis, rodesac monacemebi

ar aris dajgufebuli, medianad pirobiTad aiReba mniSvneloba, romelic

sixSireTa ganawilebaSi or SuaTana qulas Soris mdebareobs. magaliTad,

ganvixiloT gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebi: _ 34, 35, 37, 38,

44, 72. am qulebis medianaa 37.5 anu mniSvneloba, romelic xvdeba 37-isa (mesame

qula) da 38-is (meoTxe qula) SuaSi. eqvsi qulidan erTi naxevari mdebareobs 37.5-is

qvemoT da meore naxevari ki _ mis zemoT.

magaliTi N4-1

medianis moZebna

amocana

realuri samyaros vizualuri aRqmis zogierT kvlevaSi cdis pirs eZleva

suraTi, romelzec gamosaxulia raime scena, da Semdeg moeTxoveba, rom gaixsenos

am suraTis detalebi. davuSvaT, aseTi Tanmimdevrobis cda CaatareT 18 cdis

pirTan da TiToeulma maTganma gaixsena Semdegi raodenobis detalebi: _ 7, 3, 7, 9,

6, 10, 14, 3, 6, 9, 14, 9, 8, 5, 4, 6, 9 da 13. ipoveT mediana.

amoxsna

ganawileba Sedgeba Tvrameti qulisgan, amitom mediana iqneba qula, romlis

zemoT da qvemoT moTavsebulia cxra qula. Sesabamisad, rodesac qulebi

dalagebulia sixSireTa ganawilebaSi (rogorc mocemulia qvemoT), mediana

mdebareobs mecxre da meaTe qulebs Soris.

mecxre qulaa 7, xolo meaTe _ 8, amitom mediana iqneba 7-sa da 8-s Soris anu

7.5.


medianis gamoTvla procentilis formuliT

dajgufebuli monacemebisTvis aRar gamogvadgeba medianis gamoTvlis martivi

gzebi. am SemTxvevaSi medianis misaRebad SesaZlebelia formulis gamoyeneba.

mediana aris ganawilebis 50-e procentili. gavisxenoT me-3 Tavidan, rom

procentili aris mniSvneloba, romlis qvemoTac moTavsebulia ganawilebis

qulaTa garkveuli procenti. medianis SemTxvevaSi misi mniSvnelobis qvemoT xvdeba

qulaTa 50%. Sesabamisad, medianis miReba SesaZlebelia dajgufebul sixSireTa

ganawilebidan, me-3 TavSi mocemuli procentilis formulis gamoyenebiT:

if

cfNXXMdni

LL

−+===

)(50.50.mediana

sadac X.50 = mediana, romlis gamoTvlac gvsurs.

XL = 50-e procentilis Semcveli intervalis qveda realuri sazRvari.

N = ganawilebaSi qulaTa saerTo raodenoba.

cƒL = medianis Semcveli intervalis qveda realur sazRvramde qulebis

dagrovili sixSire.

ƒi = medianis Semcvel intervalSi Semaval qulaTa sixSire.

i = intervalis sigrZe.

medianis Tvisebebi

mediana, faqtobrivad, miiReba qulebis daTvliT da yuradRebis miRma rCeba

yvela qulis ricxviTi mniSvneloba, garda erTisa; gamonaklisia mxolod im qulis

mniSvneloba, romelic medianas warmoadgens. garda amisa, medianaze ar moqmedebs

ganawilebaSi Semaval qulaTa ukiduresi ricxviTi mniSvnelobebi. magaliTad,

qvemoT mocemuli TiToeuli ganawilebis medianaa 37 (gamuqebulia).

9 qula

9 qula

7.5 = mediana

qula ƒ

14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

2 1 0 0 1 4 1 2 3 1 1 2 0 0 0


mediana ar iTvaliswinebs b da g ganawilebis

boloebSi ganlagebul ukidures qulebs. es Tvisebebi

medianas sasurvel sazomad aqcevs, rodesac

asimetriuli ganawilebis centraluri tendenciis

gazomva xdeba saWiro. magaliTad, mediana xSirad

gamoiyeneba, rodesac saubaria populaciis tipur

Semosavalze, radgan Semosavlis ganawileba mkveTrad

dadebiTad asimetriulia. mxolod mcire raodenobis adamianebs aqvT

mravalmilioniani Semosavali. Tu am Semosavlebis ricxviTi mniSvnelobebi

gaTvaliswinebuli iqneba tipuri Semosavlis gazomvaSi, is Zalze maRali iqneba,

raTa saSualo adamianis Semosavlis reprezentaciuli iyos. vinaidan medianaze ar

zemoqmedebs es mcire raodenobiT didi Semosavali, is warmoadgens tipuri

Semosavlis reprezentaciul centraluri tendenciis sazoms.

SeamowmeT Tqveni codna IV.1

1. ganmarteT: _ bimodaluri ganawileba, aRweriTi statistikebi, centraluri

tendenciis sazomebi, mediana, moda, multimodaluri ganawileba,

unimodaluri ganawileba.

2. daasaxeleT aRweriTi statistikebis ori funqcia.

3. davuSvaT, mkvlevari zomavs 25 meSvide klaseli gogonas TviTSefasebas.

gogonebma TavianTi TviTSefaseba ganalages Sefasebis skalaze 10-dan

(yvelaze dabali TviTSefaseba) 70-mde (yvelaze maRali TviTSefaseba).

miRebul iqna Semdegi qulebi: _ 49, 53, 67, 20, 27, 36, 49, 27, 61, 17, 49, 50, 29,

56, 65, 27, 48, 24, 49, 15, 32, 66, 46, 24 da 38.

a) ipoveT am ganawilebis moda.

b) rogoria es ganawileba: _ unimodaluri, bimodaluri Tu

multimodaluri?

4. davuSvaT, erT-erTma gogonam, romelmac miiRo zemoT (me-3 kiTxvaSi)

dasaxelebuli 49 qula, 49-is nacvlad miiRo 27 qula.



multimodaluri?

5. davuSvaT, erT-erTma gogonam, romelmac miiRo zemoT (me-3 kiTxvaSi)

dasaxelebuli 49 qula, 49-is nacvlad miiRo 24 qula.



multimodaluri?

ganawileba

a b g

22 34 35 37 39 44 44

34 34 35 37 39 44 109

22 34 35 37 39 44 117


6. modis, rogorc aRweriTi statistikis, romeli SezRudvis magaliTs

warmoadgens me-4 da me-5 kiTxvebze Tqvens mier miRebuli pasuxebi?

7. ipoveT me-3 kiTxvaSi mocemuli ganawilebis mediana.

8. davuSvaT, eqsperimentatorma TviTSefaseba kidev erT me-7 klasel gogonas

gauzoma da miiRo 28 qula. CarTeT es qula ganawilebaSi da ipoveT am

ganawilebis mediana.

9. aviRoT me-3 kiTxvaSi mocemuli sawyisi ganawileba (26-e cdis piris

monacemis damatebis gareSe). davuSvaT, rom qulebi 61, 65, 66 da 67

Seicvala qulebiT 51, 55, 56 da 57-iT, Sesabamisad. ra iqneba qulaTa axali

ganawilebis mediana?

10. SeadareT erTmaneTs: _ (a) me-7 da me-8 kiTxvebze miRebuli pasuxebi; (b) me-7

da me-9 kiTxvebze miRebuli pasuxebi. medianis, rogorc aRweriTi

statistikis, ra SezRudvis magaliTia Tqvens mier miRebuli pasuxebi?

SerCevis saSualo

centraluri tendenciis yvelaze cnobili

sazomia SerCevis saSualo, romelsac xSirad

saSualo ariTmetikuls, an ubralod

saSualos uwodeben. SerCevis saSualo aris

miRebul qulaTa jami gayofili qulaTa mTlian raodenobaze. statistikuri

simboloebiT es ase gamoiTqmis:

N

XX

N

ii∑

=== 1saSualo SerCevis

sadac X aris saSualo.

∑ (berZnuli didi sigma) aris Sekrebis niSani. is miuTiTebs ricxvebis

erTobliobis dajamebaze anu Sekrebaze. i=1 da N aRniSvnebi niSnavs

imas, rom Sekrebilia yvela miRebuli qula, pirveli (i=1) cdis

piridan dawyebuli, bolo (me-N) cdis piriT damTavrebuli.

X i aris nebismieri cdis piris mier miRebuli qula, indeqsi i miuTiTebs

konkretul cdis pirze (magaliTad, X1 miuTiTebs pirvel cdis pirze,

X2 _ meore cdis pirze X10 _ meaTe cdis pirze da a.S).

N aris SerCevaSi cdis pirTa raodenoba.

rodesac cnobilia, rom xdeba SerCevaSi miRebuli yvela N qulis dajameba,

formulaSi aRar iwereba Sekrebis niSnis sazRvrebi da ugulebelyofilia indeqsi

i; aqedan gamomdinare, saSualos formula aseTi saxisaa:

SerCevis saSualo ( X ) (Sample mean):

_ miRebul qulaTa jami gayofili

qulaTa mTlian raodenobaze.


NX

X ∑=

umravles SemTxvevaSi am gamartivebul formulas gamoviyenebT.

SerCevis saSualos gamoTvla

davuSvaT, gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miviReT Semdegi

qulebi: _ 34, 35, 37, 38, 44 da 50. qulaTa am erTobliobisTvis SerCevis saSualo

Semdegnairad gamoiTvleba:

pirvel rigSi, vkrebT qulebs anu

∑ =+++++= 228404438373534X

qulaTa raodenoba N udris 6-s. qulaTa saSualos gamosaTvlelad miRebul

jams anu 228-s vyofT 6-ze, anu

0.386

228=== ∑

NX

X

gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli 100 qulisTvis,

romelic mocemulia cxrilSi N3-3, saSualo gamoiTvleba 100 qulis SekrebiTa da

miRebuli jamis 100-ze gayofiT. formulis saxiT:

6.4164.411004164

100374029...372842

===++++++

=X

magaliTi N4-2

saSualos gamoTvla

amocana

davuSvaT, magaliTSi N4-1 ganxilul suraTis gaxsenebis davalebaSi miiReT

Semdegi 18 qula: _ 7, 3, 7, 9, 6, 10, 14, 3, 6, 9, 14, 9, 8, 5, 4, 6, 9 da 13. gamoTvaleT am

qulebis saSualo.

saWiro statistika: N

XX ∑=

amoxsna

pirveli nabiji iqneba ∑ X -is miReba, romelic udris:

1421396458914963141069737 =+++++++++++++++++

Sekrebil qulaTa raodenobaa N, Cvens magaliTSi N udris 18-s. Sesabamisad,

gveqneba:

9.789.718

142==== ∑

NX

X


SerCevis saSualos Tvisebebi

saSualos gamoTvlisas gamoiyeneba ganawilebaSi Semavali yvela qulis

ricxviTi mniSvneloba, amitom ganawilebaSi Semaval qulaTa nebismieri cvlileba

aucileblad Secvlis am ganawilebis saSualos mniSvnelobas. magaliTad,

ganvixiloT gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli Semdegi

qulebi: _ 26, 26, 26, 30, 34, 40 da 42. am ganawilebis modaa 26, mediana _ 30, xolo

saSualo _ 32. Tu SevcvliT mxolod erT qulas, magaliTad, 40-is nacvlad

aviRebT 43-s, moda isev iqneba 26 da mediana _ 30, magram saSualo axla iqneba 32.4.

vinaidan saSualo damokidebulia TiToeuli qulis sidideze, ganawilebis

ukiduresi qulebi gavlenas axdenen saSualos sidideze da cvlian (amaxinjeben)

mis, rogorc centraluri tendenciis sazomis, mniSvnelobas. magaliTad, davuSvaT,

ganxilul ganawilebaSi qula 40 SevcvaleT ara 43-iT, aramed _ 78-iT. ganawilebis

moda da mediana isev 26 da 30 iqneba, Sesabamisad, magram gaizrdeba saSualo da

iqneba 37.4. aqedan gamomdinare, Tu ganawilebaSi Sedis ramdenime ukiduresi qula,

saSualo SeiZleba veRar warmoadgendes tipur qulas.

saSualos meore Tviseba isaa, rom ganawilebaSi Semaval TiToeul qulasa da

ganawilebaSi Semaval qulaTa saSualos Soris gansxvavebaTa jami nulis tolia,

anu

( )∑ =− 0XX

es gamosaxuleba gviCvenebs, rom ganawilebaSi Semaval TiToeul qulas

gamovakeliT saSualo, Semdeg SevkribeT miRebuli sxvaobebi da Sedegad miviReT

nuli. cxrilSi N4-4 mocemulia es mimarTeba: am cxrilis (a) svetSi mocemulia

gamocdebTan dakavSirebuli SfoTvis kiTxvarSi miRebuli Svidi qula da maTi

saSualo 35.0; (b) svetSi mocemulia Semdegi moqmedeba: _ saSualos ricxviTi

mniSvneloba akldeba (a) svetSi mocemul TiToeul qulas da am svetSi miRebuli

mniSvnelobebis jami ( )∑ − XX aris nuli.

qulaTa nebismieri erTobliobisTvis

( )∑ − XX jamis mniSvneloba nulis

toli iqneba, magram ( )2∑ − XX mxolod

im SemTxvevaSi iqneba nulis toli,

rodesac ganawilebaSi Semavali

TiToeuli qula saSualos tolia; sxva

SemTxvevaSi ( )2∑ − XX ar iqneba nulis

toli. es sidide gamoTvlilia (g)

svetSi, sadac TiToeul qulasa da

kvadratebis jami (SS) (Sum of squares): _

ricxvi, romelic miiReba ganawilebaSi

Semavali TiTouli qulidan ganawilebis

saSulos gamoklebiT, TiToeuli sxvaobis

kvadratSi ayvaniTa da miRebuli sidideebis

SejamebiT.

M _ simbolo, romelic amerikis

fsiqologiuri asociaciis publikaciebis

saxelmZRvaneloSi (1994) miTiTebuli

normebiT Sesrulebul naSromSi

gamoiyeneba SerCevis saSualos aRsaniSnad.


saSualos Soris sxvaoba (romelsac xSirad gadaxras uwodeben) kvadratSia

ayvanili (anu gamravlebulia Tavis Tavze). magaliTad, qula 22 X -sgan

gansxvavdeba -13-iT, xolo (-13)2 = 169. (g) svetSi mocemul sxvaobaTa kvadratebis

jamia 264. ( )2∑ − XX sidide xSirad gamoiyeneba statistikur gamoTvlebSi da

kvadratebis jami (SemoklebiT SS) ewodeba. amrigad,

( )2∑ −== XXSSjami skvadratebi

maTematikurad SesaZlebelia imis Cveneba, rom saSualodan qulaTa gadaxrebis

kvadratebis jami (anu ( )2∑ − XX ) ufro naklebia, vidre nebismieri sxva

statistikidan, magaliTad, medianidan an modidan, qulaTa gadaxrebis kvadratebis

jami. es Tviseba SesaZleblobas iZleva, rom saSualo gamoyenebul iqnes ramdenime

Zalze mniSvnelovan statistikur meTodSi, romelzec mogvianebiT, Semdeg TavebSi

visaubrebT.

cxrili N4.1. sveti (a): Svidi qula da maTi

saSualo 35.0; sveti (b): XX − sxvaobis

mniSvneloba TiToeuli qulisTvis; sveti (g)

XX − gadaxra TiToeuli qulisTvis

ayvanilia kvadratSi.

(a)

X

(b)

XX −

(g)

2)( XX −

22 34 35 35 37 38 44

22-35 = -13 34-35 = -1 35-35 = 0 35-35 = 0 37-35 = +2 38-35 = +3 44-35 = +9

169 1 0 0 4 9 81

X 0.35 0 jami 264

SerCevis saSualos warmodgena samecniero statiaSi

statistikosebi, rogorc wesi, SerCevis saSualos aRsaniSnad iyeneben X

simbolos, Tumca, samecniero JurnalebSi xSirad gamoiyeneba statistikaSi

miRebulisgan gansxvavebuli simboloebi. bihevioristuli mecnierebis mravali

Jurnali icavs amerikis fsiqologiuri asociaciis publikaciebis

saxelmZRvaneloSi (Publication Manual of the American Psychological Association. American

Psychological Association, 1994) mocemul saredaqcio stils. am stilis mixedviT,

SerCevis saSualo aRiniSneba laTinuri didi daxrili M-iT. amrigad, samecniero


JurnalSi gamoqveynebul statiaSi SeiZleba SexvdeT Semdegnairad warmodgenil

saSualos:

qali cdis pirebis mier qali politikosebis kompetenturobis saSualo Sefaseba (M =

6.13) ufro maRali iyo, vidre mamakaci cdis pirebis saSualo Sefaseba (M = 5.42).

sabeWd manqanaze akrefil an xelnawer teqstSi statistikuri simbolos

gamosayofad gamoiyeneba xazgasma1. amrigad, laboratoriul angariSs aseTi saxe

SeiZleba hqondes:

qali cdis pirebis mier qali politikosebis kompetenturobis saSualo Sefaseba (M =

6.13) ufro maRali iyo, vidre mamakaci cdis pirebis saSualo Sefaseba (M = 5.42).

populaciis saSualo

populaciisTvis qulaTa

saSualo SerCevis saSualos

msgavsad gamoiTvleba, im

gansxvavebiT, rom N populaciaSi

qulaTa raodenobas aRniSnavs, saSualos mniSvneloba ki berZnuli µ-iTaa

(berZnuli anbania me-12 aso, warmoiTqmis, rogorc „miu“) warmodgenili da ara _

X -iT. amrigad, populaciis saSualo gamoisaxeba Semdegi formuliT:

populacia

saSualo populaciisN

X∑== µ

umravles SemTxvevaSi µ-s Sesaxeb daskvna X -dan gamomdinare keTdeba, radgan

SeuZlebelia populaciaSi yvela qulis gazomva. SerCevis saSualo, rogorc

populaciis saSualos Sefaseba, ufro detalurad me-7 TavSia ganxiluli.


1. ganmarteT: _ M, µ, N, populaciis saSualo, ∑ , SerCevis saSualo,

kvadratebis jami, ∑ X , X .

2. davuSvaT, miiReT Semdegi qulebi: _ 97, 90, 86, 93, 76 da 86.

a) ipoveT maTi saSualo.

b) aCveneT, rom am qulebisTvis ( )∑ =− 0XX .

1 es meTodi gamoiyeneba laTinuri asoebiT (magaliTad, inglisurad an germanulad)

dawerili teqstis SemTxvevaSi; qarTulSi statistikuri simbolos xazgasma ar aris

saWiro (mTargmnelis SeniSvna).

populaciis saSualo (Population mean – µ) _

populaciaSi yvela SemTxvevis jamis fardoba

qulebis saerTo raodenobasTan.


g) am qulebisTvis gamoTvaleT ( )2∑ − XX .

3. davuSvaT, mkvlevarma gazoma me-7 klaseli gogonebis TviTSefaseba da

miiRo Semdegi qulebi: _ 49, 53, 67, 20, 27, 36, 49, 27, 61 da 28.

a) gamoTvaleT am qulebis saSualo.

b) davuSvaT, eqsperimentatorma kidev erTi me-7 klaseli gogonas

TviTSefaseba gazoma da miiRo 17 qula. CarTeT es qula zemoT

mocemul ganawilebaSi da ipoveT miRebuli ganawilebis saSualo.

g) davuSvaT, qula 67 Seicvala 27-iT da miRebuli qulebi gaxda 49, 53,

27, 20, 27, 36, 49, 27, 61 da 28. ra iqneba qulaTa am axali simravlis

saSualo?

d) SeadareT erTmaneTs am kiTxvis (a), (b) da (g) punqtebSi miRebuli

pasuxebi. saSualos romel Tvisebebs aCvenebs Tqvens mier miRebuli

pasuxebi?

centraluri tendenciis sazomebis Sedareba

romelia „saukeTeso“ saSualo qula? am kiTxvaze pasuxis gacema ar aris ioli.

centraluri tendenciis saukeTeso sazomi damokidebulia imaze, Tu risTvis aris

mowodebuli mocemuli statistika, anu risTvis viyenebT mas. centraluri

tendenciis sazomi gamoiyeneba rogorc aRsawerad, ise _ daskvnebis gasakeTeblad.

centraluri tendenciis sazomebis aRweriTi gamoyeneba

arCeva imisa, centraluri tendenciis romeli sazomi gamoviyenoT, rogorc

aRweriTi statistika, damokidebulia gazomvis skalaze, romelzec imyofeba

miRebuli qulebi da am qulebis sixSireTa ganawilebis formaze.

qulebis gazomvis skalebi

saxeldebis skala. saxeldebis skalaze pasuxebi ubralod nawildeba or

kategoriaSi: _ pasuxi aris, an ar aris. centraluri tendenciis erTaderTi sazomi,

romelic SeiZleba gamoviyenoT saxeldebis skalaze, aris moda anu ganvsazRvroT

kategoria, romelic yvelaze xSirad gvxdeba.

rigis skala. rigis skalaze qulebi lagdeba sididis mixedviT, magram ver

viqnebiT darwmunebulni, rom cvladis gverdigverd ganlagebuli mniSvnelobebi

skalaze toli intervalebiT aris daSorebuli erTmaneTisgan. Sesabamisad,

SegviZlia im qulis gansazRvra, romelic iqneba mediana. mediana qulaTa

ganawilebas or tol nawilad yofs. SesaZlebelia modaluri qulis moZebnac.

intervalebisa da Sefardebis skalebi. orive skalis SemTxvevaSi miRebulia, rom

am skalaze gverdigverd ganlagebul qulebs Soris intervali Seesabameba im

skalis intervalebs, romelzec gaizoma cvladi. amitom SesaZlebelia


intervalebisa Tu Sefardebis skalaze ganlagebul qulaTa saSualos gamoTvla.

am doneze myofi qulebisTvis medianisa da modis povnac SesaZlebelia.

qulebis ganawilebis forma

is, Tu ramdenad kargad axasiaTebs centraluri tendenciis sazomi ganawilebas,

ganawilebis formazec aris damokidebuli. suraTze N3.8 warmodgenilia qulaTa

ramdenime tipuri ganawileba. Tu ganawileba unimodaluria da simetriuli,

rogoricaa suraTze N3.8a mocemuli normaluri ganawileba, maSin moda, mediana da

saSualo erTi da igive ricxvi iqneba. TiToeuli statistika erTnairad kargad

axasiaTebs ganawilebas; Tumca, miRebuli ganawilebebi xSirad ar aris

unimodaluri da simetriuli; isini dadebiTad an uaryofiTad asimetriulia.

asimetriul ganawilebaSi modas, medianasa da saSualos mniSvnelobebi

gansxvavdeba erTmaneTisgan. vinaidan saSualo mgrZnobiarea asimetriuli

ganawilebis ukiduresi qulebis mimarT, is ufro metad gadainacvlebs ukiduresi

qulebis mimarTulebiT, vidre moda an mediana. amrigad, saSualosa da medianas

Soris sxvaobis niSani gviCvenebs ganawileba dadebiTad asimetriulia Tu

uaryofiTad. uaryofiTad asimetriul ganawilebaSi saSualo ufro naklebi iqneba,

vidre mediana. analogiurad, ramdenime ukiduresad maRali qula dadebiTad

asimetriul ganawilebaSi gazrdis saSualos mniSvnelobas da is medianaze meti

iqneba. suraTze N4.1 naCvenebia medianisa da saSualos fardobiTi adgilmdebareoba

dadebiTad da uaryofiTad asimetriul ganawilebebSi. rogorc suraTze xedavT,

Tu ganawileba sakmaod asimetriulia da didi sxvaobaa saSualosa da medianas

mniSvnelobebs Soris, mediana, rogorc wesi, centraluri tendenciis ukeTes

sazoms warmoadgens, vidre saSualo. amis magaliTia beisbolis ZiriTadi ligis

moTamaSeTa Semosavlebis ganawileba. erT-erT amerikul gundSi wliuri

Semosavalia $109.000-dan $5.155.250-mde (Cafardo, 1994). am ganawilebis saSualoa

$1.357.163, magram medianaa $500.000. ganawileba mkveTrad dadebiTad asimetriulia;

gundis 25 moTamaSidan mxolod 8 maTgani iRebs saSualoze mets. amrigad, am

gundis moTamaSeTa tipuri anazRaurebis saSualoTi daxasiaTeba araswor suraTs

gvaZlevs, radgan moTamaSeTa naxevari $500.000-s an naklebs iRebs (Fudin & Levinson,

1994).


yovelTvis, rodesac mniSvnelovania aRwera, umjobesia qulaTa ganawilebis

centraluri tendenciis samive sazomis warmodgena; Tumca, rogorc naxavT

bihevioristuli mecnierebis literaturaSi, centraluri tendenciis yvelaze

farTod gamoyenebad sazoms saSualo warmoadgens, xolo moda iSviaTad

gamoiyeneba.

SeiZleba, rom zogierT ganawilebas ar hqondes centraluri tendencia?

bimodaluri ganawileba ver aRiwereba saTanadod centraluri tendenciis

mxolod erTi sazomiT, gansakuTrebiT maSin, rodesac modaluri qulebi didi

manZiliT arian daSorebuli erTmaneTisgan. magaliTad, davuSvaT, rom mkvlevarma

miiRo gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebis iseTi ganawileba,

rogoric mocemulia cxrilSi N4.2. es ganawileba aSkarad bimodaluria; misi erTi

modaa 34, xolo meore _ 58. ganawilebis medianaa 45.5, saSualo ki _ 45.8. cxrilidan

Cans, rom arc erTma cdis pirma ar miiRo medianis an saSualos axlos mdebare

qula. am SerCevaSi arc mediana da arc saSualo ar warmoadgens tipuri pirovnebis

gamocdebTan dakavSirebul SfoTvas. moda 34 da moda 58 am SemTxvevaSi tipuri

qulebis saukeTeso aRweraa.

dadebiTad asimetriuli

mediana

saSualo

mediana

saSualo

suraTi N4.1. medianisa da saSualos fardobiTi adgilmdebareoba dadebiTad

da uaryofiTad asimetriul ganawilebaSi.

uaryofiTad asimetriuli


ganvixiloT kidev erTi magaliTi.

soflis meurneobis departamentis

bolo wlis monacemebis mixedviT,

aSS-Si yovelwliurad erT sul

mosaxleze modioda 1.33 funti

moxmarebuli saReWi Tambaqo.

Sesabamisia es aRwera TqvenTvis? Tu

Tqven ar ReWavT Tambaqos, es ricxvi

Zalze maRali iqneba, magram Tu Tqven

ReWavT Tambaqos, es sidide SeiZleba

Zalze mcirec ki iyos TqvenTvis.

saReWi Tambaqos gamoyenebis

ganawileba bimodaluria. populaciis

mcire nawili iyenebs saReW Tambaqos

da misi moxmareba mniSvnelovnad

aRemateba 1.33 funts weliwadSi.

populaciis didi nawili ar moixmars

saReW Tambaqos da maTi modaluri

gamoyeneba iqneba nuli funti

weliwadSi.

centraluri tendenciis sazomebis daskvniTi gamoyeneba

centraluri tendenciis sazomis arCevas, nawilobriv, ganapirobebs is, Tu

ramdenad gamoiyeneba mocemuli statistika SemdgomSi daskvniTi miznebisTvis.

pirvel TavSi vTqviT, rom populaciis parametrebi xSirad fasdeba SerCevisgan

miRebul statistikebze dayrdnobiT. SerCevis saSualo X populaciis saSualos

(romelic aRiniSneba berZnuli asoTi μ) saukeTeso Sefasebaa. aqedan gamomdinare,

SerCevis saSualo Sedis mniSvnelovan statistikur SefasebaTa ricxvSi. vinaidan

saSualo aseTi gamosadegia daskvnisTvis, bihevioristuli mecnierebis

warmomadgenlebi xSirad iyeneben mas monacemTa aRsawerad. SerCevis saSualos,

rogorc populaciis saSualos Sefasebis Tvisebebs ufro srulad me-7 TavSi

ganvixilavT.

cxrili N4.2. gamocdebTan dakavSirebuli

SfoTvis kiTxvaris 26 qulis bimodaluri

ganawileba.

qula Tvlis

Sedegi

ƒ

61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31

/ // ///// /// / / / /// ///// // / /

1 2 5 3 1 1 1 3 5 2 1 1

moda

8.45=X , Md = 45.5 moda



1. centraluri tendenciis romel sazoms airCevT qvemoT aRweril

TiToeul SemTxvevaSi?

a) ipoveT mniSvneloba, romelic warmoadgens ganawilebis 50-e

procentils.

b) romeli sazomisTvis aris minimaluri ganawilebaSi Semaval

TiToeul qulasa da centraluri tendenciis sazoms Soris

sxvaobebis kvadratebis jami.

g) aRwereT qula, romelic yvelaze xSirad gxvdebaT ganawilebaSi.

d) mis gamosaTvlelad gamoiyeneT TiToeuli qulis ricxviTi

mniSvneloba.

2. ganawileba mkveTrad uaryofiTad asimetriulia. rogor fiqrobT,

ganawilebis saSualo iqneba medianaze naklebi, medianis toli Tu masze

meti? axseniT Tqveni pasuxi.

3. ganawilebis saSualo mniSvnelovnad aRemateba medianas. rogor fiqrobT,

ganawileba simetriulia, uaryofiTad asimetriulia Tu dadebiTad

asimetriuli? ratom?

Sejameba

• centraluri tendenciis sazomebis gamoyeneba SesaZlebelia aRsawerad

da daskvnebis gasakeTeblad.

• centraluri tendenciis sami yvelaze xSirad gamoyenebadi sazomia: _

moda, mediana da SerCevis saSualo.

• moda aris qula, romelic yvelaze xSirad gvxvdeba qulaTa

ganawilebaSi.

• mediana aris qula ganawilebaSi, romlis zemoT da qvemoT

ganlagebulia qulaTa toli raodenoba.

• SerCevis saSulo gamoiTvleba ∑ NX / SefardebiT.

• saSualo centraluri tendenciis yvelaze xSirad gamoyenebadi sazomia

daskvniTi miznebisTvis.

• TiToeuli statistikis saTanado gamoyeneba aRweriTi miznebisTvis

damokidebulia gazomvis skalaze da qulebis sixSireTa ganawilebis

formaze.


aRweriTi statistika

bimodaluri ganawileba

mediana (Mdn)

moda

multimodaluri

ganawileba


populaciis saSualo

saSualo

unimodaluri ganawileba

SerCevis saSualo

cvalebadobis sazomebi

centraluri tendenciis

sazomebi

M

μ (miu)

∑ (dajameba)

X

∑ X

( )2∑ − XX

kiTxvebi

1. rotonma da kelim (Rotton & Kelly, 1985) Seqmnes skala, romelsac mTvaris

efeqtis rwmenis skala uwodes. am skaliT xdeba imis Sefaseba, Tu

ramdenad swams adamians, rom savse mTvare gavlenas axdens qcevaze.

skalaze qulaTa diapazonia 9-dan 81-is CaTvliT. am skalaze individTa

erTi SerCevis saSualoa 37.9, xolo mediana _ 39. SeZlebT Tu ara am

informaciaze dayrdnobiT daaskvnaT qulaTa ganawilebis Sesaxeb: _

daaxloebiT rogoria is, simetriuli Tu asimetriuli? ratom?

2. aSS-is Census Bureau-is monacemebis mixedviT, 1992 wels mamrobiTi sqesis

advokatebisa da mosamarTleebis saSualo Semosavalma $94.757 Seadgina

maSin, roca maTi Semosavlis mediana $78,052 iyo. asimetriulia es

ganawileba? Tu fiqrobT, rom asimetriulia, maSin ra mimarTulebiT aris

asimetriuli? ratom?

3. eglim da meiersma (Egli & Meyers, 1984) video TamaSebis pavilionSi

gamokiTxes 151 mozardi. gamokiTxvis Sedegebma aCvena, rom individebis

mier kviris manZilze TamaSSi gatarebuli saSualo dro iyo 5.2 saaTi,

xolo mediana _ 2.5 saaTi kviraSi. asimetriulia Tu ara kviris manZilze

TamaSis saaTebis miRebuli ganawileba? Tu fiqrobT, rom asimetriulia,

maSin ra mimarTulebiT aris asimetriuli? ratom?

4. bihevioristul medicinaSi momuSave mkvlevrebs ainteresebdaT tkivilis

kontrolis fsiqologiuri meqanizmebi. cdis pirebSi tkivilis

usafrTxod gamosawvevad maT SeeZloT civi presoris amocanis gamoyeneba.

am amocanis erT-erTi forma gulisxmobs civ wyalSi xelis Cayofasa da

im drois dafiqsirebas, ramden xansac SeZlebs cdis piri xelis

gaCerebas wyalSi. davuSvaT, eqsperimetSi monawileobda 17 cdis piri,

romelTac xeli Cayofili hqondaT 40°F-ian (4°C) wyalSi da fiqsirdeboda

dro, ramden xansac aCerebda cdis piri xels wyalSi manam, sanam tkivili

gausaZlisi gaxdeboda. eqsperimentatorma cda oTxjer gaimeora da miiRo

qulaTa oTxi jgufi, romelic qvemoT aris mocemuli (qulebi

warmoadgens wamebiT gazomil dros):

5.


jgufi qulebi

1 35, 39, 38, 30, 37, 43, 30, 38, 41, 52, 25, 44, 33, 41, 38, 56, 37.

2 20, 28, 26, 29, 60, 21, 37, 30, 28, 57, 29, 25, 63, 26, 32, 30, 53.

3 45, 23, 57, 50, 47, 52, 20, 53, 50, 30, 52, 60, 52, 49, 52, 49, 53.

4 52, 23, 40, 60, 52, 21, 20, 52, 50, 23, 55, 26, 28, 53, 26, 49, 26.

qulaTa TiToeuli jgufisTvis:

a) aageT daujgufebel martiv sixSireTa ganawileba. gamoiyeneT es

sixSireTa ganawileba qulaTa ganawilebis formis Sesaxeb daskvnis

gasakeTeblad.

b) gamoTvaleT moda, mediana da saSualo.

g) SeadareT centraluri tendenciis sazomebi erTmaneTs. asimetriulia

ganawileba? axseniT, rogor mixvediT am daskvnamde. am Sedarebis

Sedegad miRebuli pasuxi Tanxvdeba Tu ara (a) punqtSi agebul

sixSireTa ganawilebas?

d) rogor fiqrobT, centraluri tendenciis romelime sazomi aRwers

qulebs ukeTesad? Tu ki, romeli sazomi da ratom?

5. Tqveni statistikis profesori apirebs, rom xuTi gamocdis safuZvelze

dagiweroT saboloo niSani. TiToeul gamocdaSi maqsimaluri qulaa 100.

man giTxraT, rom Tqveni niSani gamoiTvleba qulebis centralur

tendenciaze dayrdnobiT da rom Tqven unda airCioT, romel qulaze

dayrdnobiT gamoTvlili Sefaseba girCevniaT: _ modalur, medianur Tu

saSualo qulaze. davuSvaT, sami studentis niSnebia:

gamocdebi

1 2 3 4 5

roberto

dimitriosi

kareni

35

10

56

40

50

56

65

65

67

90

80

78

90

85

98

Tu Tqven erT-erTi am studentTagani xarT, centraluri tendenciis romel

sazoms airCevdiT Tqveni saboloo niSnis gansasazRvrad? axseniT

TiToeuli studentis SemTxvevaSi gakeTebuli arCevanis mizezi.

6. grosma (Gross, 1993) gamokiTxa studentebi did centralur evropul

universitetSi. gamokiTxva Seexeboda kolejis mdedrobiTi da mamrobiTi

sqesis studentebSi alkoholis moxmarebas. gamokiTxvis erT-erTi kiTxvaSi

respondentebs moeTxovebodaT mieTiTebinaT, Tu ramdenjer iRebdnen

alkohols kviraSi. grosma aRmoaCina, rom 21 wlis da ufrosi asakis

studentebSi mamakacebTan kviraSi dalevis SemTxvevaTa saSualo raodenoba

iyo 23.9, qalebTan _ 7.5.


a) gaakritikeT dalevis qcevis centraluri tendenciis sazomad

saSualos gamoyeneba; kerZod, fiqrobT, rom dalevaTa

ganawileba asimetriulia? Tu ki, ra mimarTulebiT aris

asimetriuli?

b) rogor fiqrobT, centraluri tendenciis romeli sazomi iqneba

saukeTeso am SemTxvevaSi?

7. martivi reaqciis dro (dro, romelic sWirdeba reaqcias iseT martiv

gamRizianeblebze, rogoricaa sinaTle an toni) warmoadgens zogadi

anesTeziidan individis gamosvlis sazoms. davuSvaT, zomavT 8 qirurgiuli

pacientis martiv reaqciis dros, mcire qirurgiuli operaciisTvis zogadi

anesTeziis miRebidan 2 saaTis Semdeg da miiReT reaqciis Semdegi droebi

miliwamebSi: _ 782, 793, 793, 817, 832, 847, 866, 893. mecxre pacienti ar iyo

sakmarisad gamosuli anesTeziidan, rom moeca reaqciis dro, amitom am

cdis piris monacemi ganisazRvra, rogorc reaqciis usasrulo dro.

a) Tu gsurT cxrave pacientis monacemis erTad ganxilva,

centraluri tendenciis romel sazoms an sazomebs

gamoTvlidiT?

b) gamoTvaleT am sazomis ricxviTi mniSvneloba.

8. aSS-is Census Bureau-is monacemebis mixedviT, 1998 wels pirveli qorwinebis

asaki mamakacebisTvis 26.7 weli iyo. axseniT, ratom aris ukeTesi am

statistikis aRsawerad medianis gamoyeneba da ara _ saSualo

ariTmetikulis.

Tavi V. monacemTa aRwera: cvalebadobis sazomebi


diapazoni

kvartilTSorisi gabnevis diapazoni da kvartilTSorisi

gabnevis diapazonis naxevari

cvalebadobis sazomebi SerCevis saSualos mimarT

dispersia

standartuli gadaxra

populaciis dispersiisa da standartuli gadaxris

Sefasebis gamosaTvleli formula

dispersiisa da standartuli gadaxris gamoTvla

kalkulatoris gamoyenebiT

simboloebisa da formulebis mimoxilva

dispersiisa da standartuli gadaxris interpretacia

rodis aris cvalebadoba didi

cvalebadobis sazomis warmodgena samecniero statiebSi

cvalebadobis sazomebis Sedareba

aRweriTi statistikis arCeva

gazomvis skala

sixSireTa ganawilebis forma

monacemebis Semdgomi analizi

თავი V. მონაცემთა აღწერა: ცვალებადობის საზომები 2

cvalebadoba gulisxmobs imas, Tu ramdenad gansxvavdebian qulebi

erTmaneTisgan da ganawilebis centraluri tendenciis sazomisgan. rac ufro

metad gansxvavdebian qulebi erTmaneTisgan, miT metad gansxavebulni iqnebian isini

centraluri tendenciis sazomisganac da ufro meti cvalebadoba iqneba

ganawilebaSi; da, piriqiT, rac ufro naklebad gansxvavdebian qulebi

erTmaneTisgan ganawilebaSi, miT naklebad gansxvavebulni iqnebian isini

centraluri tendenciis sazomisgan da miT naklebi cvalebadoba iqneba

ganawilebaSi. amrigad, centraluri tendenciis sazomi individTa SerCevisgan

miRebuli qulebis Sesaxeb informaciis mxolod nawils gvatyobinebs. mxolod am

sazomze dayrdnobiT ver vityviT, qulebi

farTod arian gafantulni, Tu mWirdod

arian Sejgufebulni centraluri tendenciis

sazomis irgvliv. mxolod moda, mediana anda

saSualo ver gviCvenebs, Tu ramdenad didia

ganawilebaSi Semavali qulebis gafantuloba an cvalebadoba.

cvalebadobis cnebis sailustraciod, davuSvaT, rom mkvlevarma gazoma cxra

cdis pirisgan Semdgari sami SerCeva imis mixedviT, Tu ramden xans icinodnen

isini komiqsebis kiTxvisas. wamebiT gazomili qulebi mocemulia cxrilSi N5.1.

moda, mediana da saSualo TiToeuli jgufisTvis aris 41.0; magram cxadia, rom es

ganawilebebi Zalze gansxvavdeba erTmaneTisgan da centraluri tendenciis

sazomebi erTnairad kargad ver aRweren qulaTa

TiToeul jgufs. pirvel jgufSi TiToeuli qula

41-is tolia; amrigad, am jgufSi ar aris

cvalebadoba. aqedan gamomdinare, centraluri

tendenciis TiToeuli sazomi srulyofilad

warmoadgens ganawilebaSi Semaval qulebs; magram

am jgufSi miRebuli Sedegi Zalze iSviaTi iqneba,

Tu sicilis xangrZliobas realurad gavzomavT

cxra cdis pirTan. meore jgufis qulebi mcire

cvalebadobiT xasiaTdebian. qulebi Sejgufebulni

arian centraluri tendenciis sazomis irgvliv.

arc erTi qula gansxvavdeba 41 wamisgan 3 wamze

metiT. centraluri tendenciis TiToeuli sazomi

jgufSi Semavali yvela individis

reprezentaciulia, radgan mcire cvalebadobaa calkeul qulebs Soris.

gansxvavebuli situacia gvaqvs mesame jgufSi. aq ufro meti cvalebadobaa

sicilis xangrZliobis TvalsazrisiT. qulebi mniSvnelovnad gansxvavdeba

erTmaneTisgan da, meore jgufisgan gansxvavebiT, zogierTi qula 20 wamze metiTac

cvalebadoba (Variability): _ ramdenad

gansxvavdebian qulebi erTmaneTisgan

da ganawilebis centraluri

tendenciis sazomisgan.

cxrili N5.1. sicilis

xangrZliobis qulebi (wamebiT)

cxra cdis pirisgan Semdgari

sami jgufisTvis.

jgufi

1 2 3

moda Md

X

41 41 41 41 41 41 41 41 41 41 41 41

38 39 40 41 41 41 42 43 44 41 41 41

21 26 33 41 41 41 50 51 65 41 41 41


gansxvavdeba centraluri tendenciis (41 wm) sazomisgan. naTelia, rom centraluri

tendenciis verc erTi sazomi ver iqneba qulaTa am erTobliobis

reprezentaciuli, rogorc es xdeba pirveli da meore jgufis qulebis

SemTxvevaSi.

centraluri cendenciis sazomebis msgavsad, unda gavzomoT qulebis

cvalebadoba; amitom ganawilebaSi Semavali qulebis cvalebadobis aRweris

ramdenime sazomi arsebobs.


cvalebadobis erT-erTi sazomi aRwers ganawilebaSi qulebis garkveuli

procentis Semcvel intervals. Cveulebisamebr, gamoiyeneba sami aseTi sazomi;

esenia: _ diapazoni, kvartilTSorisi gabnevis diapazoni da kvartilTSorisi

gabnevis diapazonis naxevari.

diapazoni

qulaTa erTobliobis cvalebadobis umartivesi

sazomia diapazoni, romelic Semdegnairad

gamoiTvleba:

diapazoni = Xudidesi _ Xumciresi

sadac Xudidesi warmoadgens ganawilebis maqsimalur qulas, xolo Xumciresi _

minimalur qulas. cxrilSi N5.1 mocemuli mesame jgufisTvis Xudidesi = 65 da

Xumciresi = 21; amrigad, am jgufis diapazonia 65_21 anu 44. meore jgufis qulebis

diapazonia 44_38 anu 6, xolo pirveli jgufis _ 41_41 anu 0. cxrilSi 3-3

mocemuli gamocdebTan dakavSirebuli SfoTvis kiTxvaris 100 qulis SemTxvevaSi

maqsimaluri qulaa 73, minimaluri _ 22, amitom am ganawilebis diapazoni iqneba

73_22 anu Sefasebis skalis 51 qula.

diapazonis Tvisebebi

diapazoni aRwers ganawilebis maqsimalur da minimalur qulebs. maqsimaluri da

minimaluri qulis nebismieri cvlileba gavlenas axdens diapazonze, maSinac ki,

Tu yvela sxva danarCeni qula ucvleli darCeba. amrigad, diapazoni cvalebadobis

SedarebiT arastabiluri sazomia. garda amisa, rac ufro didia SerCeva, miT metia

albaToba imisa, rom erTma eqstremalurma qulam gazardos diapazonis ricxviTi

mniSvneloba. diapazonis dadebiTi Tviseba isaa, rom martivad gamoiTvleba, magram

vinaidan aklia stabiluroba, xSirad ar gamoiyeneba cvalebadobis aRsawerad.

diapazoni (Range): _ sxvaoba

ganawilebis maqsimalur da

minimalur qulebs Soris.


kvartilTSorisi gabnevis diapazoni da kvartilTSorisi gabnevis

diapazonis naxevari

kvartilTSorisi gabnevis diapazoni (IQR)

aRwers ganawilebis qulebis SuaTana 50%-s da

xSirad gamoiyeneba rigis skalaze. is

warmoadgens sxvaobas 75-e procentilis

Sesabamis qulasa (X75) da 25-e procentilis

(X25) Sesabamis qulas Soris. kvartilTSorisi

diapazoni gamoiTvleba tolobiT:

kvartilTSorisi gabnevis diapazoni =IQR =

X75 _ X25

X75-sa da X25-is mniSvnelobebi gamoiTvleba

me-3 TavSi mocemuli, ganawilebis konkretuli

procentilis gamosaTvleli formuliT.

kvartilTSorisi gabnevis diapazonis naxevari (SIQR) gamoiTvleba formuliT:

kvartilTSorisi gabnevis diapazonis naxevari = SIQR = IQR/2= (X75 _ X25)/2

IQR-isa da SIQR-is Tvisebebi

IQR-iT aRiwereba medianis irgvliv dajgufebuli individebis SuaTana 50%-is

tipur qulaTa diapazoni. vinaidan IQR-Si ar Sedis ganawilebis qulaTa zeda 25%

da qveda 25%, mis mniSvnelobaze gavlenas ar axdens asimetriuli ganawilebis

ukiduresi qulebi. naTelia, rom kvartilTSorisi gabnevis diapazoni ufro

naklebi iqneba, vidre qulaTa erTobliobis diapazoni. magaliTad, cxrilSi N3-3

mocemuli gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebisTvis X75 = 45 da

X75 = 37 (damrgvalebulia uaxloes mTel ricxvamde). amrigad, IQR iqneba

IQR = X75 _ X25 = 45 _ 37 = 8

maSin, rodesac am qulaTa diapazonia 51. miuxedavad imisa, rom cxrilSi N3-3

mocemuli qulebis sruli diapazoni didia, SedarebiT mcirea qulaTa Sua 50%-is

diapazoni.

SIQR-iT aRiwereba medianis zemoT da qvemoT ganlagebul qulaTa 25%-is

diapazoni; Tumca, mxolod simetriuli ganawilebis SemTxvevaSi aRwers SIQR

medianis zemoT da qvemoT ganlagebul qulaTa zustad 25%-is diapazons.

magaliTad, cxrilSi N3.3 mocemuli gamocdebTan dakavSirebuli SfoTvis

kiTxvaris qulebiTvis SIQR aris 8/2 anu 4. es SIQR gviCvenebs, rom ganawilebaSi

qulaTa 50% ar aris medianaze daaxloebiT 4 quliT meti da daaxloebiT 4 quliT

naklebi. yuradReba miaqcieT, rom am winadadebaSi gamoyenebulia sityva

„daaxloebiT“. vinaidan gamocdebTan dakavSirebuli SfoTvis kiTxvaris qulebis

kvartilTSorisi gabnevis diapazoni

(Interquartile range, IQR): _ ganawilebaSi

Semaval qulaTa SuaTana 50%.

X75 _ ganawilebis 75-e procentilis

Sesabamisi qula.

X25 _ ganawilebis 25-e procentilis

Sesabamisi qula.

kvartilTSorisi gabnvis diapazonis

naxevari (Semi-interquartile range, SIQR) _

kvartilTSorisi gablevis

diapazonis naxevari.


sixSireTa ganawileba mcired asimetriulia, SIQR-Si ar Sedis medianis TiToeul

mxares moTavsebul qulaTa zustad 25%.


1. ganmarteT: _ kvartilTSorisi gabnevis diapazoni, diapazoni,

kvartilTSorisi gabnevis diapazonis naxevari, cvalebadoba, X25, X75..

2. klasSi Catarebul testSi miRebuli maqsimaluri qulaa 98, xolo

minimaluri _ 54. miRebuli qulebis 75-e procentilia 92 qula, xolo 25-e

procentili _ 74. gamoTvaleT am qulaTa diapazoni, IQR da SIQR.

3. rTuli reaqciis drois testSi miRebuli yvelaze didi qulaa 929

miliwami, xolo yvelaze mcire _ 642 miliwami. qulaTa 75-e procentilia

809 miliwami, xolo 25-e procentili _ 724 miliwami.

a) gamoTvaleT am qulebis diapazoni, IQR da SIQR.

b) ra informacias iZleva qulebis Sesaxeb IQR?

g) ra informacias iZleva qulebis Sesaxeb SIQR?

4. axseniT, ratom aris diapazoni cvalebadobis SedarebiT aramdgradi

sazomi.

5. axseniT, ratom aris IQR cvalebadobis ufro mdgradi sazomi, vidre

diapazoni.

cvalebadobis sazomebi SerCevis saSualos mimarT

TiToeuli statistika, rogoricaa diapazoni, IQR da SIQR, migviTiTebs

ganawilebis centralur nawilSi ganlagebul qulaTa garkveuli procentis

Semcvel intervalze (magaliTad, diapazoni Seesabameba 100 procents), magram arc

erTi maTganis gamoTvlisas ar xdeba ganawilebaSi Semavali yvela qulis

gamoyeneba; Tumca, arsebobs cvalebadobis ori mniSvnelovani sazomi, romelic

iTvaliswinebs yvela qulis cvalebadobas saSualos mimarT; esenia: _ dispersia da

standartuli gadaxra.

me-4 TavSi gamovTvaleT ganawilebaSi Semavali TiToeuli qulis gansxvaveba anu

gadaxra ganawilebis saSualodan; Cven daviTvaleT XX − TiToeuli qulisTvis.

es gadaxra migviTiTebs imaze, Tu ramdenad gansxvavebulia qula saSualosgan. Tu

qula saSualos tolia, maSin XX − nulis toli iqneba; magram Tu qula ar aris

saSualos toli, maSin XX − miiRebs nulisgan gansxvavebul ricxviT

mniSvnelobas da rac ufro gansxvavebuli iqneba erTmaneTisgan X da X , miT meti

iqneba XX − sxvaobis mniSvneloba. Cven SegviZlia miviRoT cvalebadobis sazomi

imiT, rom SevkriboT ganawilebaSi Semavali yvela qulis gadaxra da miRebuli


jami gavyoT qulaTa raodenobaze. formulis saxiT es ase Caiwereba: ( )∑ − NXX / ;

Tumca, miRebuli sazomis SemTxvevaSi arsebobs erTi problema. rogorc naCvenebia

cxrilis N4.1 (b) svetSi, ganawilebis qulebis XX − gadaxraTa jami yovelTvis

nulis toli iqneba anu ( )∑ =− 0XX . am problemuri situaciis Tavidan

asacileblad, Sekrebamde da gayofamde Cven kvadratSi unda aviyvanoT TiToeuli

gadaxris mniSvneloba (anu ( )2XX − ). ( )∑ −2

XX gamosaxuleba mxolod maSin iqneba

nulis toli, Tu TiToeuli XX − sxvaobis mniSvneloba nulis toli iqneba. es

procedura gamoiyeneba dispersiisa da, Semdgom, standatruli gadaxris misaRebad.

Cven jer gavecnobiT dispersiisa da standartuli gadaxris formulebs da maT

gamoyenebas, Semdeg ki visaubrebT maT interpretaciaze.

dispersia

SerCevis dispersia

SerCevis dispersia (aRiniSneba S2)

ganisazRvreba, rogorc

( )N

XXS ∑ −

==2

2dispersia SerCevis

SerCevis dispersiis gamoTvlis es

formula migviTiTebs, rom:

1. TiToeul qulas vaklebT

saSualos [anu ( )XX − ];

2. TiToeuli miRebuli sxvaoba

agvyavs kvadratSi [anu ( )2XX − ];

3. SevkrebT kvadratSi ayvanil sxvaobebs [anu ( )∑ −2

XX ];

4. miRebuli jams vyofT N-ze, ganawilebaSi Semaval qulaTa ricxvze.

populaciis dispersia

Tu N populaciis tolia da populaciis saSualo (µ) cnobilia, maSin

formulaSi X-is nacvlad µ-is CasmiT populaciis dispersias miviRebT. populaciis

dispersia σ2-iT (berZnuli aso sigma, kvadratSi) aRiniSneba da miuTiTebs, rom es

dispersia populaciis parametria da ara _ SerCevidan miRebuli statistikuri

sidide. amrigad, populaciis dispersia gamoisaxeba formuliT:

( )populacia

dispersia populaciisN

X∑ −==

22 µ

σ

SerCevis dispersia (Sample variance, S2): _

cvalebadobis sazomi, romelic miiReba

ganawilebaSi Semavali ToToeuli qulisa

da am ganawilebis saSualos sxvaobebis

kvadratSi ayvaniT, miRebuli sidideebis

dajamebiTa da miRebuli jamis gayofiT

ganawilebaSi Semavali qulebis

raodenobaze.

populaciis dispersia (Population variance,

σ2) _ populaciis yvela qulis gazomviT

miRebuli dispersia.


sadac X _ populaciis calkeuli wevris qula;

µ _ populaciis qulaTa saSualo;

Npopulacia _ populaciaSi qulaTa saerTo raodenoba.

populaciis dispersiis Sefaseba

zemoT mocemuli formula SerCevis

dispersias aRwers; Tumca, rogorc pirvel

TavSi aRvniSneT, mkvlevars xSirad

sWirdeba SerCevidan miRebul dispersiaze

dayrdnobiT populaciis dispersiis

Sefaseba. Sesabamisad, S2-iT gamoTvlili

SerCevis dispersia srulad ver afasebs

populaciis dispersiis realur sidides

da iZleva σ2-is (berZnuli aso sigma,

ayvanili kvadratSi) Canacvlebis mqone Sefasebas. es Canacvleba SeiZleba

gaswordes imiT, rom formulis mniSvnelSi N-is nacvlad gamoviyenoT N_1.

amrigad, populaciis dispersiis Sefaseba aRiniSneba s2-iT da ganisazRvreba,

rogorc

( )1

s2

2

−

−== ∑

NXX

Sefaseba dispersiis populaciis .

rogorc me-4 TavSi aRvniSneT, dispersiis formulis mricxvelSi mocemul

kvadratSi ayvanil sxvaobebis jams ( )∑ −2

XX kvadratebis jami ewodeba da

aRiniSneba simboloTi SS. amrigad, populaciis dispersiis Sefaseba, aseve,

SeiZleba ganisazRvros, rogorc

12

−==

NSSs Sefaseba dispersiis populaciis

populaciis dispersiis Sefaseba _ s2 xSirad, mravali miznisTvis ufro

gamosadegia, vidre SerCevis dispersia _ S2 da ufro xSirad gamoiyeneba, vidre S2.

amrigad, am TvalsazrisiT, rodesac dispersiaze vsaubrobT, yovelTvis

igulisxmeba s2, garda im SemTxvevebisa, rodesac konkretulad vuTiTebT, rom

viyenebT S2-s. s2-isa da S2-is Tvisebebze, rogorc σ2-is Sefasebaze, ufro srulad me-

7 TavSi visaubrebT.

standartuli gadaxra

SerCevis standartuli gadaxra

qulaTa SerCevis standartuli gadaxra,

romelsac SerCevis standartul gadaxras

populaciis dispersiis Sefaseba (Estimated

population variance, s2): _ cvalebadobis

sazomi, romelic miiReba ganawilebaSi

Semavali ToToeuli qulisa da am

ganawilebis saSualos sxvaobebis

kvadratSi ayvaniT, miRebuli sidideebis

dajamebiTa da miRebuli jamis gayofiT

ganawilebaSi Semavali qulebis

raodenobaze erTiT nakleb ricxvze.

SerCevis standartuli gadaxra (Sample

standard deviation, S): _ kvadratuli fesvi

SerCevis dispersiidan.


uwodeben, S-iT aRiniSneba da warmoadgens kvadratul fesvs SerCevis dispersiidan

anu

( )N

XXSS ∑ −

===2

2 gadaxra istandartul SerCevis

populaciis standartuli gadaxra

Tu N populaciis individTa

raodenobas gamoxatavs da populaciis

saSualo µ cnobilia, maSin populaciis

standartuli gadaxra, romelic σ-iT

aRiniSneba, gamoisaxeba formuliT

populaciaNX∑ −

===2

2 )( µσσgadaxra istandartul populaciis

Tumca, vinaidan, Cveulebisamebr, µ ar aris xolme cnobili, σ-is mniSvneloba

SerCevidan gamomdinare fasdeba.

populaciis standartuli gadaxris Sefaseba

populaciis standartuli gadaxris Sefaseba _ s, aris kvadratuli fesvi

populaciis dispersiis Sefasebidan, anu

.1

)( 22

−

−=== ∑

NXX

ssSefaseba gadaxris istandartul populaciis

kvadratebis jamis gamosaxulebis formulaSi Setanis SemTxvevaSi gveqneba

1−=

NSSs .

populaciis standartuli gadaxris Sefasebis gamosaTvlelad:

1. TiToeul qulas vaklebT saSualos [anu ( )XX − ];

2. TiToeuli miRebuli sxvaoba agvyavs kvadratSi [anu ( )2XX − ];

3. SevkrebT kvadratSi ayvanil sxvaobebs [anu ( )∑ −2

XX ];

4. miRebuli jams vyofT N_1-ze, ganawilebaSi Semaval qulaTa

raodenobaze erTiT nakleb ricxvze.

5. miRebuli Sefardebidan viRebT kvadratul fesvs.

am SemTxvevaSic, populaciis standartuli gadaxris Sefaseba mravali

miznisTvis ufro gamosadegia, vidre SerCevis standartuli gadaxra da rodesac

viyenebT termins standartuli gadaxra, vgulisxmobT s-s.

populaciis standartuli gadaxra (σ) _ kvadratuli fesvi populaciis dispersiidan.


(Estimated populaation standard deviation) _

kvadratuli fesvi populaciis dispersiis

Sefasebidan.


magaliTi N5-1

populaciis dispersiisa da standartuli gadaxris Sefasebis gamoTvla

ganmartebiTi formuliT

amocana

ipoveT N5.1 cxrilidan meore jgufSi gaerTianebuli cxra cdis piris mier

miRebuli sicilis xangrZliobis qulebze dayrdnobiT populaciis dispersiisa da

standartuli gadaxris Sefaseba; gamoiyeneT s2-is ganmartebuli formula.

saWiro statistika: ( ) ( )∑ −−= 1/22 NXXs .

amoxsna

qvemoT mocemul cxrilSi warmodgenilia s2-isa da s-is gamoTvlisTvis

aucilebeli nabijebi. Tavdapirvelad viTvliT s2-s. formulaSi Casasmelad

gvWirdeba kvadratebis jamis, SS-is anu ( )∑ −2

XX gamoTvla. cxra qula mocemulia

(a) svetSi. am qulebis saSualoa 41 da misi mniSvneloba mocemulia (b) svetSi.

pirveli nabiji: _ TiToeul qulas gamovakloT saSualo, rogorc es naCvenebia (g)

svetSi. meore nabiji: _ TiToeul qulasa da saSualos Soris sxvaoba aviyvanoT

kvadratSi, rogorc es naCvenebia (d) svetSi. kvadratSi ayvanili sxvaobebi

Seikribeba da miRebuli jami, romelic udris 28-s, mocemulia (d) svetis bolos.

es jami aris ( )∑ −2

XX anu SS, romelic warmoadgens s2-is formulis mricxvels.

bolo nabiji: _ SS gavyoT (N_1)-ze. vinaidan sul gamoviyeneT cxra qula anu N=9,

amitom N_1=8. Sesabamisad, s2=28/8=3.50. yuradReba miaqcieT, rom vinaidan TiToeuli

qulis gadaxra saSualodan kvadratSia ayvanili, s2 yovelTvis iqneba nuli an

dadebiTi ricxvi. uaryofiTi s2 gviCvenebs, rom gamoTvlaSi Secdoma davuSviT.

populaciis standartuli gadaxris Sefaseba s

warmoadgens s2-idan kvadratul fesvs. amrigad,

87.150.3 ==s wams. yuradReba miaqcieT, rom s

izomeba igive erTeulebiT anu am SemTxvevaSi

wamebiT, romliTac nedli qulebi. mniSvnelovania,

davimaxsovroT, rom s-is gamoTvlisas s2-idan unda

amoviRoT kvadratuli fesvi, radgan s-is

gamoTvlisas gavrcelebuli daSvebuli Secdomaa s2-

idan kvadratuli fesvis aramoReba.

gamoTvlis safexurebis Sejameba

1. ipoveT (a) svetSi mocemuli qulebis saSualo da Semdeg SeitaneT

miRebuli mniSvneloba (b) svetSi, rogorc naCvenebia cxrilSi.

(a)

X

(b)

X

(g)

( )XX −

(d)

( )2XX −

38 39 40 41 41 41 42 43 44

41 41 41 41 41 41 41 41 41

-3 -2 -1 0 0 0 +1 +2 +3

9 4 1 0 0 0 1 4 9

jami 0 28


2. (b) svetSi mocemuli saSualo gamoakeliT (a) svetSi mocemul TiToeul

qulas. miRebuli sxvaoba mocemulia (g) svetSi.

3. (g) svetSi mocemuli TiToeuli sxvaoba aiyvaneT kvadratSi. kvadratSi

ayvanis Sedegebi mocemulia (d) svetSi.

4. SekribeT (d) svetSi mocemuli sxvaobebis kvadratebi.

5. (d) svetis jami gayaviT (N_1)-ze. miRebuli sidide aris s2.

6. s-is misaRebad amoiReT kvadratuli fesvi s2-dan.

populaciis dispersiisa da standartuli gadaxris Sefasebis

gamosaTvleli formula

dispersiis ganmartebiTi formula saukeTesod gamoxatavs, Tu ras warmoadgens

cnebis doneze s2. formula martivia gamosayeneblad, rodesac mxolod ramdenime

qulaa saWiro; amasTan, qulebi da maTi saSualo mTeli ricxvebia. rodesac qulebi

an maTi saSualo aTwiladebiT gamoixateba an rodesac gvWirdeba didi

raodenobiT qulebi, gamoTvla ufro mouxerxebeli xdeba. amitom xSirad s2-s

gamoTvla ufro martivia formuliT, romelic algebrulad tolfasia

ganmartebiTi formulisa. s2-is gamosaTvleli formulaa

( )[ ]1

/222

−

−= ∑∑

NNXX

s

populaciis dispersiis Sefaseba miiReba s2-idan kvadratuli fesvis amoRebiT,

anu

( )[ ]1

/222

−

−== ∑ ∑

NNXX

ss

magaliTi N5-2

populaciis dispersiis Sefasebisa da standartuli gadaxris gamoTvla

gamosaTvleli formuliT

amocana

ipoveT cxrilidan N5.1 meore jgufis cxra cdis piris mier miRebuli sicilis

xangrZliobis qulebze dayrdnobiT populaciis dispersiisa da standartuli

gadaxris Sefaseba; gamoiyeneT s2-is gamosaTvleli formula.

(a)

X

(b)

2X

38 39

1,444 1,521


saWiro statistika: ( )[ ]

1/22

2

−

−= ∑∑

NNXX

s

amoxsna

qvemoT mocemul cxrilSi warmodgenilia s2-is gamoTvlis

nabijebi. cxra qula mocemulia (a) svetSi. formulaSi

Casasmelad am qulebidan gvWirdeba ori sididis miReba; esaa yvela qulis jami

(anu ∑ X ) da TiToeuli qulis kvadratebis jami (anu ∑ 2X ). qulaTa jami ∑ X

aris 369 da mocemulia (a) svetis bolos. TiToeuli qulis kvadratebi mocemulia

(b) svetSi (magaliTad, (38)2 = 1,444). kvadratSi ayvanili qulebis jami ∑ 2X aris

15,157 da mocemulia (b) svetis bolos. sul gamoyenebulia cxra qula anu N = 9,

xolo N _1 = 8.

es mniSvnelobebi CavsvaT formulaSi. miviRebT:

( )[ ]( )

50.3828

8129,15157,15

89/161,136157,15

199/369157,15 2

2

==

−=

−=

−−

=s

s2-is am formulis gamoyenebiT igive 3.50 miviReT, rac _ N5-1 magaliTSi s2-is

ganmartebiTi formuliT; da isev, 87.150.32 === ss wami.

dispersiisa da standartuli gadaxris gamoTvla kalkulatoris

gamoyenebiT

mas Semdeg, rac gaecaniT dispersiasa da standartul gadaxras, mosalodnelia,

rom mcire moculobis monacemebisTvis kalkulatoris gamoyenebiT mogiwioT am

statistikebis gamoTvla. kalkulatorTa umravlesobas aqvs rogorc X -is, ise _

dispersiisa da standartuli gadaxris gamosaTvleli programebi, Tumca, maTi

gamoyenebis meTodebi gansxvavdeba sxvadasxva kalkulatorze. aqedan gamomdinare,

Tqveni sakuTari kalkulatoris instruqciis wakiTxva mogiwevT. bevri

kalkulatori ∑ X -is, ∑ 2X -isa da N-is gamoTvlis saSualebasac iZleva, Tumca,

maT aRsaniSnad gamoyenebuli simboloebi SeiZleba gansxvavdebodes winamdebare

teqstSi gamoyenebulisgan. populaciis standartuli gadaxris Sefaseba xSirad

(Tumca, ara yovelTvis) s-iT aRiniSneba. es simboloa gamoyenebuli am teqstSic.

rac Seexeba SerCevis standartul gadaxras, mas xSirad σ-iT an SD-iT aRniSnaven.

40 41 41 41 42 43 44

1,600 1,681 1,681 1,681 1,764 1,849 1,936

jami 369 15,157


cxadia, mniSvnelovania imis codna, Tu romeli statistikuri sidide rogor aris

aRiniSnuli Tqvens kalkulatorze.

simboloebisa da formulebis mimoxilva

winamdebare nawilSi warmogidgineT mTel rig simboloebsa da formulebs.

qvemoT gTavazobT maT mimoxilvas.

S2 SerCevis dispersia. S2 aris mxolod aRweriTi statistika SerCevisTvis.

S SerCevis standartuli gadaxra. S aris mxolod aRweriTi statistika

SerCevisTvis.

s2 populaciis dispersiis Sefaseba. es statistika gamoiyeneba populaciis

dispersiis Sesafaseblad.

s populaciis standartuli gadaxris Sefaseba. es statistika gamoiyeneba

populaciis standartuli gadaxris Sesafaseblad.

σ2 populaciis dispersia. es parametri ufro srulad aris ganxiluli me-6

TavSi.

σ populaciis standartuli gadaxra, ufro srulad aris ganxiluli me-6

TavSi.

populaciis dispersiis Sefaseba s2 da standartuli gadaxra s is statistikebia,

romlebic, Cveulebisamebr, SerCevis qulebiT gamoiTvleba. cxrilSi N5.2

warmodgenilia ganmartebiTi, gamoTvliTi kvadratebis jamis formulebi s2-isa da s

isTvis. qulaTa erTi da imave erTobliobis mimarT gamoyenebisas TiToeuli

formuliT statistikis erTi da igive ricxviTi mniSvneloba miiReba.

cxrili N5.2. populaciis dispersiisa da standartuli gadaxris Sefasebis ganmartebiTi,

gamoTvliTi da kvadratebis jamis formulebi.

formulebi

ganmarteba gamosaTvleli kvadratebis jami

dispersia, s2 ( )1

2

−

−∑N

XX

( )[ ]1

/22

−

−∑ ∑N

NXX 1−N

SS

strandartuli

gadaxra, 2ss =

( )1

2

−

−∑N

XX

( )[ ]1

/22

−

−∑ ∑N

NXX 1−N

SS

dispersiisa da standartuli gadaxris interpretacia

kvlevis angariSebSi standartuli gadaxra ufro xSirad aris xolme

warmodgenili rogorc aRweriTi statistika, vidre dispersia. amis erT-erTi

mizezi isaa, rom vinaidan standartuli gadaxra warmoadgens kvadratul fesvs

dispersiidan, is gvaZlevs qulaTa cvalebadobis sazoms maT sawyis erTeulebSi.


N5-1 da N5-2 magaliTebSi s2 aris 3.50 wami kvadratSi (wm2); am mniSvnelobis

interpretireba ar aris ioli; Tumca, s-is mniSvnelobaa 1.87 wami; aq gvaqvs igive

erTeuli (anu wami), romliTac gavzomeT nedli qulebi. amrigad, am qveTavSi s-is

interpretaciaze gveqneba saubari. dispersia gamoiyeneba statistikuri hipoTezebis

Semowmebis meTodebSi da ufro srulad ganixileba me-10-12 TavebSi.

standartuli gadaxra zomavs, saSualod ramdenad gansxvavdebian ganawilebaSi

Semavali qulebi saSualosgan. Tu yvela ganawilebaSi Semavali qula saSualos

tolia (aseTi ganawileba mocemulia cxrilSi N5-1, pirveli jgufi), maSin s nulis

toli iqneba. amrigad, saSualo sidide, romliTac pirveli jgufis qulebi

gansxvavdebian saSualosgan, aris nuli. am SemTxvevaSi, Tu ganawilebaSi nebismieri

qulis Sesaxeb winaswarmetyvelebisTvis gamoviyenebT X -s, Cvens

winaswarmetyvelebaSi Secdoma nulis toli iqneba. Tu viciT, rom s-is mniSvneloba

nulis tolia, maSin Cven isic viciT, rom ganawilebaSi Semavali yvela qula

SerCevis saSualos tolia da saSualo srulyofilad winaswarmetyvelebs yvela

qulis Sesaxeb.

meore mxriv, Tu ganawilebaSi, sul mcire, erTi qula mainc ar aris saSualos

toli, maSin s miiRebs nulze met mniSvnelobas da misi mniSvneloba miT ufro

meti iqneba, rac ufro gaizrdeba qulaTa cvalebadoba. s-is mateba, romelic Tan

sdevs ganawilebaSi Semavali qulebis cvalebadobis matebas, SegviZlia vnaxoT

cxrilSi N5-1 warmodgenili meore da mesame jgufebis monacemebis Sedarebisas.

yuradReba miaqcieT, rom mesame jgufis qulebi ufro cvalebadia saSualos (41

qula) mimarT, vidre meore jgufis qulebi. meore jgufis qulebisTvis s = 1.87 wams

maSin, roca mesame jgufis s = 13.44 wams. amrigad, meore jgufisTvis saSualo

sidide, romliTac qulebi gansxvavdebian X -isgan, aris 1.87 wami, xolo mesame

jgufisTvis _ 13.44 wami. Tu 41=X wams gamoviyenebT meore jgufSi Semavali

nebismieri qulis Sesaxeb winaswarmetyvelebisTvis, Cveni winaswarmetyvelebis

Secdoma daaxloebiT iqneba 1.87 wami, xolo mesame jgufisTvis aseTi

winaswarmetyvelebis Secdoma daaxloebiT 13.44 wams Seadgens.

s-is meore interpretacia gulisxobs qulebis aRweras imis mixedviT, Tu ramdeni

standartuli gadaxriT arian isini daSorebulni ganawilebis saSualosgan. bevr

ganawilebaSi qulaTa umravlesoba saSualosgan erTi standartuli gadaxris

farglebSia daSorebuli. magaliTad, cxrilSi N3-3 mocemuli gamocdebTan

dakavSirebuli SfoTvis kiTxvaris 100 qulis saSualoa 41.64, xolo am ganawilebis

s _ 8.66. X -is erTi standartuli gadaxris farglebSi moTavsebuli qulebia 32.98-

dan 50.30-is CaTvliT. es aris mniSvnelobebi, romlebic SemosazRvrulia, erTi

mxriv, erTi standartuli gadaxriT saSualos qvemoT (anu

98.3266.864.411 =−=− sX ) da, meore mxriv, erTi standartuli gadaxriT saSualos


zemoT (anu 30.5066.864.411 =+=+ sX ). cxrilSi N3-3 mocemuli qulebis daTvliT

vnaxavT, rom am intervalSi 100-dan 79 qula xvdeba. Tu ganawilebas normaluri

forma aqvs (rogorc naCvenebia suraTze N3-7a), maSin standartuli gadaxris codna

ufro met informacias gvaZlevs ganawilebaSi Semavali qulebis Sesaxeb. s-is es

interpretacia ufro srulad aris ganxiluli me-6 TavSi.

garda amisa, yuradReba miaqcieT, rom qulebi SeiZleba aRiweros imaze

miTiTebiTac, Tu saSualodan ramdeni standartuli gadaxriT zemoT an qvemoT

mdebareoben isini. magaliTad, Tu qulaTa ganawilebis saSualoa 80 da

standartuli gadaxraa 6, maSin qula 86 mdebareobs saSualodan erTi

standartuli gadaxriT zemoT, xolo qula 74 _ erTi standartuli gadaxriT

qvemoT. standartuli gadaxris es gamoyeneba ufro srulad aris axsnili me-6

TavSi.

rodis aris cvalebadoba didi

ar arsebobs martivi wesi imis gansasazRvrad, Tu ra SemTxvevaSi gamoxatavs

cvalebadobis nebismieri sazomis mniSvneloba qulaTa ganawilebis did Tu mcire

cvalebadobas. cvalebadobis sidide fardobiTia cvladis SesaZlo mniSvnelobebis

diapazonisa da qulaTa saSualos mimarT. magaliTad, senderi (Sender, 1958)

miuTiTebs, rom 100 metriani eleqtrogadamcemi mavTulisTvis 7.5 sm ar iqneba

uCveulo standartuli gadaxra, radgan misi saSualo sigrZe sakmaod didia;

Tumca, Tu Tqven adamianebis cxvirebis sigrZes zomavT, 7.5 sm-iani standartuli

gadaxra sagangaSo iqneba, vinaidan adamianis cxviris sigrZis SemTxvevaSi

standartuli gadaxra 7.5 sm Zalian didia.

cvalebadobis warmodgena samecniero statiebSi

populaciis standartuli gadaxris Sefaseba s cvalebadobis is sazomia,

romelic xSirad aris warmodgenili samecniero statiebSi ganxilul kvlevebSi.

JurnalebSi, sadac daculia amerikis fsiqologiuri asociaciis publikaciebis

saxelmZRvaneloSi (Publication Manual of the American Psychological Association. American

Psychological Association, 1994) mocemuli saredaqcio stili, standartuli gadaxris

aRsaniSnad iyeneben simbolos SD. amrigad, JurnalSi gamoqveynebul samecniero

statiaSi SegxvdebaT Semdegnairad formulirebuli nawyveti:

agresiuli qmrebis saSualo asaki iyo 34.3 weli (SD = 6.8).

sabeWd manqanaze akrefil an xelnawer teqstSi igive fraza SegxvdebaT aseTi

formiT:

agresiuli qmrebis saSualo asaki iyo 34.3 weli (SD = 6.8).


SeamowmeT Tqveni codna V.2

1. ganmarteT: _ populaciis standartuli gadaxris Sefaseba, populaciis

dispersiis Sefaseba, SerCevis dispersia, kvadratebis jami, s, s2, SS, σ, σ2.

2. homenma, topingma da holma (Homan, Toppung & Hall, 1986) ikvlies gansxvaveba

studentebis testur qulebs Soris, romelTagan erTi jgufi ismenda

maswavleblis mier wakiTxul tests, meore ki _ Tavad kiTxulobda.

davuSvaT, Tqvenc am problemiT xarT dainteresebuli da ikvlevT

studentTa or jgufs. erTi jgufi TviTon kiTxulobs tests, meores ki

Tqven ukiTxavT xmamaRla. miiReT testis qulaTa cxrilSi mocemuli ori

jgufi (minimaluri qulaa 0 da maqsimaluri _ 100).

a) TiToeuli jgufisTvis ipoveT X , s2 da s, ganmartebiTi formulis

gamoyenebiT, romelic mocemulia magaliTSi N5-1.

b) TiToeuli jgufisTvis ipoveT X , s2 da s, gamoTvliTi formulis


g) s2-isa da s-is miRebuli mniSvnelobebis mixedviT gansazRvreT, romel

jgufSi Sedis metad cvalebadi qulebi.

d) Tu meore jgufis (romelsac maswavlebeli ukiTxavda) tipuri

qulis winaswarmetyvelebisTvis X -s gamoiyenebT, ra iqneba am

winaswarmetyvelebis saSualo Secdoma?

3. mravali fsiqologi dainteresebulia pirovnuli sivrcis sakiTxiT anu

mosazrebiT imis Taobaze, rom TiToeul adamians uxilavi sazRvrebi aqvs,

romelic ar surs, rom daurRvion ucxo adamianebma. am fenomenis erT-erT

kvlevaSi pirovnebis piradi sivrcis darRveva xdeboda sazogadoebrivi

TavSeyris iseT adgilebSi, rogoricaa, magaliTad, kolejis biblioTeka

an kafe, imiT, rom eqsperimentatori jdeboda pirovnebis gverdiT

Tavisufal skamze. davuSvaT, mamakaci eqsperimentatori arRvevda kolejis

asakis 10 gogonasa da 10 vaJis pirovnul sivrces, rodesac isini

biblioTekaSi isxdnen. fiqsirdeboda drois xangrZlivoba (wuTebiT),

jgufi

kiTxulobdnen

TviTon

ukiTxavdaT

maswavlebeli

60 50 64 53 61 65 60

51 74 64 63 52 45 50


romlis manZilzec pirovneba Cerdeboda eqsperimentatoris gverdiT;

miRebul iqna Semdegi Sedegebi:

pirovnebis sqesi

mamrobiTi mdedrobiTi

10.6 19.8 4.3 21.6 15.7 8.1 26.4 17.9

8.6 4.7 9.1 6.3 2.4 7.0 7.8 10.5

a) TiToeuli jgufisTvis ipoveT X , s2 da s, gamoTvliTi formulis


b) s2-isa da s-is miRebuli mniSvnelobebis mixedviT gansazRvreT,

romel jgufSi Sedis metad cvalebadi qulebi.

g) ra iqneba saSualo Secdoma, Tu vityviT, rom mamrobiTi sqesis cdis

pirebi pirovnuli sivrcis darRvevis Semdeg kidev 15.55 wuTs

Cerdebian TavianT adgilebze?

d) Tu cdis piris tipuri qcevis aRsawerad gamoiyenebT jgufis

saSualos, romeli jgufisTvis (vaJebis Tu gogonebis) ufro zusti

iqneba winaswarmetyveleba? axseniT Tqveni pasuxi.

cvalebadobis sazomebis Sedareba

cvalebadobis sazomebis Sedarebis gamartivebis mizniT cxrilSi N5.3

warmodgenilia cvalebadobis sazomebi cxrilSi N3.3 mocemuli gamocdebTan

dakavSirebuli SfoTvis kiTxvaris qulebisTvis. naTelia, rom miRebuli sidideebi

mniSvnelovnad gansxvavdeba erTmaneTisgan. vinaidan TiToeuli sazomi qulebis

Sesaxeb gansxvavebuli informaciis matarebelia, sxvadasxva sazomis ricxviT

mniSvnelobebs pirdapir ver SevadarebT erTmaneTs. diapazoni, IQR da SIQR asaxaven

cdis pirTa qulebis garkveuli procentis Semcvel intervals; 100% _ diapazoni,

SuaTana 50% _ kvartilTSorisi gabnevis diapazoni da 25% medianis TiToeul

cxrili N5.3. cvalebadobis sazomebis Sedareba gamocdebTan

dakavSirebuli SfoTvis kiTxvaris 100 qulisTvis (ix. cxrili 3-3).

cvalebadobis sazomebi mniSvneloba

diapazoni

kvartilTSorisi gabnevis diapazoni (IQR)

semikvartilTSorisi gabnevis diapazoni (SIQR)

populaciis standartuli gadaxris Sefaseba (s)

populaciis dispersiis Sefaseba (s2)

51

8

4

8.66

74.92


mxares _ semikvartilTSorisi gabnevis diapazoni. dispresia da standartuli

gadaxra warmoadgens imis maCvenebels, saSualod ramdenad gadaxrilia qulebi

maTi saSualodan; Tumca, cvalebadobis yvela sazomis SemTxvevaSi, rac ufro

mcirea sazomis mniSvneloba, miT naklebad iqneba qulebi gafantuli. Tu yvela

individs erTi da igive qula aqvs da saerTod ar aris cvalebadoba, TiToeuli

sazomis mniSvneloba nulis toli iqneba.

aRweriTi statistikebis arCeva

Cven ganvixileT mTeli rigi statistikebisa, romelic saSualo qulisa da

SerCevaSi miRebuli qulebis cvalebadobis aRsawerad gamoiyeneba; Tumca,

mkvlevrebi miRebuli monacemebis Sesajameblad irCeven centraluri tendenciisa

da cvalebadobis mxolod TiTo-TiTo sazoms. konkretuli aRweriTi statistikis

arCeva, ZiriTadad, damokidebulia sam pirobaze: _ (1) gazomvis skalaze, romelzec

mocemulia qulebi, (2) qulebis sixSireTa ganawilebis formaze da (3) Semdgom

statistikur analizSi rogor gamoiyeneba mocemuli aRweriTi statistika.

cxrilSi N5.4 mocemulia rekomendebuli sazomebi konkretuli gazomvis skalisa

da ganawilebis formisaTvis.

cxrili N5.4. rekonendebuli sazomebi mocemuli gazomvis skalisa da ganawilebis

formisaTvis.


daaxloebiT simetriuli,

unimodaluri asimetriuli, unimodaluri

gazomvis skala

centraluri

tendenciis

sazomi

cvalebadobis

sazomi

centraluri

tendenciis

sazomi

cvalebadobis

sazomi

rigis

intervalebis an

Sefardebis

Mdn

X

SIQR

s

Mdn

Mdn

SIQR

SIQR

gazomvis skala

saxeldebis skala

saxeldebis skalaze pasuxebi dalagebulia kategoriebSi da kategoriis

farglebSi yvela pasuxi gaigivebuli erTmaneTTan. Sesabamisad, Tvisebrivi

monacemebi, Cveulebriv, mxolod sixSireebis (anu jgufidan ramdeni individi

Sedis erT, an meore kategoriaSi) an procentebis terminebiT aRiwereba. amitom

saxeldebis skala ar aris mocemuli cxrilSi N5.4.


rigis skala

rigis skalis monacemebisTvis mediana warmoadgens centraluri tendenciis

saukeTeso sazoms, ganurCevlad sixSireTa ganawilebis formisa, an imisa,

SemdgomSi Tu ra mizans isaxavs statistikuri analizi. rodesac centraluri

tendenciis sazomad gamoiyeneba mediana, maSin cvalebadobis sazomad, rogorc

wesi, gamoiyeneba diapazoni, kvartilTSorisi gabnevis diapazoni an

semikvartilTSorisi gabnevis diapazoni.

intervalebisa da Sefardebis skalebi

rodesac intervalebisa da Sefardebis skalebze qulebis sixSireTa ganawileba

daaxloebiT simetriulia, umjobesia, rom centraluri tendenciis sazomad

saSualo gamoviyenoT. saSualosTan erTad cvalebadobis sazomad, Cveulebisamebr,

gamoiyeneba populaciis standartuli gadaxris Sefaseba.


Cveulebriv, ganawilebis forma mxolod maSin gansazRvravs aRweriTi

statistikis arCevas, rodesac damokidebuli cvladi intervalebis an Sefardebis

skalaze izomeba. Tu qulaTa ganawileba daaxloebiT simetriulia, maSin

upiratesoba eniWeba SerCevis saSualosa da standartul gadaxras; magram Tu

qulaTa ganawileba mniSvnelovnad asimetriulia, maSin SeiZleba mediana da SIQR

airCion.

monacemebis Semdgomi analizi

centraluri tendenciisa da cvalebadobis konkretuli sazomebis arCeva imazec

aris damokidebuli, Tu ramdenad gamoiyeneba mocemuli aRweriTi statistikebi

statistikuri daskvnebis gakeTebisas. daskvnisTvis iyeneben saSualos, populaciis

standartuli gadaxris Sefasebasa da populaciis dispersiis Sefasebas, vinaidan

isini ukavSirdebian normaluri ganawilebis cnobil maxasiaTeblebs. am kavSiris

Sesaxeb me-6 TavSi visaubrebT. aqedan gamomdinare, bihevioristuli mecnierebis

warmomadgenlebma monacemebis sistematizaciisTvis maSinac ki SeiZleba gamoiyenon

SerCevis saSualo da standartuli gadaxra, rodesac gazomvis skalis done da

sixSireTa ganawilebis forma sawinaaRmdegoze metyveleben. xandaxan aris

SemTxvevebi, rodesac monacemebze dayrdnobiT daksvnebis gamotanas eniWeba

upiratesoba da ar xdeba am pirobebis (gazomvis skalisa da sixSireTa

ganawilebis formis) gaTvaliswineba. Tqven naxavT, rom bihevioristul kvlevebSi

yvelaze xSirad gamoyenebad aRweriT statistikebs saSualo da standartuli

gadaxra warmoadgens.


SeamowmeT Tqveni codna V.3

1. qulaTa TiToeuli qvemoT mocemuli ganawilebisTvis daasaxeleT

centraluri tendenciisa da cvalebadobis is sazomebi, romlebic

saukeTesod aRweren am qulebs.

a) intervalebis skalaze mocemuli qulebi, romelTa ganawileba

mkveTrad uaryofiTad asimetriulia.

b) rigis skalaze mocemuli qulebi, romelTa ganawileba daaxloebiT

simetriulia.

g) Sefardebis skalaze mocemuli qulebi, romelTa ganawileba

daaxloebiT simetriulia.

d) rigis skalaze mocemuli qulebi, romelTa ganawileba mkveTrad

dadebiTad asimetriulia.

e) intervalebis skalaze mocemuli qulebi, romelTa ganawileba

daaxloebiT simetriulia.

v) intervalebis skalaze mocemuli qulebi, romelTa ganawileba

mkveTrad dadebiTad asimetriulia.

z) Sefardebis skalaze mocemuli qulebi, romelTa ganawileba

mkveTrad uaryofiTad asimetriulia.

Sejameba

• cvalebadobis sazomebi gviCvenebs, Tu ramdenad gansxvavdebian qulebi

erTmaneTisgan da ganawilebis centraluri tendenciis sazomisgan.

• diapazoni gamoiTvleba Semdegnairad: Xmaqsimaluri _ Xminimaluri.

• kvartilTSorisi gabnevis diapazoni aris intervali ganawilebis 25-e da 75-e

procentilebs Soris.

• SerCevis dispersia S2 gamoiTvleba formuliT ( )∑ − populaciaNXX /2

.

• populaciis dispersiis Sefaseba s2 gamoiTvleba formuliT

( ) ( )∑ −− 1/2

NXX .

• kvadratuli fesvi dispersiidan gvaZlevs standartul gadaxras.

kvadratuli fesvi S2-dan aris SerCevis standartuli gadaxra, kvadratuli

fesvi s2-dan ki _ populaciis standartuli gadaxris Sefaseba.

• populaciis dispersiisa da standartuli gadaxris Sefaseba warmoadgens

cvalebadobis yvelaze xSirad gamoyenebad sazoms.

• cvalebadobis sazomis arCeva damokidebulia gazomvis skalaze, romelzec

mocemulia qulebi, qulaTa sixSireebis ganawilebis formasa da imaze, Tu

ra mizans isaxavs monacemebis Semdgomi analizi.



diapazoni

kvadratebis jami

kvartilTSorisi gabnevis diapazoni

populaciis dispersia (σ 2)

populaciis dispersiis Sefaseba (s2)

populaciis standartuli gadaxra (σ)

populaciis standartuli gadaxris

Sefaseba (s)

kvartilTSorisi gabnevis diapazonis

naxevari

SerCevis dispersia


cvalebadoba

interkvartiluri rangi (IQR)

SIQR

S

S2

SS

s

s2

X25

X75

σ2

kiTxvebi

1. damokidebulia Tu ara studentis mier profesoris kompetenturobis

aRqma profesoris sqesze? davuSvaT, kolejis mamrobiTi sqesis 20

moswavle afasebs mamrobiTi sqesis profesoris kompetenturobas,

romelic maT leqcias utarebs. kompetenturobis Sesafaseblad iyenebT

maswavleblis Sefasebis formas, romelic warmoadgens jamuri Sefasebis

skalas (McCarthy & Schmeck, 1982). am skalaze qulebis diapazonia 12-dan

(Zalze arakompetenturidan) 84-is (Zalze kompetenturis) CaTvliT. Tqvens

mier miRebuli 20 qulaa: _ 50, 65, 44, 47, 52, 55, 35, 43, 43, 50, 60, 40, 45, 50, 45,

57, 47, 53, 54 da 57.

a) ipoveT am ganawilebis Mdn, X , IQR da s.

b) wminda aRweriTi TvalsazrisiT, aris Tu ara Mdn an X am qulebis

centraluri tendenciis saukeTeso sazomi? axseniT Tqveni pasuxi.

2. davuSvaT, axla kolejis mamrobiTi sqesis 20 moswavle afasebs

mdedrobiTi sqesis profesoris kompetenturobas, romelic igive leqcias

atarebs, rasac wina kiTxvaSi aRwerili mamrobiTi sqesis profesori. am

SemTxvevaSi miiReT kompetenturobis Semdegi qulebi: 50, 80, 44, 47, 52, 55,

17, 43, 43, 50, 60, 40, 45, 50, 45, 57, 47, 53, 54 da 57.

a) ipoveT am ganawilebis Mdn, X , IQR da s.

b) sufTad aRweriTi TvalsazrisiT, aris Tu ara Mdn an X am

qulebis centraluri tendenciis saukeTeso sazomi? axseniT Tqveni

pasuxi.


g) qulaTa am erTobliobisTvis s-is mniSvneloba, 1 kiTxvaSi miRebuli

s-is mniSvnelobasTan SedarebiT, mniSvnelovnad gaizarda, Tumca,

ar Secvlila IQR . axseniT miRebuli Sedegi.

3. cvalebadobis romeli sazomia aRwerili qvemoT mocemul TiToeul

debulebaSi?

a) is aRwers intervals, romelic moicavs qulaTa 100%-s.

b) misi mniSvneloba gamoiTvleba 75-e procentilis qulidan 25-e

procentilis qulis gamoklebiT.

g) is aRwers intervals, romelic moicavs qulaTa daaxloebiT 25%-s

medianis zemoT da qvemoT.

d) is aRwers ganawilebis saSualodan qulis saSualo kvadratul

gadaxras da warmoadgens populaciis Sefasebas.

e) is aRwers ganawilebis saSualodan qulis saSualo kvadratuli

gadaxridan kvadratul fesvs da warmoadgens populaciis

Sefasebas.

v) is aRwera intervals, romelic moicavs ganawilebis qulaTa Sua

50%-s.

z) misi mniSvneloba damokidebulia ganawilebaSi Semaval mxolod or

qulaze.

T) misi mniSvneloba gamoiTvleba formuliT: ( )∑ − NXX /2

.

i) misi mniSvneloba gamoiTvleba formuliT: ( ) ( )∑ −− 1/2

NXX .

k) misi mniSvneloba gamoiTvleba formuliT: ( )∑ − NXX /2

.

l) misi mniSvneloba gamoiTvleba formuliT: ( ) ( )∑ −− 1/2

NXX .

m) misi mniSvneloba gamoiTvleba formuliT:. ( ) 2/2575 XX − .

n) misi mniSvneloba gamoiTvleba formuliT: ∑ − populaciaNX /)( 2µ

o) misi mniSvneloba gamoiTvleba formuliT: ∑ − populaciaNX /)( 2µ

4. me-4 Tavis bolos mocemul me-6 SekiTxvaSi warmodgenili iyo grosis

(Gross, 1993) mier Catarebuli gamokiTxvis Sedegebis nawili, romelic

Seexeboda kolejis studentebSi alkoholis miRebis sakiTxs; kerZod,

mocemuli iyo, rom 21 welze ufros mamakacebTan dalevis saSualo

raodenoba kviraSi iyo 23.9, qalebTan ki _ 7.5 daleva kviraSi. am

saSualoebis Tanmxlebi standartuli gadaxrebi iyo 15.4 daleva kviraSi

mamakacebisTvis da 8.7 daleva kviraSi qalebisTvis.


a) Tu Tqven iwinaswarmetyvelebT, rom kolejis mamrobiTi sqesis, 21

welze ufrosi tipuri studenti kviraSi 23.9-jer iRebs alkohols,

ra iqneba am winaswarmetyvelebis saSualo Secdoma?

b) SegiZliaT, TqvaT, rom es standartuli gadaxra migviTiTebs

gakeTebuli Sefasebis did an mcire cvalebadobaze?

g) ra informacias gaZlevT es standartuli gadaxra saSualos,

rogorc centraluri tendenciis sazomis, adekvaturobis Sesaxeb

am gamokiTxvaSi miRebuli qulebis ganawilebis SemTxvevaSi?

5. me-4 Tavis bolos mocemul me-7 SekiTxvaSi warmodgenili iyo cxridan

rva qirurgiuli pacientis martivi reaqciis dro zogadi anesTeziis

miRebidan ori saaTis Semdeg. mecxre pacienti ar iyo sakmarisad

gamosuli anesTeziidan, rom gaezomaT reaqciis dro, amitom am cdis

piris monacemi ganisazRvra, rogorc reaqciis usasrulo dro. Tu Tqven

gsurT cxrave qulis aRwera, cvalebadobis romel sazoms gamoiyenebdiT

maTTvis? axseniT sazomis arCevis mizezi.

Tavi VI. normaluri ganawileba, albaToba da

standartuli qulebi

normaluri ganawileba

normaluri ganawileba da bihevioristuli mecnierebebi

normaluri ganawilebis Tvisebebi

normaluri ganawilebis qveS moTavsebuli farTobi

standartuli normaluri ganawileba

standartuli normaluri ganawilebis cxrilis gamoyeneba

z qulebis gamoyenebis magaliTi

albaToba

diskretuli albaTobis ganawilebis Tvisebebi

Teoriuli da empiriuli albaToba

standartuli normaluri ganawileba, rogorc albaTuri

ganawileba

standartuli qulebi

თავი VI. ნორმალური განაწილება, ალბათობა და სტანდარტული ქულები 2

me-3-5 TavebSi SerCevidan miRebuli qulebis mowesrigebisa da aRweris sakiTxebze

visaubreT. axla ganvixiloT populaciebi da maTi maxasiaTeblebi; kerZod,

populaciis erT-erTi ganawileba _ normaluri ganawileba.Y pirvel TavSi

populacia ganvmarteT, rogorc saerTo maxasiaTeblis mqone adamianebis,

cxovelebis, sagnebisa Tu movlenebis sruli erToblioba. SerCeva aris

populaciidan arCeuli qvesimravle anu qvejgufi. me-3, me-4 da me-5 TavebSi

ganvixileT SerCevidan miRebuli qulebis mowesrigeba da aRwera sixSireTa

ganawilebisa da iseTi aRweriTi statistikebis gamoyenebiT rogoricaa mediana,

saSualo, kvartilTSorisi gabnevis diapazonis naxevari da standartuli gadaxra;

Tumca, rogorc wina TavebSi iTqva, populaciis qulebis aRwerac SesaZlebelia

saSualoTi da standartuli gadaxriT. vinaidan axla vambobT, rom vzomavT mTel

populacias, saSualos aRvniSnavT simboloTi μ (miu, berZnuli anbanis meoce aso),

xolo standartul gadaxras _ simboloTi σ (sigma, berZnuli anbanis meTvramete

aso). garda amisa, miRebul sidideebs axla ukve vuwodebT ara statistikebs,

aramed parametrebs.

parametri aris ricxvi, magaliTad, μ an σ, romelic aRwers populacias.

terminologiis aseTi Secvlis mizani isaa, rom gasagebi iyos, raze vsaubrobT: _

SerCevaze Tu populaciaze. mniSvnelovania am gansxvavebis gageba, radgan

normaluri ganawileba, romelzec qvemoT visaubrebT, xasiaTdeba μ-Ti da σ-Ti da

ara _ X da s-iT. aRniSnuli terminologia mocemulia cxrilSi N6-1.


bihevioristul mecnierebebSi populaciis erT-erT mniSvnelovan ganawilebas

warmoadgens normaluri ganawileba. normaluri ganawileba aris Teoriuli

maTematikuri ganawileba, romelic aRwers populaciis qulebis fardobiT

sixSireebs. normalurad ganawilebul qulaTa erTobliobis sixSireTa ganawileba

SeiZleba srulad aRiweros, Tu cnobilia ganawilebis saSualo (μ) da

standartuli gadaxra (σ). normaluri ganawileba, romlis μ = 70 da σ = 10,

cxrili N6.1. SerCevisa da populaciis Sesatyvisi aRniSvnebi.

SerCeva populacia

aRweriTi

maxasiaTeblebi statistikebi parametrebi

saSualo

standartuli

gadaxra

dispersia

X

s

s2

μ

σ

σ2


mocemulia suraTze N6-1. mocemuli ganawileba warmoadgens maTematikur

ganawilebas, romelic ar asaxavs romelime realurad gazomili populaciis

qulebs. realuri populaciis qulebi arasodes aris ise zustad ganawilebuli,

rom ganawileba zustad normaluri iyos; magram zogjer gamosadegia imis daSveba,

rom zogierTi populaciis qulebi daaxloebiT normalurad aris ganawilebuli;

Tumca, winamdebare TavSi normalur ganawilebas ganvixilavT, rogorc Teoriul

maTematikur ganawilebas.

normaluri ganawileba da bihevioristuli mecnierebebi

normaluri ganawileba iakob bernulis (Jakob Bernoulli, 1654-1705), abraam de

moiavris (Abraham de Moivre, 1667-1754), pier remond de monmortis (Pierre Remond de

Montmort, 1678-1719), karl fridrix gausisa (Karl Friedrich Gauss, 1777-1855) da sxvebis

SromebSi ganviTarda. maTi interesis sagani iyo sxvadasxva azartul TamaSebSi

mosalodneli Sedegebis albaTobebis gamoTvla da astronomiul da fizikur

cdebSi gazomvebis Secdomebis ganawilebebis ageba. normaluri ganawileba maleve

gadaiqca statistikosebisTvis mniSvnelovan ganawilebad, radgan statistikaSi

arsebuli mravali problemis gadaWra mxolod normaluri ganawilebis daSvebis

Semdeg gaxda SesaZlebeli.

normaluri ganawileba bihevioristuli mecnierebebis warmomadgenlebisTvisac

mniSvnelovania, radgan gamarTlebulia daSveba imisa, rom mravali damoukidebeli

normaluri ganawileba (Normal distribution): _ Teoriuli maTematikuri ganawileba, romelic

aRwers populaciis qulebis fardobiT sixSireebs.

40 -3σ

70 μ

50 -2σ

60 -1σ

100 +3σ

90 +2

80 +1σ

.10

.20

.30

.40

qulebi

fardobiTi sixSire

suraTi N6.1. normaluri ganawileba, romlis μ =70 da σ =10.


cvladiT ganpirobebuli nebismieri qcevis gazomvis Sedegebi popoulaciaSi

daaxloebiT normalurad iqneba ganawilebuli. adamianis mravali unari, iqneba es

maTematikuri Tu verbaluri unarebi, ganpirobebulia damoukidebeli faqtorebis

mTeli rigiT: _ genetikuri TandayolilobiT, mSoblebis mxridan miRebuli

waxalisebiT, ganaTlebiT, konkretuli maswavlebliT, socialuri mdgomareobiT,

kulturuli Zalebis zemoqmedebiT da a.S. Sesabamisad, aseTi unarebis

mocemulobis xarisxi mosalodnelia, rom populaciaSi daaxloebiT normalurad

iyos ganawilebuli. Cven imitom viyenebT sityvas „daaxloebiT“, rom realurad

miRebuli qulebi arasodes araa zustad normalurad ganawilebuli.

normaluri ganawilebis Tvisebebi

normaluri ganawilebis Tvisebebia: _ simetriuloba, asimptoturoba da

uwyvetoba.

simetriuloba

Teoriuli normaluri ganawileba simetriulia modis, medianisa da saSualos

mimarT; Sesabamisad, normalur ganawilebaSi moda, mediana da saSualo erTmaneTis

tolia.

asimptoturoba

yuradReba miaqcieT, rom suraTze N6.1 gamosaxuli normaluri ganawilebis

boloebi (kudebi) rac ufro daSorebulia μ-sgan, miT metad uaxlovdeba X RerZs

anu abscisas, Tumca, ganawileba asimptoturia, anu ramdenadac ar unda daSordes

μ-s, kudebi mainc ar Seexeba X RerZs.

uwyvetoba

normaluri ganawileba uwyvetia plius da minus usasrulobebs Soris

moTavsebuli yvela qulisTvis. es niSnavs, rom nebismieri ori qulisTvis

moiZebneba mesame qula, romelic maT Soris mdebareobs.

normaluri ganawilebis qveS moTavsebuli farTobi

normaluri ganawileba fardobiT sixSireTa Teoriuli ganawilebaa. gavixsenoT

me-3 Tavidan, rom qulis fardobiTi sixSire aris ganawilebaSi qulis Sexvedris

sixSire gayofili ganawilebaSi Semaval qulaTa saerTo raodenobaze. amrigad,

fardobiTi sixSire gamoisaxeba SefardebiT da ganawilebis saerTo dagrovili

fardobiTi sixSire aris 1.0 (ix. cxrili N3.4, (i) sveti). vinaidan normaluri

ganawileba fardobiTi sixSireebis ganawilebaa, ganawilebis qveS moTavsebuli

sruli farTobi 1.0-s udris; amitom normalur ganawilebaSi qulaTa fardobiTi

sixSireebi dagrovilia 1.0-mde.


yvela normaluri ganawilebaSi qulaTa garkveuli nawili moTavsebulia

saSualos garSemo garkveul intervalSi. nebismier normalurad ganawilebul

populaciaSi qulaTa .3413 wili moTavsebulia intervalSi μ-sa da μ-s plus erT

standartul gadaxras Soris, anu qulaTa 34.13% moTavsebulia intervalSi (μ, μ+1σ).

vinaidan ganawileba simetriulia, qulaTa igive procenti moTavsebulia

intervalSi (μ_1σ, μ). saSualodan erT da or standartul gadaxras Soris anu

intervalSi (μ_2σ, μ_1σ) moTavsebulia qulaTa 13.59%. intervalSi (μ_2σ, μ+2σ)

moTavsebulia qulaTa (13.59+34.13+34.13+13.59)%=95.44%. normaluri ganawilebis

intervalebSi Semaval qulaTa fardobiTi sixSirebi naCvenebia suraTze N6-2. es

damokidebuleba ZalaSia nebismieri normaluri ganawilebis SemTxvevaSi, garda im

SemTxvevebisa, rodesac μ da σ zogierT konkretul mniSvnelobas iRebs.

garda amisa, normaluri ganawilebis qveS moTavsebuli farTobebi SeiZleba

Seikribos. ase magaliTad, saSualos minus erTi standartuli gadaxra da

saSualos plius erTi standartuli gadaxra anu intervali (μ_1σ, μ+1σ) moicavs

qulaTa (.3413+.3413)-s anu .6826 wils. analogiurad, saSualos minus ori

standartuli gadaxra da saSualos plius ori standartuli gadaxra anu (μ_2σ,

μ+2σ) intervalSi Sedis qulaTa .1359+.3413+.3413+.1359 anu .9544 wili. qulaTa ufro

didi intervali, romelic Seadgens sam standartul gadaxras saSualos orive

mxares anu intervali (μ_3σ, μ+3σ), moicavs qulaTa (.0215+.1359+.3413+.3413+.1359+.0215)-s

anu .9974 wils. es damokidebulebac naCvenebia surTze N6.2. es fardoba ucvleli

rCeba μ-sa da σ-s mniSvnelobebisgan damoukideblad.


aRniSnuli mimarTebebis saCveneblad, davuSvaT, rom qulaTa populacia

normalurad aris ganawilebuli. ganawilebis μ=100 da σ =15. ganawilebaSi qula 85

aris erTi standartuli gadaxra saSualos marcxniv (anu μ_1σ = 100-15=85), 115 qula

ki _ erTi standartuli gadaxra saSualos marjvniv (anu μ+1σ = 100+15=115).

Sesabamisad, ganawilebaSi qulaTa .6826 wili Sedis intervalSi (85, 115).

analogiurad, qulaTa .9544 wili moTavsebuli iqneba 70-sa da 130 qulas Soris,

intervalSi (μ_2σ, μ+2σ), xolo qulaTa .9974 wili _ 55-sa da 145 qulas Soris,

intervalSi (μ_3σ, μ+3σ).

kidev erTi magaliTi. davuSvaT, gvaqvs qulaTa normalurad ganawilebuli

populacia, romlis μ=50 da σ =5. am ganawilebaSi qulaTa .6826 wili moTavsebulia

intervalSi 45-sa da 55-s Soris anu intervalSi (μ_1σ, μ+1σ), qulaTa .9544 wili _

intervalSi 40-sa da 60-s Soris anu intervali (μ_2σ, μ+2σ) da qulaTa .9974 wili _

intervalSi 35-sa da 65-s Soris anu intervali (μ_3σ, μ+3σ).

amrigad, normalur ganawilebaSi μ-sa da σ-s Semdegi damokidebulebaa: _

saSualos orive mxares mdebare konkretuli intervalebi ganawilebis qulaTa

konkretul wils da konkretul procents moicaven.

intervali

intervalSi

Semaval qulaTa

wili

intervalSi

Semaval qulaTa

procenti

-3σ -2σ -1σ μ +1σ +2σ +3σ

suraTi N6.2. normaluri ganawilebis konkretul

intervalebSi Semaval qulaTa fardobiTi sixSireebi.

.0013 .0013 .0215

.1359

.1334 .1334

.1359 .0215

.6826

.9544

.9974


(μ_1σ)-dan (μ+1σ)-mde (μ_2σ)-dan (μ+2σ)-mde (μ_3σ)-dan (μ+3σ)-mde

.6826

.9544

.9974

68.26 95.44 99.74

Semdeg qveTavSi viswavliT cxrilis gamoyenebas imis gansasazRvrad, qulaTa ra

nawili xvdeba normaluri ganawilebis saSualos irgvliv mdebare konkretul

intervalebSi. ase rom, ar aris saWiro aq warmodgenili mniSvnelobebis

damaxsovreba; Tumca, sasurveli da gamosadegi iqneba miaxloebiTi mniSvnelobebis

damaxsovreba, radgan amocanebis amoxsnisas SeZloT Tqvens mierve warmoebuli

gamoTvlebis sisworis Semowmeba. amrigad, amocanebis amoxsnisas gamogadgebaT, Tu

daimaxsovrebT, rom:

• intervali (μ_1σ)-dan (μ+1σ)-mde daaxloebiT moicavs qulaTa 68%-s.



SeamowmeT Tqveni codna VI.1

1. ganmarteT: _ asimptoturi, uwyveti, μ, normaluri ganawileba, parametri,

populacia, s, SerCeva, SerCeva, σ, statistika, X .

2. daasaxeleT normaluri ganawilebis sami Tviseba.

3. ratom ver iqneba adamianTa realuri populaciisgan miRebuli qulebi

zustad normalurad ganawilebuli? miniSneba: gaixsneT normaluri

ganawilebis Tvisebebi.

4. davuSvaT, gaqvT qulaTa normalurad ganawilebuli populacia, romlis μ

= 200 da σ = 20. miuTiTeT, qulaTa ra wili Seva qvemoT mocemul TiToeul

intervalSi (gamoiyeneT suraTi N6.2).

intervali

a) 220-220;

b) 200-240;

g) 200-260;

e) 180-200;

f) 160-200;

v) 140-200;

z) 180-260;

T) 160-240;

i) 140-260;

k) 220-240;

l) 220-260;

m) 140-160.


normaluri ganawilebis saSualosa da standartuli gadaxris codniT viRebT

zust informacias konkretul intervalebSi qulaTa ganawilebis Sesaxeb. vinaidan

normaluri ganawileba maTematikuri gantolebiT ganisazRvreba, Cven SegviZlia

zusti fardobiTi sixSiris gansazRvra ganawilebis qulaTa nebismieri


intervalisTvis; Tumca, aq Semdegi problemis winaSe vdgebiT: _ Cven unda

gamoviyenoT rTuli maTematikuri gamosaxuleba, sadac Seva qulebis CvenTvis

saintereso ganawilebis μ-sa da σ-is mniSvnelobebi. sabednierod, arsebobs am

problemisTvis gverdis avlis martivi gza: _ normaluri ganawileba, romlis

saSualoa μ da standartuli gadaxra _ σ (magaliTad, μ = 20, σ = 2) unda gardavqmnaT

standartuli formis normalur ganawilebad, romlis μ = 0, σ = 1. amis Semdeg, Tu

mxolod erT standartul normalur ganawilebas gamoviyenebT (romlis μ = 0, σ = 1),

Cven SevZlebT gantolebis amoxsnas qulis mocemuli mniSvnelobisTvis da misi

Sesabamisi fardobiTi sixSiris tabulirebas cxrilSi. SemdgomSi, Cven normaluri

ganawilebis gantolebis gamoyenebis nacvlad SegviZlia, ubralod cxrilSi

movZebnoT mocemuli qulis fardobiTi sixSire, an dagrovili fardobiTi sixSire.

standartuli normaluri ganawilebis

saSualo nulis (μ = 0), standatruli gadaxra

ki _ 1-is (σ = 1) tolia. nebismieri μ-sa da σ-is

mqone normalurad ganawilebuli cvladisTvis konkretuli qula (magaliTad, X)

SeiZleba gardaiqmnas standartuli normaluri ganawilebis qulad, Tu mas

SevitanT gamosaxulebaSi

( ) σµ /−= Xz ,

sadac z _ qulidan z-is realurad miRebuli1 mniSvneloba;

X _ sawyisi anu originaluri qula;

μ _ saSualo;

σ _ standartuli gadaxra im normaluri ganawilebisa, saidanac aRebulia X

qula.

mocemul formulaSi ricxviTi

mniSvnelobebis Setanis Semdeg X-is

mniSvnelobisgan miRebuli z-is

mniSvnelobas z realurs vuwodebT da zrealuri-iT aRvniSnavT. am terminologias

imis naTelsayofad viyenebT, rom z-is mniSvneloba formulaSi (zrealuri) ricxviTi

mniSvnelobebis SetaniTaa gamoTvlili Tu a.1. cxrilSia napovni. aRniSnuli

cxrilis Sesaxeb qvemoT visaubrebT. z-s xSirad standartul normalur gadaxras

uwodeben, radgan misi mniSvneloba standartul normalur ganawilebaSi qulis

mniSvnelobas gamoxatavs. standartuli normaluri gadaxra warmodgenilia

suraTze N6-3. suraTze naCvenebia qulaTa ganawileba mocemul z qulebs Soris.

yuradReba miaqcieT, rom vinaidan z = +1 miuTiTebs saSualodan erTi standartuli

1 SemdgomSi mTel teqstSi „realurad miRebulis“ nacvlad SemoklebiT gamoiyeneba

„realuri“ (mTargmnelis SeniSvna).


(Standard normal distribution): _ normaluri

ganawileba, romlis μ = 0, σ = 1.

standartuli normaluri gadaxra (Standard

normal deviate): _ normaluri ganawileba,

romlis μ = 0, σ = 1.


gadaxriT zemoT mdebare qulaze, z = 0-dan z = +1-mde SualedSi xvdeba qulaTa 3413

nawili, anu igive raodenoba, rac μ-sa da (μ+1)-s Soris suraTze 6.2.

აკლია სურათი 6-3 suraTi N6.3. standartuli normaluri ganawileba, romlis μ = 0 da σ = 1.0. naCvenebia z-is

or dadgenil mniSvnelobas Soris moxvedril qulaTa wili.

standartuli normaluri ganawilebis gamoyenebis sailustraciod ganvixiloT

magaliTi. davuSvaT, 115 qula (anu X = 115) aviReT qulaTa normalurad

ganawilebuli erTobliobidan, romlis μ = 100, σ = 15. es qula gadaviyvanoT

zrealuri-Si formuliT

( )

0.11515

15100115

+==

−=

−=

σµXzrealuri

sawyisi 115 qulis gardaqmniT miRebuli standartuli normaluri qulaa +1.0.

zrealuri = +1.0 niSnavs, rom sawyisi qula (anu 115) aris im normaluri ganawilebis

erTi standartuli gadaxriT saSualos (anu 100) zemoT, saidanac is aviReT.

qulis gadayvana standartuli ganawilebis z qulaSi gviCvenebs, Tu ramdeni

standartuli gadaxriT aris sawyisi qula (anu X) daSorebuli saSualosgan.

magaliTad, aRebuli 115 qula +1.0σ erTeuliT aris daSorebuli ganawilebis

saSualodan. ganvixiloT cxrilSi N6.2 mocemuli qulebi da maTi Sesabamisi

zrealuri-is mniSvnelobebi. es qulebi miRebulia normaluri ganawilebidan, romlis

μ = 75 da σ =10. SeecadeT, gamoTvaloT zogierTi mocemuli qula. yuradReba

miaqcieT, rom zrealuri-is mniSvneloba

nebismieri qulisTvis ubralod

warmoadgens imas, Tu standartuli

gadaxris ramdeni erTeuliT aris

daSorebuli qula standartuli

normaluri ganawilebis saSualosgan

(romelic nulis tolia) zemoT an

qvemoT.

105 100 95 90 85 80 75 70 65 60 55 50 45

+3.0 +2.5 +2.0 +1.5 +1.0 +0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0

cxrili N6.2. normaluri ganawilebis (μ =75

da σ =10) qulebis gadayvana standartuli

ganawilebis z qulebSi.


aqve unda gagafrTxiloT, rom X

qulis gadayvana z qulaSi mxolod im

SemTxvevaSi gviCvenebs sawyisi qulis zust mdebareobas standartul normalur

ganawilebaSi, Tu es sawyisi qula aRebulia qulaTa normalurad ganawilebuli

simravlidan. Tu qula aRebulia ara normaluri ganawilebidan, aramed iseTi

ganawilebidan, rogoricaa, magaliTad, marTkuTxa an bimodaluri ganawileba,

qulis gadayvana z qulaSi ver aqcevs sawyis qulebs normalurad ganawilebuls.

standartuli ganawilebis cxrilis gamoyeneba

qulebis standartuli normaluri ganawilebis z qulebSi gadayvanis

upiratesoba isaa, rom z-is mniSvnelobebs Soris moxvedrili qulebis Tanafardoba

cnobilia da mocemulia cxrilSi. es Tanafardobebi mocemulia danarTis

cxrilSi a-1. am cxrilis nawili warmodgenilia N6.3 cxrilis saxiT. (a) svetSi

mocemulia z-is mniSvnelobebi, romlebic miRebulia ricxviTi mniSvnelobebis

SetaniT formulaSi

( )σµ−

=Xzrealuri

yuradReba miaqcieT, rom Cven viyenebT aRniSvnas zrealuri, rodesac vsaubrobT z-

is nedli qulidan (magaliTad, X-dan) miRebul mniSvnelobaze, xolo aRniSvna z,

indeqsis „realuri“-is gareSe, Seesabameba a-1 cxrilis (a) svetSi mocemul z-is

mniSvnelobas. (b) svetSi mocemulia farTobi z = 0-sa da z-is (a) svetSi mocemul

mniSvnelobas Soris. svetis Tavze warmodgenil grafikze es farTobi gamuqebulia.

(g) svetSi mocemulia farTobi z-is (a) svetSi mocemuli mniSvnelobidan

usasrulobamde. aqac svetis Tavze mocemul grafikze gamuqebulia es farTobi.

vinaidan normaluri ganawileba simetriulia, z-is uaryofiTi mniSvnelobebiT

SemosazRvruli farTobebi z-is dadebiTi mniSvnelobebis identuria. amrigad,

cxrilSi ar aris warmodgenili z-is uaryofiTi mniSvnelobebi.

cxrilis gamoyenebis sailustraciod, ganvixiloT qulaTa normalurad

ganawilebuli simravle, romlis μ =100 da σ =15. davuSvaT, am ganawilebidan aviReT

qula 115. ganawilebis qulaTa ra wili iqneba 115-is toli an masze naklebi? am

amocanis amosaxsnelad, Cven 115 qula gadagvyavs z qulaSi, anu

( ) 0.115

100115+=

−=

−=

σµXzrealuri

qula (X) z


(a) (b) (g)

0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10

.3289 .3315 .3340 .3365 .3389 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621 .3643

.1711 .1685 .1660 .1635 .1611 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379 .1357

nawyveti amoRebulia II.1 cxrilidan, normaluri ganawileba da masTan dakavSirebuli ganawilebebi. CRC Handbook of Tables for Probability and Statistics (meore gamocema). COpyright 1968, CRCPress, Inc., Boca Raton, Florida. gamoiyeneba nebarTviT.

yuradReba miaqcieT, rom vinaidan z-is mniSvneloba nedli 115 qulidan miiReba,

mas zrealuri-s vuwodebT, z-is cxrilSi N6-3 mocemuli mniSvnelobisgan

gansasxvaveblad. saZiebeli qulis TvalsaCinod gamosaxvisTvis sasargebloa

standartuli normaluri ganawilebis daxazva da masze nedli qulebisa da am

qulebis Sesabamisi z qulebis miTiTeba. ganawilebis nedli qulebi da maTi

Sesabamisi z qulebi naCvenebia suraTze N6-4. amocanis amosaxsnelad saWiroa, rom

miviRoT ganawilebis farTobi z = _ ∞-dan z = +1.0-mde. ganawilebis qulaTa erTi

naxevari (anu 0.5000) z = 0-s qvemoT aris moTavsebuli. cxrilis N6.3 (b) sveti

gviCvenebs, rom ganawilebis qulaTa .3413 moTavsebulia z = 0-sa da z = +1.0-s Soris

(ix. suraTi N6.4, (g) RerZi). amrigad, amocanis pasuxia qulaTa z = _ ∞-dan z = 0-mde

moTavsebul wils (.5000) plius qulaTa z = 0-dan da z = +1.0-mde moTavsebuli wili

cxrili N6.3. farTobebi normaluri ganawilebis qveS z = 0.95-dan z = 1.10-mde.

(a) (b) (g)

0

0

0

0

+z +z

-z -z -z

z-is mniSvneloba


(.3413 a-1 cxrilis (b) svetidan), anu .8413. 115-is toli an masze nakleb qulaTa

wilia .8413 anu 84.13%.

აკლია სურათი 6-4 suraTi N6.4. normalur ganawilebaSi, romlis μ = 100 da σ =15, Semavali qulis (115)

adgilmdebareoba standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs

(a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _∞-dan z =

0-mde da z = 0-dan z = +1-mde.

a-1 cxrilis wakiTxvisas Zalze advilia Secdomis daSveba, amitom yovelTvis

sasargebloa, rom amocanis amoxsniT miRebuli paxusi SeadaroT a-1 cxrilSi

mocemul mniSvnelobebs. Secdomis Tavidan asacileblad, SegiZliaT daaxloebiT

gansazRvroT pasuxi imis mixedviT, rom normalurad ganawilebul qulaTa erT

mesamedze odnav meti (anu 33%-ze odnav meti) moTavsebulia ganawilebis saSualosa

da erT standartul gadaxras Soris. amiT gansazRvravT, rom Tqveni pasuxi

(50+33)%-ze anu 83%-ze odnav meti unda iyos. miRebuli pasuxi 84.13% Seesabameba

aseT Sefasebas.

davuSvaT, axla gvsurs, rom vicodeT, qulaTa ra nawilia 115-is toli an masze

meti. amis gamosaTvleli ori gza arsebobs:

(1) vinaidan Cven ukve viciT, rom qulaTa 84.13% tolia an naklebia 115-ze,

normaluri ganawilebis qveS darCenili farTobi unda migviTiTebdes im qulaTa

raodenobaze, romelic 115-is toli an masze meti iqneba. amitom saZiebel pasuxs

miviRebT, Tu ganawilebis mTlian farTobs 1.000-s gamovaklebT .8413-s, anu qulaTa

1.000_.8413=.1587 nawili tolia an metia 115-ze.

(2) Tu winaswar ar gvaqvs gamoTvlili, rom qulaTa .8413 nawili tolia an

naklebia 115-ze, maSin amocanis amosaxsnelad unda ganvsazRvroT, rom saZiebeli

nawili aris ganawilebis farTobi z = +1.0-is Semdeg. cxrilis N6-3 (a) svetSi

vpoulobT z = 1.00-is mniSvnelobas, Semdeg vnaxulobT mis gaswvriv (g) svetSi

mocemul mniSvnelobas, sadac warmoadgenilia z-is zemoT qulaTa wili. z = +1.0-

isTvis es mniSvnelobaa .1587. amrigad, qulaTa .1587 nawili anu 15.87% tolia an

metia 115-ze.

amave ganawilebis magaliTze ganvixiloT kidev erTi amocana. qulaTa ra

nawilia 82 qulis toli an masze naklebi? am SemTxvevaSic 82 gadagvyavs z qulaSi:

zrealuri = (82_100)/15 = _1.2. es zrealuri gviCvenebs, rom 82 qula mdebareobs

ganawilebis saSualodan 1.2 standartuli gadaxriT qvemoT. ganawilebis nedli

qulebi da zrealuri = _1.2 naCvenebia suraTze N6-5. Semdeg gamovTvliT z = _1.2-is

qvemoT mdebare qulaTa wils. yuradReba miaqcieT, rom naxazis mixedviT mxolod

imis gansazRvra SegviZlia, rom es nawili Zalze mcire iqneba. vinaidan

standartuli normaluri ganawileba simetriulia, imdeni qula iqneba


moTavsebuli z = _1.2-is qvemoT, ramdenic z = +1.2-is zemoT; amitom viTvliT z = +1.2-

is zemoT; Tumca, es mniSvneloba ar aris mocemuli cxrilSi N6-3. amitom viyenebT

normaluri ganawilebis qveS mdebare farTobis wilebis srul cxrils (ix.

cxrili a-1). (a) svetSi vpoulobT z = 1.2-s da vnaxulobT mis gaswvriv (g) cxrilSi

mocemul mniSvnelobas. amrigad, qulaTa .1151 nawili anu 11.51% tolia an naklebia

82-ze. standartuli normaluri ganawilebis gamoyenebis kidev erTi magaliTi

warmodgenilia qveTavSi „magaliTi N6.1“.

აკლია სურათი 6-5 suraTi N6.5. normalur ganawilebaSi, romlis μ = 100 da σ =15, Semavali qulis (82)


(a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _∞-dan z =

_1.2-mde.

magaliTi N6.1

standartuli normaluri ganawilebis gamoyeneba

vTqvaT, gvaqvs normalurad ganawilebuli populacia, romlis μ = 80, da σ =5.

amocana

qulaTa ra nawilia 88 qulis toli an masze meti?

saWiro statistika: ( ) σµ /−= Xzrealuri

amoxsna

88 qula unda gadaviyvanoT zrealuri-Si. Semdeg cxrilis Na-1 gamoyenebiT

vipoviT zrealuri-is tol an masze met qulaTa wils. z-is formulaSi CavsvaT

ricxviTi mniSvnelobebi:

6.158

58088

+==−

=realuriz .

a-1 cxrilis (g) sveti gviCvenebs, rom standartuli normaluri ganawilebis z =

+1.6-is toli an masze meti farTobia .0548. aqedan gamomdinare, 88-is tol an masze

met qulaTa wili iqneba .0548 anu 5.48%. amrigad, populaciis yoveli 1000 qulidan

54.8 tolia an metia 88-ze.

amocana

ganxiluli ganawilebis qulaTa ra nawilia moTavsebuli 83-sa da 87-s Soris?

amoxsna


orive qula _ 83 da 87 unda gadaviyvanoT zrealuri qulebSi. Semdeg viTvliT

standartuli normaluri ganawilebis farTobs zrealuri-is or mniSvnelobas

Soris. ricxviTi mniSvnelobebis Casmis Semdeg miviRebT:

zrealuri 83-isTvis 6.053

58083

+==−

= ;


58087

+==−

= .

nedli qulebi da maTi Sesabamisi zrealuri qulebi mocemulia suraTze N6-6.

zrealuri = +0.6-isTvis a-1 cxrilis (b) svetSi vnaxulobT, rom z = 0-sa da z = 0.6-s

Soris mdebare qulaTa wilia .2257. zrealuri= +1.4-isTvis z = 0-sa da z = +1.4-s Soris

mdebare qulaTa wilia .4192. ganawilebis farTobi, romelic z-is am or

mniSvnelobas Soris mdebareobs, gamoxatavs 83 da 87 qulebs Soris moTavsebul

qulaTa wils. am farTobis gamosaTvlelad .4192-s vaklebT .2257-s. miRebuli

pasuxia .1935 (ix. suraTi N6.6, (g) RerZi). amrigad, qulebis wili, romelic Tavsdeba

83-sa da 87-s Soris intervalSi, aris .1935 anu 19.35%. yoveli 1000 qulidan 193.5

moTavsebulia 83 qulasa da 87 qulas Soris.

აკლია სურათი 6-6 suraTi N6.6. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semavali qulebis (83 da 87)


(a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z =

+0.6-mde da z = 0-dan z = +1.4-mde.

amocana

qulaTa ra nawilia moTavsebuli 65-sa da 74-s Soris?

amoxsna

es amocana wina amocanis msgavsad ixsneba. qulebi 65 da 74 unda gadaviyvanoT

zrealuri qulebSi. Semdeg viTvliT standartuli normaluri ganawilebis farTobs

zrealuri-is or mniSvnelobas Soris. ricxviTi mniSvnelobebis Casmis Semdeg

miviRebT:


58065

−=−

=−

= ;


58074

−=−

=−

= .

nedli qulebi da maTi Sesabamisi zrealuri qulebi mocemulia suraTze N6.7. a-1

cxrilis (b) svetSi vnaxulobT, rom qulaTa .4987 moTavsebulia z = 0-sa da z = _3.0-

s Soris. zrealuri = _1.2-isTvis qulaTa .3849 nawili moTavsebulia z = 0-sa da z =


_1.0-s Soris. z = _3.0-sa da z = _1.2-s Soris moTavsebul qulaTa wilis

gamosaTvlelad .4987-s gamovaklebT .3849-s da miviRebT .1138-s (ix. suraTi N6.7, (g)

RerZi). qulaTa wili 65-dan 74-mde intervalSi aris .1138 anu 11.38%. populaciaSi

yoveli 1000 qulidan 113.8 moTavsebulia 65-sa da 74-s Soris.

აკლია სურათი 6-7 suraTi N6.7. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semavali qulebis (65 da 74)


(a) nedl qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z =

-3.0-mde da z = 0-dan z = -1.2-mde.

amocana

qulaTa ra wili Seadgens ganawilebis Sua 80%-s?

amoxsna

amocanis amosaxsnelad a-1 cxrilSi unda vnaxoT (b) sveti, sadac

warmodgenilia saSualos orive mxares mdebare qulaTa .40 nawili anu 40%; am

qulebisTvis ganvsazRvravT z-is mniSvnelobas da Semdeg z-is formulis amoxsniT

viRebT X-is mniSvnelobebs. suraTze N6-8 naCvenebia qulaTa saZiebeli wilebi. z-is

sapovnelad, romelic moicavs z = 0-sa da mis mniSvnelobas Soris moTavsebul

qulaTa .40 wils, (b) svetSi veZebT .40-Tan yvelaze miaxloebul wils. es aris .3997,

romelic Seesabameba z = 1.28-s. amrigad, z = 0-sa da z = + 1.28-s Soris moTavsebuli

qulebis Sexvedris fardobiTi sixSirea .3997 da z = 0-sa da z = _ 1.28-s Soris

moTavsebuli qulebic igive fardobiTi sixSiriT (.3997) gvxvdeba. z = _ 1.28-sa da z =

+ 1.28-s Soris moTavsebuli qulebis saerTo raodenobaa .3997+.3997=.7994 anu

daaxloebiT .80. bolo nabijia X-is mniSvnelobebis gamoTvla, romlebic z = _ 1.28-

sa da z = + 1.28-s Seesabameba. am mniSvnelobebis gamosaTvlelad z-is, μ-sa da σ-s

ricxviTi mniSvnelobebi CavsvaT formulaSi zrealuri=(X_μ)/σ da gamovTvaloT X-is

mniSvnelobebi.

აკლია სურათი 6-8 suraTi N6.8. normalur ganawilebaSi, romlis μ = 80 da σ =5, Semaval qulaTa Sua 80%-is

povna standartul normalur ganawilebaSi. X RerZis warwerebi gviCvenebs (a) nedl

qulebs, (b) nedli qulebis Sesabamis z qulebs, (g) qulaTa wils z = _0-dan z = -1.28-mde

da z = 0-dan z = +1.28-mde.

zrealuri= +(1.28)-isTvis


58028.1 −

=+X

gantoleba amovxsnaT X-is mimarT

( )( )

4.86804.6

8028.1580528.1

=+=

+=−=+

XX

zrealuri= _1.28-isTvis

58028.1 −

=−X

gantoleba amovxsnaT X-is mimarT

( )( )

6.73804.6

8028.1580528.1

=+−=

+−=−=−

XX

Sesabamisad, qulaTa 80% moTavsebulia 73.6-sa da 86.4-s Soris.

z qulebis gamoyenebis magaliTi

xSirad iTvleba, rom didi raodenobiT adamianebze Catarebul standartizebul

testebSi miRebuli qulebi daaxloebiT normalurad aris ganawilebuli da z-is

mniSvnelobebiT gamoixateba. amitom cnobilia xolme, ganawilebis danarCen

qulebTan mimarTebaSi zustad sad mdebareobs mocemuli qula. magaliTad,

iarnoldma, grimma da miuserma (Yarnold, Grimm & Mueser, 1986) ikvlies, Tu ramdenad

ukavSirdeba A da B tipis qcevis paternebi socialur konformulobas. adamianebis

azriT, A tipis qceva gulisxmobs Sejibrianobas, agresiulobas, mouTmenlobasa da

swrafad gaRizianebadobas. B tipis individebi xasiaTdebian rogorc

araSejibrianebi, araagresiulebi, momTmenebi, romlebic mSvidad xvdebian

imedgacruebebsa da frustraciebs. A da B tipis qcevis paternebis Sefasebis erT-

erTi sazomia jenkinsis aqtiurobis kiTxvari (Jenkins, Zyzanski & Rosenman, 1979),

romelic tipur aqtivobebs Seexeba. masSi Sedis iseTi kiTxvebi, rogoricaa,

magaliTad:

dagisignalebiaT Tu ara frustraciis gamosaxatad avtomanqaniT sacobSi

moxvedrisas?

A tipis individebi pasuxoben „xSirad“ an „TiTqmis yovelTvis“, xolo B tipis

individebi _ „xandaxan“ an „arasodes“.

iarnoldma da sxvebma gamoTqves hipoTeza, rom xazis sigrZis Sefasebisas A

tipis individebi ufro naklebad daeqvemdebarebodnen jgufis gavlenas da

arasodes xandaxan xSirad TiTqmis yovelTvis


wavidodnen kompromisze, vidre _ B tipis individebi. kvlevis am hipoTezis

Sesamowmeblad maT SearCies individTa ori jgufi: _ jenkinsis aqtiurobis

kiTxvaris qulebis mixedviT, ukiduresad A tipis individebi da ukiduresad B

tipis individebi. maTi monacemebiT, jenkinsis aqtiurobis kiTxvaris saSualo z

qula A tipis individebiTvis iyo +1.22 (dadebiTi z qula gviCvenebs A tipis qcevas),

xolo B tipis individebisTvis _ _1.38 (uaryofiTi z qula gviCvenebs B tipis qcevas).

TiToeuli jgufis saSualo z qulebis Sesaxeb informacia saSualebas gvaZlevs,

zustad ganvsazRvroT TiToeuli jgufis mdebareoba jenkinsis aqtiurobis

kiTxvaris A _ B tipis wrfeze. a-1 cxrilis (g) svetSi vnaxavT, rom standartul

normalur ganawilebaSi z-is mniSvnelobaTa mxolod .11 nawili tolia z = +1.22-is,

an aRemateba mas. amrigad, z = +1.22 A tipis qcevis jgufis saSualos qulaTa 89-e

procentilze ayenebs. analogiurad, z = _1.38-is SemTxvevaSi vxedavT, rom B tipis

wrfeze qulaTa mxolod .08 nawilia moTavsebuli am mniSvnelobis qvemoT.

ukiduresad A tipis qcevis da B tipis qcevis mqone individebi sxvadasxva jgufSi

Tavsdebodenen. miRebuli Sedegebi adasturebs hipoTezas: _ B tipis individebi 2-

jer metad eqvemdebarebian jgufis gavlenas da midian konsensusze, vidre A tipis

cdis pirebi.


1. ganmarteT: _ standartuli normaluri gadaxra, standartuli normaluri

ganawileba, z, zrealuri.

2. X-is, μ-sa da σ-s TiToeuli mniSvnelobisTvis gamoTvaleT zrealuri. Semdeg

cxrilis Na-1 gamoyenebiT ipoveT farTobi: _ (1) z = 0-sa da z-s Soris; (2)

z-is zemoT.

3. amoxseniT Semdegi amocanebi „magaliTi N6-1“-

Si warmodgenili midgomis gamoyenebiT.

davuSvaT, Tqven CaatareT farTomasStabiani

kvleva studentebSi da ikvlieT verbaluri

analogiebiT msjelobis unari. vTqvaT,

testis qulebi normalurad aris

ganawilebuli, ganawilebis μ = 72 da σ = 8.

a) ganawilebis qulaTa ra wilia 86-is toli

an masze meti?

b) ganawilebis qulaTa ra wilia moTavsebuli 80-sa da 90-s Soris?

g) ganawilebis qulaTa ra wilia moTavsebuli 60-sa da 70-s Soris?

μ σ X

a. 73 4.2 81

b. 127 11.6 121

g. 17 1.3 18

d. 50 3.0 39

e. 5 0.4 5.7

v. 256 35.0 212


d) qulaTa ra diapazons moicavs testSi miRebul qulaTa ganawilebis

Sua 90%?

albaToba

Cven ganvixileT normaluri ganawileba, rogorc fardobiT sixSireTa

Teoriuli ganawileba, Tumca, is, amavdroulad, uwyveti cvladis albaTur

ganawilebasac warmoadgens. es cnebebi statistikis mniSvnelovan komponents _

albaTobas _ ukavSirdeba; Tumca, saxelmZRvanelos Semdgom TavebSi albaTobis

mxolod yvelaze mniSvnelovani aspeqtebia warmodgenili. am qveTavSi gavecnobiT

albaTobis cnebas, romelic saWiroa albaTuri ganawilebis gagebisa da

albaTobis statistikaSi gamoyenebisaTvis.

yoveldRiur cxovrebaSi ganxorcielebuli qmedebebi albaTobis intuitur

wvdomas uzrunvelyofs. magaliTad, migiRiaT rodesme qvemoT ganxilul magaliTSi

aRwerilis msgavsi werili?

saswrafo Setyobineba _ upasuxeT dauyovnebliv. Tqven moigeT qvemoT CamoTvlili

prizebidan erT-erTi:

principulad axali modelis manqana, fasi $36.000.

orkviriani ufaso mogzauroba havaize, fasi $6.000.

farTo ekraniani stereo televizori, fasi $1995.

mikrotalRovani Rumeli, fasi $375.

umaRlesi xarisxis fotoaparati, fasi $75.

kompiuteri, fasi $749.

mogebuli prizis wasaRebad Tqven ubralod unda mobrZandeT CvenTan da waiRoT

kuTvnili prizebi.

mdidruli manqanisa Tu orkviriani dasvenebis perspeqtivam SeiZleba

dagafiqroT, Tumca, lamazi ocneba gagiqrebaT, Tu waikiTxavT meore gverdze

wvrilad dabeWdil teqsts, sadac naxavT TiToeuli saCuqris mogebis Sanss,

romelic ase gamoiyureba:

davuSvaT, TiToeul monawiles

erTi prizis mogebis toli Sansi

aqvs. rogor fiqrobT, romeli

prizis mogeba gendomebodaT

yvelaze metad? romeli prizis

mogeba gendomebodaT yvelaze

naklebad? albaTobis Teoria

saerTod rom ar gqondeT naswavli,

prizi

saCuqrebis

raodenoba

(1,000,000 monawile)

1. manqana

2. dasveneba

3. televizori

4. mikrotalRovani Rumeli

5. fotoaparati

6. kompiuteri

1

20

34

66

999,845

34


Tqven, albaT, mainc ityodiT, rom manqanaze fiqrs azri ar aqvs da isev

fotoaparatis arCeva sjobia. milioni monawilidan mxolod erTi moigebs manqanas

maSin, roca milionidan 999,845-s Sexvdeba fotoaparati. cxadia, rom, manqanasTan

SedarebiT, ufro metia fotoaparatis mogebis Sansi.

davuSvaT, gekiTxebian, Tu ras udris albaToba imisa, rom televizori moigoT.

mosalodnelia, rom daaxloebiT ase upasuxebT: _ „34 Sansi milionidan“.

faqtobrivad, Tqven am pasuxiT miuTiTeT, rom konkretuli prizis mogebis

albaToba gamoiTvleba am prizis raodenobis gayofiT mosageb prizTa saerTo

raodenobaze, anu

raodenoba saerTo prizTa gasacem

raodenoba prizis konkretulialbaToba mogebis prizis konkretuli = .

Tu am formulas rigiT mesame prizis mimarT gamoviyenebT, miviRebT

televizoris mogebis albaTobas anu 34/1,000,000=.000034. Tu am formulas TiToeuli

prizis mimarT gamoviyenebT, maTTvis albaTobebis Semdeg ganawilebas miviRebT:

Tqven, albaT, SeamCnevT, rom prizis

mogebis albaTobis formula Zalian hgavs

fardobiTi sixSiris formulas, romelic me-3

TavSi ganvixileT. SegviZlia vTqvaT, rom es

formula fardobiTi sixSiris formulis

gamoyenebis specifikuri SemTxvevaa.

Tu prizebis ganawilebaSi TiToeul

SesaZlo xdomilobas Sedegs vuwodebT, maSin

am ganawilebaSi 1,000,000 SesaZlo Sedegi

gveqneba. TiToeuli Sedegi Tanabrad albaTuria da TiToeulis ganxorcielebis

albaToba 1/1,000,000 anu .000,001 iqneba. CvenTvis saintereso Sedegebis erTobliobis

ganxorcielebas xdomilobas uwodeben. magaliTad, Tu ganvixilavT me-5 prizis

mogebis xdomilobas, mosalodnelia, rom es xdomiloba 999,845 SemTxvevaSi

ganxorcieldes. Tu me-3 an me-4 prizis mogebis xdomiloba gvainteresebs, maSin

vnaxavT, rom me-3 prizisTvis 34 SemTxvevaa, me-4 prizisTvis ki _ 66, rac jamSi am

xdomilobis ganxorcielebis 100 SemTxvevas, SesaZlo Sedegs Seadgens. yuradReba

miaqcieT, rom me-5 prizis mogebis xdomiloba 1,000,000 SesaZlo SemTxvevidan 999,845

Sedegia, magram imave 1,000,000 mxolod 20 SemTxvevaSia mosalodneli me-2 prizis

mogeba. ufro zogadad, Tu populaciaSi xdomilobis ganxorcieleba ar aris

damokidebuli sxva xdomilobebis ganxircielebaze, maSin xdomilobis

ganxorcielebis albaToba (p) ganisazRvreba, rogorc

( )raodenoba saerTo axdomilobaT SesaZlo ipopulaciaS

raodenoba aSemTxvevaT xelSemwyob xdomilobis ipopulaciaSxdomiloba =p

prizi

saCuqrebis

raodenoba

(1,000,000 monawile)

mogebis

albaToba

1

2

3

4

5

6

1

20

34

66

999,845

34

.000001

.000020

.000034

.000066

.999845

.000034


simbolo p gamoiyeneba xdomilobis

ganxorcielebis albaTobis aRsaniSnad.

am formulis gamoyenebiT warmoebuli

gamoTvlis Sedegi, rogorc wesi,

aTwiladiT gamoixateba. magaliTad,

p(prizi 4) = .000066

miuxedavad amisa, albaTobebi

procentebiT an wiliTac SeiZleba

gamoisaxos. es formula gansazRvravs

diskretuli xdomilebebis Sedegebs,

anu iseTi xdomilebebis albaTobas,

romelTac aqvT mniSvnelobaTa

sasruli raodenoba, rogoricaa,

magaliTad, milion monawileze prizis

saxiT gacemul manqanaTa raodenoba. amrigad, TiToeuli prizis miRebis

albaTobebi qmnian diskretul albaTur ganawilebas.

diskretuli albaTuri ganawilebis kidev erTi magaliTis saxiT cxrilSi N3.3

mocemuli gamocdasTan dakavSirebuli SfoTvis 100 qula ganvixiloT. davuSvaT,

gvainteresebs, ramdenia albaToba imisa, rom am qulebidan SemTxveviT arCeuli

erTi qula moTavsebuli iqneba 45-sa da 49-s Soris SualedSi. es sakiTxi 45-sa da

49-s Soris qulis moxvedris xdomilobis albaTobis gansazRvras gulisxmobs.

amrigad, am xdomilobaSi moiazreba is Sedegebi, rodesac aRebuli qula iqneba 45,

46, 47, 48 da 49. cxomilobis am albaTobis gamosaTvlelad Cven unda vipovoT am

qulebTan dakavSirebuli Sedegebis saerTo raodenoba, risi gakeTebac am qulebis

cxrilSi N3.4 mocemuli ganawilebis mixedviT SegviZlia. cxrili gviCvenebs, rom

45-49 intervalSi 15 qulaa. vinaidan cnobilia, rom mocemul ganawilebaSi sul 100

qulaa, albaToba imisa, rom SemTxveviT aRebuli qula moTavsebuli iqneba 45-sa da

49-s Soris, aris

.15.100/15)4945( ==−− Soris s da sa qulap

aseve, SeiZleba vTqvaT, rom es albaToba:

.15.)4945( =≤≤ Xp

sadac ≤ _ niSnavs „naklebia an tolia“. ≤ simbolos win mdgomi ricxvi

naklebia an tolia ricxvis, romelic mosdevs am simbolos.

X _ SemTxveviT arCeuli qula.

mocemuli gamosaxuleba ase ikiTxeba: „albaToba imisa, rom X tolia an metia

45-ze (anu 45≤X) da tolia an naklebia 49-ze (anu X≤49) aris .15.“ es albaToba

pirdapir cxrilSi N3.4 SeiZleba moviZioT am qulebisTvis v svetSi mocemuli

Sedegi (outcome) _ albaTobis ganawilebaSi

TiToeuli SesaZlo varianti.

xdomiloba (event) _ albaTobis ganawilebaSi

Sedegebis konkretuli simravlis

ganxorcieleba.

xdomilobis ganxorcielebis albaToba

(probability of occurrence of en event – p) _

raodenoba saereTo

Sedegebis SesaZlo

raodeoba Sedegebis

iSemadgenel xdomilobis

xdomiloba =)(p

diskretuli Sedegebi (discrete outcomes) _

mniSvnelobebaTa sasruli raodenobis mqone

Sedegebi ganawilebaSi.


fardobiTi sixSireebis gamoyenebiT, romelic gviCvenebs, rom 45-sa da 49-s Soris

moTavsebuli qulebis fardobiTi sixSire .15-ia. yuradReba miaqcieT, rom es

albaToba gamocdasTan dakavSirebuli SfoTvis kiTxvarSi miRebuli qulebis

fardobiTi sixSiris histogramidan da fardobiTi sixSiris poligonidanac

(Sesabamisad, suraTi 3.5 da 3.6) SeiZleba vipovoT.

diskretuli albaTuri ganawilebis Tvisebebi

albaTobebis ganawilebas aqvs Tvisebebi, romelTac Tqven, albaT, ukve icnobT.

pirveli Tviseba isaa, rom xdomilebis albaToba meryeobs 0.00-dan +1.00-mde.

xdomileba, romlis albaToba udris nuls, mosalodnelia, rom namdvilad ar

ganxorcieldeba da xdomileba, romlis albaToba udris erTs, mosalodnelia,

rom namdvilad ganxorcieldeba. suraTze N6.9 warmodgenilia albaTobis skala da

individis cxovrebaSi zogierTi xdomilebis ganxorcielebis albaTobebi.

zogierTi xdomileba sruliad mosalodnelia cxovrebaSi da misi albaToba 1.00-is

tolia, zogierTi ki _ sruliad SeuZlebeli da maTi albaToba 0.00-is tolia.

albaTobis diapazoni 0.00-dan 1.00-mde Seesabameba fardobiTi sixSireebis

ganawilebis Tvisebas _ dagrovili fardobiTi sixSireebi (crƒ) icvleba 0-dan 1.0-

mde diapazonSi (ix. cxrili N3.4, sveti (i)).

meore Tviseba isaa, rom albaTur ganawilebaSi Sedegebis saerTo

raodenobasTan dakavSirebuli calkeuli albaTobebis jami 1.00-s Seadgens.

albaTuri ganawilebis es maxasiaTebeli iolad SeiZleba davinaxoT cxrilSi N3.4:

_ gamocdasTan dakavSirebuli SfoTvis kiTxvaris qulebis dagrovili fardobiTi

sixSireebi 1.00-s Seadgenen (ix. i sveti). Tu Tqven Lazy Acres Vacation Resort-s ewveviT,

prizis mogebis labaToba 1.00 iqneba.


suraTi 6.9. albaTobebis cxrili da zogierTi xdomilebis ganxorcielebis albaToba.

albaToba

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

0

albaToba imisa, rom Tqven mokvdebiT.

.86 albaToba imisa, rom qali cocxali iqneba 60 wlis asakSi (yoveli 100 dabadebuli qalidan 86 cocxalia 60 wlis asakSi, p=86/100=.86).1

.81 albaToba imisa, rom mamakaci cocxali iqneba 60 wlis asakSi (yoveli 100 dabadebuli mamakacidan 81 cocxalia 60 wlis asakSi, p=81/100=.81).1

.64 albaToba imisa, rom aSS-is SemTxveviT SearCeul zrdasrul moqalaqes eqneba, sul mcire, erTi sakredito baraTi.2

albaToba imisa, rom agdebisas 50-TeTriani borjRalze dajdeba (p=1/2=.5)

.077 banqos dastidan valetis amoRebis albaToba (p=4/52=.077).

.000000005 totalizatorSi mogebis miaxloebiTi albaToba.3

1 1980 Commissioners Standard Ordinary (CSO) Mortality Table (The Wizards, 1989). 2 The Way, 1985. 3 Artis, 1993


mesame Tviseba isaa, rom

urTierTgamomricxavi Sedegebis momcveli

xdomilobis albaToba SeiZleba am

xdomilobis Semadgeneli Sedegebis albaTobebis SekrebiT vipovoT.

urTierTgamomricxavi Sedegebi is Sedegebia, romlebic ver ganxorcieldeba

erTdroulad. zemoT ganxilul magaliTSi individis mier me-2 da me-5 prizebis

mogeba urTierTgamomricxavi xdomilobebia. mas SeuZlia moigos erTi an meore

prizi, magram ara _ orive erTdroulad. analogiurad, gamocdasTan dakavSirebuli

SfoTvis kiTxvarSi individi erTdroulad ver miiRebs 53 da 67 qulas. es Sedegebi

urTierTgamomricxavia. magaliTad, davuSvaT, rom gvainteresebs, ramdenia

albaToba imisa, rom gamocdasTan dakavSirebuli SfoTvis kiTxvaris qulebis

ganawilebidan SemTxveviT aRebuli erTi qula 45 an masze meti iqneba? am kiTxvaze

pasuxis gasacemad cxrilis N3.4 e sveti unda vnaxoT, sadac martiv sixSireTa

ganawilebaa mocemuli. yuradReba miaqcieT, rom am svetSi 45-is toli an masze

meti 28 qulaa. es ricxvi e svetSi mocemuli, 45-is toli an masze meti qulebis

sixSireebis dajamebiTaa miRebuli (anu 15+5+3+2+1+2=28). amrigad, 45-is toli an

masze meti qulis SemTxveviT arCevis albaTobaa 28/100 anu .28. albaTobis tolobis

gamoyenebiT:

.28.)45( =≥Xp

es albaToba N3.4 cxrilidanac SeiZleba miviRoT: _ 45-is toli anda masze meti

qulebisTvis v svetSi mocemuli fardobiTi sixSireebis SekrebiT (anu

.15+.05+.03+.02+.01+.02=.28).

es magaliTi gviCvenebs, rom Tu ori an meti Sedegi urTierTgamomricxavia,

maSin erT-erTi SedegiT xdomilobis ganxorcielebis albaToba TiToeuli Sedegis

albaTobebis dajamebiT gamoiTvleba. albaTobis terminebiT Tu vityviT, Tu A da

B albaTuri ganawilebidan aRebuli ori urTierTgamomricxavi Sedegia, maSin

erTis an meoris ganxorcielebis albaToba A-s ganxorcielebis albaTobisa da B-s

ganxorcielebis albaTobis jamis tolia, anu:

).()()( BpApBAp +=an

es damokidebuleba cnobilia, rogorc aditiurobis wesi (anu damatebis,

Sevsebis) urTierTgamomricxavi Sedegebis albaTobisTvis.

am formulis gamoyenebis kidev erTi magaliTi: _ davuSvaT, isev SemTxveviTad

virCevT gamocdasTan dakavSirebuli SfoTvis kiTxvarSi miRebul erT qulas da

gvainteresebs, ramdenia albaToba imisa, rom moTavsebuli iqneba 20-sa da 24-s, an

70-sa da 74-s Soris. cxrilSi N3.4 mocemuli sixSireTa ganawilebis gamoyenebiT

vnaxavT, rom 20-sa da 24-s Soris, an 70-sa da 74-s Soris SemTxveviT qulis arCevis

albaToba 20-24 intervalSi moTavsebuli qulebis fardobiTi sixSirisa (anu .01-is)

urTierTgamomricxavi Sedegebi (Mutually

exclusive outcomes) _ Sedegebi, romelic ver

ganxorcieldeba erTdroulad.


da 70-74 intervalSi moTavsebuli qulebis fardobiTi sixSiris (anu .02-is) jamia

anu sruli albaToba .03-s Seadgens. albaTobis gamosaxulebiT:

.03.02.01.)70()2420()702420(

=+=−−+−−=

=−−−−

mde74 danmde dan

qula arCeuli Soris s74 da sa an s, da sa

ppp

aditiurobis es wesi Zalze mniSvnelovani gaxdeba SemdgomSi _ am Tavis

bolosa da Semdeg TavebSi, sadac hipoTezebis Semowmebaze iqneba saubari.

Teoriuli (klasikuri) da empiriuli albaToba

xdomilebis albaTobis gansazRvris ori gza arsebobs: _ Teoriuli da

empiriuli. zogierTi xdomilobisTvis Cven SegviZlia davaskvnaT misi xelSemwyob

SemTxvevaTa albaTobis Sesaxeb. magaliTad, 50-TeTriani monetis agdebisas

borjRalis gamosvlis Sesaxeb SeiZleba msjelobiT davaskvnaT. 50-TeTriani

monetis agdebisas ori SesaZlo Sedegi gvaqvs: _ borjRalis an safasuris gamoCena.

Tu moneta sworia, TiToeuli maTgani erTnairad mosalodnelia. Sesabamisad,

borjRalis mosvliT dasrulebuli erTi agdebis albaToba, ori SesaZlo

Sedegidan, erT-erTia, anu p(borjRalis mosvla) = 1/2 = .5. Teoriuli albaTobis

modelebi xSirad am magaliTze gacilebiT rTulia. Semdeg qveTavSi ganvixilavT

standartul normalur ganawilebas, rogorc albaTur Teoriul ganawilebas.

albaTuri Teoriuli ganawileba (Theoretical probability distribution) _ Teoriuli albaTobis

modelis gamoyenebiT miRebuli albaTuri ganawileba.

albaTuri klasikuri ganawileba (Empirical probability distribution) _ xdomilobis realurad

ganxorcielebis sixSiris daTvliT miRebuli albaTuri ganawileba.

zogierTi xdomilebis SemTxvevaSi Teoriulad ver vimsjelebT misi

ganxorcielebis albaTobis Sesaxeb; magaliTad, Teoriuli msjelobiT ver

davaskvniT, Tu ras udris albaToba imisa, rom Tu qalbatoni xarT, Tqven

cocxali iqnebiT 60 wlis asakSi. am albaTobis gamosaTvlelad gamoviyenebT

xdomilobis ganxorcielebis xangrZlivi SeswavliT miRebul empiriul

albaTobebs. magaliTad, sicocxlis dazRvevis industriaSi aSS-is moqalaqeTa

sicocxlis xangrZliobis sixSiris gansasazRvrad specialuri komisiis mier

SemuSavebul bunebrivi sikvdilianobis standartul cxrilebs iyeneben (The Wizards,

1989). es cxrili empiriulad aris Sedgenili xangrZlivi drois manZilze

sxvadasxva asakis mamakacebisa da qalebis sikvdilianobis sixSiris Canawerebis

warmoebis Sedegad. cxrili gviCvenebs, rom yoveli 10,000,000 cocxali qalidan

8,603,801 jer kidev cocxalia 60 wlis asakSi. amrigad, dabadebisas albaToba imisa,

rom aSS-is moqalaqe qali jer kidev cocxali iqneba 60 wlis asakSi, iqneba

p(60 wlis asakis cocxali qali) 86.000,000,10801,603,8

== .


cxadia, p-is es mniSvneloba Seicvleba, Tu Seicvleba sikvdilianobis sixSire

janmrTelobis dacvisa da kvebis gaumjobesebis Sedegad.

bihevioristul mecnierebaSi aris orive (Teoriuli da empiriuli) albaTobis

gamoyenebis SemTxvevebi, Tumca, winamdebare saxlmZRvaneloSi aqcenti keTdeba iseT

Teoriul ganawilebaze, rogoricaa standartuli normaluri ganawileba.

Semdgom TavebSi statistikuri daskvnis iseT problemebs SevexebiT rogoricaa

populaciidan aRebuli SerCevidan miRebul sazomebze dayrdnobiT am populaciis

Sesaxeb daskvnebis gakeTeba; Tumca, aseTi daskvnebis gamotanisas verasodes

viqnebiT absoluturad darwmunebulni maT sisworeSi; amrigad, Cveni daskvnebi

albaTobis terminebiT iqneba formulirebuli.

standartuli normaluri ganawileba, rogorc albaTuri ganawileba

normaluri ganawileba fardobiTi sixSireebis Teoriuli ganawilebaa. vinaidan

normalur ganawilebaSi warmodgenili fardobiTi sixSireebi populacias

efuZneba, normaluri ganawileba albaTobebis ganawilebac aris; Tumca, is odnav

gansxvavebuli albaTuri ganawilebaa, vidre axlaxans ganxiluli diskretuli

albaTuri ganawileba. normalur ganawilebaSi Semavali X qula uwyveti cvladia,

romelmac SeiZleba miiRos mniSvnelobaTa usasrulo raodenoba Tavisi

cvalebadobis diapazonSi. magaliTad, davuSvaT, rom gvaqvs sxeulis wonis

amsaxveli qulebis Teoriuli normaluri ganawileba. ras udris albaToba imisa,

rom vipoviT individs, romelic zustad 123 funts (58 kg-s) iwonis? wona uwyveti

cvladia da zogierTi adamiani SeiZleba iwonides 123.3 funts, 123.36 funts, an

123.364 funtsac ki. Teoriulad, Cven SeiZleba iseTi individic vipovoT, romelic

123.3642193 funts iwonis, Tu SevZlebT ase zustad gazomvas. SeiZleba, rom 123.3

funti ganvixiloT, rogorc 123 funti? am TvalsazrisiT, ra SeiZleba iTqvas

123.00001 funtze? principSi, Cven verasodes miviRebT zustad 123 funts, Tu

yovelTvis ufro zustad gazomvas vcdiT da yovel jerze vnaxavT, rom gazomvis

Sedegi ar aris zustad 123 funts.

Tu am mimarTulebiT vimsjelebT, uwyveti cvladis zusti mniSvnelobis miRebis

albaToba nulis toli iqneba, magram garkveul intervalSi monacemis moxvedris

albaToba ar aris nulis toli. miuxedavad imisa, rom ar SegviZlia zustad 123

funtis miReba, SegviZlia miviRoT wona 122.5 funtsa da 123.5 funts Soris

intervalSi. magaliTad, ganurCevlad imisa, aTwiladSi mZimis Semdeg ramdeni

Tanrigis sizustiT SegviZlia gazomva, 123.36 funti (anda 123.3642193 funtic ki)

yovelTvis moxvdeba 122.5 funtsa da 123.5 funts Soris intervalSi. amrigad,

normaluri ganawilebis saxiT mocemuli albaTobis funqcia uzrunvelyofs ara X-

is diskretuli mniSvnelobebis albaTobebs, aramed imis albaTobas, rom X-is


mocemuli mniSvneloba moxvdeba konkretul intervalSi. albaToba gamoisaxeba

ganawilebis qveS mdebare farTobiT, romelic SemosazRvrulia intervaliT.

normaluri ganawilebis albaTuri ganawilebis saxiT gamoyenebis

sailustraciod ganvixiloT Semdegi amocana. davuSvaT, iciT, rom qulaTa

erToblioba normalurad aris ganawilebuli da ganawilebis μ = 150 da σ = 8. Tu

Tqven ganawilebidan SemTxveviT airCevT qulas, ras udris albaToba imisa, rom

aRebuli qula moTavsebuli iqneba 150-sa da 158-s Soris intervalSi? am kiTxvaze

pasuxis gasacemad, pirvel rigSi, mocemuli ganawileba unda gardavqmnaT

standartul normalur ganawilebad, romlis μ = 0 da σ = 1; amisaTvis viyenebT

formulas z = (X_ μ)/ σ. amrigad, 150 qula gaxdeba z = 0, xolo 158 qula _ z = +1.0.

Sesabamisad, dasmuli kiTxva gadaiqceva kiTxvad imis Sesaxeb, Tu ramdenia z = 0-sa

da z = +1.0-s Soris intervalSi moTavsebuli qulis miRebis albaToba. am kiTxvis

pasuxia standartuli normaluri ganawilebis qveS mdebare da z = 0-sa da z = +1.0-s

Soris intervaliT SemosazRvruli farTobi anu .3413 (ix. a-1 cxrilis (b) sveti).

Sesabamisad, 150-sa da 158-s Soris intervalSi moTavsebuli qulis miRebis

albaTobaa .3413.

magaliTi N6.2

standartuli normaluri ganawilebis rogorc albaTuri ganawilebis

gamoyeneba

davuSvaT, Tqven gaqvT magaliTSi N6.1 aRwerili normalurad ganawilebuli

populacia, romlis μ = 80 da σ = 5. qvemoT mocemuli amocana N6.1 magaliTSi

mocemuli amocanis msgavsia, Tumca, aq mogeTxovebaT ara qulaTa wilis, aramed

maTi albaTobebis gamoTvla. amocanis amoxsna gagiadvildebaT, Tu magaliTis N6.1

Tanmxleb suraTs mimarTavT.

amocana

ras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula 88-is

toli an masze meti iqneba?

saWiro statistika: ( ) σµ /−= Xzrealuri

amoxsna

88 qula unda gadaviyvanoT zrealuri-Si. Semdeg a-1 cxrilis gamoyenebiT vipoviT

qulaTa wils, romelic tolia an metia zrealuri-ze. es wili warmoadgens saZiebel

albaTobas. zrealuri-is formulaSi ricxviTi mniSvnelobebis Setanis Semdeg

miviRebT:

6.158

58088

+==−

=realuriz .


a-1 cxrilis (g) sveti gviCvenebs, rom standartuli normaluri ganawilebis

qveS mdebare farTobi, romelic tolia an metia z = +1.6-ze, aris .0548. amrigad,

albaToba imisa, rom qula toli an meti iyos 88-ze, aris .0548 anu, Tu albaTobis

saxiT CavwerT, p(X≥88) = .0548.

amocana

ras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula

moTavsebuli iqneba 83-sa da 87-s Soris?

amoxsna

orive qula _ 83 da 87 unda gadaviyvanoT zrealuri qulebSi. Semdeg vpoulobT

standartuli normaluri ganawilebis qveS farTobs zrealuri-is or mniSvnelobas

Soris (ix. suraTi N6.6). ricxviTi mniSvnelobebis Setanis Semdeg miviRebT:


58083

+==−

= ;


58087

+==−

= .

cxrilis Na-1 (b) svetidan vxedavT, rom zrealuri = +0.6-isTvis z = 0-sa da z =

+0.6-s Soris intervalSi moTavsebul qulaTa wilia .2257. zrealuri = +1.4-isTvis z =

0-sa da z = +0.6-s Soris intervalSi moTavsebul qulaTa wilia .4192. ganawilebis

farTobi z-is am or mniSvnelobas Soris gvaZlevs 83-sa 87-s Soris moTavsebul

qulaTa wils. am farTobis misaRebad .4192-s vaklebT .2257-s da viRebT .1935-s.

miRebuli wili warmoadgens saZiebel albaTobas. amrigad, 83-sa da 87-s Soris

intervalSi qulis moxvedris albaTobaa .1935; albaTobis terminebiT,

p(83≤X≤87)=.1935.

amocana

ras udris albaToba imisa, rom mocemul ganawilebaSi Semavali qula


amoxsna

orive qula _ 65 da 74 unda gadaviyvanoT zrealuri qulebSi. Semdeg vpoulobT

standartuli normaluri ganawilebis farTobs zrealuri-is or mniSvnelobas Soris

(ix. suraTi N6.7). ricxviTi mniSvnelobebis Setanis Semdeg miviRebT:


58065

−=−

=−

= ;


58074

−=−

=−

= .


Tu vnaxavT a-1 cxrilis (b) svets da gavixsenebT, rom standartuli normaluri

ganawilebis simetriulobidan gamomdinare, erTnairi mimarTebaa z-is rogorc

dadebiTi, ise _ uaryofiTi qulebis SemTxvevaSi, vnaxavT, rom z = 0-sa da z = _3.0-s

Soris SualedSi moTavsebulia qulaTa .4987 nawili. zrealuri = _1.2-isTvis ki

qulaTa .3849 nawili moTavsebulia z = 0-sa da z = _1.2-s Soris SualedSi.

Sesabamisad, Cven miviRebT z = _3.0-sa da z = _1.2-s Soris moTavsebul qulaTa wils,

Tu .4987-s gamovaklebT .3849-s, rac .1138-is tolia. es wili warmoadgens saZiebel

albaTobas; aqedan gamomdinare, p(65≤X≤74)=.1138.


1. ganmarteT: _ diskretuli xdomileba, Sedegi, empiriuli albaToba,

xdomiloba, urTierTgamomricxavi Sedegebi, xdomilobis albaToba,

Teoriuli albaToba.

2. romeli formuliT gamoiTvleba diskretuli xdomilobis ganxorcielebis

albaToba?

3. velosipedis mosagebad SeiZineT latareis bileTi. bileTidan SeityveT,

rom yovel 700 gayidul bileTze gaTamaSdeba mxolod erTi velosipedi.

ras udris albaToba imisa, rom Tqven pirvelive gaTamaSebisas moigoT

velosipedi?

4. X-is, μ-sa da σ-s qvemoT mocemuli TiToeuli mniSvnelobisTvis

gamoTvaleT z-is mniSvneloba. Semdeg a-1 cxrilis gamoyenebiT moZebneT

albaToba imisa, rom qula toli an meti

iqneba mocemul qulaze.

5. Tqven CaatareT farTomasStabiani kvleva

studentebSi da ikvlevdiT verbaluri

analogiebiT msjelobis unars. vTqvaT,

testis qulebi normalurad aris

ganawilebuli, ganawilebis μ = 72 da σ = 8.

a) ras udris albaToba imisa, rom

ganawilebaSi Semavali qula 84-is toli an masze meti iqneba?

b) ras udris albaToba imisa, rom ganawilebaSi Semavali qula


g) ras udris albaToba imisa, rom ganawilebaSi Semavali qula


standartuli qulebi

μ σ X

a. 73 4.2 81

b. 127 11.6 121

g. 17 1.3 18

d. 50 3.0 39

e. 5 0.4 5.7

v. 256 35.0 212


Cven vnaxeT, rom SegviZlia normalurad ganawilebuli nedli qulis aReba da

misi gadayvana standartuli normaluri ganawilebis qulad Semdegi formulis

gamoyenebiT:

σµ−

=Xz .

z aris standartuli qula, romlis gamoyeneba SesaZlebelia ori sxvadasxva

normaluri ganawilebidan aRebuli qulebis Sesadareblad. magaliTad, vTqvaT,

individma erT testSi miiRo 75 qula, romlis μ = 70 da σ = 5, xolo meore testSi,

romlis μ = 100 da σ = 10, miiRo 110 qula. davuSvaT, rom orive testSi miRebuli

qulebi normalurad aris ganawilebuli. TiToeuli qulisTvis (75 da 110) zrealuri

aris +1. orive testSi individma miiRo saSualoze erTi standartuli gadaxriT

meti qula. Sesabamisad, misi mdebareoba sxvebis mier miRebul qulebTan

mimarTebaSi erTnairi iqneba orive testis SemTxvevaSi: _ man miiRo saSualoze

erTi standartuli gadaxriT meti qula.

SerCevidan miRebuli qulebis msgavsi gardaqmnis ganxorcieleba maSinac

gamosadegia, rodesac qulebi ar aris normalurad ganawilebuli, rac Semdegi

formulis gamoyenebiT aris SesaZlebeli:

SXXz −

= ,

sadac X = saZiebeli qula.

X = qulaTa SerCevis saSualo.

S = SerCevis standartuli gadaxra, romelic, Tavis mxriv,

( )N

XXS ∑ −=

2

.

am formuliT gamoTvlil z-is

mniSvnelobas xSirad standartul qulas

uwodeben. es formula qulas gardaqmnis

ricxvad, romelic gviCvenebs, Tu ramdeni standartuli gadaxriT aris

daSorebuli mocemuli qula SerCevis saSualodan. amis sailustraciod, davuSvaT,

rom xuT students miveciT ori _ statistikis da istoriis _ testi da miviReT

N6.4 cxrilSi mocemuli nedli da z qulebi. standartuli qulebi miRebul iqna

zemoT mocemuli formulis gamoyenebiT. magaliTad, anas mier statistikaSi

miRebuli nedli qulaa 91. testis saSualo iyo 72.0, xolo S _ 14.2. amitom anas z

qula iqneba

standartuli qula (Standard score): _ qula,

romelic miiReba formuliT ( ) SXXz /−= .


34.12.140.19

2.140.7291

+==−

=−

=S

XXzrealuri

rodesac viciT standartuli

qula, warmodgena gvaqvs klasis

saSualosTan anas qulis

mimarTebaze. es aris 1.34

standartuli gadaxra saSualos

zemoT.

standartuli qulebis

gamoyenebiT Cven martivad

SevadarebT erTmaneTs or

sxvadasxva testSi miRebul

qulebs, risi gakeTebac

SeuZlebelia nedli qulebis

SemTxvevaSi. magaliTad, joni statistikis testis mixedviT, moTavsebulia

saSualodan 1.06 standartuli gadaxriT zemoT, xolo istoriis testis mixedviT _

0.50 standartuli gadaxriT qvemoT SerCevis saSualodan. dadebiTi standartuli

qulebi gviCvenebs SerCevis saSualos zemoT mdebare qulebs, nulis toli

standartuli qulis (rogoricaa daviTis mier istoriaSi miRebuli qula)

SemTxvevaSi qula SerCevis saSualos tolia, xolo uaryofiTi standartuli

qulebi migviTiTebs SerCevis saSualos qvemoT mdebare qulebze; Tumca, yuradReba

unda miaqcioT, rom vinaidan es z SerCevis qulebidan aris miRebuli, mas ar eqneba

igive Tvisebebi normaluri ganawilebis TvalsazrisiT, rac z = (X_μ)/σ formuliT

miRebul z-s axasiaTebs. miuxedavad amisa, standartuli qulebi xSirad gamoiyeneba

testis Sedegebis warmodgenisas, rodesac aRiwereba individis fardobiTi

mdgomareoba im sxva individebTan mimarTebaSi, vinc igive testi gaiara. unda

aRiniSnos, rom standartul qulebSi gadayvana ar cvlis qulaTa ganawilebis

formas. Tu nedli qulebis ganawileba asimetriulia, maSin standartuli qulebis

ganawilebac asimetriuli iqneba.

Sejameba

• normaluri ganawileba fardobiTi sixSireebis Teoriuli ganawilebaa.

gazomvis Sedegad miRebul qulaTa arc erTi erToblioba ar aris zustad

normalurad ganawilebuli, Tumca, xSirad gamosadegia imis daSveba, rom

qcevis gazomvis zogierTi Sedegi daaxloebiT normalurad aris

ganawilebuli.

cxrili 6.4. xuTi studentis mier statistikisa

da istoriis testebSi miRebuli nedli qulebi

da standartuli qulebi.

testi

statistika istoria

pirovneba nedli

qula z qula

nedli

qula z qula

amalia

denisi

jefi

hansi

maTTi

91

62

87

56

64

+1.34

-0.70

+1.06

-1.13

-0.56

93

79

73

90

60

+1.18

0.00

+0.50

+0.92

-1.60


• normaluri ganawilebidan aRebuli qula SeiZleba standartuli normaluri

ganawilebis, romlis µ=0 da σ=1, qulad gardaiqmnas. amisaTvis gamoiyeneba

formula

z = (X_μ)/σ.

• diskretuli xdomilebis albaToba gamoiTvleba formuliT

raodenoba saerTo axdomilobaT ipopulaciaS

raodenoba aSemTxvevaT xelSemwyob xdomilobis ipopulaciaSxdomiloba =)(p

• nebismier albaTuri ganawilebisTvis Sedegis albaToba .00-sa da 1.00-s Soris

unda iyos moTavsebuli.

• calkeuli albaTobebis jami, romlebic albaTur ganawilebaSi Sedegebis

saerTo raodenobas ukavSirdeba, 1.00-s Seadgens.

• Tu A da B urTierTgamomricxavi Sedegebia, maSin ).()()( BpApBAp +=an

• standartuli normaluri ganawileba albaTuri ganawilebaa, romelic

gviCvenebs ganawilebis garkveul intervalSi qulebis moxvedris

albaTobebs. misi gamoyeneba SesaZlebelia normalurad ganawilebuli

qulebis SemTxvevaSi rogorc konkretul intervalSi moxvedrili qulebis

wilis, ise _ intervalebSi maTi moxvedris albaTobebis gansasazRvrad.

• SerCevidan miRebuli nedli qulebis standartul qulebSi gadayvana

SesaZlebelia formuliT

SXXz −

= .

es formula qulas gardaqmnis ricxvad, romelic gviCvenebs, Tu ramdeni

standartuli gadaxriT aris is daSorebuli saSualosgan. am dros ar

icvleba nedli qulebis ganawilebis forma.


asimptoturi ganawileba

albaToba

diskretuli xdomiloba

diskretuli xdomilobis

empiriuli albaToba

empiriuli albaTuri

ganawileba

Teoriuli albaToba

Teoriuli albaTuri

ganawileba


parametri

populacia

simetriuli ganawileba

standartuli normaluri

gadaxra

standartuli normaluri

ganawileba

standartuli qula

statistika

urTierTgamomr

icxavi

Sedegebi

uwyveti

ganawileba

Sedegebi

SerCeva

xdomiloba

xdomilobis

ganxorcie


lebis albaToba

μ

σ

s

X

z

zrealuri

kiTxvebi

1. inteleqtis testebSi xSirad Sedis ricxvebis Tanmimdevrobis

dalagebis, instruqciebis Sesrulebis, leqsikonis, analogiebis

danaxvis, suraTis dasrulebis, logikuri msjelobisa da

maTematikuri gamoTvlebis warmoebis unarebis Sefaseba. axseniT,

ratom aris mosalodneli, rom inteleqtis testis qulebi

populaciaSi daaxloebiT normalurad iqneba ganawilebuli.

2. reaqciis (anu gamRizianeblis miwodebasa da masze reaqciis

ganxorcielebas Soris) drois gazomvis Sedegebi, rogorc wesi, ar

aris normalurad ganawilebuli. metic, miRebul qulaTa ganawilebas

xSirad Sebrunebuli J-is forma aqvs, sadac qulaTa umravlesoba

SedarebiT mcirea, umciresoba ki _ didi. axseniT, ratom aris

mosalodneli, rom qulebi swored ase iqneba ganawilebuli.

3. wlebis manZilze dagrovda zrdasrulTa veqsleris inteleqtis

skalis monacemebi. davuSvaT, miRebuli qulebi normalurad aris

ganawilebuli da ganawilebis μ = 100 da σ = 15.

a) inteleqtis testis qulaTa ra nawili iqneba 115-is toli an masze

meti?

b) inteleqtis testis qulaTa ra wili iqneba 70-ze naklebi?

g) inteleqtis testis qulaTa ra wili iqneba 60-ze naklebi?

d) inteleqtis testis qulaTa ra wili iqneba moTavsebuli 100-sa da

120-s Soris?

e) inteleqtis testis qulaTa ra wili iqneba moTavsebuli 75-sa da

100-s Soris?

v) romel intervalSi iqneba moTavsebuli testis qulaTa Sua 80%?

z) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,

rom misi inteleqtis qula moTavsebuli iyos 90-sa da 120-s

Soris?

T) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,

rom misi inteleqtis qula 80-is toli an masze naklebi iyos?

i) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,

rom misi inteleqtis qula 125-is toli an masze meti iyos?

k) Tu SemTxveviT airCevT cdis pirs, ramdeni iqneba albaToba imisa,

rom misi inteleqtis qula 140-is toli an masze meti iyos?


4. Tqven sxva 64 adamianTan erTad gamogiZaxes sasamarTloSi, rogorc

nafici msajulobis erT-erTi kandidati. nafic msajulTa SesarCevad

furclebze dawerili saxelebi yuTSi Cayares da saTiTaod amoiRes.

a) ras udris albaToba imisa, rom pirvelive jerze Tqveni saxeli

amoiRon?

b) Tu Tqveni saxeli ar aRmoCnda pirvel rva amoRebul furcelze,

ras udris albaToba imisa, rom mecxre amoRebul furcelze

Tqveni saxeli eweros, Tu gaviTvaliswinebT, rom pirveli rva

amoRebuli furceli aRar daabrunes yuTSi?

5. nacionalurma gazeTma USA Today-m gamoaqveyna statia saTauriT

„Sansebi“ (31 dekemberi, 1985), sadac mocemuli iyo Semdegi wlis

manZilze individis cxovrebaSi garkveuli xdomilebebis

ganxorcielebis Sansebi. qvemoT CamoTvlili xdomilebebisTvis

gamoTvaleT TiToeuli maTganis ganxorcielebis albaToba.

a) doRis dros mravali firma iRebs fsonebs. erT-erTi garbenze

yovel 3000 ganacxadze erTi prizi gaicemoda. ras udrida

albaToba imisa, rom am garbenze nebismieri erTi monawile

moigebda prizs?

b) bolo wlebi mZime gamodga fermerebisTvis. weliwadSi yoveli

5314 fermeridan erTi tovebda Tavis bizness. ras udris

albaToba imisa, rom mimdinare wels fermerma miatovos biznesi?

g) mraval Cvengans uocnebia popularuli romanis daweraze, Tumca,

gamomcemlobaSi erTi wlis manZilze Sesuli yoveli 4637 wignis

moculobis mqone xelnaweridan mxolod erTi ibeWdeba. Tqven

dawereT romani. ras udris albaToba imisa, rom is daibeWdeba

mimdinare wels?

6. 6-1 magaliTSi Cven ganvixileT iarnoldisa da sxvebis (Yarnold et al.,

1986) kvleva, A da B tipis individebis Sedarebis Sesaxeb. A da B

tipis individebis misaRebad iarnoldma da misma kolegebma cdis

pirebi jenkinsis aqtiurobis kiTxvaris qulebis mixedviT SearCies.

davuSvaT, rom jenkinsis aqtiurobis kiTxvarSi populaciis saSualoa

0 da populaciis standartuli gadaxra _ 10. vTqvaT, Tqven

dainteresdiT da ganagrZeT iarnoldis kvleva. A tipis cdis

pirebidan Tqven gsurT iseTi cdis pirebis SerCeva, romelTac

jenkinsis aqtiurobis kiTxvarSi miRebuli aqvT +13-ze meti qula,

xolo B tipis cdis pirebidan _ visac miRebuli aqvT -13-ze naklebi

qula. davuSvaT, jenkinsis aqtiurobis kiTxvaris qulebi normalurad

aris ganawilebuli.


a) ras udris albaToba imisa, rom SemTxveviT arCeul individs

jenkinsis aqtiurobis kiTxvarSi eqneba +13-ze meti qula?

b) ras udris albaToba imisa, rom SemTxveviT arCeul individs

jenkinsis aqtiurobis kiTxvarSi eqneba -13-ze naklebi qula?

g) ras udris albaToba imisa, rom SemTxveviT arCeul individs

jenkinsis aqtiurobis kiTxvarSi eqneba -13-sa da+13-s Soris

intervalSi moTavsebuli qula?

7. davuSvaT, amerikeli 21-30 wlis mamakacebis saSualo wonaa 156 funti

(70.36 kg), standartuli gadaxra ki _ 8 funti (4.63 kg).

a) 21-30 wlis mamakacTa ra nawili iwonis 144 funts (65 kg) an ufro

naklebs?

b) 21-30 wlis mamakacTa ra nawili iwonis 170 funts (77 kg) an ufro

mets?

g) 21-30 wlis mamakacTa ra nawilis wona iqneba 140-dan (63.5 kg) da

172 funtamde (78 kg)?

d) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba

imisa, rom misi wona iyos 144-dan (65 kg) 168 funtamde (76 kg)?

e) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba

imisa, rom misi wona 150 funti (68 kg) an ufro naklebi iyos?

v) Tu SemTxveviT airCevT 21-30 wlis mamakacs, ras udris albaToba

imisa, rom misi wona 160 funti (72.6 kg) an ufro meti iyos?

8. davuSvaT, ikvlieT eqvsi studenti; gainteresebdaT kviraSi video

TamaSebze saSualod daxarjuli dro (saaTebSi) da semestruli

niSnebis saSualo. vTqvaT, miiReT Semdegi monacemebi:

a) gamoTvaleT saSualo da SerCevis

standartuli gadaxra qulaTa

TiToeuli jgufisTvis da Semdeg

TiToeuli miRebuli qula

gadaiyvaneT standartul qulaSi.

b) lelas video TamaSebisTvis

daxarjuli drois standartuli

qulaa nuli. ras geubnebaT es qula

yoveli kviris manZilze video TamaSebisTvis daxarjuli drois Sesaxeb?

g) ra informacias gawvdiT ias semestruli niSnebis saSualos

standartuli qula misi semestruli niSnebis saSualos Sesaxeb?

d) ras gviCvenebs semestruli niSnebis saSualos uaryofiTi

standartuli qula?

studenti

video

TamaSebze

daxarjuli

dro

semestruli

niSnebis

saSualo

aleqsi loreni ivona lavali boni iasoni

6 8 0 15 3 16

2.5 2.0 3.9 3.4 3.1 1.3


e) ras gviCvenebs video TamaSebze daxarjuli drois dadebiTi

standartuli qula?

9. cota xnis win literaturasa da istoriaSi gamocda CaabareT. Tqveni

niSnebi da klasis saSualo da standartuli gadaxra (S) aseTia:

literatura istoria

Tqveni qula

klasis saSualo

klasis S

88

83

6

70

65

5

TanaklaselebTan SedarebiT romel gamocdaSi gaqvT ukeTesi Sedegi?

ratom?

10. cxrilSi warmodgenilia aSS-is Census Bureau-is (marti, 1998) monacemebi,

romlis mixedviTac, ojaxSi wevrebis raodenobis fardobiTi

sixSireebi Semdegnairad nawildeba:

a) Tu SemTxveviT avarCevT erT

ojaxs, ramdenia albaToba imisa,

rom masSi zustad oTxi wevri

iqneba?

b) Tu SemTxveviT avarCevT erT

ojaxs, ramdenia albaToba imisa,

rom masSi eqvsi an meti wevri

iqneba?

11. qvemoT mocemulia aSS-Si 1998 wel amerikeli qmrebis asakis sixSireTa

ganawileba (U.S. Census Bureau, 1998, marti).

a) Tu SemTxveviT avarCevT erT ojaxs,

ramdenia albaToba imisa, rom qmari

44 wlis an ufro axalgazrda iqneba?

b) Tu SemTxveviT avarCevT erT ojaxs,

ramdenia albaToba imisa, rom qmari

34 wlis an ufri axalgazrda, anda 75

wlis da ufro xnovani iqneba?

ojaxis

wevrTa

raodenoba

fardobiTi

sixSire

2 3 4 5 6

7 da meti

.427

.230

.206

.092

.029

.016

asaki fardobiTi

sixSire

24 an naklebi 25-34 35-44 45-54 55-59 60-61 62-64 65-69 70-74 75-79 80-84

85 an meti

.023

.178

.257

.217

.079

.029

.040

.062

.049

.039

.019

.008

Tavi VII. statistikis gamoyeneba daskvnisa da

SefasebisTvis

X rogorc μ-s wertilovani Sefaseba

Caunacvlebeli Sefaseba

Zalmosili Sefaseba

Sefasebis sizuste

populaciis dispersiisa da standartuli gadaxris

Sefaseba

dispersia

standartuli gadaxra

saSualos SerCeviTi ganawileba: _ Sefasebis sizustis

gansazRvra

saSualos SerCeviTi ganawilebis Tvisebebi

saSualos standartuli Secdomis Sefaseba

faqtorebi, romlebic gavlenas axdenen Xs -is

mniSvnelobaze

Xs -is gamoyeneba

ganawilebis tipebis mimoxilva

populaciis saSualos intervaluri Sefaseba

ndobis intervalis ageba

თავი VII. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 2

winamdebare TavSi ganvixilavT populaciis parametrebis SefasebisTvis

aRweriTi statistikis gamoyenebas. rogorc pirvel TavSi aRvniSneT,

qcevaTmecnierebebis warmomadgenlebs xSirad surT SerCevaSi miRebuli Sedegebis

ufro farTo populaciaze ganzogadeba. magaliTad, saarCevno kampaniaSi CarTul

organizaciebs surT arCevnebis Sedegebis prognozireba, magram isini ver

gamokiTxaven amomrCevelTa did populaciaSi Semaval yvela amomrCevels. isini

zomaven amomrCevelTa populaciidan aRebul SerCevas da Semdeg axdenen miRebuli

Sedegebis populaciaze ganzogadebas. analogiurad, bevr SemTxvevaSi SeuZlebelia

did populaciaSi alkoholis moxmarebis, Sobadobis kontrolisadmi

damokidebulebis, avtomanqanaSi usafrTxoebis Rvedebis gamoyenebis an nebismieri

sxva qcevis gazomva. Sefaseba unda moxdes sakvlevi populaciidan aRebuli

SerCevidan miRebuli statistikebis safuZvelze. aRweriTi statistikebis

safuZvelze populaciis parametrebis SefasebisTvis saWiroa SerCevis monacemebze

dayrdnobiT daskvnebis gakeTeba. daskvna aris raRac cnobilis safuZvelze raRac

ucnobze msjelobis procesi. statistikuri daskvna aris populaciis ucnobi

parametrebis Sefaseba SerCevis cnobil statistikebze dayrdnobiT. winamdebare

TavSi Tavdapirvelad ganvixilavT SerCevis saSualos ( )X , rogorc populaciis

saSualos (μ) Sefasebis, Tvisebebs; Semdeg gavecnobiT populaciis dispersiasa (s2)

da standartul gadaxras (s), rogorc, Sesabamisad, populaciis dispersiisa (σ2) da

standartuli gadaxris (σ) Sefasebebs; bolos ki SevexebiT X -is mier μ-s

Sefasebis sizustis sakiTxs.

statistikuri daskvna (Statistical inference) _ SerCevisgan miRebuli statistikis mixedviT

populaciis Sefaseba.

X , rogorc μ-s wertilovani Sefaseba

SerCevis saSualo ( )X xSirad

gamoiyeneba, rogorc populaciis saSualos

(μ) wertilovani Sefaseba. wertilovani

Sefaseba aris parametris mniSvnelobis

Sefaseba statistikis mniSvnelobiT (anu erTi ricxviT). davubrundeT me-3 TavSi

ganxilul SfoTvis gazomvas. konsultantma Tavisi universitetis studentTa

populaciidan SearCia 100 studenti da gazoma maTi gamocdebTan dakavSirebuli

SfoTva. SerCevis qulaTa saSualo warmodgenilia cxrilSi N3.3 da Seadgens 41.64-

s, xolo standartuli gadaxra _ 8.66-s. SerCevis saSualo 41.64 SeiZleba

gamoyenebul iqnas, rogorc μ-s (gamocdebTan dakavSirebuli SfoTvis im

wertilovani Sefaseba (Point estimation): _

parametris mniSvnelobis Sefaseba

statistikis mniSvnelobiT (anu erTi

ricxviT).


populaciis saSualo, saidanac SerCeva iyo aRebuli) Caunacvlebeli da

Zalmosili Sefaseba.

Caunacvlebeli Sefaseba

Caunacvlebeli Sefaseba aris statistika,

romelic monacemTa usasrulod didi

raodenobisTvis Sesafasebeli parametris

mniSvnelobis tolia. amrigad, Caunacvlebel

SefasebasTan saqme gvaqvs maSin, rodesac

garkveuli moculobis SemTxveviT SerCevaTa

usasrulo raodenobis aRebisas statistikis mniSvnelobaTa saSualo Sesafasebeli

parametris tolia. SerCevis saSualo ( )X imitom aris μ-is Caunacvlebeli

Sefaseba, rom Tu Cven populaciidan aviRebT N moculobis yvela SesaZlo

SemTxveviT SerCevas, maSin SerCevaTa saSualoebis saSualo μ-s toli iqneba.

SerCevis dispersia S2 imitom ar aris σ2-is Caunacvlebeli Sefaseba, rom is

sistematurad gadaixreba misgan, Tumca, populaciis dispersiis Sefaseba (s2) σ2-is

Caunacvlebel Sefasebas warmoadgens.

Zalmosili Sefaseba

Zalmosili Sefaseba aris statistika,

romlisTvisac albaToba imisa, rom es

statistika ufro axlos iqneba

Sesafasebel parametrTan, izrdeba

SerCevis moculobis zrdasTan erTad.

vinaidan is Zalmosili Sefasebaa, SerCevis 25 quliT gamoTvlil saSualos ufro

meti albaToba aqvs ufro axlos iyos μ-sTan, vidre SerCevis 5 qulaze

dayrdnobiT gamoTvlil saSualos. am maxasiaTeblis gacnobiT intuiciurad

gveqneba warmodgena SerCevis saSualoze. rac ufro didia populaciidan aRebuli

SerCevis moculoba, miT meti qulaa CarTuli X -is gamoTvlaSi da, Sesabamisad,

mosalodnelia, rom X ukeTesad Seafasebs μ-s. Tu Cven SevZlebT iseTi SerCevis

aRebas, romelic moicavs mTel sakvlev populacias, maSin X igive iqneba, rac μ

da, aqedan gamomdinare, Sefasebisas ar gveqneba Secdoma.

Sefasebis sizuste

SerCevis saSualo μ-s Caunacvlebeli da Zalmosili Sefasebaa, magram Cven

uyuradRebod ar unda davtovoT is faqti, rom Sefaseba mxolod miaxloebiTi

gamoTvlaa. nebismieri Sefasebisas SeuZlebelia, rom X zustad μ-s toli iyos. is,

Caunacvlebeli Sefaseba (Unbiased

estimation): _ statistika, romlis

mniSvnelobaTa saSualo,

gamoTvlili usasrulod bevri

SerCevidan, Sesafasebeli

parametris mniSvnelobis tolia.

Zalmosili Sefaseba (Consistent estimation): _

statistika, romlisTvisac albaToba imisa,

rom es statistika ufro axlos iqneba

Sesafasebel parametrTan, izrdeba SerCevis

moculobis zrdasTan erTad.


Tu ramdenad kargad afasebs X μ-s, damokidebulia SerCevis meTodze, SerCevis

moculobasa da populaciaSi qulaTa cvalebadobaze.

SerCevis meTodi

populaciidan aRebuli nebismieri SerCeva garkveul azrs gviqmnis am

populaciis maxasiaTeblebis Sesaxeb. faqtobrivad, informaciis uqonlobis

pirobebSi X -is mixedviT SegviZlia „saukeTeso varaudi gamovTqvaT“ Sesabamisi μ-s

mniSvnelobis Sesaxeb, magram raTa darwmunebuli viyoT, rom X warmoadgens μ-s

karg Sefasebas, SerCeva unda xasiaTdebodes igive maxasiaTeblebiT, riTac

sakvlevi populacia. arsebobs reprezentaciuli SerCevis meTodebis mTeli rigi,

romlebic mowodebulia imis uzrunvelsayofad, rom SerCeva iyos im populaciis

reprezentaciuli, saidanac is iqna aRebuli. Cven ganvixilavT or meTods.

martivi SemTxveviTi SerCeva. martivi

SemTxveviTi SerCeva (romelsac xSirad

SemTxveviT SerCevasac uwodeben)

ganisazRvreba, rogorc populaciidan

wevrebis SerCevis iseTi procesi,

rodesac:

1. populaciis yvela wevrs

SerCevaSi moxvedris toli Sansi aqvs.

2. erTi wevris arCeva populaciis nebismieri sxva wevris arCevisgan

damoukideblad xdeba.

SemTxveviTi SerCeva miukerZoebeli SerCevaa. Sesabamisad, SerCevis

maxasiaTeblebi ar unda gansxvavdebodes im populaciis maxasiaTeblebisgan raime

sistematuri da mudmivi gziT, saidanac es SerCeva iqna aRebuli; Tumca,

SemTxveviTi SerCeva ar iZleva imis garantias, rom konkretuli SerCeva zustad

populaciis reprezentaciuli iqneba. vinaidan SerCevebi, Cveulebriv, populaciaze

gacilebiT mcire moculobisaa, Cven ar unda movelodeT, rom SerCevis

maxasiaTeblebi zustad iseve iqneba ganawilebuli, rogorc populaciaSi.

zogierTi SemTxveviTi SerCeva ufro reprezentaciuli iqneba populaciisa, vidre

sxvebi. Tumca, SemTxveviTi SerCeva uzrunvelyofs, rom, zogadad (anu usasrulod

didi raodenobis SerCevaTa SemTxvevaSi), aseTi SerCevebi populaciis

reprezentaciuli iqneba.

martivi SemTxveviTi SerCeva (Simple random

sampling): _ populaciidan wevrebis SerCevis

iseTi procesi, rodesac populaciis

TiToeul wevrs SerCevaSi moxvedris toli

Sansebi aqvT da erTi wevris arCeva

populaciis nebismieri sxva wevris

arCevisgan damoukideblad xdeba.

stratificirebuli SemTxveviTi SerCeva (Stratified random sampling): _ SerCevis meTodi,

rodesac populaciis wevrebi erTiandeba homogenur qvejgufebSi, romelTac stratebi

ewodeba. populaciis wevrebi SemTxveviT airCevian stratidan im proporciiT, romliTac

es strata populaciaSi gvxvdeba.


stratificirebuli SemTxveviTi SerCeva. stratificirebuli SemTxveviTi

SerCevisas populaciis weverebi erTiandebian homogenur qvejgufebSi, romelTac

stratebi ewodebaT. strata gulisxmobs msgavsi obieqtebis Sres an dones. aqedan

gamomdinare, stratebSi Semavali wevrebi homogenurebi anu msgavsni arian

zigierTi iseTi maxasiaTeblis mixedviT rogoricaa sqesi, asaki, anda ganaTlebis

done. Semdeg populaciis wevrebs SemTxveviTad irCeven stratidan im proporciiT,

romliTac es strata populaciaSi gvxvdeba. amis sailustraciod ganvixiloT

erTi universitetis studentebis populacia. am populaciaSi sxvadasxva SesaZlo

homogenuri qvejgufia, magaliTad, qvejgufebi swavlis xangrZliobis mixedviT: _

swavlis pirveli, meore, mesame da meoTxe wlis studentebi. TiToeul am

qvejgufSi Semavali wevri homogenuri anu erTmaneTis msgavsia iseTi

maxasiaTeblis mixedviT, rogoricaa swavlis xangrZlivoba (wlebiT). cxadia, rom

mravali sxva SesaZlo qvejgufi arsebobs, romelic dajgufebulia iseTi

maxasiaTeblebis mixedviT rogoricaa, magaliTad, sqesi, ZiriTadi sagani an naxevar

ganakveTze muSaoba.

davuSvaT, studentTa stratificireba swavlis xangrZlivobis (wlebiT)

mixedviT movaxdineT da TiToeul strataSi (anu doneze an qvejgufSi) moxvda

studentTa Semdegi raodenoba: _ pirveli weli _ 40%, meore weli _ 25%, mesame

weli _ 20 % da meoTxe weli _ 15%. am populaciidan martivi SemTxveviTi SerCeva

gvaZlevs imas, rom swavlis TiToeuli wlisTvis populaciidan arCeul studentTa

proporcia analogiuri proporciiTve iqneba warmodgenili mraval SerCevaSi,

magram aseTi proporcia ver gveqneba nebismier erT konkretul SerCevaSi.

stratificirebuli SemTxveviTi SerCevisas individebis SerCeva xdeba stratis

populaciaSi Sexvedris sixSiris proporciulad. Tu meore wlis studentebi

populaciis 25%-s Seadgenen, maSin SerCevis 25% SemTxveviT SeiCeva am stratidan.

rodesac SesaZlebelia populaciaSi stratebis gamoyofa da Semdeg rodesac

konkretuli stratis wevroba ukavSirdeba gasazom qcevas, maSin

stratificirebuli SemTxveviTi SerCeva mosalodnelia, rom ufro

reprezentaciul SerCevas gvaZlevdes, vidre SemTxveviTi SerCeva.

SerCevis moculoba

Zalmosili Sefasebis ganxilvisas aRvniSneT, rom rac ufro didia SerCeva, miT

metia albaToba imisa, rom SerCevis saSualo ufro axlos iyos populaciis

saSualos mniSvnelobasTan. amrigad, Zalmosili Sefasebisas SerCevis moculoba

miRebuli Sefasebis sizustes ukavSirdeba.

populaciis sazomvis cvalebadoba

ganvixiloT mcired albaTuri SesaZlebloba, rodesac populaciis yvela

wevris qula erTmaneTis tolia. am SemTxvevaSi TiToeuli qula populaciis

saSualos toli, xolo populaciis σ nulis toli iqneba. am pirobebSi naTelia,


rom nebismieri moculobis SerCevidan miRebuli X μ-s toli iqneba. SerCevis

SefasebaSi ar gveqneba Secdoma. meore mxriv, ganvixiloT populacia, romelic

gazomvis Sedegad miRebuli qulebis didi cvalebadobiT xasiaTdeba da romlis σ

Zalian didia. miuxedavad imisa, rom es iSviaTad SeiZleba moxdes, erT SemTxveviT

SerCevaSi SeiZleba SemTxveviT mTlianad dabali qulebi miviRoT. am dros

miRebuli X mniSvnelovnad gansxvavebuli iqneba μ-isgan. analogiurad, μ-is

SedarebiT cudi Sefaseba gveqneba, Tu populaciidan SemTxveviT, ZiriTadad,

maRali qulebi avirCieT. amrigad, X -is, rogorc μ-is wertilobani Sefasebis,

sizuste populaciaSi cvalebadobis zomas ukavSirdeba. rac ufro didia

populaciis standartuli gadaxra, miT naklebia albaToba imisa, rom konkretuli

SerCevis saSualo populaciis saSualos kargi Sefaseba iqneba.

araSemTxveviTi SerCeva

rogorc vnaxeT, SerCevis meTodi gadamwyvetia imisaTvis, Tu ramdenad iqneba

SerCeva populaciis reprezentaciuli; Tumca, sazogadoebrivi azris mraval

gamokiTxvaSi araSemTxveviTi SerCevebi gamoiyeneba. xanmokle komunikaciebis

epoqaSi, rogorc Cans, mravali Cvengani dainteresebulia imiT, Tu ras fiqroben

adamianebi ama Tu im sakiTxis Sesaxeb. am interesis dasakmayofileblad xanmokle

gamokiTxvebis sruliad axali industria Seiqmna. gazeTebi da televiziebi

mimdinare dRis aqtualur Temebze mokle satelefono gamokiTxvebs gvTavazoben.

TiToeuli aseTi gamokiTxviT informacia SerCevidan _ anu im adamianebisgan,

romlebic gamokiTxvaSi monawileoben _ grovdeba, magram ramdenad gamosadegia

aseTi gamokiTxvebi im populaciis Sesaxeb daskvnebis gasakeTeblad, romlidanac

es SerCevaa aRebuli? samwuxarod, arc ise gamosadegia ramdenime mosazrebis gamo.

pirveli sirTule isaa, rom populacia zustad unda iqnes gansazRvruli.

gavixsenoT pirveli Tavidan, rom populacia saerTo maxasiaTeblis mqone

adamianTa, cxovelTa, saganTa an movlenaTa sruli erTobliobaa. SerCevis

mixedviT populaciaze daskvnis gasakeTeblad aucileblad unda vicodeT, Tu ra

saerTo maxasiaTebeli aqvs mocemul populacias, raTa davrwmundeT, rom SerCevis

yvela wevrs aqvs es maxasiaTebeli. meore: _ SerCeva populaciis reprezentaciuli

unda iyos, rac SerCevis sistematuri procesis ganxorcielebas gulisxmobs. arc

erTi es moTxovna ar sruldeba, rodesac individebi TviTon iReben gazeTsa Tu

televiziaSi darekvis an internetiT azris gamoTqmis gadawyvetilebas. magaliTad,

erT-erT internet-gamokiTxvaSi amerikis moqalaqeebs ekiTxebodnen, Tu ramdenad

Sors apirebdnen samogzaurod wasvlas Svebulebis dros; azris gamoTqma

paraskevobiT an dasvenebis dReebSi iyo SesaZlebeli. gamokiTxvaSi daaxloebiT

20,000 adamianma miiRo monawileoba, romelTagan 78% miuTiTebda, rom ar hqondaT

dagegmili, Tu rogor gaatarebda Svebulebas. ramdenad SegviZlia, darwmunebulni


viyoT, rom es ricxvi realurad asaxavs amerikelTa Svebulebis dros samogzauro

gegmebs? samwuxarod, am gamokiTxvis SerCeva TviTSerCeulia da mxolod im

adamianebisgan Sedgeba, romlebic dainteresdnen pasuxis gacemiT. metic, vinaidan

bevri adamiani ukve wasuli iyo samogzaurod manamde, sanam gamokiTxva

Catardeboda, isini verafers upasuxebdnen Svebulebis samomavlo gegmebis Sesaxeb

dasmul SekiTxvas. amrigad, Cven ver ganvsazRvravT im populacias, romlidanac es

SerCevaa aRebuli; Sesabamisad, am SerCevidan populaciis Sesaxeb daskvnis gakeTeba

aramizanSewonilia. aseTi gamokiTxvebi mecnierulad gamousadegaria.

SeamowmeT Tqveni codna VII.1

1. ganmarteT: _ Zalmosili Sefaseba, martivi SemTxveviTi SerCeva,

statistikuri daskvna, stratificirebuli SemTxveviTi SerCeva,

Caunacvlebeli Sefaseba.

2. davuSvaT, SfoTviTi aSlilobis mqone individTa populaciaSi 27%-s aqvs

generalizirebuli SfoTviTi aSliloba, 23%-s _ agorafobia, 19%-s _

martivi fobia, 22%-s _ panikuri aSliloba da 9%-s _ obsesiur-

kompulsuri aSliloba. Tqven am populaciidan SemTxveviTi SerCevis

stratificirebas da stratad aSlilobebis tipis gamoyenebas apirebT.

a) ra proporciiT iqneba warmodgenili TiToeuli strata Tqvens

SerCevaSi?

b) Tu Tqven aiRebT 100 individisgan Semdgar SerCevas, ramdeni adamiani

moxvdeba TiToeul strataSi?

g) Tu Tqven aiRebT erT SeTxveviT SerCevas, ramdeni adamiani moxvdeba

TiToeul strataSi?

populaciis dispersiisa da standartuli gadaxris Sefaseba

dispersia

populaciis dispersia (σ2) mexuTe TavSi ganisazRra, rogorc:

( )populaciaN

X∑ −=

22 µ

σ ,

Tumca, rogorc ukve aRvniSneT, Cveulebriv, SeuZlebelia populaciis yvela

wevris gazomva. aqedan gamomdinare, populaciis dispersiis (σ2) Sefaseba, rogorc

wesi, SerCevaze dayrdnobiT xdeba. rogorc me-5 TavSi vnaxeT, σ2-is Caunacvlebeli

da Zalmosili Sefaseba gamoiTvleba formuliT

( )1

2

2

−

−= ∑

NXX

s .


yuradReba miaqcieT, rom wiladis mniSvneli aris ara _ N, aramed _ (N_1).

rogorc me-5 TavSi vTqviT, SerCevis dispersia, romelic gamoiTvleba formuliT

( )N

XXS ∑ −

=2

2,

aris σ2-is Canacvlebuli Sefaseba. rogorc wesi, is bolomde ver (naklebad)

afasebs σ2-is mniSvnelobas. aqedan gamomdinare, Tu calke ar aRvniSneT, dispersiis

saxiT yovelTvis s2-s gamoviyenebT.

standartuli gadaxra

σ-is Sesafaseblad populaciis standartuli gadaxris Sefaseba s gamoiyeneba. s2-

isgan, rogorc σ2-is Sefasebisgan, gansxvavebiT, s ar aris σ-is Caunacvlebeli

Sefaseba. metic, s odnav arazustia σ-s Sefasebisas; magram vinaidan s-is

SemTxvevaSi Canacvleba mcirea da kidev ufro mcirdeba SerCevis moculobis

zrdasTan erTad, mkvlevarTa umravlesoba σ-is Sefasebis saWiroebisas s-s iyenebs.

saSualos SerCeviTi ganawileba: _ Sefasebis sizustis gansazRvra

populaciis saSualos wertilovani Sefasebis sizuste damokidebulia rogorc

SerCevis moculobaze, ise _ populaciis standartul gadaxraze. garda im

SemTxvevebisa, rodesac SerCeva moicavs populaciis yvela qulas an rodesac

populaciis qulebi ar xasiaTdeba cvalebadobiT, μ-s Sefasebisas X -s yovelTvis

Tan sdevs Secdoma. miuxedavad imisa, rom Cven darwmunebiT arasodes viciT, Tu

ramdenad zustia populaciis parametris konkretuli Sefaseba, Sefasebisas

mosalodneli Secdomis zomis gansazRvra SerCeviTi ganawilebis cnebis

gamoyenebiT SegviZlia.

SerCeviTi ganawileba aris populaciidan

aRebuli N moculobis mqone yvela SesaZlo

SerCevidan miRebuli statistikis

mniSvnelobebis albaTuri Teoriuli

ganawileba. amrigad, saSualos SerCeviTi

ganawileba aris X -is mniSvnelobebis

ganawileba, rodesac populaciidan

aRebulia N moculobis mqone yvela

SesaZlo SerCeva. SerCeviTi ganawileba

Teoriulad ganisazRvreba maTematikuri

gantolebiT, vinaidan Cven verasodes

SevZlebT populaciidan yvela SesaZlo SerCevis aRebas. Semdeg TavebSi

warmodgenili TiToeuli statistikuri kriteriumi am statistikis SerCeviTi

SerCeviTi ganawileba (Sampling

distribution): _ populaciidan aRebuli N

moculobis mqone yvela SesaZlo

SerCevidan miRebuli statistikis

mniSvnelobebis albaTuri Teoriuli

ganawileba.

saSualos SerCeviTi ganawileba

(Sampling distribution of the mean): _ X -is

mniSvnelobebis ganawileba, rodesac

populaciidan aRebulia N moculobis

mqone yvela SesaZlo SerCeva.


ganawilebis codnas saWiroebs. es SerCeviTi ganawilebebi maTematikuri

gantolebebiT miRebuli Teoriuli SerCeviTi ganawilebebia.

miuxedavad imisa, rom statistikuri kriteriumebi Teoriul SerCeviT

ganawilebas iyeneben, masSi CarTul cnebaTa sailustraciod Cven saSualos

ramdenime empiriul SerCeviT ganawilebas ganvixilavT. sityva „empiriuli“ niSnavs

dakvirvebads an dakvirvebis Sedegad miRebuls. aqedan gamomdinare, saSualos

empiriuli SerCeviTi ganawileba SeiZleba miviRoT populaciidan N moculobis

mqone ramdenime SerCevis aRebiT, TiToeuli SerCevisTvis X -is gamoTvliTa da

Semdeg miRebuli saSualoebis sixSireTa ganawilebis agebiT. amis sailustraciod,

aviRoT 100 qulisgan Semdgari populacia, sadac qulebi meryeobs 8-dan 20-mde

diapazonSi da romlis μ = 14.0 da σ = 2.14. populaciis 100 qulis sixSireTa

poligoni warmodgenilia suraTze N7.1. rogorc suraTze xedavT, qulaTa

ganawileba simetriuli da unimodaluria.

0

2

4

6

8

10

12

14

16

18

20

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

qulaTa mniSvnelobebi

six

Sir

e

saSualos empiriuli SerCeviTi ganawileba sami sxvadasxva moculobis mqone

SerCevidan miviReT: _ N = 2, N = 5 da N = 10. es SerCevebi kompiuteris daxmarebiT

populaciidan martivi SemTxveviTi SerCevis gziT miviReT. SerCevis TiToeuli

moculobisTvis SerCeul iqna 100 SemTxveviTi SerCeva; TiToeuli maTganisTvis

gamovTvaleT saSualo. saSualos empiriuli SerCeviTi ganawilebebi miRebuli iyo

sami sxvadasxva zomis SerCevisgan: _ N = 2, N = 5 da N = 10, rogorc naCvenebia

suraTze N7.2. TiToeuli (a), (b) da (g) ganawileba warmoadgens konkretuli

moculobis mqone SerCevis (N = 2, N = 5 da N = 10, Sesabamisad) saSualos

empiriul SerCeviT ganawilebas.

სურათი 7-2 არასრულია

suraTi N7.1. sixSireTa ganawileba 100 qulisgan Semdgari

populaciisTvis, romlis μ = 14.0 da σ = 2.14.


saSualos SerCeviTi ganawilebis Tvisebebi

suraTze N7.2 mocemuli empiriuli SerCeviTi ganawileba saSualos Teoriuli

SerCeviTi ganawilebis ramdenime Tvisebis ilustraciaa. gavixsenoT, rom sixSireTa

ganawilebis aRwera formis, centraluri tendenciisa da cvalebadobis sazomebis

terminebiT aris SesaZlebeli. vinaidan saSualos SerCeviTi ganawileba sixSireTa

ganawilebas warmoadgens, isic SeiZleba aRiweros formis, saSualosa da

standartuli gadaxris terminebiT.

saSualos SerCeviTi ganawilebis forma

suraTze N7.2 mocemuli sami SerCeviTi ganawileba SedarebiT simetriuli da

unimodaluria. simetriidan gadaxra ubralod saSualo mniSvnelobebis

SemTxveviTi cvalebadobiT aixsneba. yuradReba miaqcieT, rom ganawileba miT ufro

wamaxvilebuli da naklebad gafantulia, rac ufro izrdeba SerCevis moculoba

(SeadareT SerCevebi, romelTa moculobebia N = 2 da N = 10). principi, romelsac

centraluri zRvris Teorema ewodeba, saSualos SerCeviT ganawilebas Seexeba:

SerCevis moculobis zrdasTan erTad saSualos SerCeviTi ganawileba normalur

ganawilebas uaxlovdeba. saSualos SerCeviTi ganawilebis normalur

ganawilebasTan aseT miaxloeba xdeba maSin, rodesac populaciis qulaTa

ganawilebis forma simetriuli an asimetriulia.

centraluri zRvris Teorema (Central limit theorem): _ maTematikuri Teorema, romlis Tanaxmad,

SerCevis moculobis zrdasTan erTad saSualos SerCeviTi ganawileba normalur

ganawilebas uaxlovdeba.

suraTi N7.2. 7.1 suraTze mocemuli populaciis qulebidan aRebuli 100

SerCevis saSualoebis sami empiriuli SerCeviTi ganawileba. SerCevis

moculobaa: _ (a) N = 2, (b) N = 5 da (g) N = 10.

(a) N = 2 (b) N = 5 (g) N = 10

SerCevis saSualos mniSvneloba ( )X

sixSire


ramdenad didi unda iyos N imisaTvis, rom saSualos SerCeviTma ganawilebam

normalur ganawilebasTan miaxloeba daiwyos? am kiTxvaze pasuxi damokidebulia

populaciaSi qulaTa ganawilebis formaze. populacia, saidanac Cven SerCevebi

aviReT, ar aris normalurad ganawilebuli, magram is unimodaluri da

simetriulia (ix. suraTi N7.1). aseTi ganawilebebisTvis saSualos SerCeviTi

ganawileba, rodesac SerCevis moculoba iseTi mcirea, rogoricaa N = 5 an N = 10,

normaluri ganawilebis formasTan miaxloebas iwyebs (ix. suraTi N7.2, (b) da (g)).

Tu populaciaSi qulaTa ganawileba asimetriulia, maSin normalur

ganawilebasTan misaaxloeblad SerCevis moculoba saSualos SerCeviTi

ganawilebisTvis ufro didi unda iyos. mravali statistikosi Tvlis, rom Tu

populaciis ganawileba ar aris Zalian asimetriuli, SerCevis moculoba N = 30

aris is moculoba, rodesac saSualos SerCeviTi ganawileba axlos iqneba

normalur ganawilebasTan.

centraluri zRvris Teoremaze dafuZnebuli saSualos Teoriuli SerCeviTi

ganawileba warmodgenilia suraTze N7.3. yuradReba miaqcieT, rom es ganawileba

simetriuli, unimodaluri da normaluria. Semdeg TavebSi daskvniTi statistikis

kriteriumebis ganxilvisas swored am da sxva Teoriul SerCeviT ganawilebebs

gamoviyenebT. empiriuli SerCeviTi ganawilebebi, rogoricaa, magaliTad, suraTze

N7.2 warmodgenili ganawilebebi, mxolod SerCeviTi ganawilebis cnebis

asaxsnelad gamoiyeneba.

Xµ _ saSualos Teoriuli SerCeviTi ganawilebis saSualo.

XX _ saSualos empiriuli SerCeviTi ganawilebis saSualo.

saSualos SerCeviTi ganawilebis saSualo

Teoriul SerCeviT ganawilebaSi SerCevis saSualoebis saSualo (aRiniSneba

Xµ -iT) populaciaSi Semaval calkeul qulaTa saSualos (μ) toli iqneba, xolo

saSualos empiriul SerCeviT ganawilebaSi SerCevis saSualoebis saSualo

(aRiniSneba XX -iT) ar iqneba zustad populaciis saSualos toli. es mosazreba

naCvenebia suraTze N7.2 warmodgenil empiriul SerCeviT ganawilebaSi. sam

SerCeviT ganawilebaSi SerCevis saSualoebis saSualo ar aris zustad 14.0-is (anu

sawyisi populaciis saSualos) toli. (a), (b) da (g) ganawilebebisTvis saSualos

mniSvnelobebia 96.13=XX , 14.14=XX da 99.13=XX .


SerCevis Secdoma da saSualos standartuli Secdoma

SerCevis Secdoma. suraTze N7.2 mocemuli

TiToeuli SerCevis saSualo warmoadgens im

populaciis saSualos Sefasebas, saidanac isini

iqnen aRebuli (am magaliTSi μ = 14). rogorc

xedavT, zogierTi SerCevis saSualo ufro zustad afasebs μ-s, vidre sxvebi.

SerCevis Secdoma aris is sidide, romliTac konkretuli SerCevis saSualo

gansxvavdeba populaciis saSualosgan (anu µ−X ). Tu X zustad imdenia,

ramdenic μ anu Tu µ=X da amitom 0=− µX , maSin SefasebaSi ar gveqneba

SerCevis Secdoma. suraTi N7.2 gviCvenebs, rom SerCevis Secdoma SerCevis

moculobas ukavSirdeba: _ rac ufro didia SerCevis moculoba, miT naklebia

SerCevis Secdoma. suraTze N7.2a mocemuli 100 SerCevis saSualos mniSvnelobebi

(sadac TiToeuli saSualo or qulas efuZneba) ufro cvalebadia, vidre N7.2b da

N7.2g suraTze mocemuli SerCevis saSualoebi, romelic, Sesabamisad, 5 an 10

quliT aris gamoTvlili. amrigad, SerCevis Secdoma ufro TvalsaCinoa suraTze

N7.2a, vidre N7.2b anda suraTze N7.2g.

saSualos SerCeviT

ganawilebaSi SerCevis

saSualoebis cvalebadobis

sidide aRiwereba SerCevis

saSualoebis mniSvnelobebiT

gamoTvlili standartuli

gadaxriT. am standartul

gadaxras saSualos

standartul Secdomas an ubralod standartul Secdomas uwodeben; amiT mas

nedli qulebis populaciiT gamoTvlili standartuli gadaxrisgan ganasxvaveben.

SerCevis Secdoma (Sampling error): _

sidide, romliTac konkretuli

SerCevis saSualo gansxvavdeba

populaciis saSualosgan.

saSualos standartuli Secdoma, Xσ (Standard error of

the mean): _ saSualos SerCeviTi ganawilebis

standartuli gadaxra, romelic miiReba σ-as SerCevis

moculobis kvadratul fesvTan SefardebiT.

saSualos standartuli Secdomis Sefaseba, Xs

(Estimated standard error of the mean) _ s-is gamoyenebiT σ-as,

saSualos standatruli Secdomis Sefaseba.

-3 Xσ -2 Xσ -1 Xσ Xµ +1 Xσ +2 Xσ +3 Xσ

suraTi N7.3. saSualos Teoriuli SerCeviTi ganawileba,

romlis saSualoa Xµ da standartuli Secdoma Xσ


saSualos standartuli Secdoma, romelic aRiniSneba simboloTi Xσ , populaciis

standartuli gadaxrisa da SerCevis moculobidan kvadratuli fesvis Sefardebis

tolia. formulis saxiT:

NX

σσ ==Secdoma istandartul saSualos .

sadac σ aris populaciaSi Semaval qulaTa standartuli gadaxra, xolo N _

SerCevis moculoba. saSualos standartuli Secdoma Xσ iseTive standartuli

gadaxraa, rogoric σ, im gansxvavebiT, rom Xσ SerCevis saSualoebis ganawilebis

standartuli gadaxraa da ara _ populaciaSi Semaval nedl qulaTa standartuli

gadaxra (rasac σ warmoadgens).

SexedeT suraTze N7.3 mocemul saSualos Teoriul SerCeviT ganawilebas.

yuradReba miaqcieT, rom misi saSualo aRniSnulia simboloTi Xµ , xolo misi

standartuli gadaxra _ simboloTi Xσ . me-6 TavSi ganvixileT normaluri

ganawilebis qveS mdebare farTobi da avxseniT, Tu rogor unda vipovoT qulaTa

wili ganawilebis qveS mdebare farTobebiT. vinaidan saSualos SerCeviTi

ganawileba normaluri ganawilebaa, aqac ZalaSi rCeba es damokidebuleba, Tumca,

am SemTxvevaSi Cven gvaqvs ara intervalebSi moxvedrili nedli qulebi, aramed _

SerCevis saSualoebi. ase magaliTad, SerCeviTi ganawilebis saSualos qvemoT erT

standartul Secdomasa da SerCeviTi ganawilebis saSualos zemoT erT

standartul Secdomas Soris intervali (anu intervali XX σµ 1− -dan XX σµ 1+ -mde)

moicavs SerCevaTa saSualoebis .6826-s, intervali XX σµ 2− -dan XX σµ 2+ -mde _

SerCevaTa saSualoebis .9544-s, xolo intervali XX σµ 3− -dan XX σµ 3+ -mde _

SerCevaTa saSualoebis .9974-s. yuradReba miaqcieT, rom am intervalebis

aRniSvnisas gamoviyeneT simbolo Xµ , radgan saSualos SerCeviT ganawilebasTan

gvaqvs saqme. vinaidan µµ =X , Cven SegviZlia, rom Xµ -is nacvlad µ gamoviyenoT,

rogorc es Semdeg magaliTSia naCvenebi.

am damokidebulebis gamoyenebis sailustraciod, davuSvaT, rom qulaTa

populaciis saSulo μ = 50 da σ = 6, xolo am populaciidan SemTxveviT aRebulia

N = 9 moculobis SerCevebi. am SerCeviTi ganawilebisTvis saSualos standartuli

Secdoma gamoiTvleba formuliT

239

96

====NX

σσ ,

miRebuli mniSvneloba 2=Xσ warmoadgens σ = 6 mqone populaciidan aRebuli

SerCevebis (romelTa N = 9) saSualos SerCeviTi ganawilebis standartul


gadaxras. amrigad, populaciidan aRebuli N = 9 SerCevebisTvis saSualos

SerCeviT Teoriul ganawilebaSi SerCevaTa saSualoebis .6826 xvdeba XX σµ 1− -dan

(anu 50-2=48) XX σµ 1+ -mde (anu 50+2=52), anu 48-dan 52-mde intervalSi; XX σµ 2− -dan

(anu 50-4=46) XX σµ 2+ -mde (anu 50+4=54), anu 46-dan 54-mde intervali moicavs

SerCevaTa saSualoebis .9544-s, xolo XX σµ 3− -dan (anu 50-6=44) XX σµ 3+ -mde (anu

50+6=56), anu 44-dan 56-mde intervali _ X -is mniSvnelobaTa .9974-s. es informacia

X -is, rogorc μ-s Sefasebis, sizustis Sesafaseblad gamogvadgeba.

standartuli normaluri sxvaobis gamoyeneba saSualos SerCeviT ganawilebaSi

saSualos SerCeviTi ganawileba normaluri ganawilebaa, romlis saSualoa Xµ ,

xolo standartuli Secdoma _ Xσ . Cven saSualos SerCeviTi ganawilebis X -is

mniSvnelobis standartuli normaluri ganawilebis qulad gardaqmna SegviZlia

formuliT

X

XXz

σµ−

=realuri .

am z qulis gamoyeneba SerCevis saSualos mimarT isevea SesaZlebeli, rogorc

calkeuli qulis mimarT (ix. Tavi 6). amis ilustracias warmoadgens magaliTi N7-1.

magaliTi N7.1

saSualoebis SerCeviTi ganawilebis gamoyeneba

amocana

davuSvaT, gvaqvs qulaTa normalurad ganawilebuli populacia, romlis μ = 80

da σ = 12. Cven vgegmavT, rom am populaciidan aviRoT didi raodenobiT

SemTxveviTi SerCevebi, romelTa moculoba N = 16 da gamovTvaloT TiToeuli

SerCevis saSulo. SerCevaTa saSualoebis ra nawili iqneba 82-is toli an masze

meti?

amoxsna

am amocanis amosaxsnelad saWiroa aRvniSnoT, rom saSualos SerCeviTi

ganawileba normaluri ganawilebaa, romlis 80=Xµ da 0.34/1216/12 ===Xσ .

Teoriuli SerCeviTi ganawilebis nebismier intervalSi moxvedrili SerCevaTa

saSualoebis wili Semdegnairad gamoiTvleba: _ SerCevis saSualos mniSvneloba

(anu 82) gadaiyvaneba standartuli normaluri ganawilebis mniSvnelobad da a-1


cxrilSi moiZebneba misi Sesatyvisi wili. SerCevis saSualo zrealuri-is

mniSvnelobad gardaiqmneba formuliT

67.0.3

20.38082

+==−

=−

=X

XXz

σµ

realuri .

.

SerCevebis saSualoebisa da zrealuri-is mniSvnelobebi, rodesac SerCevis

saSualoa 82, mocemulia suraTze N7.4. a-1 cxrilis (g) sveti gviCvenebs, rom z =

+0.67-is zemoT mdebare farTobia .2514 (ix. suraTi N7.4, (g) RerZi). amrigad,

mosalodnelia, rom SerCevaTa saSualoebis .2514 nawili 82-is toli an masze meti

iyos. es daskvna albaTobis terminebiT ase Caiwereba:

( ) 2514.82 =≥Xp

აკლია სურათი 7.4

saSualos standartuli Secdomis Sefaseba

saSualoebis SerCeviTi ganawileba warmoadgens SerCevaTa saSualoebis

mosalodneli cvalebadobis Sefasebis meTods, magram naTlad Cans mocemuli

codnis gamoyenebis sazRvari. populaciis standartuli gadaxra σ populaciis

parametria, populaciis nebismieri parametri ki, rogorc wesi, ucnobia. maS, rogor

ganisazRvreba saSualos standartuli Secdoma, Tu ar viciT σ?

am kiTxvaze pasuxi SerCevis monacemebze dayrdnobiT σ-s SefasebaSi Zevs

(rogorc μ SevafaseT X -iT). am SemTxvevaSi saSualos standartuli Secdomis

Sefasebis misaRebad populaciis standartuli gadaxris Sefaseba (s) gamoiyeneba.

saSualos standartuli Secdomis Sefaseba, romelic aRiniSneba simboloTi Xs ,

efuZneba populaciis standartul gadaxras da formulis saxiT gamoisaxeba:

NssXX ===σ Sefaseba Secdomis istandartul saSualos ,

sadac s gamoiTvleba SerCevis qulebidan, xolo N SerCevis moculobaa. amrigad,

Xs warmoadgens saSualos SerCeviTi ganawilebis standartuli gadaxris

Sefasebas. samecniero statiaSi standartul Secdomas xSirad SE (da ara Xs )

simboloTi aRniSnaven.

suraTi N7.4. SerCevis saSualos (82) mdebareoba saSualos SerCeviT ganawilebaSi,

romlis μ = 80 da Xσ = 3. X RerZis warwerebi gviCvenebs: _ (a) SerCevebis

saSualos mniSvnelobebs, (b) SerCevebis saSualos mniSvnelobebis Sesabamis z

qulebs, (g) SerCevaTa saSualoebis wils, romelic 82-is tolia an metia masze.


mniSvnelovania Xσ -sa da Xs -is Soris arsebuli gansxvavebis gageba. Xσ

fiqsirebuli mniSvnelobaa, romelic SeiZleba ganisazRvros maSin, rodesac

cnobilia σ; Tumca, radgan CvenTvis iSviaTad aris σ cnobili, amitom, Cveulebriv,

Xσ -is Sefaseba Xs -iT xdeba. vinaidan Xs SerCevis s-is mniSvnelobas eyrdnoba,

sxvadasxva SerCevaSi misi mniSvneloba sxvadasxva iqneba. erTi da imave

populaciidan aRebuli sxvadasxva SemTxveviTi SerCeva sxvadasxva standartul

gadaxras mogvcems, sxvadasxva SerCevaSi qulaTa SemTxveviTi gansxvavebis gamo.

miuxedavad amisa, Xσ -is Sefasebis erTaderT safuZvels SerCevis qulebi

warmoadgenen.

faqtorebi, romlebic gavlenas axdenen Xs -is mniSvnelobaze

Xs -is formula Ns / gviCvenebs, Tu rogor ukavSirdeba SerCeviT ganawilebaSi

SerCevaTa saSualoebis cvalebadoba (1) populaciaSi Semavali qulebis

cvalebadobas da (2) SerCevis moculobas. rac ufro cvalebadia populaciis

qulebi, miT meti iqneba SerCevis qulaTa s; amasTan, s-is gazrdasTan erTad Xs -ic

izrdeba; Tumca, SerCevis moculobis (N) gazrdiT Xs -is formulis mniSvnelic

izrdeba, rac, Tavis mxriv, standartuli Secdomis Semcirebas ganapirobebs.

amrigad, qulaTa mcire cvalebadobis mqone populaciidan aRebuli didi

moculobis SemTxveviTi SerCevebis pirobebSi mcire Xs -is miRebas unda

movelodeT. miuxedavad imisa, rom mkvlevars SeuZlia SerCevis moculobis

kontroli, xSirad praqtikulad SeuZlebelia populaciis qulebis cvalebadobis

Semcireba. amrigad, qcevaTmecnierebis warmomadgenlebi Xs -is mniSvnelobis

Sesamcireblad xSirad didi zomis SerCevebs iyeneben.

Xs -is gamoyeneba

standartuli Secdoma saSualoebis SerCeviT ganawilebaSi SerCevis Secdomis

sididis gasazomad gamoiyeneba. Sesabamisad, Xs SeiZleba gamoviyenoT imis

gansasazRvrad, Tu ramdenad kargad afasebs SerCevis miRebuli saSualo

populaciis saSualos. rac ufro mcirea Xs , miT ufro darwmunebulni SegviZlia

viyoT, rom X mniSvnelovnad ar gansxvavdeba populaciis saSualosgan. me-8 TavSi,

Xs -is gamoyenebiT, detalurad aRvwerT X -iT μ-is Sefasebis sizustes; axla ki,

qvemoT ganxiluli sakiTxebis ganxilvis gasaioleblad, davuSvaT, rom CvenTvis

Xσ cnobilia, Tumca, unda gvaxsovdes, rom praqtikaSi Xσ ar aris cnobili da

mis nacvlad Xs unda gamoviyenoT.


ganawilebis tipebis mimoxilva

cxrilSi N7-1 gaerTianebulia aqamde ganxiluli ganawilebis tipebi da maTi

maxasiaTeblebis aRmniSvneli simboloebi. ganawileba SeiZleba aigos nedli

qulebiT, an populaciidan aRebuli SerCevebis saSualoebiT.

• nedli qulebis ganawileba populaciis yvela wevris gazomvis Sedegad

miiReba. Tu Cven gvaqvs populaciis gazomvis saSualeba, maSin SevZlebT

populaciis saSualos (μ), standartuli gadaxrisa (σ) da dispersiis (σ2)

gamoTvlas.

• populaciidan aRebuli SerCevis nedli qulebis ganawileba SerCevis

saSualos _ X -is, standartuli gadaxris _ s-isa da dispersiis _ s2-is

gamoyenebiT aRiwereba. es statistikuri sidideebi im populaciis

Sesabamisi parametris aRsawerad SeiZleba iqnes gamoyenebuli,

romlidanac es SerCevaa aRebuli.

cxrili N7.1. ganawilebis tipebi da maTi maxasiaTeblebi.

sazomi

ganawilebis

tipi miiReba saSualo gadaxra dispersia

nedli qulebi

populacia populaciis yvela wevris

gazomvis Sedegad.

μ σ σ2

SerCeva populaciidan aRebuli

SerCevis gazomviT. X s s2


Teoriuli Teoriulad maTematikuri

gantolebiT. Xµ Xσ X

2σ

empiriuli empiriulad populaciidan

SerCevebis aRebiT. XX Xs Xs2

• saSualoebis SerCeviTi ganawileba aris populaciidan aRebul SerCevaTa

saSualoebis ganawileba. SerCevis mocemuli moculobis SemTxvevaSi

saSualoebis Teoriuli SerCeviTi ganawileba matematikuri gantolebiT

miiReba. saSualoebis Teoriuli SerCeviTi ganawilebis saSualo

gamoixateba Xµ -iT, romelic im populaciis μ-is tolia, romlidanac

SerCeva iqna aRebuli. saSualoebis SerCeviTi ganawilebis standartuli

gadaxra gamoixateba Xσ -iT, xolo dispersia _ X2σ -iT. rogorc wesi, μ-

sa da Xσ -is Sefaseba X -isa da Xs -is safuZvelze xdeba.


• saSualoebis empiriuli SerCeviTi ganawileba populaciidan SerCevebis

realurad aRebiTa da Semdeg am SerCevaTa saSualoebis sixSireTa

ganawilebis agebiT miiReba. empiriuli SerCeviTi ganawileba, ZiriTadad,

mowodebulia SerCeviTi ganawilebis cnebis gagebis gasamartiveblad. maT,

Cveulebriv, ar iyeneben realur kvlevaSi.


1. ganmarteT: _ centraluri zRvariTi Teorema, saSualos standartuli

Secdomis Sefaseba, Xµ , SerCeviTi ganawileba, saSualos SerCeviTi

ganawileba, SerCevis Secdoma, Xσ , saSualos standartuli Secdoma.

2. aris Tu ara ( )∑ −= NXXS /22 formuliT miRebuli S2-is mniSvneloba σ2-

is Caunacvlebeli Sefaseba? Tu ar aris, ratom?

3. aris Tu ara ( ) ( )∑ −−= 1/22 NXXs formuliT miRebuli s2-is mniSvneloba

σ2-is Caunacvlebeli Sefaseba? Tu ar aris, ratom?

4. aris Tu ara ( ) ( )∑ −−= 1/2

NXXs formuliT miRebuli s-is mniSvneloba

σ-is Caunacvlebeli Sefaseba? Tu ar aris, ratom?

5. ra gansxvavebaa saSualos empiriul da Teoriul SerCeviT ganawilebebs

Soris?

6. CamoayalibeT centraluri zRvariTi Teorema.

7. davuSvaT, inteleqtis testSi miRebuli qulebis populaciidan, romlis μ

= 100 da σ = 15, Tqven 100 SemTxveviTi SerCevis aRebas apirebT, romelTa

moculoba N = 25.

a) SerCevaTa saSualoebis ra nawili iqneba moTavsebuli 97-sa da 103-s

Soris?

b) SerCevaTa saSualoebis ra wili iqneba 103-ze meti?

g) SerCevaTa saSualoebis ra wili iqneba 103-ze naklebi?

d) SerCevaTa saSualoebis ra wili iqneba 105-ze meti?

e) SerCevaTa saSualoebis ra wili iqneba moTavsebuli 100-sa da 105-s

Soris?

8. davuSvaT, gaqvT qulaTa SerCeva, romelic monitorze moZravi

obieqtisTvis kursoris midevnebis Sefasebis Sedegad miiReT; Tqven

afiqsirebdiT cdis piris mier samizne obieqtze kursoris SeCerebis dros.

am qulebis ganawilebis s = 18.0 wami.

a) gamoTvaleT Xs -is mniSvneloba, Tu SerCevis moculobaa 4.


b) gamoTvaleT Xs -is mniSvneloba, Tu SerCevis moculobaa 9.

g) gamoTvaleT Xs -is mniSvneloba, Tu SerCevis moculobaa 36.

d) Xs -is mniSvnelobasa da SerCevis moculobas Soris arsebul ra

damokidebulebasTan gvaqvs saqme?

2. davuSvaT, wina kiTxvaSi aRweril amocanaSi miRebul qulaTa sami

gansxvavebuli SerCeva gaqvT. maTTvis s-is mniSvnelobebia 10, 15 da 20,

Sesabamisad. SerCevis moculobaa 25 (N =25).

a) TiToeuli SerCevisTvis gamoTvaleT Xs -is mniSvneloba.

b) qulaTa cvalebadobasa da Xs -is mniSvnelobas Soris arsebul ra

damokidebulebasTan gvaqvs saqme?

3. qulaTa ori sxvadasxva _ A1 da A2 _ populaciidan erTnairi moculobis

ori SerCevis aRebas apirebT. A1 populaciis σ = 17.0 da A2 populaciis σ

= 30.0. rogor fiqrobT, romeli populaciidan aRebuli SerCevis Xs -is

mniSvneloba iqneba meti? axseniT Tqveni pasuxi.

populaciis saSualos intervaluri Sefaseba.

SerCevis saSualo μ-is Caunacvlebeli da

Zalmosili Sefasebaa. Tu Cven im

populaciis saSualos vafasebT, saidanac

SerCeva iqna aRebuli, Sefasebis saukeTeso

wertilovani Sefaseba X iqneba; magram,

rogorc ukve aRvniSneT, wertilovan

Sefasebas Tan axlavs Secdoma; μ-is

mniSvneloba ar iqneba X -is mniSvnelobis toli. Sefasebis sizusteSi rom ufro

metad viyoT darwmunebulni, Cven SegviZlia, SevafasoT ara marto μ-is

wertilovani mniSvneloba, aramed avagoT ndobis intervalic, romelSic Seva μ-is

mosalodneli mniSvnelobebi. magaliTad, gamocdebTan dakavSirebuli SfoTvis

kiTxvaris qulebis populaciis saSualos zustad 41.64-quliT Sefasebis nacvlad,

mkvlevarma is SeiZleba μ-is Semcveli, 39.94-dan 43.34-mde intervaliT Seafasos.

ndobis done anu albaTobis sidide SeiZleba Tan axldes am Sefasebas; magaliTad,

mkvlevars SeuZlia 95%-iT darwmunebuli iyos, rom es intervali moicavs μ-s.

intervalis qveda (magaliTad, 39.9) da zeda (magaliTad, 43.3) qulebs ndobis

sazRvrebi ewodebaT. axla vnaxoT, rogor aigeba aseTi intervalebi da ras

niSnavs, rodesac vambobT, rom eqsperimentatori 95%-iT darwmunebulia, rom

intervali moicavs μ-is mniSvnelobas.

ndobis intervali (Confidence interval): _

qulis mniSvnelobaTa diapazoni,

romelic, savaraudod, moicavs μ-is

mniSvnelobas ndobis garkveul doneze.

ndobis sazRvrebi (Confidence limits): _

ndobis intervalis SemomsazRvreli

qveda da zeda qulebi.


ndobis intervalis ageba

davubrundeT suraTs N7.3, romelzec mocemulia saSualos SerCeviTi Teoriuli

ganawileba. gavixsenoT, rom am SerCeviTi ganawilebis standartuli gadaxraa Xσ

da rom is normaluri ganawilebaa. am codnaze dayrdnobiT zemoT vnaxeT, rom

XX σµ 1− -dan XX σµ 1+ -mde intervali moicavs am populaciidan aRebul SerCevaTa

saSualoebis .6826 nawils. davuSvaT, axla vsvamT kiTxvas: _ Tu Cven aviRebT erT

SerCevas da misTvis X -s gamovTvliT, ras udris albaToba imisa, rom X moxvdeba

XX σµ 1− -dan XX σµ 1+ -mde intervalSi? pasuxia .6826 anu .6826-isTvis SerCevaTa

saSualoebis mniSvnelobebi moTavsebulia XX σµ 1− -sa da XX σµ 1+ -s Soris.

Sesabamisad, Tu Cven aviRebT SerCevas, albaToba imisa, rom misi saSualo moxvdeba

XX σµ 1− -sa da XX σµ 1+ -s Soris, aris .6826-is toli. yuradReba miaqcieT, rom

kiTxvaze pasuxis gasacemad Cven intervali Xµ -is irgvliv (anu μ-is irgvliv,

radgan µµ =X ) avageT da gamoviyeneT Xσ -is mniSvneloba, Semdeg ki normaluri

ganawilebis TvisebebiT visargebleT; magram SemTxvevaTa umravlesobaSi μ ucnobia

da amitom mas X -iT vafasebT. davuSvaT, μ-s irgvliv intervalis asagebad igive

procedura gamoviyeneT, magram μ-is Sesafaseblad X -s mivmarTeT. amrigad, miviReT

XX σ1− -sa da XX σ1+ -s Soris intervali. axla ukve aRar vsvamT kiTxvas: _ Sedis

Tu ara X -is mniSvneloba am intervalSi? Cven ukve viciT X . amjerad kiTxva

aseTia: _ moicavs Tu ara es intervali μ-s? am kiTxvaze pasuxis gacema albaTobis

terminebiTac SegviZlia. Tu albaToba imisa, rom XX σµ 1− -dan XX σµ 1+ -mde

intervali moicavs X -s, aris .6826, maSin, marTebulia isic, rom albaToba imisa,

rom XX σ1− -dan XX σ1+ -mde intervali moicavdes μ-s, aris .6826.

XX σ1− -dan XX σ1+ -mde intervali aris 68%-iani ndobis intervali (.6826

davamrgvaleT da procentis misaRebad 100-ze gavamravleT). am intervalis

sazRvrebi anu ukiduresi mniSvnelobebi ndobis sazRvrebs warmoadgenen. Cven am

intervalis interpretireba Semdegnairad SegviZlia: _ Tu Cven mravaljer aviRebT

SerCevebs, TiToeuli maTganisTvis gamovTvliT saSualos da Semdeg TiToeuli X -

sTvis avagebT XX σ1− -dan XX σ1+ -mde intervals, maSin am intervalTa 68%-Si Seva

μ-is mniSvneloba. yuradReba miaqcieT, rom vinaidan μ populaciis parametria da

amitom konkretuli mniSvneloba aqvs, is an Sedis intervalSi, an _ ara.

ganvixiloT am msjelobis ricxviTi magaliTi.


davuSvaT, Cven gvaqvs populacia, romlis μ = 40 da σ = 10. am populaciidan

aviReT SerCeva, romlis moculobaa 25 (N = 25) da gamovTvaleT 5.41=X . Semdeg

gamovTvaloT 0.225/10/ === NX σσ . μ-sTvis 68%-iani ndobis intervali miviReT

XX σ1− -dan XX σ1+ -mde intervalis povniT. ricxviTi mniSvnelobebis 5.41=X da

2=Xσ Setanis Semdeg miviRebT ndobis intervals 39.5-dan (anu

5.3925.411 =−=− XX σ ) 43.5-mde (anu 5.4325.411 =+=+ XX σ ). Cven SegviZlia 68%-iT

darwmunebulni viyoT, rom es intervali Seicavs μ-s mniSvnelobas da am

SemTxvevaSi es asea. axla davuSvaT, rom X 41.5-is nacvlad 44.0-is tolia, maSin

68%-iani ndobis intervali iqneba 42.0-dan (anu 0.4220.441 =−=− XX σ ) 46.0-mde (anu

0.4620.441 =+=+ XX σ ). axlac SegviZlia 68%-iT darwmunebulni viyoT, rom es

intervali Seicavs μ-s, magram am SemTxvevaSi intervali ar Seicavs μ = 40-s.

cxadia, rom praqtikaSi zustad ar viciT, Seicavs ndobis intervali μ-s Tu ara,

vinaidan Cven ar viciT μ-is mniSvneloba. CvenTvis μ rom cnobili yofiliyo, aRar

dagvWirdeboda misi mniSvnelobis Sefaseba. Cven mxolod garkveuli doniT

SegviZlia viyoT darwmunebulni, rom intervali Seicavs μ-s mniSvnelobas.

kvlevaSi, rogorc wesi, 95%-iani an 99%-iani (da ara 68%-iani) ndobis

intervalebi gamoiyeneba. am intervalebis sazRvrebis dadgena standartuli

normaluri ganawilebis (romlis μ = 0 da σ = 1) Tvisebebis gamoyenebiT xdeba. a-1

cxrilSi vnaxulobT, rom z = _1.96-dan 0-mde intervali Seicavs z-is mniSvnelobaTa

.4750-s da z = 0-dan z = +1.96-mde intervali, aseve, Seicavs z-is mniSvnelobaTa .4750-s.

amrigad, z = _1.96-dan z = +1.96-mde intervalSi moTavsebuli iqneba z-is

mniSvnelobaTa .95 (anu .4750+.4750=.95). Sesabamisad, populaciisTvis 95%-iani ndobis

intervali iqneba

mde dan −+−− XX XX σσ 96.196.1 .

99%-iani ndobis intervalisTvis vnaxavT, rom z = _2.58-dan 0-mde intervali

Seicavs z-is mniSvnelobaTa .4951-s da z = 0-dan z = +2.58-mde intervali, aseve,

Seicavs z-is mniSvnelobaTa .4951-s. amrigad, z = _2.58-dan z = +2.58-mde intervalSi

moTavsebuli iqneba z-is mniSvnelobaTa .99 (anu .4951+.4951=.9902 anu .99). Sesabamisad,

populaciisTvis 99%-iani ndobis intervali iqneba

mde dan −+−− XX XX σσ 58.258.2

magaliTSi, sadac 5.41=X da 2=Xσ , 95%-iani ndobis intervali iqneba:

( ) ( ) mde21.961.5 dan −+−+ 4296.15.41 ,

rac udris


mde3.921.5 dan −+−+ 492.35.41 anu mde5.4 dan −− 458.37 .

99%-iani ndobis intervali iqneba:

( ) ( ) mde22.581.5 dan −+−+ 4258.25.41

rac udris

mde51.5 dan −+−+ 16.416.55.41 anu mde.7 dan −− 4634.36 .

am msjelobis erTma aspeqtma SeiZleba dagabnioT. Cven imitom vafasebT

populaciis parametrs μ-s, rom ar SegviZlia am parametris mniSvnelobis misaRebad

mTeli populaciis gazomva; Tumca, ndobis intervalebis agebisas Xσ -is

gamosaTvlelad viyenebT meore parametrs σ-s. saidan unda vicodeT σ-is

mniSvneloba, Tu ar viciT μ? pasuxia: _ ar unda vicodeT. Xσ unda SevafasoT Xs -

iT. Cven intervaluri Sefasebis axsnis gasamartiveblad davuSviT, rom viciT Xσ ,

magram praqtikaSi ndobis sazRvrebis dasadgenad Xs ufro gamoiyeneba, vidre _

Xσ . Xs -is gamoyenebis SemTxvevaSi ndobis intervalebi odnav didia, vidre Xσ -is

gamoyenebisas, Tumca, intervaluri Sefasebis cneba ar icvleba ganurCevlad imisa,

Xσ -s viyenebT,YTu Xs -s. ndobis intervalebis agebisas Xs -is gamoyenebaze me-8

TavSi visaubrebT.

magaliTi N7.2

ndobis intervalis ageba, rodesac Xσ cnobilia

amocana

populaciidan, romlis σ = 13.0 aviReT 30-quliani SerCeva. SerCevis saSualoa

71.0 μ-sTvis ipoveT 95%-iani ndobis intervali.

amoxsna

vpoulobT Xσ -s da a-1 cxrilSi veZebT z-is mniSvnelobas, romelic

gansazRvravs standartuli normaluri ganawilebis farTobis 95%-is Semcvel

intervals.

37.2477.5

0.1330

0.13====

NX

σσ .

a-1 cxrilSi vnaxulobT, rom standartul normalur ganawilebaSi z = _1.96-dan

z = +1.96-mde intervali Seicavs ganawilebis farTobis 95%-s. amrigad, μ-sTvis 95%-

iani ndobis intervalia

mde dan −+−− XX XX σσ 96.196.1 .


X -isa da Xσ -is ricxviTi mniSvnelobebis Casmis Sedegad miviRebT intervals

( ) ( ) mde dan −+−− 37.296.10.7137.296.10.71

anu

mde dan −+−− 65.40.7165.40.71

rac 66.35-dan 75.65-mde intervalis tolia. μ-sTvis 95%-iani ndobis intervalia

66.35-dan 75.65-mde.


1. ganmarteT: _ ndobis intervali, ndobis sazRvrebi, intervaluri Sefaseba.

2. populaciidan, romlis σ=10.0 aiReT 25-quliani SerCeva. X -is mniSvnelobaa

73.0.

a) ipoveT μ-s wertilovani Sefaseba.

b) aageT 95%-iani ndobis intervali μ-sTvis.

g) aageT 99%-iani ndobis intervali μ-sTvis.

(d), (e) da (v) kiTxvebisTvis davuSvaT, rom SerCevis moculoba (25-is nacvlad)

iyo 100 (N = 100).

d) ipoveT μ-s wertilovani Sefaseba.

e) aageT 95%-iani ndobis intervali μ-isTvis.

v) aageT 99%-iani ndobis intervali μ-isTvis.

z) N = 100-ze dafuZnebuli ndobis intervali ufro mcirea, vidre _ N =

25-ze dafuZnebuli ndobis intervali. axseniT, ra ganapirobebs am

gansxavebas.

Sejameba

• statistikuri daskvna aris populaciis ucnobi parametrebis Sefaseba

SerCevis cnobil statistikebze dayrdnobiT.

• X aris μ-s Caunacvlebeli da Zalmosili Sefaseba.

• zusti Sefasebis uzrunvelsayofad SerCeva im populaciis reprezentaciuli

unda iyos, saidanac is iqna aRebuli. reprezenaciulobas ganapirobebs

SerCevis aRebis meTodi.

• s2 aris σ2-is Caunacvlebeli da Zalmosili Sefaseba.

• s gamoiyeneba σ-is Sesafaseblad, miuxedavad imisa, rom is odnav bolomde

ver afasebs mas.

• saSualos SerCeviTi ganawileba mocemuli moculobis SerCevebis saSualos

Teoriuli ganawilebaa.


• saSualos standartuli Secdoma aris saSualos SerCeviTi ganawilebis

standartuli gadaxra.

• Xσ gamoiTvleba formuliT NX /σ . umravles SemTxvevaSi is fasdeba

formuliT NssX /= .

• SerCevis moculobis zrdasTan erTad saSualos SerCeviTi ganawileba

normalur ganawilebas uaxlovdeba. amrigad, Xσ SeiZleba gamoyenebul iqnas

X -ze dayrdnobiT μ-s Sefasebis ndobis intervalebis dasadgenad.

• ndobis intervali aris monacemTa diapazoni, romelic, savaraudod, moicavs

μ-s mniSvnelobas ndobis garkveul doneze. ndobis intervalis

SemomsazRvreli qveda da zeda qulebs ndobis sazRvrebi ewodebaT.


dispersia

martivi SeTxveviTi SerCeva

ndobis intervali

ndobis sazRvrebi

saSualos standartuli

Secdoma


statistikuri daskvna

SerCevis Secdoma

Caunacvlebeli

Sefaseba

centraluri zRvariTi

Teorema

Zalmosili Sefaseba

intervaluri Sefaseba

wertilovani Sefaseba

Xµ

Xσ

2σ

Xs

s

s2

X

XX

Tavis damateba: ZiriTadi simboloebi da formulebi

me-4-dan me-7 Tavis CaTvliT mTeli rigi mniSvnelovani statistikuri

simboloebi da formulebi warmogidgineT. aRniSnuli masalis gagebisa da

daswavlis gasamartiveblad mas cxrilis saxiT gTavazobT.

simbolo formula ganmarteba

N SerCevaSi Semaval qulaTa raodenoba anu SerCevis

moculoba.

Npopulacia populaciaSi Semaval qulaTa raodenoba anu populaciis

moculoba.

X i an X SerCevis an populaciis calkeuli qula.

X

NX∑

SerCevis saSualo. X aRniSnavs SerCevaSi Semaval

calkeul qulas.

თავი 7. სტატისტიკის გამოყენება დასკვნისა და შეფასებისთვის 25

μ

populaciaNX∑

populaciis saSualo. X aRniSnavs populaciaSi Semaval

calkeul qulas.

S2 ( )N

XX∑ − 2

SerCevis dispersia. es dispersia ar aris populaciis

dispersiis Caunacvlebeli Sefaseba.

S ( )N

XX∑ − 2

SerCevis standartuli gadaxra. es standartuli gadaxra

SerCevis aRweriTi statistikaa, magram is ar gamoiyeneba

poulaciis standartuli gadaxris SefasebisTvis.

s2 ( )1

2

−

−∑N

XX

populaciis dispersiis Sefaseba. es dispersia

populaciis dispersiis Caunacvlebeli Sefasebaa.

s ( )1

2

−

−∑N

XX

populaciis standartuli gadaxris Sefaseba.

standartuli gadaxris es formula yvelaze xSirad

gamoiyeneba standartuli gadaxris gamosaTvlelad. Tu

gansakuTrebulad ar aris aRniSnuli, saxelmZRvanelos

teqstSi swored es standartuli gadaxraa gamoyenebuli.

SS ( )∑ −2

XX kvadratebis jami.

σ2 ( )populaciaN

X∑ − 2µ

populaciis dispersia. X aRniSnavs populaciaSi Semaval

calkeul qulas.

σ ( )populaciaN

X∑ − 2µ

populaciis standartuli gadaxra. X aRniSnavs

populaciaSi Semaval calkeul qulas.

Xµ saSualos SerCeviTi Teoriuli ganawilebis saSualo. es

saSualo populaciis saSualos (μ) tolia.

XX saSualos SerCeviTi empiriuli ganawilebis saSualo.

Xσ

Nσ

N moculobis SerCevis saSualos standartuli Secdoma.

saSualos standartuli Secdoma aris saSualos

SerCeviTi ganawilebis standartuli gadaxra.

Xs

Ns

N moculobis SerCevis saSualos standartuli Secdomis

Sefaseba.

z

σµ−X

standartuli normaluri gadaxra. am formuliT xdeba

normalurad ganawilebuli nedli qulebis gadayvana

standartuli normaluri ganawilebis z qulebSi.

z

SXX −

standartuli qula. am formuliT xdeba nedli qulis

gadayvana z qulaSi, romliTac aRiwereba, SerCevis

saSualodan ramdenad zemoT an qvemoT imyofeba nedli


qula. am formuliT miRebul z qulebs ver gamoviyenT

cxrilSi Na-1 mocemul standartul normalur

ganawilebasTan.

z

X

Xσ

µ−

am formuliT xdeba SerCevis saSualos gadayvana

standartuli normaluri ganawilebis z qulaSi. am

formuliT miRebuli zrealuri-is mniSvneloba SeiZleba

gamoyenebul iqnes cxrilSi Na-1 mocemul standartul

normalur ganawilebasTan.

kiTxvebi

1. romeli statistikaa romeli parametris Sefaseba?

statistika parametri

a) s μ

b) X Xσ

g) s2 σ

d) Xs σ2

2. μ saSualos mqone populaciidan aiReT ori SerCrva. pirveli SerCevis

moculoba N = 10, meore SerCevis moculoba N = 24. rogor fiqrobT,

romeli SerCevis saSualo iqneba μ-sTan ufro axlos? ratom?

3. arCevnebis Sedegebis prognozirebis mizniT mkvlevari qalaqSi

gamokiTxvas atarebs.man icis, rom qalaqis amomrCevelTa populaciis 34%

Seadgens registrirebul demokrats, 29% _ registrirebul

respublikels, xolo 37%-s ara aqvs politikuri mikuTvnebuloba.

SerCevis romeli tipi (martivi Tu stratificirebuli SemTxveviTi

SerCeva) uzrunvelyofs amomrCevelTa ufro metad reprezentaciul

SerCevas? ratom?

4. XXI saukunis mecnierebs globaluri daTbobisa da am movlenaSi

adamianis qcevis SesaZlo wvlilis sakiTxebi ainteresebT; Tumca,

ramdenad sainteresoa es Temebi Cveulebrivi saSualo adamianisTvis?

CvenTvis ukve cnobili internet-gamkiTxvis gverdze ganTavsda SekiTxva

imis Sesaxeb, Tu ramdenad ainteresebT gverdis damTvaliereblebs

globaluri daTbobis Tema. daaxloebiT 25,000 adamianma upasuxa dasmul

SekiTxvas da maTgan 56%-ma aRniSna, rom ainteresebs, danarCenma

respondentebma ki miuTiTes, rom maTTvis es Tema uinteresoa.

yoveldRiurad, sul mcire asobiT aTasi adamiani aTvalierebs am veb


gverds. ra daskvnis gakeTeba SegiZliaT am gamokiTxvis mixedviT imis

Sesaxeb, Tu ramdenad ainteresebT adamianebs globaluri daTbobis

problema? axseniT Tqveni pasuxi.

5. kolejSi mwevelebis populaciaSi yoveldRiurad moweuli sigaretis

raodenobis Sesafaseblad, Tqven SemTxveviT SearCieT 16 studenti im

studentebidan, vinc sakuTar Tavs mwevelad miiCnevs. Tqven naxeT, rom am

jgufis SerCevis saSualoa 9.0 sigareti dReSi.

a) gansazRvreT populaciis saSualos wertilovani Sefaseba.

b) davuSvaT, rom σ = 3.1 sigareti dReSi. ipoveT saSualos 95%-iani

ndobis intervali.

6. davuSvaT, rom kolejis studentebis populaciaSi saSualo Zilis

xangrZlivobaa 7.1 sT, xolo σ = 1.2 sT. Tqven SemTxveviT airCieT 16

studenti da gansazRvreT am SerCevis Zilis saSualo xangrZlioba

(saaTebSi).

a) ra iqneba N = 16 moculobis SerCevis saSualos standartuli

Secdoma?

b) ras udris albaToba imisa, rom Tqveni SerCevis saSualo 8.0 sT-is

toli an masze meti iyos?

g) ras udris albaToba imisa, rom Tqveni SerCevis saSualo 6.8 sT-is

toli an masze naklebi iyos?

d) ras udris albaToba imisa, rom Tqveni SerCevis saSualo

moTavsebuli iyos 6.5-sa da 7.7 sT-s Soris?

7. ra gansxvavebaa s-sa da Xs -s Soris?

8. cukermanma (Zuckerman, 1979) SegZnebebis Ziebis gasazomi skala Seqmna.

romelTac SegrZnebebis Ziebis maRali maCveneblis mqone adamianebs male

bezrdebaT dawyebuli aqtivobebi da Zalian esaWiroebaT axal da zogjer

riskian saqmianobebSi CarTva. skalaze qulebi 10-dan 40-mde meryeobs.

maRali qula gviCvenebs SegrZnebebis Ziebis maRal xarisxs. davuSvaT,

kolejis aTleti studentebis poulaciis saSualoa 23.0, xolo σ = 4.0.

Tqven am skalaze kolejis aTleti studentebis xuTi (a, b, g, d da e)

SerCeva gazomeT. SerCevebi sxvadasxva moculobis iyo, rogorc qvemoT

aris mocemuli:

SerCeva N

a

b

g

d

4

9

16

25


e 100

TiToeuli SerCevisTvis

a) gamoTvaleT Xσ .

b) ipoveT albaToba imisa, rom Tqveni SerCevis saSualo 25-is toli

an masze meti iqneba.

g) Xσ -isa da N-is ra kavSiris ilustraciaa es magaliTi.

d) TiToeuli SerCevisTvis ipoveT albaToba imisa, rom Tqveni

SerCevis saSualo moTavsebuli iqneba 23-sa da 25-s Soris.

Tavi VIII. statistikuri hipoTezebis Semowmebis

Sesavali: z kriteriumi da erTi SerCevis t kriteriumi


parametruli da araparametruli kriteriumebi

statistikuri hipoTezebi

kriteriumis statistika: z

mniSvnelovnobis doneebi: statistikuri iSviaToba

gadawyvetilebis miReba statistikuri hipoTezebis Sesaxeb

kriteriumis statistikisa da mniSvnelovnobis donis

gamoyenebiT

statistikuri hipoTezis Semowmebis etapebis Sejameba

erTi SerCevis t kriteriumi

t statistikis SerCeviTi ganawileba

statistikuri hipoTezis Semowmeba erTi SerCevis t

kriteriumiT

erTi SerCevis t kriteriumis Sedegebis prezentacia

mniSvnelovani sakiTxebi statistikuri hipoTezis

SemowmebaSi

ratom ar viRebT nulovan hipoTezas?

I da II gvaris Secdomebi statistikuri kriteriumebis

gamoyenebisas

statistikuri da mecnieruli mniSvnelovnoba

ndobis intervalebis ageba populaciis saSualosaTvis t

statistikis gamoyenebiT

wina TavebSi ganvixileT statistikis gamoyeneba monacemTa organizebis, aRweris

mizniT da populaciis parametrebis Sesafaseblad. am TavSi ki ganvixilavT

statistikis gamoyenebas hipoTezis Semowmebis mizniT.

ganvixiloT Semdegi problema. adamianuri faqtori aris bihevioristuli

mecnierebis sfero, romelic adamianis SesaZleblobebisadmi manqanisa da teqnologiis

Sesatyvisobas swavlobs. adamianuri faqtoris Semswavleli fsiqologebi xSirad

atareben adamianTa fizikuri maxasiaTeblebis gazomvebs, raTa Seqmnan skamebis,

kontrolis panelisa da monitoris iseTi dizaini, romelic adamians moergeba.

magaliTad, warmoidgineT, rom xarT satvirTo manqanebis mwarmoebeli kompaniis

TanamSromeli, romelic pasuxismgebelia Sida dizainze. wlebis manZilze skamebi

Tqvens satvirTo manqanebSi iseTnairad iyo mowyobili, rom Seesatyviseboda mZRolebis

populacias, romelTa saSualo muxlqveSa simaRle iyo 42 santimetri (sm) da σ _ 3 sm.

fexis muxlqveSa simaRle aris vertikaluri manZili mjdomare mZRolis muxlidan

iatakamde.

axlaxan Tqvenma kompaniam daiwyo satvirTo manqanebis gayidva mTel msoflioSi

da miiRo saCivrebi skamebis arakomfortulobis Taobaze. Tqven ivaraudeT, rom

SesaZloa zogierT qveyanaSi mZRolebis muxlqveSa simaRle gansxvavdeba im

populaciisagan, romlisTvisac Seqmnili iyo es skamebi. am hipoTezis Sesamowmeblad,

Tqven aiyvaneT satvirTo manqanis 25 mZRolisagan Semdgari SerCeva, romelic im qveynis

mZRolebis populaciisagan damoukideblad da SemTxveviTad iyo SerCeuli,

romelTaganac iyo miRebuli saCivrebi. am SerCevis mZRolebis saSualo muxlqveSa

simaRle aris 44 sm. aSkaraa, rom am SerCevis saSualo _ 44 sm ar aris igive, rac im

populaciis saSualo, romelic 42 sm-is tolia. magram ras gveubneba es gansxvaveba

populaciis Sesaxeb? arsebobs ori SesaZlebloba. pirveli aris is, rom populacias,

romlidanac iyo aRebuli realurad SerCeva, saSualo muxlqveSa simaRle 42 sm-is

toli aqvs, xolo Tqveni SerCevis saSualo _ 44 sm gansxvavdeba am µ-gan mxolod

SerCevis Secdomis gamo. SemTxveviT Tqven SearCieT iseTi SerCeva, romlis saSualo

muxlqveSa simaRle iyo 42 sm-ze meti.

meore SesaZlebloba isaa, rom SerCevis saSualo asaxavs populacias iseTi µ-iT,

romelic gansxvavdeba 42 sm-isagan. mZRolebis populacias am qveyanaSi aqvs saSualo

muxlqveSa simaRle, romelic gansxvavdeba 42 sm-sagan. am ori SesaZleblobidan romeli

unda airCioT rogorc im situaciis amsaxveli, romelic arsebobs im qveyanaSi,

saidanac iyo aRebuli SerCeva?

Cven gvaqvs problema, romelic moiTxovs statistikuri hipoTezis Semowmebas,

raTa gavakeToT arCevani, Tu romelia ufro metad mosalodneli im ori SesaZlo

situaciidan. statistikuri terminebiT, es problema moiTxovs statistikuri hipoTezis

Semowmebas imis gadasawyvetad, mniSvnelovnad gansxvavdeba Tu ara SerCevis saSualo _

44 sm populaciis saSualosagan, romelic 42 sm-is tolia. suraTi #8.1 am problemis

ilustracias warmoadgens.

suraTi #8.1. ori SesaZlo situacia, romelic SeiZleba ganapirobebdes SerCevis realurad miRebul saSualos.

SerCevis realuri saSualo = 44 sm qvemoT mocemuli ori SesaZleblobidan romeli ganapirobebs SerCevis am saSualos?

.


parametruli da araparametruli kriteriumebi

parametruli kriteriumebi

parametrulia kriteriumebi Tu hipoTezaSi aris

gamoTqmuli varaudi populaciis iseTi parametrebis

Sesaxeb rogoricaa, magaliTad, µ da σ2. parametruli

kriteriumebi xSirad garkveul pirobebs

warmoadgenen, rogorc WeSmarit daSvebas im

populaciis monacemebisaTvis, romlidanac aris

aRebuli SerCeva. roca gavecnobiT konkretul

parametrul kriteriumebs, agreTve, unda ganvsazRvroT am kriteriumebisaTvis

gakeTebuli daSvebebi. am TavSi gavecnobiT statistikuri hipoTezis Semowmebas,

romelic iyenebs or gansxvavebul parametrul kriteriums, erTi SerCevis z kriteriums

parametruli kriteriumi

(parametric test) _ statistikur

kriteriums ewodeba parametruli,

Tu hipoTezaSi aris gamoTqmuli

varaudi populaciis parametris

Sesaxeb.

populaciis saSualo, romlidanac SerCeva iyo aRebuli, aris 42 sm. SerCevis saSualo gansxvavdeba populaciis saSualosagan SerCevis Secdomis gamo.

populaciis saSualo, romlidanac SerCeva iyo aRebuli, ar aris 42sm. SerCevis saSualo gansxvavdeba populaciis saSualosagan ara SerCevis Secdomis gamo, aramed sxva mizeziT.

rogor vupasuxoT SekiTxvas: gamoviyenoT statistikuri kriteriumi, raTa ganvsazRvroT,

gansxvavdeba Tu ara X =44 sm mniSvnelovnad µ=42 sm-sagan.

da erTi SerCevis t kriteriums. sxva parametrul kriteriumebs gavecnobiT IX-XII

TavebSi.

araparametruli kriteriumebi

araparametrulia kriteriumebi, Tu

hipoTezaSi ar aris gamoTqmuli

varaudi populaciis parametris

Sesaxeb. Cveulebriv, araparametruli

kriteriumebi ar akeTeben daSvebebs im populaciis monacemebis ganawilebis Sesaxeb,

romlidanac aris aRebuli SerCeva. am mizeziT, isini xSirad miekuTvnebian e.w.

ganawilebisagan Tavisufal kriteriumebs. araparametruli kriteriumebi xSirad

gamoiyeneba, roca parametruli kriteriumebisaTvis aucilebeli daSvebebi ar

Seesabameba monacemebs. parametruli kriteriumebis didi nawilisaTvis, magram ara

yvela kriteriumisaTvis, arsebobs alternatiuli araparametruli kriteriumebi igive

monacemebis gasaanalizeblad. ramdenime araparametruli kriteriumi warmodgenilia me-

13 TavSi. statistikuri hipoTezis Semowmebisas gamoyenebuli ZiriTadi cnebebis

umravlesoba erTnairia parametruli da araparametruli kriteriumisaTvis.


statistikuri hipoTezis

Semowmeba, iqneba is parametruli

Tu araparametruli, iwyeba


CamoyalibebiT. parametruli

kriteriumisaTvis statistikuri

hipoTeza aris debuleba

populaciis parametris Sesaxeb. es

debuleba SeiZleba WeSmariti iyos

da SeiZleba _ ara. is ubralod

imisaTvis keTdeba, raTa safuZveli

daudos Semowmebas. magaliTad, Cven SegviZlia CamovayaliboT hipoTeza imis Sesaxeb,

rom muxlqveSa simaRlis saSualo im populaciisaTvis, romlidanac ayvanili gvyavda

25 kaciani SerCeva, aris 42 sm. realurad populaciis saSualo SesaZloa iyos 42 sm an

ar iyos. Cven ubralod vvaraudobT, rom is 42 sm-ia da am hipoTezas SemdgomSi

SevamowmebT. statistikuri kriteriumebi moiTxovs, rom formulirebuli iyos ori

hipoTeza, nulovani da alternatiuli.

araparametruli kriteriumi (nonparametric test) _

statistikur kriteriums ewodeba araparametruli,

Tu hipoTezaSi ar aris gamoTqmuli varaudi

populaciis parametris Sesaxeb.

statistikuri hipoTeza (statistical hypothesis) _

debuleba populaciis parametris

Sesaxeb(parametruli kriteriumis Sesaxeb).

nulovani hipoTeza (null hypothesis) - aris im pirobis

Camoyalibeba, romelic mecniers populaciis

Sesaxeb apriorulad WeSmaritad miaCnia. es aris

hipoTeza, romelic mowmdeba statistikuri

kriteriumiT.

alternatiuli hipoTeza (statistical alternative hypothesis)

_ aris debuleba imis Sesaxeb, Tu ra unda iyos

WeSmariti, Tu nulovani hipoTeza mcdaria.

nulovani hipoTeza

nulovani hipoTeza aris im pirobis Camoyalibeba, romelic mecniers populaciis

Sesaxeb apriorulad WeSmaritad miaCnia. es aris hipoTeza, romelic mowmdeba

statistikuri kriteriumiT. nulovani hipoTeza, Cveulebriv, aRiniSneba simboluri an

pirobiTi formiT H0-is gamoyenebiT imis aRsaniSnavad, rom SemoTavazebulia iseTi

hipoTeza, rogoricaa µ=42 smis. nulovani hipoTezis _ µ=42 sm gamoxatva statistikur

CanawerSi Semdegnairad SegviZlia H0:µ=42sm, da es gamoiTqmis aseTnairad: `aS nuli: miu

tolia 42 sm-is”.

alternatiuli hipoTeza

alternatiuli hipoTeza aris debuleba imis Sesaxeb, ra unda iyos WeSmariti, Tu

nulovani hipoTeza mcdaria. magaliTad, Tu H0 : µ=42 sm ar aris WeSmariti, maSin

alternatiuli unda iyos, rom µ ar aris 42 sm-is toli. es alternatiuli hipoTeza

aRiniSneba H1-iT da iwereba Semdegnairad: H1: µ ≠ 42 sm. es gamoiTqmis aseTnairad: `aS

erTi: miu ar aris 42 sm-is toli“.

statistikuri hipoTezebis Tvisebebi

TiToeuli statistikuri kriteriumi moiTxovs nulovani da alternatiuli

hipoTezebis formulirebas. es hipoTezebi iseTnairad unda iyos warmodgenili, rom

akmayofilebdnen Semdeg pirobebs:

• isini arian urTierTgamomricxavni: urTierTgamomricxavi niSnavs, rom orive

hipoTeza ar SeiZleba iyos WeSmariti erTdroulad. magaliTad, SeuZlebelia H0 :

µ= 42 sm da H1 : µ ≠ 42 sm iyos erTdroulad WeSmariti erTi da imave

populaciisaTvis.

• isini moicaven parametris yvela SesaZlo mniSvnelobas, romelic CarTulia

hipoTezaSi. H0 : µ=42 sm da H1 : µ ≠42 sm akmayofilebs am moTxovnas; maT Soris

moTavsebulia µ-s yvela SesaZlo mniSvneloba CvenTvis saintereso

populaciisaTvis.

am Tvisebebis gamo mocemuli ori hipoTezidan mxolod erTi unda warmoadgendes

populaciisaTvis WeSmarit pirobas. magaliTad, populaciis saSualo an aris 42 sm-

is toli da H0 : µ= 42 sm aris WeSmariti, an ar aris toli da maSin alternatiuli

hipoTeza H1 : µ≠ 42 sm-s aris WeSmariti.

amrigad, ganxiluli problemisaTvis statistikuri hipoTezebia:

H0 : µ= 42 sm (nulovani hipoTeza)

H1 : µ ≠ 42 sm (alternatiuli hipoTeza)

statistikuri hipoTezebis funqcia

nulovani hipoTezis funqciaa statistikur kriteriumSi daadginos is pirobebi,

romlis drosac statistikis SerCeviTi ganawileba iyo miRebuli. me-7 TavSi Cven

ganvsazRvreT saSualos SerCeviTi ganawileba, rogorc X -is mniSvnelobebis

ganawileba, roca N moculobis yvela SesaZlo SerCeva populaciidan iyo aRebuli.

magaliTad, Tu vivaraudebT, rom H0:µ=42 sm aris WeSmariti da σ=3 sm, Cven SegviZlia

miviRoT X -is SerCeviTi ganawileba am populaciis iseTi SerCevebisaTvis, romelTa

moculoba aris 25. suraTze #8.2 mocemuli es SerCeviTi ganawileba aris normaluri

ganawileba populaciis saSualoTi µ X =42.0 sm, standartuli SecdomiT _ Xσ -iT, sadac

6.053

253

====NXσσ

am ganawilebaSi SerCeviTi saSualo _ 44 sm µ X -isagan daSorebulia saSualos sami

standartuli SecdomiT. Sesabamisad, im pirobebSi, roca H0:µ=42 sm-s, iseTi SerCevis

albaToba, romlis saSualoa 44 sm, aris Zalian mcirea, Tuki populaciidan aRebuli

SerCevis moculoba iqneba 25. amitomac Cven ufro mivemxrobodiT nulovani hipoTezis _

H0:µ=42 sm-is uaryofas, Tuki aRebuli SerCevis saSualo iqneboda 44 sm. X -is aseTi

mniSvneloba aris statistikurad iSviaTi xdomiloba, roca H0 aris WeSmariti. magram

sanam gadavwyvetdeT sabolood ganvixiloT Tu ara H0 rogorc WeSmariti hipoTeza,

Cven unda:

• davadginoT albaToba imisa, rom SeiZleba miviRoT X -is iseTi mniSvneloba,

romelic iqneba 44 sm-is toli an masze meti, Tu H0:µ=42 sm iqneba WeSmariti. es

nabiji moiTxovs kriteriumis statistikis gamoyenebas;

• ganvsazRvroT ras vgulisxmobT statistikurad iSviaT xdomilobaSi. es nabiji

moiTxovs mniSvnelovnobis donis gansazRvras.

suraTi #8.2. saSualos

Teoriuli SerCeviTi

ganawileba agebuli N=25

moculobis toli

SerCevebisaTvis µ X =42.0 sm-

is da _ Xσ =0.6-s mqone

populaciidan.

-3σ X -2σ X -1σ X µ X +1σ X +2σ X +3σ X 40.2 40.8 41.4 42.0 42.6 43.2 43.8

muxlqveSa simaRle (sm)

kriteriumis statistika: z

kriteriumis statistika aris ricxvi, romelic nulovani hipoTezis Semowmebis

saSualebas gvaZlevs da gamoiTvleba SerCevis monacemebze dayrdnobiT. magaliTad,

Cven gvinda kriteriumi, raTa davadginoT albaToba imisa, rom SerCeviTi saSualo

iqneba 44 sm-is toli an masze meti, Tu H0 : µ=42 sm aris WeSmariti. radgan suraTze #8.2

ilustrirebuli SerCeviTi saSualos ganawileba aris normaluri ganawileba µ=42 sm-

iTa da Xσ =0.6, Cven SegviZlia X -is nebismieri mniSvnelobis standartizacia:

z =X

Xσ

µ−

sadac:

X =SerCeviTi saSualo

µ= populaciis hipoTeturi saSualo

Xσ =saSualos standartuli Secdoma

z-is mniSvneloba gansazRvravs SerCeviTi saSualos mdebareobas standartul

normalur ganawilebaSi. aRsaniSnavia, rom es

formula identuria im formulisa, romelic

mocemuli iyo z-isaTvis me-7 TavSi.

Tu X =44 sm µ-s hipoTeturi mniSvneloba _ 42 sm,

da Xσ =0.6, maSin zrealuris mniSvnelobaa

zrealuri= 33.36.0

26.04244

+==−

=−

X

Xσ

µ

z statistikuri sidide gansazRvrulia rogorc zrealuri, radgan is miRebulia SerCevis

X mniSvnelobebze dakvirvebiT. z statistikis SerCeviTi ganawilebidan, romelic

mocemulia cxrilSi #a-1, Cven vpoulobT, rom albaToba imisa, rom z iqneba toli an

meti +3.33-ze aris 0.0004 (ix. g sveti cxrilSi #a-1 z=3.33-saTvis). Sesabamisad X =44 sm an

masze meti mniSvnelobis miRebis albaToba 25-is toli moculobis SerCevisaTvis aris

0.0004, Tu H0:µ=42sm-is aris WeSmariti. sxva sityvebiT rom vTqvaT, Tu Cven SemTxeviT

SevarCevT 25-is toli moculobis 10000 SerCevas populaciidan, romlis µ=42 sm da

σ=3, maSin mosalodnelia, rom 10000 SerCevidan mxolod oTxs eqneboda SerCeviTi

saSualo 44-is toli an masze meti. amrigad, Tuki H0:µ=42sm aris WeSmariti, iseTi

SerCevis arseboba, romlis X tolia 44 sm-is, iqneboda statistikurad iSviaTi

xdomiloba. Tuki Cven miviRebdiT am X , Cven namdvilad uarvyofdiT nulovan hipoTezas,

radgan is miRebulia iseTi populaciidan, romlis saSualo muxlqveSa simaRle aris

42 sm da miviRebdiT alternatiul hipoTezas, rom µ ≠ 42 sm.

kriteriumis statistika (test

statistic) aris ricxvi, romelic

nulovani hipoTezis Semowmebis

saSualebas gvaZlevs da

gamoiTvleba SerCevis monacemebze

dayrdnobiT.

aseTi gadawyvetilebis miRebis azri isaa, rom Tu populaciis saSualo aris 42 sm,

maSin iseTi SerCevis arseboba, romlis saSualo aris 44 sm, SesaZlebeli iqneboda

mxolod 4-jer yoveli aRebuli 10 000 SerCevisaTvis. marTalia, X =44 sm aris

SesaZlebeli xdomiloba, Tuki H0 : µ=42 sm-a, magram is mainc ar aris maRali albaTobis

mqone xdomiloba. misi miReba ufro metad mosalodneli iqneboda, Tuki populaciis

saSualo gansxvavebuli iqneboda 42sm-isagan. amrigad, Cven uarvyaviT H0 da miviReT

H1. es gadawyvetileba SeiZleba mcdaric iyos. Cven SegveZlo migveRo X =44 sm, maSinac

ki, roca µ=42 sm-ia, magram es iqneboda naklebad mosalodneli xdomiloba.

davuSvaT, miviReT SerCeviTi saSualo, romelic 42.5 sm-is tolia, nacvlad 44 sm-

isa. aq z toli iqneba:

zrealuri= 83.06.05.0

6.0425.42

+==−

cxrili #a-1-is g svetSi vipoviT, rom albaToba imisa, rom z iqneba toli an meti

+0.83-ze aris 0.2033. amrigad, Tu H0 aris WeSmariti da Cven aviRebT 10000 SerCevas,

romlis moculoba aris 25-is toli, amaTgan 2033-s eqneba X -is mniSvneloba 42.5-is

toli an masze meti. aris Tu ara 42.5 sm-is toli SerCeviTi saSualo statistikurad

iSviaTi xdomiloba, Tu H0 : µ=42sm aris WeSmariti? ra gadawyvetilebas miiRebT am

SemTxvevaSi nulovani hipoTezis Sesaxeb? uaryofT H0 : µ=42sm hipoTezas da miiRebT H1 :

µ ≠ 42sm hipoTezas? Tu ver uaryofT H0-s da amitom ar miiRebT H1-s? imisaTvis rom

vupasuxoT am SekiTxvas, Cven unda ganvsazRvroT zrealuris romeli mniSvneloba unda

miviCnioT statistikurad iSviaT Sedegad, UTuki H0 aris WeSmariti . statistikuri

iSviaToba ganisazRvreba im mniSvnelovnobis donis dadgeniT, romelic iqneba

miRebuli.

mniSvnelovnobis doneebi: statistikuri iSviaToba

mniSvnelovnobis done – esaa albaTobis is mniSvneloba,

romelic iZleva nulovani hipoTezis uaryofis pirobas

statistikur kriteriumSi. statistikur kriteriumSi

mniSvnelovnobis doneebi mocemulia rogorc alfas anu

α-s mniSvneloba (berZnuli anbanis pirveli aso).

Cveulebriv, bihevioristebi iyeneben alfas doneebs,

romlebic tolia 0.05 an 0.01. amrigad, Tu α=0.05, maSin

zrealuris mniSvnelobebi, romlebic 100-dan 5-jer an

naklebad xorcieldeba, iTvleba iSviaT xdomilobebad, Tu H0 aris WeSmariti, amitom

Cven varCevT H0-is uaryofasa da H1-is miRebas.

mniSvnelovnobis done

(significance level) _ esaa

albaTobis is mniSvneloba,

romelic iZleva nulovani

hipoTezis uaryofis pirobas

statistikur kriteriumSi.

alfa (α): mniSvnelovnobis

donis sidide (albaToba).

statistikuri hipoTezebis Sesaxeb gadawyvetilebis miReba kriteriumis

statistikisa da mniSvnelovnobis donis gamoyenebiT

mas Semdeg, rac avirCevT kriteriumis statistikas da mniSvnelovnobis dones, Cven

ukve mzad varT miviRoT gadawyvetileba statistikuri hipoTezis Sesaxeb. es moicavs

sam safexurs:

1. kriteriumis statistikis SerCeviT ganawilebaSi uaryofis aris an areebis

mdebareobis gansazRvra.

2. kriteriumis statistikis mniSvnelobis gamoTvla SerCevidan miRebuli

monacemebis safuZvelze.

3. statistikuri hipoTezebis Sesaxeb gadawyvetilebis miReba imis safuZvelze,

xvdeba Tu ara kriteriumis statistika uaryofis areSi.

uaryofis areebis mdebareobis

gansazRvra

uaryofis are warmoadgens

kriteriumis statistikis (SerCeviTi

ganawilebis) im mniSvnelobebs,

romelTa albaToba tolia an naklebia

α-ze nulovani hipoTezis WeSmaritebis

pirobebSi. magaliTad, uaryofis are

gansazRvravs z-is mniSvnelobebs,

romlebic uzrunvelyofen H0-is

uaryofas.

z statistikis SerCeviTi

ganawileba Tavisi uaryofis areebiT, romlebic ganlagebulni arian ganawilebis

TiToeul kudTan , naCvenebia suraTze #8.3. z-is statistikurad iSviaTi mniSvnelobebi

SeiZleba Segvxvdes ganawilebis orive kudTan. magaliTad, z-is mniSvnelobebi +3 an -3

Tanabrad SeuZlebelia, roca H0 WeSmaritia. z-is rogorc didi dadebiTi, aseve didi

uaryofiTi mniSvneloba ganxiluli problemisaTvis SeiZleba saeWvod xdides H0 : µ=42

sm-is WeSmaritebas. amrigad, uaryofis areebi z statistikisaTvis mdebareobs SerCeviTi

ganawilebis orive mimarTulebiT. kriteriumis statistika uaryofis areebiT

kriteriumis statistikis SerCevis ganawilebis orive mimarTulebiT ganisazRvreba

rogorc ormxrivi an aramimarTulebiTi kriteriumi. zrealuris mniSvnelobis

ganTavseba romelime uaryofis areSi araalbaTuri, an iSviaTia, Tuki H0 WeSmaritia,

magram amis albaToba ufro didia H1-is WeSmaritebis pirobebSi. amitom, Tuki zrealuri

moxvdeba uaryofis romelime areSi, H0 iqneba uaryofili da H1 miRebuli.

uaryofis are (rejection region) - kriteriumis

statistikis (SerCeviTi ganawilebis) is

mniSvnelobebi, romelTa albaToba tolia an

naklebia α-ze nulovani hipoTezis

WeSmaritebis pirobebSi. Tuki kriteriumis

statistika moxvdeba uaryofis areSi, maSin H0

uaryofili iqneba.

ormxrivi kriteriumi (two-tailed test) - statistikuri kriteriumi, roca uaryofis

aris asagebad gamoyenebulia kriteriumis

statistikis SerCevis ganawilebis orive kudi.

calmxrivi kriteriumi (one-tailed test)-

SeiZleba iseTi SemTxvevac moxdes, rom Cven garkveuli mizeziT uaryofis are z-is

SerCeviTi ganawilebis mxolod erTi mimarTulebiT ganvaTavsoT. am SemTxvevaSi

gamoiyeneba calmxrivi kriteriumi an mimarTulebis mqone kriteriumi.

suraTi #8.3. z statistikis SerCeviTi ganawileba gviCvenebs uaryofis areebs ormxrivi

kriteriumisaTvis, roca α=0.05. Tu zrealuris mniSvneloba xvdeba erT-erT

uaryofil areSi, H0 unda uarvyoT.

calmxrivi kriteriumis upiratesoba isaa, rom zrealuris ufro mcire mniSvneloba

SeiZleba moxvdes uaryofis areSi. calmxriv kriteriums ufro detalurad mogvianebiT

ganvixilavT am TavSi. am SemTxvevaSi ki ganvixilavT ormxrivi kriteriumis magaliTs.

Kkritikuli mniSvnelobebi. Kkritikuli mniSvnelobebi

konkretuli ricxviTi mniSvnelobebia, romlebic adgenen

uaryofis aris sazRvrebs. z-is kritikuli mniSvnelobebi

mocemulia cxrilSi #a-1 g sveti. pirveli nabiji am

kritikuli mniSvnelobis povnaSi aris mniSvnelovnobis

donis SerCeva, rogoricaa α=0.05. ormxrivi kriteriumisaTvis uaryofis areebi

ganlagebulia z-is SerCeviTi ganawilebis orive kudTan. amitomac Cven gvWirdeba are

ganawilebis TiToeul kudTan, romelSic moxvedra tolia α-s naxevrisa an 0.5/2, rac

tolia 0.025. z-is ra mniSvnelobebi akmayofileben moTxovnas, rom z-is SerCeviTi

ganawilebis 0.025 wili moxvdes am mniSvnelobebis miRma? cxrilSi #a-1 Cven gavyvebiT

g svets, sanam ar vipoviT ares, romelic mdebareobs z-is im mniSvnelobis gaswvriv,

romelic tolia 0.025. roca vipoviT am ares, wavikiTxavT z-is Sesatyvis mniSvnelobas

a svetSi. z-is es mniSvneloba iqneba 1.96. amrigad, α=0.05-saTvis ormxrivi kritikuli

mniSvnelobebia z = -1.96 da z=+1.96. z-is es kritikuli mniSvnelobebi (simbolurad

aRvniSnoT zkritikuli) gansazRvraven uaryofis areebis mdebareobas z-is SerCeviT

ganawilebaSi, rogorc es naCvenebia suraTze #8.3.

kritikuli mniSvnelobebi

(critical value) _ konkretuli

ricxviTi mniSvnelobebi,

romlebic adgenen

uaryofis aris sazRvrebs.

-3 -2 -1 0 +1 +2 +3 z-is mniSvnelobebi

H0 uaryofa uaryofis are 0.025


H0 miRebis zona

z=-1.96 z=+1.96 z-is kritikuli z-is kritikuli mniSvneloba mniSvneloba

gadawyvetilebebi statistikuri hipoTezebis Sesaxeb

H0 hipoTezis uaryofisa Tu miRebis Sesaxeb gadawyvetilebis miReba xdeba

zrealursa da zkritikuli mniSvnelobaTa SedarebiT. zrealuris is mniSvneloba, romelic

tolia an naklebia zkritikuli =-1.96 an tolia an metia zkritikuli=+1.96 xvdeba uaryofis

areSi. zrealuris mniSvnelobebi, romlebic xvdebian uaryofis areSi, arian

statistikurad iSviaTni, Tuki H0 aris WeSmariti , magram arian ufro mosalodneli im

SemTxvevebSi, roca H1 aris WeSmariti. amrigad, roca zrealuri xvdeba uaryofis areSi,

amas mivyavarT H0-is uaryofamde. Tuki uarvyofT H0-s, maSin unda miviRoT H1. Tu

zrealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da Tuki H0-s ver

uarvyofT, maSin ver miviRebT H1-s.

aRsaniSnavia, rom Cven viyenebT zrealuris mniSvnelobas, raTa miviRoT H0-is Sesaxeb

gadawyvetileba; uarvyoT H0, TuU CavTvaloT, rom H0-s ver uarvyofT. gadawyvetileba,

romelic miiReba H0-is Sesaxeb, SemdgomSi gansazRvravs gadawyvetilebas, romelic

miiReba H1-is Sesaxeb. Tu Cven uarvyofT H0-s, maSin unda miviRoT H1. piriqiT, Tu H0-s

ver uarvyofT, ver miviRebT H1-s.

muxlqveSa simaRlis magaliTs Tu davubrundebiT, zrealuri= + 3.33 aris ufro meti,

vidre zkritikuli=+1.96, amitom is xvdeba maRal uaryofis areSi. Sesabamisad, Cven unda

uarvyoT H0 : µ=42sm da miviRoT H1 : µ≠42sm. Cven uarvyofT nulovan hipoTezas imis

Sesaxeb, rom SerCeva iyo aRebuli im populaciidan, romlis saSualo 42.0 sm-is tolia.

radgan SerCevis saSualo, X =44.0 sm, aris ufro didi, vidre populaciis hipoTeturi

saSualo _ 42 sm, Cven gamogvaqvs daskvna, rom im qveyanaSi, romlidanac iyo aRebuli

SerCeva, mZRolebis saSualo muxlqveSa simaRle aris ufro meti, vidre X =42 sm.

saukeTeso wertilovani Sefaseba am populacis saSualosaTvis aris SerCevis saSualo,

X =44.0. erT-erTi mizezi imisa, rom satvirTo manqanebis skamebi SeiZleba yofiliyo

arakomfortuli, isaa, rom zogierT qveyanaSi mZRolebis populacias ufro grZeli

muxlqveSa simaRle aqvs, vidre im skamebis simaRle, romelic maTTvis iyo Seqmnili.

SevajamoT, rogor unda miviRoT gadawyvetileba statistikuri hipoTezebis

Sesaxeb:

• Tu zrealuri xvdeba uaryofis areSi, maSin uarvyofT H0-s da viRebT H1-s.

• Tu zrealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da ar

viRebT H1-s.

statistikurad mniSvnelovani gansxvaveba

statistikurad mniSvnelovania gansxvaveba, roca

kriteriumis statistikis realuri mniSvneloba xvdeba

uaryofis areSi da uarvyofT H0-s. z kriteriumisaTvis

stastikurad mniSvnelovani gansxvaveba gviCvenebs, rom

Cven miviReT gadawyvetileba imis Sesaxeb, rom SerCevis

saSualo aRebulia im populaciidan, romlis saSualo

gansxvavdeba populaciis hipoTeturi saSualosagan.

statistikurad mniSvnelovani gansxvaveba xSirad warmodgenilia Semdegi formiT:

zrealuri =+3.33, p≤0.05. simbolo ≤ niSnavs naklebia an tolia, amrigad, p≤0.05 ikiTxeba

rogorc “p naklebia an toli 0.05”. xSirad naklebobis niSani < gamoiyeneba ≤ niSnis

nacvlad, ase rom z aris warmodgenli, rogorc zrealuri=+3.33, p<0.05. orive SemTxvevaSi

naCvenebia, rom H0 uaryofilia 0.05 mniSvnelovnobis doneze, radganac zrealuris miRebis

albaToba (p), roca H0 WeSmaritia, tolia an naklebia 0.05-s. es gadawyvetilebebi da

daskvnebi, romelic miRebulia statistikurad mniSvnelovani gansxvavebidan z

kriteriumis saSualebiT Sejamebulia cxrili #8.1-is marcxena svetSi.

cxrili #8.1. SerCevis saSualosa da populaciis hipoTeturi saSualos Sefasebisas

gadawyvetilebebisa da daskvnebis Sejameba statistikuri hipoTezebis Semowmebis Sesaxeb z

kriteriumis gamoyenebiT. gamoyenebulia 0.05 mniSvnelovnobis done.

Tu zrealuri xvdeba

uaryofis areSi α=0.05-saTvis, maSin:

Tu zrealuri ar xvdeba


zrealuris miRebis albaToba naklebia an toli

0.05, an p<0.05.

H0 uaryofilia.

H1 miRebulia. SerCevis saSualo aRebulia

populaciidan, romlis saSualo gansxvavebulia

populaciis hipoTeturi saSualosagan.

gansxvaveba X -sa da hipoTetur µ-s Soris

statistikurad mniSvnelovania 0.05 doneze.

miiReba gadawyvetileba imis Sesaxeb, rom X -sa

da hipoTetur µ-s Soris gansxvavebis mizezi ara

SerCevis Secdomaa, aramed sxva ram.

zrealuris miRebis albaToba metia an toli 0.05,

an p>0.05

H0 ar aris uaryofili

H1 ar aris miRebuli


aris statistikurad aramniSvnelovani 0.05

doneze

miiReba gadawyvetileba imis Sesaxeb, rom

SerCevis Secdoma yvelaze damajerebelad

xsnis gansxvavebas X -sa da hipoTetur µ-s

Soris.

statistikurad

mniSvnelovani gansxvaveba

(statistically significant difference)

_ kriteriumis statis-tikis

realuri mniSvneloba

xvdeba uaryofis areSi da

H0 uaryofilia.

aramniSvnelovani gansxvaveba

gansxvaveba SerCevis saSualosa da populaciis

saSualos Soris aris aramniSvnelovani, Tuki

kriteriumis statistikis realuri mniSvneloba ar

xvdeba uaryofis areSi. nulovani hipoTeza ar aris

uaryofili da realurad miRebuli gansxvaveba X -sa

da hipoTetur µ-s Soris miewereba mxolod SerCevis

Secdomas. aramniSvnelovani gansxvaveba aRiniSneba p>0.05, rac mosdevs z realuris

ricxviT mniSvnelobas. gadawyvetilebebi da daskvnebi, romelic miRebulia z

kriteriumis saSualebiT statistikurad aramniSvnelovani gansxvavebidan, Sejamebulia

cxrilis #8.1 marcxena svetSi.

statistikuri hipoTezis Semowmebis etapebis Sejameba

yvela statistikuri kriteriumi moiTxovs Semdegs:

• kriteriumis statistikis arCevas, rogoricaa magaliTad, z;

• ori statistikuri hipoTezis formulirebas, H0 da H1;

• kriteriumis statistikis SerCeviTi ganawilebis miRebas H0-is WeSmaritebis

pirobebSi;

• mniSvnelovnobis donis SerCevas;

• kriteriumis statistikis kritikuli mniSvnelobis an mniSvnelobebis povnas;

• kriteriumis statistikis SerCeviT ganawilebaSi uaryofis aris an areebis

mdebareobis gansazRvras;

• statistikuri hipoTezebis Sesaxeb gadawyvetilebis miRebis wesebis

formulirebas;

• kriteriumis statistikis mniSvnelobis gamoTvlas SerCevis monacemebiT;

• gadawyvetilebis miRebas H0-is uaryofisa Tu misi ver uaryofis Sesaxeb imis

safuZvelze, rom kriteriumis statistikis realuri mniSvneloba xvdeba Tu

ara uaryofis areSi;

• gadawyvetilebis gamotanas H1-is miRebisa Tu ar miRebis Sesaxeb H0-isaTvis

miRebuli gadawyvetilebis safuZvelze.

statistikuri hipoTezebis Semowmebis Sesavali SeiZleba rTuli mogeCvenoT, magram

is male gaxdeba martivi, radgan TiToeuli statistikiuri kriteriumisaTvis Semdeg

erTi da igive etapebia Sesasrulebeli.

aramniSvnelovani gansxvaveba (nonsignificant difference) _

kriteriumis statistikis

realuri mniSvneloba ar xvdeba

uaryofis areSi da nulovani

hipoTeza ar aris uaryofili

SeamowmeT Tqveni codna VIII-1

1. gansazRvreT: alfa done, α, alternatiuli hipoTeza, kritikuli mniSvneloba, H0,

H1, aramniSvnelovani gansxvaveba, nulovani hipoTeza, erTmxrivi kriteriumi,

uaryofis are, mniSvnelovnobis done, statistikuri hipoTeza, statistikurad

mniSvnelovani gansxvaveba, kriteriumis statistika, ormxrivi kriteriumi,

zkritikuli, zrealuri.

2. gansazRvreT statistikuri hipoTezebis ori Tviseba.

3. CawereT H0 da H1 pirobiTi niSnebiT Semdegi debulebebisaTvis:

a) veqsleris inteleqtis testis qulebisaTvis µ aris 100.

b) 25-dan 54 wlamde qalebis populaciisaTvis televizoris yurebaze

daxarjuli saSualo dro dReSi 4.4 saaTis tolia.

g) martiv stimulze 19 wlis mamakacis reaqciis saSualo dro aris 423.0

miliwami.

4. qvemoT moyvanili TiToeuli monacemisaTvis µ warmoadgens populaciis saSualos,

Xσ saSualos standartul Secdomas am populaciisaTvis da X _ SerCevis

saSualos, romelic aRebulia am populaciidan. qvemoT mocemul mniSvnelobaTa

TiToeuli rigisaTvis gansazRvreT H0 da H1 da ipoveT zrealuris mniSvneloba.

Semdeg cxrili #a-1-is gamoyenebiT gansazRvreT, xvdeba Tu ara zrealuri ormxriv

uaryofis areSi 0.05 mniSvnelovnobis doneze da miuTiTeT Tqveni gadawyvetileba

H0-isa da H1-saTvis.

5. gadawyviteT Semdegi amocana 0.05

mniSvnelovnobis donisa da ormxrivi

kriteriumis gamoyenebiT. populaciis

saSualo inteleqtis skalisaTvis aris

100 da σ=15. Tqven xarT dawyebiTi skolis

xelmZRvaneli soflis skolaSi. Tqveni

skolis 121 moswavlis X am skalaze tolia 103.4-is. gansxvavdeba Tu ara Tqveni

skolis moswavleTa saSualo mniSvnelovnad populaciis saSualosagan? Tqvens

pasuxSi miuTiTeT:

a) romel statistikas gamoiyenebT;

b) statistikuri hipoTezebi;

g) mniSvnelovnobis done;

d) zkritikuli;

e) uaryofis areebis mdebareoba;

v) zrealuri;

z) Tqveni gadawyvetilebani H0-is da H1-is Sesaxeb;

µ Xσ X

a) 50 4.7 40

b) 100 25 70

g) 143 0.4 143.9

d) 87 2.9 92

T) Tqveni gadawyvetilebani Tqveni moswavleebis saSualos Sesaxeb populaciis

saSualosTan _ 100 mimarTebaSi.


praqtikaSi σ xSirad fasdeba s-is _ SerCevaSi qulebis standartuli gadaxris

saSualebiT. amitom Xσ -c fasdeba Xs -is saSualebiT. roca Cven Xσ -s Xs -is

saSualebiT vafasebT, viRebT axal statistikas _ t-s, romelic ganisazRvreba, rogorc

t =Xs

X µ−

Tu Xs -is naclad CavsvamT s/√N ,

t =Ns

X/

µ−

sadac X = SerCevis saSualoa

µ= populaciis hipoTeturi saSualoa

Xs = saSualos standartuli Secdomis Sefaseba, romelic miRebuli iyo

populaciis standartuli gadaxris Sefasebis _ s-is saSualebiT

N= SerCevis moculoba.

t kriteriumi SeiZleba iyos gamoyenebuli im

gansxvavebis statistikuri mniSvnelovnobis

Sesamowmeblad, romelic arsebobs SerCevis saSualosa

da populaciis hipoTetur saSualos Soris, roca Xσ

Sefasebulia Xs -is saSualebiT. radgan mxolod erTi

SerCevis saSualos gulisxmobs, statistikas uwodeben

erTi SerCevis t-s, iseTi t kriteriumisagan gansxvavebis

mizniT, romelic ori SerCevis saSualos moicavs, rac

me-9 TavSi iqneba ganxiluli.

t kriteriumi SemoRebuli iyo britaneli maTematikosis uiliam sili gosetis mier

me-20 saukunis dasawyisSi. gosetma gamosca Tavisi naSromi fsevdonim “studentis”

gamoyenebiT. amitomac t kriteriums xSirad uwodeben `studentis t kriteriums”.

t-s SerCeviTi ganawileba

t statistikis gamoyenebisaTvis unda vicodeT misi SerCeviTi ganawileba. t-s

SerCeviTi ganawileba gansxvavdeba z-is SerCeviTi ganawilebisagan. radgan z statistika

iyenebs populaciis parametrs Xσ -s, mas aqvs mxolod erTi SerCeviTi ganawileba,


(one-sample t test) _

t kriteriumi, romelic

gamoiyeneba SerCevis

saSualosa da populaciis

hipoTetur saSualos Soris

gansxvavebis statistikuri

mniSvnelovnobis

Sesamowmeblad, roca σ X

Sefasebulia s X -is saSualebiT.

romelic warmodgenilia cxrilSi #a-1. t kriteriumi ki, radgan iyenebs populaciis s

standartul gadaxris Sefasebas, ganisazRvreba, rogorc

s=1

)( 2

−

−∑N

XX

s-is gamoTvla gulisxmobs SerCevis N moculobas. amitom t-s SerCeviTi ganawileba

icvleba SerCevis moculobasTan erTad, an ufro zustad rom vTqvaT, SerCevis

Tavisuflebis xarisxTan erTad.

Tavisuflebis xarisxi

Tavisuflebis xarisxi (romlis abreviaturaa df)

ganisazRvreba rogorc qulebis raodenoba, romelic

Tavisufalia cvalebadobis dros statistikis

gamoTvlisas. moviyvanT ramdenime magaliTs imis ukeT

gasagebad, Tu ras niSnavs `qulebis raodenoba, romelic

Tavisufalia cvalebadobis dros” statistikis gamoTvlisas. davuSvaT, rom

Tqven gTxoves amoirCioT sami qula testSi. Tqven SeiZleba airCioT

nebismieri sami qula, vTqvaT, 75, 62 da 97. am davalebisas ar arsebobda raime

SezRudva qulebis arCevanSi; TiToeuli qula SeiZleba Tavisufali iyos

Tavisi cvalebadobis dros. Sesabamisad qulebis am simravlisaTvis

Tavisuflebis xarisxi df aris N=3. ganvixiloT meore davaleba, roca Tqven

gTxoven airCioT sami qula, magram iseTi, romelTa jami unda iyos 254.

davuSvaT Tqven airCieT 81 da 78, rogorc pirveli ori qula; maT SeiZleba

hqondeT nebismieri mniSvneloba, romelsac Tqven isurvebT. es ori qula

SeiZleba Tavisufali iyos Tavisi cvalebadobis dros. SeiZleba mesame qula

aseve Tavisuflad Seicvalos? ara. radgan sami qulis jami unda iyos 254,

mesame qula unda iyos 95; arc erTi sxva qulis SemTxvevaSi es jami ar

miiReba. amitom, Tu Tqven giTxres, rom qulaTa jami aris 254, ori qula

SeiZleba Tavisuflad Seicvalos, xolo mesame aris gansazRvruli im ori

qulis mniSvnelobiT, romlebic Tavisuflad SeiZleba Seicvalos da am sami

qulis jamiT. Sesabamisad, Tu Tqven iciT sami qulis jami, maSin arsebobs

N_1=3_1=2 Tavisuflebis xarisxi am sami qulisaTvis.

Tu gamoviyenebT am argumentacias SerCeviTi saSualos gamoTvlisas, X -s

aqvs N df. yvela qula qulaTa erTobliobaSi aris Tavisufali

cvalebadobisas, roca viTvliT X -s. magaliTad, Tqven gTxoves gamoiTvaloT

SerCevis saSualo 5 qulis erTobliobisaTvis, aqedan 4 aris mocemuli: 7, 10,


(degrees of freedom): qulaTa raodenoba, romelic Tavisufalia cvalebadobis dros statistikis gamoTvlisas.

3 da 6. mxolod am informaciis mocemulobisas Tqven ver SesZlebTY

saSualos povnas. mexuTe qulad SeiZleba CaTvaloT nebismieri mniSvneloba

da Tqven ar gecodinebaT mexuTe qulis mniSvneloba mxolod am oTxi

mniSvnelobis codnis safuZvelze. amrigad, yvela N qula (sadac N=5

rogorc am magaliTSi) aris Tavisufali cvalebadobisas, roca xdeba

SerCevis saSualos gamoTvla.

s-is Tavisuflebis xarisxi aris N_1. amis sailustraciod davuSvaT, rom

Tqven gTxoves gamoTvaloT s 5 qulis mqone SerCevis safuZvelze.

populaciis standartuli gadaxris Sefaseba ganisazRvreba, rogorc

1

)( 2

−

−= ∑

NXX

s

davuSvaT, rom xuTidan oTx qulas aqvs Semdegi mniSvnelobebi: 13, 15, 13 da 12

da am xuTi qulis saSualo aris 13.4. Cven viciT, agreTve, rom 0)( =−∑ XX

Tuki Cven gamovaklebT X am oTxi qulidan TiToeuls, maSin Cven miviRebT

)( XX −

13 _ 13.4= - 0.4

15 _ 13.4= +1.6

13 _ 13.4= - 0.4

12 _ 13.4= -1.4

jami = -0.6

am oTxi )( XX − sxvaobis jami tolia -0.6. roca mexuTe qula iqneba CarTuli

am erTobliobaSi, es jami toli iqneba nulis. amrigad, mexuTe qulam unda

mogvces )( XX − =+0.6, +0.6 rom davumatoT -0.6, jami toli iqneba nulis. Tuki

X5 _ 13.4= +0.6, maSin X5 toli unda iyos 14. Sesabamisad, Tu X cnobilia am

xuTi qulis erTobliobisaTvis, maSin am xuTi qulidan mxolod oTxi qulis

mniSvnelobaa Tavisufali cvalebadobisas. mexuTe qula ver Seicvleba

Tavisuflad. misi mniSvneloba aris mkacrad gansazRvruli im SezRudviT,

romelic saSualos mniSvnelobiT iqna Semotanili. radgan X unda iyos

cnobili, raTa gamoviTvaloT s, yvela qula, garda erTisa, aris Tavisufali

cvalebadobis dros s-is gamoTvlisas. amrigad, roca viTvliT s an s2 NN

qulaTa erTobliobas aqvs N-1 Tavisuflebis xarisxi df.

Cven SegviZlia Tavisuflebis xarisxis Sesaxeb es codna s-is gamosaTvlel

formulaSi gamoviyenoT, raTa is ufro mokled CavweroT. radgan ∑ − 2)( XX

, agreTve, kvadratebis jams (SS) uwodeben, da N-1 asaxavs Tavisuflebis

xarisxs SS-is gamoTvlisas, populaciis standartuli gadaxris Sefaseba

SeiZleba gamovsaxoT rogorc kvadratuli fesvi kvadratebis jamidan

gayofili Tavisuflebis xarisxze:

dfSSs = .

momdevno TavebSi Cven vnaxavT, rom s-is es gamosaxuleba sakmaod xSirad

gamoiyeneba.

erTi SerCevis t kriteriumis Tavisuflebis xarisxi

axla gamoviyenoT df-is Sesaxeb Cveni codna t kriteriumisaTvis. saSualos

standartuli Secdomis Sefaseba, Xs gamoiTvleba rogorc s/√N. radgan s-s aqvs

N-1 df, Xs -s, agreTve, aqvs N-1 df. amrigad, erTi SerCevis t statistikis SerCeviTi

ganawilebisaTvis Tavisuflebis xarisxi aris, agreTve N-1.

t SerCeviTi ganawileba

TiToeuli df-isaTvis arsebobs t-s sxvadasxva Teoriuli ganawileba. radgan

potenciurad qulaTa nebismieri raodenoba SeiZleba iyos miRebuli SerCevaSi,

df SeiZleba icvlebodes 1-dan usasrulobamde. rogorc potenciuri SerCeviTi

ganawilebis magaliTi, suraTi #8.4 gviCvenebs t ganawilebebs, roca df aris 2, 4,

da 60.

Tumca t ganawilebis forma icvleba Tavisuflebis xarisxis

cvalebadobasTan erTad, is normaluri ganawilebis msgavsia ramdenime aspeqtSi.

TiToeuli t ganawileba simetriuli da unimodaluria da aqvs saSualo,

romelic nulis tolia. ganvixiloT, ratom Cndeba es Tavisebureba. roca

nulovani hipoTeza, iseTi rogoricaa: H0:µ=populaciis saSualos raime

hipoTeturi mniSvnelobisa, WeSmaritia, t-s mosalodneli mniSvneloba iqneba

nuli, radgan ( X _µ)-s saSualo im SerCeviT ganawilebaSi, romelic iyo aRebuli

aseTi populaciidan iqneba nuli. magram, maSinac roca H0 WeSmaritia, µ−_X -s

mniSvnelobebi Seicvleba nulis toli saSualos garSemo SerCevis Secdomis

gamo. ufro metic, mosalodnelia, rom _X iqneba µ-ze meti, TiTqmis iseve xSirad,

rogorc, is iqneba µ-ze naklebi. mosalodnelia, rom, roca nulovani hipoTeza

WeSmaritia, µ−_X mniSvnelobebi iseve xSirad iqneba dadebiTi, rogorc

uaryofiTi. amrigad, t-s mosalodneli mniSvnelobebi simetriulad iqneba

ganlagebuli nulis toli saSualos garSemo.

Tavisuflebis xarisxis usasrulo raodenobisaTvis (Teoriuli da ara realuri

SesaZlebloba) t SerCeviTi ganawileba aris normaluri ganawilebis msgavsi.

yvela sxva t ganawilebebi arian ufro farTod gaSlilebi, vidre normaluri

ganawileba.

ratom aris ufro swori Tavisuflebis xarisxis da ara N-is gamoyeneba? Tu

df erTi SerCevis t kriteriumisaTvis aris mocemuli NN-1 saSualebiT, ratom ar

SeiZleboda t SerCeviTi ganawilebis warmodgena N-is terminebiT df-is magivrad?

erTi SerCevis t statistikisaTvis Cven ver miviRebT verc erT upiratesobas df-

is gamoyenebiT N-is magivrad, im df-saTvis, romelic yovelTvis erTiT naklebia

N-ze. df-is gamoyenebis upiratesoba gamoCndeba me-9 TavSi, sadac Cven gagacnobT

sxva t statistikas, romelic gamoiyeneba ori SerCevis saSualoTa Sesadareblad.

am t-saTvis df N-2 da ara N-1-ia. toli df-saTvis orive t kriteriums aqvs erTi

da igive SerCeviTi ganawileba. igive magaliTebs sxva statistikebis

ganxilvisas momdevno TavebSi SevxvdebiT.


1. dawereT erTi SerCevis t statistikis formula.

2. ratom uwodeben t statistikas zogjer studentis t-s?

3. Tqven mocemuli gaqvT 7 qula: 73, 42, 51, 68, 49, 62 da 57. ramdeni

Tavisuflebis xarisxi aqvs am qulebis simravles, raTa gamoiTvaloT

a) X b) s da g) s2

4. Tqven gegmavT gamoTvaloT erTi SerCevis t statistika Semdegi

moculobebis SerCevebze. TiToeuli SerCevisaTvis, ipoveT df: a) N=10, b) N=16,

g) N=7, d) N=54, e) N= 120, v) N=30

5. Tqven gaqvT t-s ori Teoriuli ganawileba, erTi 13 df da meore 27 df.

romeli ganawileba iqneba ufro dabali da gaSlili?

6. ra pirobebSi iqneba t statistikis SerCeviTi ganawileba z SerCeviTi

ganawilebis identuri?

statistikuri hipoTezebis Semowmeba t-s saSualebiT erTi SerCevis

SemTxvevaSi

erTi SerCevis t saSualebas gvaZlevs ganvsazRvroT, gansxvavdeba Tu ara X

hipoTeturi µ-sagan ufro metad, vidre es iqneboda mosalodneli mxolod

SerCevis Secdomis SemTxvevaSi, roca σ aris gamoTvlili s-is saSualebiT. t-s

saSualebiT statistikuri hipoTezis Semowmebis procedura identuria imisa,

rac iyo z statistikisaTvis. aucilebeli nabijebi moicavs Semdegs:

• H0-is da H1-is formulireba;

• t statistikis SerCeviTi ganawilebis miReba H0-is WeSmaritebis

pirobebSi;

• mniSvnelovnobis donis SerCeva;

• t-s kritikuli mniSvnelobis an mniSvnelobebis povna;

• uaryofis aris an areebis mdebareobis dadgena t-s SerCeviT ganawilebaSi;

• gadawyvetilebis miReba statistikuri hipoTezebis Sesaxeb;

• trealuris mniSvnelobis gamoTvla SerCevis monacemebis safuZvelze;

• gadawyvetilebis miReba H0-is da H1-is Sesaxeb trealuris mniSvnelobaze

dayrdnobiT.

am nabijebs asaxavs Semdegi amocanis magaliTi. davuSvaT, bavSvTa fsiqologma,

romelic ikvlevda sakiTxs, rogor atarebdnen dros mesame klaseli bavSvebi,

romlebic CarTulni iyvnen gakveTilebis Semdgom bavSvebis movlis programaSi,

aRmoaCina, rom kviris dReebSi bavSvebi uyureben televizors saSualod 83

wuTis ganmavlobaSi, romlis s 26 wuTis tolia. davuSvaT, agreTve, cnobilia,

rom dReSi televizoris yurebaze daxarjuli drois saSualo mesame klaselTa

populaciisaTvis Seadgens 95 wuTs. aris Tu ara SerCevis saSualo, romelic

tolia 83 wuTis, mniSvnelovnad gansxvavebuli populaciis saSualosagan _ 95

wuTisagan? radgan σ aris gamoTvlili s-is saSualebiT, am kiTxvaze pasuxis

gasacemad unda iyos gamoyenebuli erTi SerCevis t kriteriumi.

statistikuri hipoTezebis formulireba

statistikuri hipoTezebi erTi SerCevis t statistikisaTvis identuria imis, rac

iyo gamoyenebuli z statistikis SemTxvevaSi. nulovani hipoTeza H0

gansazRvrulia rogorc populaciis saSualos hipoTeturi mniSvneloba.

alternatiuli hipoTeza, H1 ganisazRvreba rogorc debuleba, rom populaciis

saSualo ar aris is, rac navaraudevia H0-is SemTxvevaSi. magaliTad,

statistikuri hipoTezebis pirobiTi aRniSvna aris

H0:µ=95 wuTi

H1:µ≠95 wuTi

t statistikis SerCeviTi ganawilebis miReba

t-s SerCeviTi ganawileba damokidebulia t-s Tavisuflebis xarisxebze, romelic

gamoiTvleba SerCevis qulebis safuZvelze. df erTi SerCevis t-saTvis mocemulia

N-1 saxiT. im saSualosaTvis, romelic efuZneba 30 monacems, df t-saTvis aris

30-1=29. mniSvnelovnobis donisa da uaryofis areebis arCevis Semdeg,

Tavisuflebis xarisxis am maCvenebels gamoviyenebT t-s SerCeviTi ganawilebis

cxrilSi tkritikulis sapovnelad.

mniSvnelovnobis donis arCeva

mniSvnelovnobis done aris albaTobis mniSvneloba, romelic uzrunvelyofs

H0-is uaryofas. statistikuri hipoTezis testirebisas Cveulebriv gamoiyeneba

0.05 an 0.01 mniSvnelovnobis done. magaliTad, Cven viyenebT α=0.05.

uaryofis areebis gansazRvra t kriteriumisaTvis: ormxrivi kriteriumi

roca H0 WeSmaritia uaryofis are moicavs t ganawilebis iseT mniSvnelobebs,

romlis albaToba aris α-s toli an masze naklebi, amrigad, uaryofis are

gansazRvravs trealuris iseT mniSvnelobebs, romlebic uaryofis areSi moxvdebian.

ormxriv anu aramimarTulebiT kriteriumSi t-s uaryofis areebi

mdebareoben Teoriuli SerCeviTi ganawilebis orive kudze, rogorc es

naCvenebia suraTze #8.4. roca H0 WeSmaritia, X -is mniSvnelobaTa umetesoba

mxolod mcired iqneba gansxvavebuli µ-sagan. amrigad, t-s mniSvnelobaTa

umravlesoba dajgufebuli iqneba ganawilebis SuaSi nulis garSemo. SerCevis

Secdomis gamo, t-s zogierTi mniSvneloba iqneba ganawilebis romelime boloSi,

maSinac ki roca H0 WeSmaritia. es mniSvnelobebi miiReba, roca t-s mricxveli

(e.i. µ−_X sxvaoba) metia mniSvnelze (anu Xs ). magram t-s aseTi didi

mniSvnelobebi ufro iSviaTi xdomilobebia nulovani hipoTezis WeSmaritebis

pirobebSi.

meore mxriv, Tuki H1 aris WeSmariti da X asaxavs SerCevas im

populaciidan, romlis saSualo gansxvavdeba µ-sagan, maSin ufro xSirad

mosalodnelia µ−_X -is da, maSasadame t-s didi mniSvnelobebi. amrigad,

uaryofis are mdebareobs t-s Teoriuli SerCeviTi ganawilebis TiToeul kudze

da mTeli α iyofa or nawilad kudebs Soris. Tuki α =0.05, maSin TiToeuli

uaryofis are moicavs albaTobas α/2, an 0.025. Tuki SerCevis qulebis mixedviT

gamoTvlili t-s mniSvneloba, (gansazRvruli, rogorc trealuri ) xvdeba romelime

uaryofis areSi, maSin nulovani hipoTeza iqneba uaryofili da miiReba

alternatiuli hipoTeza. uaryofis aris mdebareobis gansazRvriT t ganawilebis

orive kudze, eqsperimentatori iTvaliswinebs orive SesaZleblobas, romelic

SeiZleba iyos nagulisxmevi alternatiul hipoTezebSi: roca X metia µ-ze da

roca X naklebia µ-ze. uaryofis am or areSi moxvedris sruli albaToba

udris 0.05, roca H0 WeSmaritia.

suraTi #8.4

t ganawilebis ormxrivi uaryofis are, roca df aris 29 da α=0.05. trealuri

mniSvneloba xvdeba uaryofis erT-erT areSi, rasac mivyevarT H0 –is uaryofamde

da viRebT H1-s.

t-s SerCeviTi ganawilebis uaryofis ori are, roca df aris 29, naCvenebia

suraTze #8.4. Tu df tolia 29-is da H0 WeSmaritia, maSin albaToba imisa, rom t

iqneba toli an naklebi, vidre -2.045, aris 0.025 da albaToba imisa, rom t iqneba

t=-2.045 t=+2.045 t-s kritikuli t-s kritikuli mniSvneloba mniSvneloba

-3 -2 -1 0 +1 +2 +3 t-s mniSvneloba



H0 miRebis zona

toli an meti +2.045-ze isev 0.025-ia. trealuris nebismieri mniSvneloba, romelic

tolia an naklebia -2.045-ze, an tolia, an metia +2.045-ze xvdeba uaryofis areSi.

es Sedegi miiReba xdomilobaTa mxolod 5 procentis SemTxvevaSi, roca

nulovani hipoTeza WeSmaritia. mniSvnelobebi -2.045 da +2.045 miRebuli iyo t-s

SerCeviTi ganawilebis kritikul mniSvnelobaTa cxrilidan. am cxrilis

gamoyeneba SemdgomSi iqneba ganmartebuli.

t kritikul mniSvnelobaTa cxrilis gamoyeneba

FmniSvnelovnobis calkeuli donisaTvis saWiroa t-s kritikul mniSvnelobebis

gansazRvra (aRiniSneba, rogorc tkritikuli.), romlebic damokidebulni arian

trealuris df-ze. t ganawilebis grafikebi naTlad gviCvenebs, rom t ganawilebis

forma icvleba Tavisuflebis xarisxebis cvalebadobasTan erTad, magram am

grafikebidan Znelia ipovo tkritikulis zusti mniSvneloba. amitom tkritikuli

mniSvnelobebi 0.05 da 0.01 mniSvnelovnobis doneebisaTvis gamoTvlilia da

Setanilia cxrilebSi.

cxrili #8.2 asaxavs tkritikulis mniSvnelobebs α=0.05 da 0.01. ormxrivi

kriteriumisaTvis. cxrili #8.2 aris danarTSi mocemuli cxrili #a-2-is

nawili. pirvel svetSi mocemulia Tavisuflebis xarisxebi t

ganawilebebisaTvis. α=0.05 da 0.01 svetebis nebismier striqonSi mocemulia

tkritikuli mniSvnelobebi am mniSvnelovnobis donis Sesatyvisi Tavisuflebis

xarisxisaTvis. magaliTad, roca α=0.05 tkritikuli ormxrivi mniSvneloba im

trealurisaTvis, romelsac aqvs 29-is toli Tavisuflebis xarisxi, aris 2.045.

miaqcieT yuradReba, rom am mniSvnelobas ara aqvs win + an _ niSani. is

warmodgenilia rogorc t-s absoluturi mniSvneloba. Tumca uaryofis areebis

mdebareobis gansazRvrisaTvis, es mniSvneloba unda iyos warmodgenili rogorc

-2.045 da +2.045. analogiurad α=0.01-saTvis ormxrivi tkritikuli mniSvneloba,

romlis Tavisuflebis xarisxi 29-is tolia udris 2.756. aRsaniSnavia, rom 0.01

mniSvnelovnobis done ufro mkacri kriteriumia H0-is uaryofisaTvis, vidre

α=0.05. nebismieri realuri µ−_X sxvaoba unda iyos ufro didi, raTa uarvyoT

H0, roca 0.05-s nacvlad miRebuli iqneba 0.01 mniSvnelovnobis done.

cxrili #8.2. t ganawilebis kritikuli mniSvnelobebi α=0.05 da α=0.01-saTvis.

mniSvnelobebi mocemulia ormxrivi kriteriumisaTvis.

df α=0.05 α=0.01

1 12.607 63.657

2 4.3030 9.925

3 3.182 5.841

4 2.776 4.604

5 2.571 4.032

6 2.447 3.707

7 2.365 3.499

8 2.306 3.355

9 2.262 3.25

10 2.228 3.169

11 2.201 3.106

12 2.179 3.055

13 2.160 3.012

14 2.145 2.977

15 2.131 2.947

16 2.120 2.921

17 2.110 2.898

18 2.101 2.878

19 2.093 2.861

20 2.086 2.845

21 2.080 2.831

22 2.074 2.819

23 2.069 2.807

24 2.064 2.797

25 2.060 2.787

26 2.056 2.779

27 2.052 2.771

28 2.048 2.763

29 2.045 2.756

30 2.042 2.750

40 2.021 2.704

60 2.000 2.660

120 1.980 2.617

∞ 1.960 2.576

30-ze ufro meti Tavisuflebis xarisxisaTvis mxolod calkeuli

kritikuli mniSvnelobebi aris mocemuli cxrilSi. roca df trealurisaTvis metia

30-ze, girCevT, rom tkritikulis mniSvneloba gamoTvlili iyos im df-s safuZvelze,

romelic cxrilSi yvelaze axlos imyofeba, magram amasTanave naklebia am

trealuris df-ze. magaliTad, TuU df=35, cxrili #8.2 gviCvenebs trealuris

mniSvnelobebs 30 da 40 df-isaTvis, magram ara maT Soris mdebare

mniSvnelobebisaTvis. amitomac tkritikulis gansazRvrisaTvis, roca df=35, Cven

viyenebT 30-is tol df (is tolia 2.042 α=0.05) nacvlad imisa, rom gamogveyenebina

40 df. am proceduris Sedegad, mniSvnelovnobis done iqneba odnav dabali, vidre

0.05. df=40 rom gamogveyenebina, maSin tkritikulis mniSvnelovnobis done iqneboda

odnav maRali, vidre 0.05. Tu trealuri xvdeba uaryofis areSi df=30-saTvis, maSin

is, agreTve, xvdeba uaryofis areSi df=40-saTvisac. Tuki trealuri (magaliTad,

trealuri =2.033) moxvdeba cxrilSi im mniSvnelobaTa Soris, romlebic

Seesabamebian df-is momdevno dabal (mag., tkritikuli=2.045, roca df aris 30) da

momdevno maRal mniSvnelobebs (mag. tkritikuli=2.021, roca df aris 40), maSin

tkritikulis ufro zusti mniSvneloba SeiZleba gamoviTvaloT cxrilis

mniSvnelobebidan.

t uaryofis areebis mdebareobis gansazRvra: calmxrivi kriteriumi

ormxriv SemowmebaSi H0-s uaryofen roca trealuri xvdeba t-s ganawilebis

romelime boloSi. magram, roca sakvlevi hipoTeziT nawinaswarmetyvelebia

saSualoTa Soris mimarTulebiTi kavSiri, (mag., Tu Cvens magaliTSi mkvlevari

varaudobs, rom X iqneba ufro naklebi, vidre µ) da ar arsebobs interesi

navaraudevis sapirisipiro SedegSi (mag., Tu X aris ufro meti, vidre µ), maSin

zogierTi mkvlevari iyenebs calmxriv kriteriums. calmxriv anu mimarTulebis

mqone kriteriumSi uaryofis are, romelic SemosazRvrulia α-s sruli

mniSvnelobiT, ganlagebulia t ganawilebis im kudis mxareze, romelic

Seesatyviseba kvlevis hipoTeziT navaraudevi Sedegis mimarTulebas. tkritikuli

mniSvnelobebi calmxrivi kriteriumisaTvis 0.01 da 0.05 mniSvnelobaTa

doneebiT warmodgenilia, agreTve, cxrilSi #a-2. df=29-saTvis, calmxrivi

tkritikuli 0.05 mniSvnelovnobis doneze aris 1.699. calmxrivi uaryofis are, roca

tkritikuli=-1.699 naCvenebia suraTze #8.5, t ganawilebis marcxena kudze. es

uaryofis are am magaliTisaTvis Sesatyvisi iqneboda, Tu davuSvebdiT Semdeg

statistikur hipoTezebs:

H0 : µ ≥42

H1 : µ<42

suraTi aklia (Reject _ H0 uaryofa, Do not reject _ H0 miRebis zona, Rejection region - uaryofis are, value of t_ t-s mniSvneloba, critical value of t _ t-s kritikuli mniSvneloba) suraTi # 8.5

t ganawilebis calmxrivi uaryofis are, roca df aris 29 da α=0.05. trealuris

mniSvneloba xvdeba uaryofis areSi, rasac mivyavarT H0 –is uaryofamde da

viRebT H1.

rodis unda iyos gamoyenebuli calmxrivi kriteriumi?

calmxrivi uaryofis aris upiratesoba isaa, rom tkritikuli iqneba ufro mcire,

radgan mTeli uaryofis are mdebareobs ganawilebis mxolod erT kudSi.

amitomac ufro advilia H0-is uaryofa, radgan X -sa da µ Soris mcire

gansxvavebisas trealuri xvdeba uaryofis areSi. TumcaRa am upiratesobas Tan

axlavs SezRudvac. gadawyvetileba imis Sesaxeb, Tu romeli uaryofis are

iqneba gamoyenebuli, calmxrivi Tu ormxrivi, miiReba im dros, roca xdeba

kvlevis dagegmva da sanam daiwyeba monacemTa Segroveba da analizi. Tuki

miRebuli iqneba calmxrivi uaryofis are, maSin eqsperimentatori ver SeZlebs

uaryos H0 saSualoTa Soris miRebuli im sxvaobis safuZvelze, romelic am

hipoTezis sapirispiro mimarTulebisaa, radgan aseTi SedegisaTvis uaryofis

are ar gvaqvs. calmxriv da ormxriv kriteriumebs ufro dawvrilebiT me-9

TavSi ganvixilavT. praqtikaSi ormxrivi kriteriumebi yvelaze farTod

gamoyenebadia; amitomac, isini gamoyenebulni arian am Tavis yvela magaliTSi.

trealuris gamoTvla

trealuris gamoTvla moiTxovs X -is, µ da Xs mniSvnelobaTa Casmas formulaSi.

magaliTad, 30 mesameklaselisaTvis televizoris yurebis X =83 wuTs, s=26wuTs

da N=30. µ-s hipoTeturi mniSvneloba tolia 95 wuTis. Xs -is mniSvneloba

miRebulia s/√N-dan; amrigad,

Xs =477.526

3026

= ,

rac tolia 4.75. amis Sedegad,

trealuri =X

Xσ

µ−

= 526.275.412

75.49583

−=−

=−

roca df aris N-1=29 .

gadawyvetilebani statistikuri hipoTezebis Sesaxeb

statistikur hipoTezaTa Sesaxeb gadawyvetilebis miRebis wesebi erTi da igivea

yvela statistikuri kriteriumisaTvis. Sesabamisad, erTi SerCevis t

kriteriumisaTvis:

• Tuki trealuri xvdeba uaryofis areSi, maSin H0 iqneba uaryofili da

H1 miRebuli;

• Tuki trealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da

H1 ar iqneba miRebuli.

tkritikulis ormxrivi mniSvneloba, roca df=29 da α=0.05, aris 2.045. Sesabamisad,

rogorc naCvenebia suraTze #8.5, trealuri, romelic tolia an naklebia -2.045-ze

an tolia, an metia +2.045-ze, xvdeba uaryofis areSi. trealuri _2.526 aris ufro

Sors mdebare (eqstremaluri), vidre tkritikuli =-2.045 da amitomac xvdeba

ganawilebis marcxena kudze mdebare uaryofis areSi. Cven uarvyofT H0:µ ≠ 95

wuTs da viReb H1:µ ≠ 95. gansxvaveba SerCeviT saSualosa da populaciis

saSualos Soris aris statistikurad mniSvnelovani 0.05 doneze. Cven vaskvniT,

rom SerCevis saSualo, romelic tolia 83 wuTis aris im populaciidan,

romlis µ naklebia 95 wuTze. im mesameklaselebis mier televizoris yurebaze

daxarjuli dro, romlebic CarTulni iyvnen gakveTilebis Semdgom bavSvebis

movlis programaSi, aris mniSvnelovnad naklebi, vidre populaciis saSualo,

romelic tolia 95 wuTisa dReSi.

Tuki trealuri xvdeba or kritikul mniSvnelobas _ (-2.045) da +2.045 Soris

(magaliTad, X aris 90 wuTi da amrigad, trealuri =-1.056), maSin H0 -s ver uarvyofT

da ver miviRebT H1-s. realuri gansxvaveba SerCeviT saSualosa da populaciis

hipoTetur saSualos Soris iqneboda aramniSvnelovani. am SemTxvevaSi ar

gveqneboda sakmarisi mtkicebuleba imisaTvis, raTa CagveTvala, rom H0 ar aris

WeSmariti.

gadawyvetilebani statistikuri hipoTezebisa da im daskvnebis Sesaxeb,

romlebic miRebuli iqneboda romelime statistikurad mniSvnelovani an

aramniSvnelovani erTi SerCevis t kriteriumisaTvis aris Sejamebuli cxrilSi

#8.3.

daSvebebi erTi SerCevis t kriteriumisaTvis

erTi SerCevis t kriteriumi ayalibebs Semdeg daSvebebs SerCevis

monacemebisaTvis:

• monacemebi SerCevaSi SemTxveviTad arian miRebuli populaciis

monacemebidan da TiToeuli monacemi damoukidebelia nebismieri sxva

monacemisagan;

• monacemebi populaciis SerCevaSi normaluradaa ganawilebuli.

praqtikaSi, SerCevidan aRebulma monacemebma SeiZleba srulyofilad ar

daakmayofilos es daSvebebi, Tumca mkvlevrebi ganagrZoben am

monacemebisaTvis t kriteriumis gamoyenebas. am sakiTxs ufro

dawvrilebiTY me-9 TavSi ganvixilavT.

cxrili #8.3

erTi SerCevis t kriteriumis gamoyenebiT statistikuri hipoTezebis Semowmebis

Sesaxeb gadawyvetilebebisa da daskvnebis Sejameba. gamoyenebuli iyo 0.05

mniSvnelovnobis done.

magaliTi #8-1

erTi SerCevis t kriteriumis gamoyeneba

amocana

fsiqologebma wamoayenes hipoTeza, rom zogierTi individi Sejibrebis dros

gaurbis did warmatebas, raTa Tavi aaridos misi Tanatolebis

kritikulobasa da jgufidan gariyvas (gandevnas). TviTSefasebis kiTxvari,

romelic zomavs warmatebisagan garidebas, SemuSavebuli iyo Ggudisa da

gudis mier (Good and Good, 1973). davuSvaT cnobilia, rom kolejis

damamTavrebeli klasis studenti gogonebisaTvis populaciis saSualo am

kiTxvaris mixedviT iyo 8.6 qula. qulebis diapazoni, romelic SeiZleboda

yofiliyo miRebuli am kiTxvaris mixedviT icvleboda 0 qulidan, rac

niSnavda warmatebisaTvis Tavis aridebis dabal motivacias, 28 qulamde, rac

niSnavda warmatebisaTvis Tavis aridebis maRal motivacias. konsultantma

moawyo konkurentuli situacia. man misca warmatebisaTvis Tavis aridebis

motivaciis sazomi skala kolejis damamTavrebeli klasis student 33

gogonas, vinc gaira es seminari da miiRo saSualo 8.1 da s=1.31. gansxvavdeba

Tu trealuri xvdeba


Tu trealuri ar xvdeba


trealuris miRebis albaToba naklebia an

toli 0.05, an p≤0.05

H0 aris uaryofili

H1 aris miRebuli. SerCevis saSualo

aRebulia populaciidan, romlis

saSualo gansxvavebulia populaciis

hipoTeturi saSualosagan.


aris statistikurad mniSvnelovani 0.05

doneze.


X -sa da hipoTetur µ-s Soris

gansxvavebis mizezi aris ara SerCevis

Secdoma, aramed sxva mizezi.

trealuris miRebis albaToba metia an

toli 0.05, an p>0.05


H1 ar aris miRebuli

gansxvaveba X -sa da hipoTetur µ-s

Soris aris statistikurad

aramniSvnelovani 0.05 doneze

miiReba gadawyvetileba imis Sesaxeb,

rom SerCevis Secdoma yvelaze

damajerebelad xsnis gansxvavebas

X -sa da hipoTetur µ-s Soris.

Tu ara es X mniSvnelovnad µ=8.6-sagan 0.05 mniSvnelovnobis doneze?

gamoiyeneT ormxrivi kriteriumi.

saWiro statistika: radganac σ gamoiTvleba s-is saSualebiT, am SemTxvevaSi

unda gamoviyenoT erTi SerCevis t kriteriumi.

daSvebebi:

1. individebi, romlebmac gamokiTxvaSi miiRes monawileoba, SemTxveviTad

iyvnen SerCeulebi kolejis damamTavrebeli klasis studenti gogonebis

populaciisagan;

2. warmatebis Tavidan aridebis motivaciis qulebi arian normalurad

ganawilebulni populaciis SerCevaSi.

statistikuri hipoTezebi: H0: µ =8.6; H1: µ ≠8.6.

mniSvnelovnobis done: α=0.05

df t-saTvis: N-1=33-1=32

t-s kritikuli mniSvneloba: tkritikuli =2.042. t-s kritikuli mniSvneloba, roca

df=32 ar aris warmodgenili cxrilSi #a-2. Cven gamoviyeneT tkritikuli df-is

ufro dabali yvelaze axlo mdgomi mniSvneloba, df=30.

uaryofis areebi: trealuris mniSvnelobebi, romlebic arian toli an naklebi -

2.042, an arian toli, an meti +2.042.

gamoTvla:

228.0745.5

31.13331.1

=

=

=Xs

amitom, 193.2228.0

5.0228.0

6.81.8−=

−=

−=

−=

xsXt µ

realuri

gadawyvetileba: trealuri ufro eqstremaluria, vidre t kritikuli = -2.042 da

xvdeba uaryofis areSi. amrigad, Cven uarvyofT H0-s da viRebT H1-s.

daskvna: warmatebisagan garidebis motivaciis saSualo kolejis damamTavrebeli

klasis studenti gogonebis im SerCevisaTvis, vinc daeswrnen seminarebs

(kerZod, X =8.1) mniSvnelovnad naklebia, vidre kolejis damamTavrebeli klasis

studenti gogonebis populaciis saSualo _ 8.6.

erTi SerCevis t kriteriumis Sedegebis prezentacia

amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvanelos (1994)

stili moiTxovs daskvniTi statistikis kriteriumebis Sedegebis

warmodgenas, raTa specialuri forma mieces statistikuri kriteriumis,

kriteriumis df-is, statistikis realuri mniSvnelobis, gamoyenebuli alfa

donis da statistikis realuri mniSvnelobis albaTobisa da eqstremaluri

mniSvnelobis Caweras nulovani hipoTezis WeSmaritebis pirobebSi. aseve

saWiroa, Sesatyvisi aRweriTi statistikis _ SerCeviTi saSualosa da

standartuli gadaxris warmodgena. Cven giCvenebT Sedegebis prezentaciis am

stils erT-erTi sanimuSo magaliTis saxiT, romelic exeba kvlevas

mesameklaseli moswavleebis mier televizoris yurebaze daxarjuli drois

Sesaxeb.

saSualo dro, romelsac 30 mesameklaseli xarjavda yoveldRiurad

televizoris yurebaze iyo 83 wuTi (SD=26 wuTis). alfa tolia 0.05-is da

ormxrivi kriteriumisaTvis erTi SerCevis t kriteriumi gviCvenebs, rom

SerCeviTi saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi

saSualosagan, romelic tolia 95 wuTis; t(29)= -2.53, p<0.05.

am Sedegebis prezentaciisas:

α=0.05 gviCvenebs am kriteriumis mniSvnelovnobis dones;

t(29) gansazRvravs testis statistikas rogorc t-s. es t

gansazRvravs trealurs, Tumca indeqsi `realuri” SeiZleba ar

axldes. radgan t ganisazRvreba, rogorc erTi SerCevis t, Cven

viciT, rom kriteriumi gulisxmobs SerCevis saSualos

Sedarebas populaciis hipoTetur saSualosTan. Tavisuflebis

xarisxi trealurisaTvis naCvenebia frCxilebSi. amrigad, Cven viciT,

rom t-s gamoTvlisas gamoyenebuli iyo 30 monacemi (radgan

df=N-1, amitom 29=30-1);

=-2.53 trealuris mniSvneloba (ara tkritikulisa, romelsac vpoulobT

cxrilSi #a-2). t-s mniSvneloba unda iyos gamosaxuli ricxviT,

romelsac eqneba mZimis Semdeg sul cota ori cifri;

p<0.05 gviCvenebs Semdegs:

a) p da naklebobis niSani (p<) gviCvenebs, rom trealuris miRebis

albaToba H0-is WeSmaritebis pirobebSi aris naklebi an toli

0.05-s. es mniSvneloba aris trealuris albaToba an ufro Sors

mdebare trealuris albaToba, roca H0 aris WeSmariti;

b) H0: µ =95.0 wuTi aris uaryofili,

H0: µ ≠95.0 wuTi aris miRebuli;

g) gansxvaveba SerCevis saSualosa da hipoTetur µ-s Soris

aris statistikurad mniSvnelovani;

d)SerCevis saSualo ganixileba rogorc aRebuli im

populaciidan, romlis saSualo naklebia hipoTetur µ=95

wuTze;

e) Tu p>0.05 iqneba SedegebSi warmodgenili, maSin metobis

niSani aRniSnavs imas, rom H0 ar aris uaryofili da X ar

gansxvavdeba mniSvnelovnad hipoTeturi µ-sagan

mniSvnelovnobis 0.05 doneze.


1. dawereT H0 da H1 erTi SerCevis t kriteriumisaTvis qvemoT mocemuli

TiToeuli debulebisaTvis.

a) Ramis Zilis saSualo xangrZlivoba populacisaTvis aris 493 wuTi.

b) depresiisaken midrekilebis Sefasebis kiTxvarSi populaciis saSualo aris

36.1.

g) populaciis saSualo standartizebuli inteleqtis testisaTvis aris 100.

2. cxrilis #a-2 gamoyenebiT, ipoveT tkritikulis mniSvnelobebi miTiTebuli

df-sa da α gaTvaliswinebiT. gamoiyeneT ormxrivi kriteriumi.

df α

a. 6 0.05

b. 17 0.01

g. 23 0.05

d. 32 0.05

e. 39 0.05

v. 75 0.01

z. 103 0.05

T. 256 0.05

3. miuTiTeT, xvdeba Tu ara trealuris Semdegi mniSvnelobebi uaryofis

areSi. Semdeg miuTiTeT Tqveni gadawyvetileba H0 da H1-saTvis. gaxsovdeT, rom

Tqven unda airCioT Semdegi: an ver uaryofT H0-s, an uaryofT H0-s, da ar

miiRebT H1-s, an miiRebT H1-s. gamoiyeneT α=0.05 da ormxrivi kriteriumi. df

TiToeuli trealurisaTvis mocemulia frCxilebSi.

a. trealuri(19)= +3.014

b. trealuri(19)=-1.989

g. trealuri(16)=-2.179

d. trealuri(16)=+2.04

e. trealuri(30)=-3.961

b. trealuri(30)=+2.743

z. trealuri(6)=-2.364

T. trealuri(6)=+2.447

i. trealuri(9)=-2..262

k. trealuri(37)=+2.938

l. trealuri(42)=-2.41

4. gansazRvreT daSvebebi, romlebic safuZvlad udevs erTi SerCevis t

kriteriums.

5. amoxseniT Semdegi amocana da upasuxeT qvemoT mocemul kiTxvebs. gamoiyeneT

ormxrivi kriteriumi da 0.05 mniSvnelovnobis done. Tqven gainteresebT Tu

ramdenad kargad SeuZlia adamians Seafasos dro saaTis gareSe. Tqven sTxoveT

11 cdis pirs SeefasebinaT drois 7 wuTiani intervali da eniSnebinaT TqvenTvis

misi gasvlis dro saaTis gareSe. am 11 Sefasebis saSualo iyo 5.8 wuTi, s=0.8

wuTi. aris Tqveni SerCevis saSualo mniSvnelovnad gansxvavebuli populaciis

saSualosagan, romelic tolia 7 wuTis? Tqvens pasuxSi miuTiTeT Semdegi:

a. kriteriumi, romelsac gamoiyenebT

b misi gamoyenebisas gakeTebuli daSvebebi

g. statistikuri hipoTezebi

d. mniSvnelovnobis done

e. kriteriumis statistikis kritikuli mniSvneloba

v. uaryofis areebi

z. trealuris mniSvneloba

T. Tqveni gadawyvetilebani statistikuri hipoTezebis Sesaxeb

i. Tqveni daskvnebi SerCevis saSualos Sesaxeb

6. es savarjiSo warmoadgens t kriteriumis gamoyenebis Sedegebs, romelic

adaptirebulia gamoqveynebuli kvlevis Sedegebidan. TiToeuli warmodgenili

SedegisaTvis upasuxeT Semdeg SekiTxvebs:

a. ras udris SerCevis saSualos mniSvneloba?

b. ras udris populaciis standartuli gadaxris Sefasebis mniSvneloba?

g. romeli t kriteriumi gamoiyeneba am kvlevaSi?

d. ra aris H0 da H1 am t kriteriumisaTvis?

e. ramdeni monacemi iyo miRebuli am kvlevaSi?

v. risi tolia trealuris mniSvneloba am kvlevaSi?

z. risi tolia df am trealurisaTvis?

T. risi tolia tkritikuli α=0.05-sa da ormxrivi uaryofis aresaTvis?

i. uaryofT H0-s, Tu ver uaryofT H0-s?

k. miiRebT H1-s, UTu ar miiRebT H1-s?

l. daaskvniT Tu ara, rom SerCevis saSualo mniSvnelovnad gansxvavdeba

populaciis hipoTeturi saSualosagan?

angariSi I: morbenali mamakacis saSualo wona iyo 74 kg (SD=7.6 kg), α=0.05 da

ormxrivi t kriteriumisaTvis erTi SerCevis t kriteriumma gviCvena,

rom SerCevis saSualo ar gansxvavdeba mniSvnelovnad populaciis

hipoTeturi saSualosagan, romelic tolia 75 kg-is 25 wlis

mamakacebisaTvis; t(5)= -1.48, p<0.05.

angariSi II: araletaluri daavadebebis mqone 65 wlis da zemoT pacienti

mamakacebis saavadmyofoSi gatarebuli dReebis saSualo Seadgenda 10.4 dRes

(SD=2.3). α=0.05 da ormxrivi kriteriumisaTvis erTi SerCevis t kriteriumma

gviCvena, rom SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis

hipoTeturi saSualosagan, romelic tolia 8.7 dRisa; t(24) = +3.7, p<0.01.

mniSvnelovani sakiTxebi statistikuri hipoTezis SemowmebaSi

ratom ar viRebT nulovan hipoTezas?

Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin Cven ar

uarvyofT nulovan hipoTezas. am gadawyvetilebas xSirad ase gamovTqvamT:

`Cven ver uarvyofT nulovani hipoTezas”. Tqven SeiZleba gainteresebT, Tu

ratom vxmarobT am rTul (Znelad gasageb) frazas; rogorc wesi, Tu Cven ar

uarvyofT raimes, maSin Cven viRebT mas. ratom ar SegviZlia ubralod miviRoT

nulovani hipoTeza, Tuki kriteriumis statistika ar xvdeba uaryofis areSi? am

kiTxvaze pasuxis gasacemad Cven unda davubrundeT magaliTs, romelic

satvirTo manqanebSi skamis simaRles exeboda. skami morgebuli iyo im

mZRolebis populaciisaTvis, visi muxlqveSa simaRlec iyo 42 sm da σ=3.0sm.

davuSvaT, rom Tqven gyavT satvirTo manqanis mZRolebis 25 kaciani SerCeva da

aRmoCnda, rom am mZRolebis saSualo muxlqveSa simaRle aris 42,5 sm. zrealuris

mniSvneloba am SerCevisaTvis iqneba +0.83.

zrealuri = 83.06.05.0

25/3425.42

+=+=−

es zrealuri ar xvdeba uaryofis areSi, romelic gansazRvrulia zkritikuls plus an

minus 1.96 α=0.05-is mniSvnelovnobis doneze, amitom is aris aramniSvnelovani.

Cven ver uarvyofT H0:µ=42sm-is. magram niSnavs Tu ara es gadawyvetileba imas,

rom im populaciis saSualo, romlidanac aris aRebuli es SerCeva unda iyos

zustad 42sm? ra Tqma unda, ara; SerCevis saSualo 42.5 sm ar niSnavs imas, rom

populaciis saSualo, romlidanac SerCeva aris aRebuli, unda iyos 42 sm-is

toli. meore mxriv, X =42.5 sm, agreTve, ar gvaZlevs imis mtkicebis saSualebas,

rom µ ar SeiZleba iyos 42 sm toli, anu X =42.5 sm ar gvaZlevs imis mtkicebis

saSualebas, rom Cven unda uarvyoT H0: µ =42 sm, magram arc imis mtkicebis

saSualebas gvaZlevs, rom µ zustad 42 sm-is tolia. is gvaCvenebs mxolod imas,

rom Cven ar SegviZlia uarvyoT H0: µ =42 sm.

davuSvaT, rom X toli iyo 44 sm, ase rom zrealuri =+3.33. es zrealuri aris

statistikurad mniSvnelovani 0.05 doneze da Cven uarvyofT H0: µ =42 sm-s da

miviRebT H1: µ ≠42sm-s. am SemTxvevaSi sworia miviRoT H1, radgan H1 amtkicebs,

rom µ aris garkveuli mniSvneloba, romelic gansxvadeba 42sm-sagan da X =44sm

gviCvenebs, rom µ marTlac aris garkveuli mniSvneloba, romelic gansxvavdeba

42sm-sagan. am SemTxvevaSi, SerCevis saSualo 44sm uzrunvelyofs µ-s saukeTeso

wertilovan Sefasebas mZRolebis im populaciisaTvis, saidanac iyo aRebuli

SerCeva.

swori daskvnebi statistikuri hipoTezebis Sesaxeb SeiZleba Semdegnairad

CamovayaliboT:

• TuU kriteriumis statistika xvdeba uaryofis areSi, maSin H0 uaryofili

iqneba da miRebuli iqneba H1;

• Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin H0-s ver

uarvyofT da ver miviRebT H1-s.

pirveli da meore gvaris Secdomebi statistikuri kriteriumebis

gamoyenebisas

statistikuri kriteriumi mkvlevars aZlevs saSualebas

uaryos H0 (1) an ver uaryos H0. (2) gadawyvetilebis

siswore damokidebulia imaze, aris Tu ara nulovani

hipoTeza faqtiurad WeSmariti populaciisaTvis. pirveli

gvaris Secdoma - Secdoma statistikuri gadawyvetilebis

miRebis dros, romelic gamoixateba nulovani hipoTezis

uaryofaSi, maSin roca is realurad WeSmaritia. meore

gvaris Secdoma - Secdoma statistikuri gadawyvetilebis

miRebis dros, romelic gamoixateba imiT, rom ar xdeba

nulovani hipoTezis uaryofa, maSin roca is realurad

mcdaria da alternatiuli H1 hipoTeza ki WeSmariti.

cxrili #8.4 gviCvenebs am Secdomebs da SesaZlo swor

gadawyvetilebebs, romelic SeiZleba miviRoT

statistikuri kriteriumis gamoyenebisas.

pirveli da meore gvaris Secdomebi xdeba imis gamo, rom statistikuri

gadawyvetilebis miReba albaTuri xasiaTiaa. TuU H0 aris WeSmariti, pirveli

gvaris Secdomis albaToba udris α-s. amitom, swori daskvnis albaToba da

imis albaToba, rom H0 ar iqneba uaryofili, maSin roca is WeSmaritia, aris

mocemuli 1-α-s saSualebiT. meore gvaris Secdomis albaToba udris β-s. aqedan

pirveli gvaris Secdoma

(Type I error) Secdoma statistikuri gadawyvetilebis miRebis dros, romelic gamoixateba nulovani hipoTezis uaryofaSi, maSin roca is realurad WeSmaritia. meore gvaris Secdoma (Type II error) - Secdoma statistikuri gadawyvetilebis miRebis dros, romelic gamoixateba imiT, rom ar xdeba nulovani hipoTezis uaryofa, maSin roca is realurad mcdaria da alternatiuli H1 hipoTeza ki WeSmariti.

beta(β): meore gvaris Secdomis albaToba.

gamomdinare, H1 -is sworad miRebis albaToba, roca H1 WeSmaritia aris

mocemuli 1-β-s saSualebiT. mniSvneloba 1-β aris statistikuri kriteriumis

simZlavre. amitomac statistikuri kriteriumis simZlavre an ubralod

simZlavre

aris H0-is uaryofis albaToba, roca H0 ar aris

samarTliani da H1 aris WeSmariti. β mniSvneloba

damokidebulia mTel rig faqtorebze da Cven am

faqtorebs me-9 TavSi ganvixilavT. TiToeuli es

albaToba gansazRvrulia cxrilSi #8.4.

I gvaris Secdoma xdeba mxolod maSin, roca H0 aris WeSmariti. Aamitom, I

gvaris Secdomis daSvebis albaToba pirdapir damokidebulia arCeuli

mniSvnelovnobis doneze. gaixseneT, rom mniSvnelovnobis done gviCvenebs

kriteriumis statistikis mniSvnelobis miRebis albaTobas (mag., an z an t),

romelsac mivyavarT H0-is uaryofamde, roca H0 aris realurad WeSmariti.

Sesabamisad, roca statistikuri kriteriumisaTvis aris arCeuli α, iqmneba I

gvaris Secdomis daSvebis riski. roca α=0.05 WeSmariti H0-is SecdomiT

uaryofis albaToba tolia 0.05. analogiurad, roca α=0.01, maSin arsebobs

mxolod 1 Sansi 100-dan, rom daSvebul iqnes I gvaris Secdoma H0-is uaryofiT,

roca is realurad WeSmaritia. mecnierebi, Cveulebriv, irCeven ufro frTxil

gzas, raTa garantirebuli iyos I gvaris Secdomis daSvebis dabali albaToba

maT kvlevebSi. Sesabamisad, α-s mniSvnelobad irCeven xolme mis dabal

mniSvnelobebs, rogoricaa, α=0.05 an α=0.01.

cxrili #8.4

Secdomebi da swori gadawyvetilebani statistikur gadawyvetilebebSi da

TiToeul SedegTan dakavSirebuli albaToba

Eeqsperimentatoris gadawyvetileba

H0-s ver uarvyofT H0 uaryofilia, H1

miRebulia

H0 WeSmaritia swori gadawyvetileba:

p=1-α

I gvaris Secdoma:

p=α

WeSmariti situacia

populaciaSi

simZlavre (Power) – H0-is uaryofis albaToba, roca H0 ar aris samarTliani da H1 ki aris WeSmariti. statistikuri kriteriumis

simZlavre mocemulia 1-β-s saSualebiT.

H1 WeSmaritia II gvaris Secdoma:

p=β

swori gadawyvetileba:

p=1-β

statistikuri da mecnieruli mniSvnelovnoba

statistikur kriteriumSi mniSvnelovnoba gviCvenebs mxolod imas, rom H0 aris

uaryofili. Tu H0 uaryofilia, maSin statistikuri kriteriumi arc miRebuli

Sedegis mizezs xsnis, arc imas gviCvenebs, Tu ramdenad mniSvnelovania,

mecnieruli TvalsazrisiT, es gansxvaveba. mTeli pasuxismgebloba rCeba

mkvlevars, romelmac yuradRebiT unda Seamowmos kvlevis pirobebi da

miRebuli monacemebis xasiaTi.

kriteriumis statistikis realuri mniSvnelobebi, romlebic ar xvdeba

uaryofis areSi iTvleba aramniSvnelovnad da ara umniSvnelod. umniSvnelo

gulisxmobs rom raimes ara aqvs mniSvneloba. maSin, roca aramniSvnelovani

gansxvaveba SeiZleba mecnierulad iyos Zalian mniSvnelovani. magaliTad,

davuSvaT, rom gakveTilebis Semdeg bavSvebis movlis centris mier

SemoTavazebuli iyo programa, romelic miznad isaxavda Seemcirebina is dro,

romelsac bavSvebi atarebdnen televizoris yurebisas. Tuki X da µ

Sedarebisas aRmoCnda aramniSvnelovani gansxvaveba, es niSnavs, rom programa

araefeqturia da ver cvlis im dros, romelsac isini atareben televizoris

yurebisas da es SeiZleba mkvlevrisaTvis iyos mniSvnelovani Sedegi.

populaciis saSualosaTvis ndobis intervalebis ageba t statistikis gamoyenebiT

ndobis intervalebi populaciis saSualosaTvis gansazRvruli iyo me-7

TavSi rogorc intervalebi, romlebSic SeiZleba iyos moTavsebuli populaciis

saSualo Cveni darwmunebulobis garkveuli xarisxiT. im TavSi Cven avageT es

intervalebi X -is, z-is da Xσ -is gamoyenebiT. magaliTad, Tu Xσ cnobilia, 95-

procentiani ndobis intervali populaciis saSualosaTvis mocemulia

intervaliT

X -1.96 Xσ -dan X +1.96 Xσ -mde.

xSirad Xσ ucnobia da misi mniSvneloba gamoiTvleba Xs -iT. roca Xs

gamoiyeneba Xσ -is gamosaTvlelad, ndobis intervali agebulia t statistikis

da ara z-is gamoyenebiT. t-sa da Xs -is gamoyenebiT 95-procentiani ndobis

intervali populaciis saSualosaTvis mocemulia intervaliT

X _(t0.05)( Xs )-dan X + (t 0.05)( Xs )-mde,

sadac X =realuri SerCevis saSualos

Xs = im saSualos gamoTvlili standartuli Secdoma, romelic

miRebulia s/√N-dan

t 0.05= tkritikulis ormxrivi mniSvnelobisa 0.05 mniSvnelovnobis doneze N-1

Tavisuflebis xarisxiT.

analogiurad, 99%-iani ndobis intervali populaciis saSualosaTvis

mocemulia

X - (t0.01)( Xs )-dan X + (t0.01)( Xs )-mde.

amis sailustraciod, Cven davadgineT 95-procentiani ndobis intervali 30

mesameklaselis mier televizoris yurebisas gatarebuli drois

monacemebisaTvis X =83 wuTisa da s=26 wuTisaTvis. radgan N=30, am SerCevisaTvis

Tavisuflebis xarisxi iqneba 29 da ormxrivi tkritikulis(29) mniSvneloba 0.05

mniSvnelovnobis doneze aris 2.045. saSualos standartuli Secdoma, Xs , tolia

26/√30, anu 4.75 wuTis. amrigad, 95-procentiani ndobis intervali µ-saTvis iqneba

83-(2.045)(4.75)-dan 83+(2.045)(4.75),

an Tu CavatarebT maTematikur gamoTvlebs, miviRebT ndobis intervals

83-9.7-dan 83+9.7 wuTamde,

rac 73.3-dan 92.7 wuTamde intervalis tolia. albaToba imisa, rom µ

moTavsebulia am intervalSi aris 0.95.

Tumca ndobis intervalebi ar aris farTod ganxiluli kvlevebSi, amJamad

mkvlevrebs urCeven xolme CarTon isini TavianT kvlevebSi. (Streiner, 1996; Wilkinson

da sxva, 1999). magaliTad, gvartnei-gibma (Gwartney-Gibb, 1986) Seiswavla wyvilebis

qorwinebamde Tanacxovreba saarqivo masalebis gamoyenebiT, romelic aRebuli

iyo qorwinebaze Tanxmobis ganacxadebidan, raTa mieRo demografiuli

informacia im wyvilebze, romlebic qorwinebamde erTad cxovrebdnen da

romlebic ar cxovrobdnen. misi kvlevis erT nawilSi, man warmoadgina

qorwinebis asakis saSualo maCvenebeli im mamakacebisaTvis da qalebisaTvis,

romlebic qorwinebamde Tanacxovrobdnen da imaTTvisac, vinc qorwinebamde ar

Tanacxovrobdnen. asakis saSualos Tan axlda 95-procentiani ndobis intervali

populaciis saSualosaTvis. rogorc magaliTi, gvartnei-gibma warmoadgina

Semdegi Sedegebi im qalebisaTvis, romlebic qorwinebamde Tanacxovrobdnen da

ar Tanacxovrobdnen da amasTanave ar iyvnen adre namyofi qorwinebaSi.

Qqorwinebamde

Tanacxovrobdnen

Qqorwinebamde ar

Tanacxovrobdnen

Qqorwinebis saSualo asaki 806 877

22.8 21.7

s 4.6 4.4

95-procentiani ndobis

intervali µ-saTvis

22.48-dan 23.12-mde 21.41-dan 21.29-mde

Cven SegviZlia viyoT 95 procentiT darwmunebulebi, rom qorwinebis

asakis µ im qalebisaTvis, romlebic qorwinebamde Tanacxovrobdnen, xvdeba 22.48-

dan 23.12 wlebis SualedSi im populaciisaTvis, romelic SerCeuli iyo

gvartnei-gibis mier. xolo im qalebisaTvis, romlebic qorwinebamde ar

Tanacxovrobdnen, µ-s 95- procentiani ndobis intervali aris 21.41-dan 21.29

wlamde. ndobis es ori intervali ar faravs erTmaneTs, amitom gvartnei-gibsma

daaskvna, rom is qalebi, romlebic qorwinebamde Tanacxovroben, Cveulebriv

qorwindebian erTi wliT gvian, vidre is qalebi, romlebic qorwinebamde ar

Tanacxovroben im populaciaSi, romelic mis mier iyo Seswavlili.


1. axseniT, Tu ratom ar iRebs mkvlevari H0-s, Tuki trealuri ar xvdeba

uaryofis areSi.

2. upasuxeT Semdeg kiTxvebs:

a. ra aris I gvaris Secdoma?

b. risi tolia I gvaris Secdomis daSvebis albaToba, roca H0 aris

WeSmariti?

g. risi tolia I gvaris Secdomis daSvebis albaToba, roca H0 aris

mcdari?

d. ra aris II gvaris Secdoma?

e. SesaZlebelia Tu ara, rom eqsperimentatorma dauSvas orive I da II

gvaris Secdomebi erTi da imave statistikur kriteriumSi? axseniT

Tqveni pasuxi.

3. ra gansxvavebaa statistikur mniSvnelovnobasa da mecnierul

mniSvnelovnobas Soris?

4. eqsperimentatori swavlobda, Tu ramden dros xarjavdnen mozardebi

video TamaSebze video salonebSi. aRmoCnda, rom TamaSis saSualo dro dReSi

Seadgenda 97 wuTs 23 mozardi biWisaTvis da standartuli gadaxra iyo 39

wuTi. aageT 95- procentiani ndobis intervali mozardi biWebis mier TamaSze

daxarjuli drois populaciis saSualosaTvis.

Sejameba

• statistikuri kriteriumebi SeiZleba iyo parametruli da

araparametruli.

• statistikuri hipoTezis Semowmeba moicavs Semdeg nabijebs:

kriteriumis statistikis arCeva (mag. z an t)

• H0-is da H1-is formulirebas. H0 gamoiyeneba kriteriumis statistikis

SerCeviTi ganawilebis dasadgenad.

• mniSvnelovnobis donis dadgena. mniSvnelovnobis done,

gansazRvruli α mniSvnelobiT, aris albaTobis mniSvneloba,

romelic adgens nulovani hipoTezis uaryofis kriteriums.

• kriteriumis statistikis kritikuli mniSvnelobis an mniSvnelobebis

gansazRvra cxrilis gamoyenebiT. kriteriumis statistikis

kritikuli mniSvnelobebi gansazRvraven uaryofis aris sazRvars an

sazRvrebs.

• uaryofis aris an areebis mdebareobis gansazRvra. es uaryofis are

warmoadgens SerCevis ganawilebaSi kriteriumis statistikis

mniSvnelobebs, romelTa miRebis albaToba α-s tolia an masze

naklebia, Tu H0 WeSmaritia.

• statistikuri hipoTezebis Sesaxeb gadawyvetilebis miRebis wesebis

formulireba. es wesebia:

_ Tu kriteriumis statistika xvdeba uaryofis areSi, maSin unda

uarvyoT H0 da miviRoT H1. gansxvaveba SerCevis saSualosa da

populaciis hipoTetur saSualos Soris aris statistikurad

mniSvnelovani.

_ Tu kriteriumis statistika ar xvdeba uaryofis areSi, maSin ver

uarvyofT H0-s da ver miviRebT H1-s. gansxvaveba SerCevis saSualosa da

populaciis hipoTetur saSualos Soris aramniSvnelovania.

• kriteriumis statistikis realuri mniSvnelobis gamoTvla SerCevis

monacemebidan.

• H0-isa da H1-is Sesaxeb gadawyvetilebis miReba damokidebulia imaze,

kriteriumis statistikis realuri mniSvneloba moxvdeba Tu ara

uaryofis areSi.

• z statistika, mocemuli z=( X -µ)/ Xσ formuliT, gamoiyeneba saSualosa

da populaciis hipoTetur saSualos Sesadareblad, roca σ

cnobilia.

• t statistika, mocemuli t=( X -µ)/ Xs formuliT, gamoiyeneba saSualosa

da populaciis hipoTeturi saSualos Sesadareblad, roca Xσ

gamoiTvleba Xs saSualebiT.

• t statistikis SerCeviTi ganawileba dmokidebula t-s df-ze, sadac

df=N-1.

• statistikuri daskvnebis gakeTeba imTaviTve albaTuria. I gvaris

Secdoma xdeba WeSmariti H0-is uaryofisas. II gvaris Secdoma xdeba

maSin, roca ar aris uaryofili mcdari H0 .

• I gvaris Secdomis albaToba aris α, xolo II gvaris Secdomis

albaToba aris β.

• statistikuri kriteriumis simZlavre aris H0-is .uaryofis albaToba,

roca H0 .aris mcdari da H1 aris WeSmariti.

• t ganawileba da Xs SeiZleba gamoyenebuli iyos populaciis

saSualosaTvis ndobis intervalebis dasadgenad.

mniSvnelovani terminebi da simboloebi

alfa (α)

alternatiuli hipoTeza (H1)

beta (β)

kritikuli mniSvneloba



aramniSvnelovani gansxvaveba

nulovani hipoTeza


calmxrivi kriteriumi

paramatruli kriteriumebi

simZlavre

uaryofis are

t-s SerCeviTi ganawileba

mniSvnelovnobis done

statistikuri hipoTeza

statistikurad mniSvnelovani gansxvaveba

tkritikuli

trealuri

kriteriumis statistika

ormxrivi kriteriumi

I gvaris Secdoma

II gvaris Secdoma

z

zrealuri

zkritikuli

kiTxvebi

1. ra moTxovnebi unda daakmayofilos nebismierma statistikurma

hipoTezam?

2. ra daniSnulebas emsaxureba nulovani hipoTeza statistikur

kriteriumSi?

3. ra aris mniSvnelovnobis done?

4. ra aris kriteriumis statistikis kritikuli mniSvneloba?

44 5. ra gadawyvetileba miiReba H0-sa da H1-is mimarTebaSi, Tuki:

a. kriteriumis statistikis realuri mniSvneloba xvdeba uaryofis areSi?

b. kriteriumis statistikis realuri mniSvneloba ar xvdeba uaryofis

areSi?

6. Tqven gTxoves SeamowmoT, gansxvavdeba Tu ara SerCevis saSualo _ 73

mniSvnelovnad populaciis hipoTeturi saSualosagan, romelic tolia

75-s. ra gansazRvravs Tqvens arCevans _ gamoiyenebT z kriteriums, Tu

erTi SerCevis t kriteriums?

7. ra gadawyvetilebas miiRebT SerCevis saSualosa da populaciis

hipoTeturi saSualos mimarTebis Sesaxeb, Tuki:

a. zrealuri xvdeba uaryofis areSi?

b. zrealuri ar xvdeba uaryofis areSi?

8. zrealuris TiToeuli mniSvnelobisaTvis gansazRvreT, moxvdeba Tu ara

zrealuri ormxriv uaryofis areSi mniSvnelovnobis 0.01 doneze. Semdeg

miuTiTeT Tqveni gadawyvetilebis Sesaxeb H0-isa da H1-saTvis. gaxsovdeT,

rom Tqven an (1) H0-s ver uaryofT an unda uaryoT H0 da, an (2) ar miiRoT,

an miiRoT H1. (a) zrealuri = +3.71,

(b) zrealuri = -2.6, (g) zrealuri = +1.74, (d) zrealuri=-3.05, (e) zrealuri = -1.96,

(v) zrealuri= +2.58.

9. populaciis saSualo inteleqtis skalaze jgufisaTvis aris 100 da σ=16.

skolis konsultantma am skalaze Seafasa 17 aTi wlis bavSvi da miiRo

X =94.1. SeasruleT statistikuri Semowmeba am monacemebis gamoyenebiT,

raTa gansazRvroT, aris Tu ara gansxvaveba populaciis saSualosa da

SerCevis saSualos Soris statistikurad mniSvnelovani. gamoiyeneT 0.05

mniSvnelovnobis done da ormxrivi kriteriumi.

10. z kriteriumis Sedegebze dayrdnobiT daadgines, rom arsebobs

mniSvnelovani gansxvaveba SerCevis saSualosa da populaciis saSualos

Soris. axseniT, ras niSnavs statistikuri mniSvnelovnoba am SemTxvevaSi.

11. ra gadawyvetilebas miiRebT SerCevis saSualosa da populaciis

hipoTeturi saSualos mimarTebis Sesaxeb, Tuki:

a. trealuri xvdeba uaryofis areSi?

b. trealuri ar xvdeba uaryofis areSi?

45 12. trealuris TiToeuli mniSvnelobisaTvis gansazRvreT, moxvdeba Tu ara is

uaryofis areSi. CaTvaleT, rom α=0.01 da gamoiyeneT ormxrivi kriteriumi.

df trealuris TiToeuli mniSvnelobisaTvis mocemulia frCxilebSi.

a. trealuri(18) = +3.713

b. trealuri(7) = -3.015

g. trealuri(35) = +2.794

d. trealuri(44) = -3.0.58

e. trealuri(74) = -1.96

v. trealuri(15) = -2.368

13. dedebi, romelTac patara bavSvebi hyavT xSirad Civian, rom maT akliaT

Zili. davuSvaT, rom Tqven gazomeT im cxra dedis Zilis xangrZlivobis

maCveneblebi, romelTac hyavT wlamde bavSvebi da miiReT Semdegi monacemebi

(saaTebSi): 6.4, 7.5, 6.9, 7.3, 7.6, 7.1, 6.5, 7.7, 7.8.

a. gansxvavdeba Tu ara mniSvnelovnad am dedebis Zilis xangrZlivobis

saSualo populaciis hipoTeturi saSualosagan, romelic tolia 7.7 saaTisa

RameSi? gamoiyeneT ormxrivi kriteriumi da 0.05 mniSvnelovnobis done.

b. aRwereT Tqveni Sedegebi im nimuSis Sesabamisad, romelic naCvenebi iyo

am Tavis erT-erT qveTavSi _ erTi SerCevis t kriteriumis Sedegebis

prezentacia.

g. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier miRebul

gadawyvetilebaSi iyos I gvaris Secdoma?

d. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier miRebul

gadawyvetilebaSi iyos II gvaris Secdoma?

e. gamoiTvaleT 95 procentiani ndobis intervali populaciis

saSualosaTvis am monacemebis mixedviT. xvdeba Tu ara populaciis

hipoTeturi saSualo _7.7 saaTi RameSi am intervalSi?

14. mkvlevari samedicino sferoSi swavlobda sakiTxs, axlavs Tu ara

avadmyofobas sxeulis temperaturis mateba. man gazoma im 12 adamianis

sxeulis temperatura, romelsac dausves daavadebis diagnozi da miiRo

Semdegi monacemebi (farengeitebSi): 99.3, 99.1, 100.4, 98.4, 98.2, 98.9, 99.7, 100.1,

100.7, 99, 98.8, 99.2.

a. gansxvavdeba Tu ara am individebis sxeulis temperatura populaciis

hipoTeturi saSualo temperaturisagan _ 98.6F? gamoiyeneT ormxrivi

kriteriumi da 0.05 mniSvnelovnobis done.

46 b. SeiZleba Tu ara statistikuri hipoTezis Sesaxeb Tqvens mier

miRebul gadawyvetilebaSi iyos II gvaris Secdoma? ganmarteT Tqveni

pasuxi.

15. TiToeuli qvemoT mocemuli debulebisaTvis miuTiTeT, aris

gadawyvetileba swori, Tu Secdomas Seicavs? Tuki is Seicavs Secdomas,

miuTiTeT, is I gvaris, Tu II gvaris Secdomaa?.

a. mocemuli populaciisaTvis H0 aris WeSmariti, magram

eqsperimentatorma zrealuris safuZvelze uaryo H0.

b. mocemuli populaciisaTvis H1 aris WeSmariti, magram

eqsperimentatorma zrealuris safuZvelze CaTvala, rom H0-s ver uaryofs.

g. mocemuli populaciisaTvis H0 aris WeSmariti da

eqsperimentatorma zrealuris safuZvelze CaTvala, rom H0-s ver uaryofs.

d. mocemuli populaciisaTvis aris WeSmariti da eqsperimentatorma

zrealuris safuZvelze uaryo H0.

1 IXFTavi. eqsperimentis safuZvlebi da saSualoTa Soris gansxvavebis Semowmeba

kvlevis procesis mimoxilva

mecnieruli kiTxvis dasma da kvlevis hipoTezis formulireba

kvlevis sqemis arCeva

cdis pirTa SerCeva

ekvivalenturi jgufebis Seqmna

damoukidebeli cvladis manipulacia

damatebiTi cvladebis kontroli

damokidebuli cvladis gazomva

aqvs Tu ara gavlena eqsperimentul Carevas statistikuri

Semowmebis saWiroebaze?

SerCevis Secdomis Sedegebi eqsperimentisaTvis

or saSualos Soris gansxvavebis statistikuri hipoTezis

Semowmebis mimoxilva

t kriteriumi ori damoukidebeli jgufisaTvis

saSualoTa Soris sxvaobebis SerCeviTi ganawileba

statistikuri Semowmeba tdamoukidebelis gamoyenebiT

simZlavre da tdamoukidebeli

eqsperimentuli Carevis zemoqmedebis siZlieris gazomva

ndobis intervalebis ageba ori damoukidebeli populaciis

saSualoTa Soris gansxvavebisaTvis t statistikis gamoyenebiT

ras amowmebs t kriteriumi realurad

statistikuri mniSvnelovnobisa da mecnieruli mniSvnelovnebis

mimoxilva

diskusia statistikuri hipoTezebis Semowmebis Sesaxeb

t kriteriumis Sedegebis prezentacia

cdis pirTa Sida sqema da t kriteriumi damokidebuli

qulebisaTvis

2 t kriteriumi damokidebuli qulebisaTvis

ndobis intervalebis ageba ori damokidebuli populaciis

saSualoTa Soris gansxvavebisaTvis

me-8 TavSi gavecaniT statistikuri hipoTezis Semowmebas, romlis

saSualebiTac ganvsazRvreT, gansxvavdeba Tu ara mniSvnelovnad SerCevis

saSualo populaciis hipoTeturi saSualosagan. am TavSi gamoviyenebT

statistikuri hipoTezis Semowmebas, raTa erTmaneTs ori SerCevis saSualo

SevadaroT da ganvsazRvroT, gansxvavdebian Tu ara isini erTmaneTisagan

ufro metad, vidre mxolod SerCevis Secdomis Sedegad miRebuli

gansxvavebisas. eqsperimentuli kvlevis Sedegebis analizi statistikuri

hipoTezis Semowmebis am tipis erT-erT gamoyenebas warmoadgens. pirvel da

meore TavebSi eqsperimentis ZiriTadi cnebebi mokled iyo warmodgenili, am

TavSi ki es Tema ufro farTod aris gaSlili. dasawyisSi ganixileba

mxolod cdis pirTa Soris sqemebi, Semdgom ki _ cdis pirTa Sida sqemebi.

kvlevis procesis ganxilva

yvela mecnieruli kvleva iwyeba kiTxviT, romelzec SeiZleba empiriulad

gavceT pasuxi. Cveulebriv, mecnieri am kiTxvaze savaraudo pasuxs kvlevis

hipoTezis saxiT gvTavazobs. kvlevis hipoTeza aris debuleba or an met

cvlads Soris mosalodneli, Tu prognozirebuli kavSiris Sesaxeb. kvlevis

hipoTeza eqsperimentSi aris prognozirebuli mimarTeba damoukidebelsa da

damokidebul cvladebs Soris. gaixseneT, rom damoukidebuli cvladi is

cvladia, romlis manipulirebas gegmavs eqsperimentatori, raTa daadginos

misi gavlena damokidebul cvladze. damokidebuli cvladi Cveulebriv

subieqtis gazomili qcevaa. kvlevis hipoTezis forma uCvenebs mecniers,

kvlevis romeli sqema unda iyos gamoyenebuli. Tuki kvlevis hipoTezaSi

nawinaswarmetyvelebia, rom damoukidebeli cvladis manipulireba gavlenas

axdens damokidebul cvladze, maSin eqsperimenti kvlevis sqemis swori

formaa da mas SeuZlia empiriulad Seamowmos kvlevis es hipoTeza.

martivi eqsperimenti iwyeba cdis pirTa ekvivalenturi jgufebis SeqmniT.

cdis pirebis TiToeul jgufs uqmnian zustad erTnair pirobebs, garda

imisa, rom maT damoukidebeli cvladis sxvadasxva done miewodebaT.

3 damoukidebeli cvladis manipulaciis Sedegad miiReba damokidebuli

cvladis qulebi. Semdeg xdeba maTi analizi, raTa ganisazRvros, axdens Tu

ara damoukidebeli cvladi gavlenas damokidebul cvladze. ganvixiloT am

procesis magaliTi.

yovel wels milionobiT recepti iwereba damamSvidebeli preparatebis

dasaniSnad, romlebic sedatiur saSualebas _ diazepams Seicaven. rogorc

uamrav sxva medikaments, diazepamsac sxvadasxva gverdiTi efeqti aqvs,

rogorc fiziologiuri, ise fsiqologiuri. erT-erTi fsiqologiuri efeqti

isaa, rom diazepami cvlis pirovnebis mier drois xangrZlivi Sualedis

aRqmas (Kleinknecht&Donaldson, 1975). unrig-neervurtma, kaizerma da koenenma

(Unrig-Neervoort, Kaiser, Coenen, 1992) wamoayenes hipoTeza, rom SesaZloa diazepams

igive efeqti aqvs drois xanmokle Sualedebzec. kvlevis am hipoTezaSi

nawinaswarmetyvelebia, rom damoukidebeli cvladi _ medikamentis piroba,

gavlenas moaxdens damokidebul cvladze _ pirovnebis mier drois

xanmokle Sualedis xangrZlivobis Sefaseba. am hipoTezis Sesamowmeblad

unrig-neervurtma da sxvebma cdis pirTa Soris eqsperimenti Caatares.

eqsperimentSi Seqmnili iyo cdis pirTa ori ekvivalenturi jgufi. erT

jgufs, placebo sakontrolo jgufs aZlevdnen abs, romelic ar Seicavda

aqtiur ingredientebs. meore jgufs, eqsperimentul jgufs, aZlevdnen 15

miligram (mg) diazepams. TiToeuli jgufi DSedgeboda 12 cdis pirisagan.

damokidebuli cvladi iyo pirovnebis mier TiTis kakunis amocanis

xangrZlivobis Sefaseba. cdis pirebs sTxovdnen TiTi daekakunebinaT

specialur mricxvelze, sanam eqsperimentatori ar miscemda gaCerebis

brZanebas. kakunis damTavrebis Semdeg cdis pirebs sTxovdnen SeefasebinaT,

Tu ra xnis ganmavlobaSi asrulebdnen isini am davalebas. Tumca orive

jgufi davalebas erTi da imave drois ganmavlobaSi asrulebda, placebo

sakontrolo jgufma es dro Seafasa rogorc 19.8 wm (s=7.6wm), maSin roca

eqsperimentuli jgufis mier Sefasebuli dro iyo 13.6 wm (s=4.8 wm). unrag-

neervurtma da sxvebma statistikurad gaaanalizes es saSualoebi

damoukidebeli jgufebis t kriteriumis gamoyenebiT da aRmoaCines, rom

eqsperimentuli jgufis mier Sefasebuli xangrZlivoba mniSvnelovnad

naklebia placebo sakontrolo jgufis mier Sefasebul xangrZlivobaze.

diazepami gavlenas axdens drois xanmokle intervalebis aRqmaze.

eqsperimentis Catareba monacemTa statistikur gaanalizebamde garkveuli

safexurebis gavlas moiTxovs. es safexurebi erTnairia yvela

4 eqsperimentisaTvis da Sedgeba mecnieruli kiTxvis dasmis, kvlevis hipoTezis

formulirebis, kvlevis sqemis arCevis, cdis pirebis SerCevis, cdis pirTa

ekvivalenturi jgufebis formirebis, gverdiTi cvladebis kontrolis,

damoukidebeli cvladis manipulirebis da damokidebuli cvladis

gazomvisagan.

mecnieruli kiTxvis dasma da kvlevis hipoTezis formulireba

mecnierulia kiTxva, romelic saSualebas iZleva empiriuli monacemebis

Segrovebis safuZvelze miviRoT pasuxi. Cveni magaliTi eyrdnoba Semdeg

mecnierul kiTxvas: `ra gavlenas axdens diazepami drois xanmokle

intervalebis aRqmaze?” aSkaraa, rom es kiTxva saSualebas iZleva masze

pasuxis gasacemad sistematuri dakvirveba Catardes. kvlevis hipoTeza

gvTavazobs savaraudo pasuxs am kiTxvaze. is acxadebs, rom diazepami

gavlenas axdens pirovnebis mier drois xanmokle intervalebis Sefasebaze.

kvlevis sqemis arCeva

kvlevis sqema aris gegma, romelic gamoyenebulia monacemTa Segrovebis

mimarTulebis gansazRvrisaTvis. am magaliTSi kvlevis hipoTezis Semowmeba

moiTxovs cdis pirTa im jgufis Sedarebas, romelic iRebda diazepams, meore

jgufTan, romelic ar iRebda diazepams. pirvel TavSi Cven ganvsazRvreT es

sqema, rogorc cdis pirTa Soris sqema. cdis pirTa Soris sqema umartivesia,

roca arsebobs cdis pirTa mxolod ori jgufi, ise, rogorc ganxilul

eqsperimentSi. es sqema moiTxovs cdis pirebis SerCevas eqsperimentSi

monawileobisaTvis da Semdeg maT ganawilebas ekvivalentur jgufebSi.

cdis pirebis SerCeva

SemTxveviTi SerCeva

cdis pirTa Soris sqema(Between–subject design) – eqsperimenti, romelSic ori an meti jgufi

SemTxveviTi SerCeva (Random sampling)- SerCevis meTodi, romelSic individebi iseTnairad arian arCeulni, rom populaciis TiToeul wevrs aqvs am SerCevaSi moxvedris erTnairi Sansi da erTi wevris SerCeva ar aris damokidebuli populaciis nebismieri sxva wevris SerCevaze.

5 eqsperimenti tardeba cdis pirebze, romlebic populaciidan aris

miRebuli. imisaTvis, rom SegveZlos eqsperimentis Sedegebis ganzogadeba

Cvens mier gamoyenebuli SerCevis farglebs gareT, rekomendebulia cdis

pirTa arCevisaTvis SemTxveviTi SerCevis gamoyeneba. magram, radgan

WeSmaritad SemTxveviTi SerCevebis miReba rTulia, SemTxveviTi SerCeva

iSviaTad miiRweva realur eqsperimentirebisas. ufro xSirad gamoiyeneba

xelmisawvdomi SerCeva.

xelmisawvdomi (SesaZleblobis) SerCeva

xelmisawvdom SerCevaSi xdeba cdis pirebis iseTi SerCeva, romlebic

mkvlevrisaTvis ufro xelmisawvdomi an xelsayrelia. SesaZleblobis SerCeva

aSkarad ar aris adekvaturi, Tuki Cven vcdilobT zustad SevafasoT

populaciis saSualo, rogoricaa, magaliTad, arCevnebis Sedegebis Sefaseba

gamokiTxviT. magram eqsperimentis mizani ar aris populaciis saSualos

konkretuli mniSvnelobis Sefaseba; misi mizania gansazRvros, gasxvavdeba Tu

ara ori an meti jgufi erTmaneTisagan damoukidebeli cvladis gavlenis

Sedegad. Tu damoukidebeli cvladi axdens gavlenas, es gavlena

gavrceldeba nebismier cdis pirze, iqnebian isini SemTxveviTad SerCeulni

Tu ara. amrigad, Tu Seqmnilia ekvivalenturi jgufebi, xelsayreli SerCeva

Cveulebriv ar zRudavs eqsperimentis interpretacias da mniSvnelobas.

ekvivalenturi jgufebis Seqmna

ekvivalenturi jgufebi aris cdis pirTa jgufebi, romlebic ar

gansxvavdebian erTmaneTisgan raime mudmivi an sistematuri maxasiaTebliT

eqsperimentis damoukidebeli cvladis miwodebamde. ekvivalenturi jgufebi

ar gulisxmoben imas, rom TiToeuli jgufis cdis pirebi zustad erTnairebi

arian damoukidebeli cvladis gamoyenebamde. is gulisxmobs imas, rom

individebis damoukidebeli cvladis doneebze gadanawilebisaTvis

xelmisawvdomi (SesaZleblobis) SerCeva (Convenience sampling): cdis pirebis iseTi SerCeva, romelic mkvlevrisaTvis ufro xelmisawvdomi an xelsayrelia.

ekvivalenturi jgufebi (Equivalent groups): cdis pirTa jgufebi, romlebic, savaraudod, ar gansxvavdebian erTmaneTisgan raime mudmivi an sistematuri maxasiaTebliT eqsperimentis damoukidebeli cvladis miwodebamde.

6 gamoyenebuli unda iyos miukerZoebeli procedura. jgufebs Soris arsebuli

nebismieri gansxvaveba damoukidebeli cvladis manipulaciamde mxolod

SemTxveviTi gansxvaveba iqneba. rogorc mogvianebiT avxsniT, t kriteriumi

saSualebas iZleva SevafasoT aseTi SemTxveviTi xdomilobis albaToba.

SemTxveviTi ganawileba

ekvivalenturi jgufebis Seqmnis tipuri meTodi aris cdis pirebis SemTxveviTi

ganawileba jgufebSi. SemTxveviTi ganawileba niSnavs, rom eqsperimentisaTvis

SerCeul nebismier individs aqvs erTnairi Sansi mikuTvnebuli iyos nebismieri

jgufisadmi. kerZod, SemTxveviTi ganawileba akmayofilebs or kriteriums:

1. yoveli individisaTvis nebismier jgufSi moxvedris albaToba

Tanabari unda iyos;

2. erTi cdis piris ganawileba jgufSi ar moqmedebs nebismieri sxva

individis ganawilebaze imave jgufSi.

SemTxveviTi ganawileba uzrunvelyofs imas, rom iseTi cvladebis gavlena,

rogoricaa motivacia, sqesi, SfoTva, inteleqti da asaki Tanabrad iqneba

gadanawilebuli jgufebSi. SemTxveviTi ganawilebis Semdeg jgufebi ar

gansxvavdebian raime sistematuri maxasiaTebliT eqsperimentuli pirobis

Semotanamde.

praqtikaSi, mkvlevrebi sxva moTxovnebsac uyeneben SemTxveviT ganawilebas.

maT SeiZleba surdeT, rom hyavdeT cdis pirTa toli raodenoba TiToeul

jgufSi, iseve, rogorc zemoT ganxilul magaliTSi, an surdeT, rom hyavdeT

qalebisa da mamakacebis toli raodenobisagan Semdgari jgufebi. am

moTxovnebis miuxedavad, es jgufebi mainc erTmaneTis ekvivalenturi rCeba.

damoukidebeli cvladis manipulacia

SemTxveviTi ganawileba (Random assignment): cdis pirTa jgufebSi ganawilebis meTodi, romlis mixedviT, eqsperimentisaTvis SerCeul nebismier individs aqvs nebismier jgufSi ganawilebis toli albaToba da erTi cdis piris ganawileba jgufSi ar

moqmedebs nebismieri sxva individis ganawilebaze imave jgufSi.

damoukidebeli cvladi ( Independent variable) _ cvladi, romliTac manipulireben eqsperimentSi, damokidebul cvladze misi gavlenis gansazRvris mizniT.

7

damoukidebeli cvladi _ cvladi, romliTac manipulireben da romelsac

iyeneben sxvadasxva jgufis cdis pirebis mimarT, raTa gansazRvron misi

gavlena qcevaze. damoukidebel cvlads unda hqondes sul cota ori

gansxvavebuli done. damoukidebeli cvladis done aris damoukidebeli

cvladis erTi mniSvneloba. damoukidebeli cvladi xSirad gansazRvrulia

rogorc faqtori, magaliTad, A faqtori da misi doneebi _ A1 da A2. cdis

pirTa raodenoba TiToeul jgufSi aRvniSnoT nA1 da nA2-iT aqamde cdis

pirTa raodenobas jgufSi N-iT aRvniSnavdiT xolme. radgan amjerad ori

jgufi gvyavs, TiToeul jgufSi cdis pirTa raodenobis aRsaniSnavad

gamoviyenoT nA, xolo cdis pirTa mTeli raodenobis aRsaniSnavad

eqsperimentSi gamoviyenebT N-s.

damoukidebeli cvladis TiToeuli done eZleva TiTo jgufs. magaliTis

saxiT moyvanil eqsperimentSi damoukidebeli cvladi (e.i. faqtori A) aris

medikamentis miRebis piroba; misi ori donea: A1 done _ sakontrolo,

placebos miReba; da A A2 done _ eqsperimentuli _ diazepamis miReba.

amrigad, cdis pirTa erT jgufs (nA1=12) eZleoda placebo _ sakontrolo

medikamenti, xolo meore jgufs (nA2=12) eZleoda diazepami. es cdis pirTa

Soris erTfaqtoriani sqemaa, romelSic xdeba erTi damoukidebeli cvladis

manipulacia da cdis pirTa ori gansxvavebuli jgufis Sedarebaa saWiro

imis dasadgenad, axdens Tu ara damoukidebeli cvladi gavlenas.

damatebiTi cvladebis kontroli

damoukidebeli cvladis done (Level of an independent variable) _ damoukidebeli cvladis erTi mniSvneloba. imisaTvis, rom cvladi gaxdes, damoukidebeli cvlads unda hqondes sul cota ori gansxvavebuli done.

faqtori (Factor) _ damoukidebeli cvladis alternatiuli saxelwodeba.

cdis pirTa Soris erTfaqtoriani sqema (One-factor between subject design): kvlevis sqema, romelSic erTi damoukidebeli cvladis manipulacia xdeba da iqmneba ori an meti jgufi.

damatebiTi cvladebi (Extraneous variables) - damoukidebeli cvladisagan gansxvavebuli nebismieri cvladi, romelsac SeuZlia imoqmedos damokidebul cvladze eqsperimentSi.

8

imisaTvis, raTa davrwmundeT, rom or jgufs Soris arsebuli nebismieri

gansxvaveba mxolod damoukidebeli cvladis moqmedebis Sedegia,

aucilebelia damatebiTi cvladebis kontroli. damoukidebeli cvladisagan

gansxvavebuli nebismieri cvladi, romelsac SeuZlia eqsperimentSi

damokidebul cvladze imoqmedos, aris damatebiTi cvladi. is SeiZleba

warmoiqmnas cdis pirebis iseTi maxasiaTeblebisagan, rogoricaa inteleqtis

done, SfoTvis done, medikamentebisadmi sensitiuroba an motivacia; an im

fizikuri garemos cvlilebebis Sedegad, romelSic tardeba eqsperimenti; an

eqsperimentuli proceduris variaciis Sedegad. imisaTvis, raTa davrwmundeT,

rom sakontrolo da eqsperimentul jgufebs Soris napovni nebismieri

gansxvaveba mxolod damoukidebeli cvladis manipulaciis Sedegia,

damatebiTi cvladebi ar unda icvlebodes sistematurad an mudmivad

damoukidebel cvladTan erTad. Sereul eqsperimentSi xdeba damatebiTi

cvladis sistematuri cvlileba damoukidebel cvladTan erTad. aseT

eqsperimentSi SeuZlebelia damoukidebeli cvladis gavlenis gamijvna

Sereuli damatebiTi cvladis SesaZlo gavlenisagan.

zemoT ganxilul eqsperimentSi jgufebi erTmaneTis identuria da

erTmaneTisagan mxolod damoukidebeli cvladis moqmedebiT gansxvavdebian.

unrig-neervortma da sxvebma placebo sakontrolo jgufi gamoiyenes, raTa

cdis pirebi identur pirobebSi moeTavsebinaT. kvlevebSi, sadac xdeba

medikamentebis gamoyeneba, placebos kontroli warmoadgens samkurnalo

pirobebis simulacias, roca cdis pirebs miewodebaT abi an raime nivTiereba,

romelic ar Seicavs aqtiur ingredientebs. amrigad, orive jgufs

_placebosa da sakontrolo jgufebs miewodebodaT abi. Tu abs miscemdnen

mxolod eqsperimentuli jgufis cdis pirebs, maSin eqsperimenti Sereuli

gaxdeboda. adamianebi SeiZleba gansxvavebulad moiqcnen mxolod imitom, rom

medikamentis zemoqmedebas elodebian. amrigad, roca medikaments vawodebT,

qceva SeiZleba Seicvalos ori mizeziT, (1) Tavad medikamentis moqmedebis

Sereuli eqsperimenti (Confounded experiment) aris eqsperimenti, romelSic damatebiTi cvladi sistematurad icvleba damoukidebel cvladTan erTad.

placebos kontroli (Placebo control) - simulirebuli eqsperimentuli (Carevis) piroba.

9 Sedegad da (2) im molodinis Sedegad, rom medikaments efeqti eqneba. roca

unrig-neervortma da sxvebma placebo sakontrolo jgufs abi misces,

romelic ar Seicavda aqtiur ingredientebs, maT gaakontroles es molodini,

amitom orive jgufs erTnairi molodini hqonda medikamentis efeqtis mimarT.

analogiurad, orive jgufs misces identuri instruqciebi da maT mier

Sefasebuli dro erTnairi meTodiT izomeboda. Serevis Tavidan acileba

uaRresad mniSvnelovania eqsperimentis Catarebis dros.

damokidebuli cvladis gazomva

damokidebuli cvladi aris cvladi, romelic imyofeba damoukidebeli cvladis

gavlenis qveS. damoukidebeli cvladis manipulaciis Semdeg TiToeuli cdis

pirisaTvis unda gaizomos damokidebuli cvladi. am gazomvis Sedegebi

TiToeuli cdis pirisaTvis qulis saxiT gamoixateba. rogorc vnaxeT me-2-5

TavebSi, qulebi Sejamebulia centraluri tendenciisa da cvalebadobis

sazomebiT.

unrig-neervortisa da sxvaTa kvlevis hipoTezis mixedviT, diazepamis

miReba cvlis pirovnebis mier drois aRqmas. damokidebuli cvladi iyo drois

aRqma da izomeboda cdis pirebis mier im drois SefasebiT, romelsac isini

xarjavdnen garkveuli davalebis Sesasruleblad. es Sefasebebi warmodgenili

iyo TiToeuli jgufis saSualosa da standartuli gadaxris saSualebiT.

radgan ori saSualo da ori standartuli gadaxra gvaqvs, maTi erTmaneTisagan

gansasxvaveblad unda gamoviyenoT qveviT miwerili indeqsebi. amrigad, A1

jgufis saSualo iqneba 1AX , xolo s 1A iqneba populaciis standartuli gadaxra

A1 jgufisaTvis, 2AX iqneba A2 jgufis saSualo, xolo 2As ki populaciis

standartuli gadaxra A2 jgufisaTvis. am magaliTSi, 1AX =19.8wm da 1As =7.6wm,

placebo sakontrolo jgufisaTvis, xolo 2AX =13.6wm da 2As =4.8wm eqsperimentuli

jgufisaTvis. saSualoTa Soris gansxvavebis statistikuri mniSvnelovnoba

gaanalizebulia damoukidebeli jgufebis t kriteriumis saSualebiT. am

kriteriums Cven mogvianebiT ganvixilavT amave TavSi.

damokidebuli cvladi (Dependent variable): aris cvladi eqsperimentSi, romelic

damokidebulia damoukidebel cvladze.

10 suraTi #9.1 gviCvenebs eqsperimentis dagegmvisa da Catarebis sxvadasxva

etapebs ganxiluli magaliTis safuZvelze. ganixileT es etapebi am suraTis

gamoyenebiT.

SeamowmeT Tqveni codna IX-1

1. gansazRvreT: Sereva, SesaZleblobis SerCeva, damokidebuli cvladi,

ekvivalenturi jgufebi, damatebiTi cvladi, SemTxveviTi ganawileba,

SemTxveviTi SerCeva.

2. ra aris ekvivalenturi jgufebis Seqmnis mizani?

3. ra gansxvavebaa cdis pirebis SemTxveviT SerCevasa da jgufebSi

SemTxveviT ganawilebas Soris?

4. rodis xdeba Sereva eqsperimentSi?

5. ra sirTuleebs qmnis Sereuli eqsperimenti damoukidebeli cvladis

gavlenis Sesaxeb daskvnis gamotanis dros?

6. eqsperimentatori ikvlevs xmauris gavlenas Tvalis midevnebis amocanis

Sesrulebaze. amocana isaa, rom cdis pirs unda gaeyolebina saCvenebeli joxi

mbrunavi diskis garkveuli wertilisaTvis (warmoidgineT, rom gaayolebT

fanqars mbrunav musikalur firfitas). damokidebuli cvladi aris drois is

odenoba, romlis ganmavlobaSi cdis pirs SeuZlia SeinarCunos saCvenebeli

joxi erT wertilze. eqsperimentatori varaudobs, rom im cdis pirebs,

romlebic am davalebas uxmauro garemoSi asruleben, ufro didi xnis

ganmavlobaSi SeuZliaT SeinarCunon saCvenebeli joxi mbrunav wertilze, vidre

im cdis pirebs, romlebic davalebas xmauris maRali donis pirobebSi

asruleben. qvemoT aRwerili eqsperimentatoris TiToeuli moqmedeba xels

uSlis am eqsperiments damatebiTi cvladis SemotaniT, romelic sistematurad

icvleba damoukidebel cvladTan _ xmauris pirobasTan _ erTad.

eqsperimentatoris TiToeuli moqmedebisaTvis gansazRvreT damatebiTi cvladi

da Semdeg axseniT, ratom aris es eqsperimenti Sereuli.

a. eqsperimentatorma gaanawila aTi 20 wlis mamakaci uxmauro

garemoSi da aTi 60 wlis mamakaci xmaurian pirobebSi.

11 b. eqsperimentatorma waaxalisa uxmauro pirobebSi myofi cdis pirebi,

rom davaleba Zalian kargad SeesrulebinaT da ar waaxalisa

xmaurian pirobebSi momuSave cdis pirebi.

g. eqsperimentatorma mamakacebi uxmauro garemoSi gaanawila, xolo

qalebi _ xmaurian garemo pirobebSi.

d. eqsperimentatorma fizikuri kulturis kolejis studentebi

uxmauro garemoSi gaanawila da socialur mecnierebaTa fakultetis

studentebi _ xmaurian garemo pirobebSi.

e. cdis pirebi, romlebic davalebas uxmauro garemoSi asruleben,

eqsperimentSi 9.00 saaTidan 10.00 saaTamde monawileobdnen da cdis

pirebi, romlebic davalebas xmaurian pirobebSi asruleben,

eqsperimentSi 4.00 saaTidan 5.00 saaTamde monawileobdnen.

suraTi #9.1

eqsperimentis dagegmvisa da Catarebis aspeqtebi cdis pirTa Soris sqemis

gamoyenebiT. es suraTi gviCvenebs, Tu rogor gansazRvravs magaliTis saxiT

ganxiluli eqsperimenti diazepamis gavlenas drois aRqmaze.

24 cdis pirisagan Semdgari SerCevis ayvana, romelic miiRebs monawileobas am eqsperimentSi

cdis pirebi SemTxveviTad arian ganawilebulni, raTa Seiqmnas ekvivalenturi jgufebi.

jgufi

A1 placebo sakontrolo jgufi: 12 cdis piri

A2 eqsperimentuli jgufi: 12 cdis piri

ekvivalentruri jgufebi

12

aqvs Tu ara gavlena eqsperimentul Carevas statistikuri

Semowmebis saWiroebaze?

SerCevis Secdomis Sedegebi eqsperimentisaTvis

Tumca eqsperimentis Catarebis safexurebi Tanmimdevruli da naTelia,

sirTuleebi mainc Cndeba. damoukidebeli cvladis gavlenis Sesaxeb nebismieri

gadawyvetileba unda iyos miRebuli SerCevis saSualoebis erTmaneTTan

SedarebiT. principSi, Tuki SerCevis saSualoebi erTmaneTis ekvivalenturia,

Carevas ar hqonia gavlena. meore mxriv, Tuki SerCevis saSualoebi ar arian

erTmaneTis ekvivalenturni, arsebobs imis mtkicebis safuZveli, rom Carevas

mouxdenia gavlena. magram SerCevis Secdomis gamo am daskvnas ase martivad ver

gamovitanT. me-7 TavSi SerCevis Secdoma ganvsazRvreT rogorc raodenoba,

damatebiTi cvladebi kontrolirdeba

xdeba damoukidebeli cvladis manipulacia

jgufi: A1 miewodeba placebo abi

jgufi: A2 miewodeba diazepamis abi

izomeba damokidebuli cvladi: yvela cdis piri afasebs, Tu ra dro dasWirda amocanis Sesasruleblad

gamoiTvleba centraluri tendenciisa da cvalebadobis sazomi:

1AX =19.8 wm da s 1A =7.6 wm,

gamoiTvleba centraluri tendenciisa da cvalebadobis sazomi:

2AX =13.6 wm da s 2A =4.8 wm

cdis pirTa Soris Sedareba

damoukidebeli cvladis gavlena dadgenilia centraluri tendenciis sazomebis SedarebiT statistikuri kriteriumis saSualebiT.

13 romliTac SerCevis saSualo gansxvavdeba populaciis saSualosagan (anu, X _

µ). Tu aviRebT cdis pirTa or jgufs erTi da imave populaciidan, gavzomavT

qcevas da gamoviTvliT X TiToeuli SerCevisaTvis, SerCevis Secdomis gamo

naklebia imis albaToba, rom am ori jgufis saSualoebi erTmaneTis toli an

populaciis saSualos toli iqneba. suraTze #7.2 warmodgenili saSualos

empiriuli SerCeviTi ganawilebebi kargad aCvenebs am faqts.

ras migvaniSnebs SerCevis Secdoma eqsperimentSi damoukidebeli cvladis

gavlenis Sesaxeb gadawyvetilebis miRebaSi? ubralod erTi da imave

populaciidan aRebuli ori gansxvavebuli jgufis saSualoTa Soris

SemTxveviTi gansxvaveba mosalodnelia nebismier eqsperimentSi, sul erTia, aqvs

Tu ara damoukidebel cvlads gavlena damokidebul cvladze. maSinac ki, roca

damoukidebel cvlads araviTari gavlena ara aqvs, or jgufs Soris

saSualoebi erTmaneTisagan mxolod SerCevis Secdomis gamo gansxvavdebian.

maSin, rogorRa axerxeben bihevioristi mecnierebi Semdeg kiTxvaze pasuxs:

SerCevis saSualoTa Soris gansxvaveba ramdenad didi unda iyos, raTa

sakmarisi safuZveli gvqondes davaskvnaT, rom eqsperimentSi SerCevis Secdomis

garda sxva raRac muSaobs?

am kiTxvaze pasuxi gacema SeiZleba or saSualos Soris gansxvavebis

statistikuri hipoTezis SemowmebiT.

or saSualos Soris gansxvavebis statistikuri hipoTezis

Semowmebis mimoxilva

statistikuri hipoTezis Semowmeba gulisxmobs, rom gamoyenebul meTodSi 1AX

aris SerCeva, romelic aRebulia µ1-is mqone populaciidan, xolo 2AX aris

SerCeva, romelic aRebulia µ2-is mqone populaciidan. Tu davuSvebT, rom ori

populaciis saSualo tolia (anu µ1=µ2), roca damoukidebeli cvladi ar axdens

gavlenas, maSin SegviZlia ganvsazRvroT saSualoTa Soris mosalodneli

gansxvavebis sidide, romelic SecdomiTaa gamowveuli. Semdeg unda SevadaroT

saSualoTa Soris gansxvaveba, romelic miRebuli iyo realur eqsperimentSi, im

gansxvavebas, romelic mosalodnelia mxolod SerCevis Secdomis gamo. Tu

realurad miRebuli gansxvaveba imdenad didia, rom SeuZlebelia is mxolod

SerCevis Secdomis gamo mimeRo, maSin gadavwyvet, rom es gansxvaveba

14 ganpirobebulia ara SerCevis SecdomiT, aramed damoukidebeli cvladis

moqmedebiT.

statistikur SemowmebaSi uaRresad mniSvnelovani sakiTxi gagacaniT. imis

gadasawyvetad, imdenad gansxvavdeba Tu ara erTmaneTisagan ori jgufis

saSualoebi (romelTac damoukidebeli cvladis gansxvavebuli doneebi

miewodebaT), rom es gansxvaveba mivaweroT damoukidebel cvlads, pirvel

rigSi, unda davadginoT saSualoTa Soris SemTxveviTi gansxvavebis sidide, im

gansxvavebisa, romelic mosalodnelia mxolod SerCevis Secdomis gamo. Tuki

saSualoTa Soris realuri sxvaoba Tavsdeba Cveulebrivi SemTxveviTi

gansxvavebis farglebSi, maSin Cven ganvixilavT realur sxvaobas, rogorc

SemTxveviT sxvaobas. damoukidebuli cvladi ar axdens gavlenas damokidebul

cvladze. meore mxriv, Tu saSualoTa Soris realuri sxvaoba sakmarisad didia

imisaTvis, rom misi SemTxveviT miRebis albaToba iSviaTad iyos mosalodneli,

maSin gadavwyvetT, rom SerCevis saSualoebi ar gansxvavdebian erTmaneTisagan

mxolod SerCevis Secdomis gamo; gansxvaveba unda mieweros damoukidebel

cvlads.

gansxvavebuli tipis monacemebisa da kvlevis sqemebisaTvis sxvadasxva

statistikuri kriteriumi gamoiyeneba.UTu ar miviRebT mxedvelobaSi

konkretuli kriteriumis gamoyenebas, statistikur hipoTezaTa Semowmebis

ZiriTadi principebi yvela statistikuri kriteriumis mimarT aris gamosadegi.

SemdgomSi gagacnobT t kriteriums ori damoukidebeli jgufisaTvis. es t

kriteriumi saSualebas gvaZlevs statistikurad SevamowmoT gansxvaveba ori

SerCevis saSualos Soris, roca TiToeuli saSualo miRebulia cdis pirTa

gansxvavebuli jgufidan.

t kriteriumi ori damoukidebeli jgufisaTvis

t kriteriumi ori damoukidebeli jgufisaTvis ganisazRvreba Semdegi

formuliT:

21

2121 )()(AxAx

AAAA

sXXt

−

−−−=

µµelidamoukideb

elidamoukidebt mricxveli gviCvenebs, ramdenad gansxvavdeba ori realuri SerCevis

saSualoebs Soris (anu 21 AA XX − ) sxvaoba ori populaciis saSualoebs Soris

sxvaobisagan (anu )21 AA µµ − , im populaciebisa, romelTa warmomadgenlebic es

15 SerCevebi arian. ufro xSirad varaudoben, rom populaciis saSualoebi tolia.

amrigad, maT Soris gansxvaveba (anu )21 AA µµ − aris nulis toli da

elidamoukidebt mricxveli martivdeba 21 AA XX − -mde. mniSvneli ( 21 AxAxs − ) aris Secdomis

cvalebadobis sazomi, romelsac ewodeba saSualoTa Soris sxvaobis

standartuli Secdoma. saSualoTa Soris sxvaobis standartuli Secdoma aris

saSualoTa Soris sxvaobebis SerCeviTi ganawilebis standartuli gadaxra.

imisaTvis, rom gaviazroT Secdomis cvalebadobis es sazomi, unda SevimuSavoT

or saSualos Soris sxvaobis SerCeviTi ganawilebis koncefcia.

saSualoTa Soris sxvaobis SerCeviTi ganawileba

ganvixiloT eqsperimenti, romelSic damoukidebeli cvladis ori donis (A1 da

A2) manipulacia xdeba. davuSvaT, rom damoukidebel cvlads ara aqvs gavlena

damokidebul cvladze. radgan damoukidebel cvlads ara aqvs gavlena

damokidebul cvladze, SegviZlia SemovitanoT xelovnuri magaliTi ori

populaciidan (populaciebi A1 da A2), romelTac aqvT toli saSualoebi (µ1=µ2).

qulebis SerCevis saSualebiT. warmoidgineT, rom vatarebT am eqsperiments da

TiToeuli populaciidan (Sesabamisad A1 da A2-dan) aRebulia ori SerCeva,

TiToeulis moculoba xuTis tolia (anu nA1=5 da nA2=5). SerCevis es moculoba

aris nebismieri; SerCevis nebismieri moculoba SegveZlo agveRo am magaliTis

ganxilvisaTvis. jer unda gamovTvaloT TiToeuli jgufis saSualo da Semdeg

1AX -s unda gamovakloT 2AX . davuSvaT, es hipoTeturi eqsperimenti 100-jer

gavimeoreT da miviReT 100 21 AA XX − sxvaoba. Tu avagebT 21 AA XX − mniSvnelobebis

sixSireTa ganawilebas, miviRebT saSualoTa Soris sxvaobebis SerCeviT

empiriul ganawilebas. saSualoTa Soris 100 aseTi gansxvavebis SerCeviTi

ganawileba mocemulia suraTze #9.2. es ganawileba aris saSualoTa Soris

sxvaobebis empiriuli SerCeviTi ganawileba, radgan Cven miviReT da

gamoviTvaleT 1AX da 2AX Soris sxvaobebis 100 mniSvneloba. praqtikaSi

mkvlevari ar iTvlis da ar agebs aseTi tipis empiriul SerCeviT ganawilebas.

is iyenebs saSualoTa Soris sxvaobis SerCevis Teoriul ganawilebas, romelic

naCvenebia suraTze #9.3. saSualoTa Soris sxvaobis SerCevis Teoriuli

saSualoTa Soris sxvaobis SerCeviTi ganawileba (Sampling distribution of the difference between means): SerCevis saSualoebs Soris sxvaobis ganawileba, roca populaciidan amoRebulia n moculobis SerCevis yvela SesaZlo wyvili.

16 ganawileba aris SerCevis saSualoTa Soris sxvaobebis ganawileba, roca n

moculobis SerCevebis yvela wyvili populaciidanaa aRebuli. gaixseneT me-7

Tavidan, rom saSualos SerCevis Teoriul ganawilebas aqvs mniSvnelovani

maxasiaTeblebi, rogoricaa misi forma, saSualo da standartuli Secdoma. es

maxasiaTeblebi saSualoTa Soris sxvaobis SerCevis Teoriul ganawilebazec

vrceldeba.

suraTi aklia (Frequency -sixSire, value of _. . .-is mniSvneloba)

suraTi #9.2 ( 21 AA XX − ) saSualoTa Soris sxvaobis SerCevis empiriuli

ganawileba 100 SerCevisaTvis, romelic aRebulia toli saSualoebis mqone

populaciebidan (µ1=µ2)

suraTi aklia ( value of _. . .-is mniSvneloba)

suraTi #9.3 saSualoTa Soris sxvaobis Teoriuli ganawileba, roca µ1=µ2.

saSualoTa Soris sxvaobebis SerCeviTi ganawilebis forma

daakvirdiT, rom SerCevis empiriuli ganawileba suraTze #9.2 SedarebiT

simetriuli da unimodaluria. centraluri zRvris Teoremis Tanaxmad, SerCevis

moculobis zrdasTan erTad saSualoTa Soris sxvaobis SerCeviTi ganawileba

uaxlovdeba normalur ganawilebas, imis miuxedavad, Tu rogoria mis

safuZvlad mdebare populaciis ganawilebis forma. es damaxasiaTebeli

Tavisebureba warmodgenilia Teoriul ganawilebaSi, romelic naCvenebia

suraTze #9.3.

saSualoTa Soris sxvaobebis SerCeviTi ganawilebis saSualo

21 AA XX − sxvaobebis usasrulo simravlis ganawilebis saSualo, X X A1- X A2,

populaciis saSualoebs Soris sxvaobis toli iqneba _ µA1-µA2. amitomac, roca

µA1=µA2 da, aqedan gamomdinare, µA1-µA2=0, realuri 21 AA XX − mniSvnelobebis

saSualo iqneba saboloo jamSi nulis toli da SerCevis saSualoebis realuri

sxvaobebi dajgufdeba populaciis saSualoebs Soris nulovani gansxvavebis

garSemo. es Tviseba kargad Cans ganawilebaSi, romelic naCvenebia suraTze #9.2.

21 AA XX − sxvaobis saSualo aris +0.07. mxolod 100 SerCevis pirobebSic 21 AA XX −

sxvaoba Zalian axlos iqneba populaciis nulovan gansxvavebasTan. Teoriuli

ganawilebisaTvis, romelic naCvenebia suraTze #9.3, 21 AA XX − sxvaoba nulis

tolia.

17 saSualoTa Soris sxvaobebis SerCeviTi ganawilebis standartuli Secdoma

21 AA XX − mniSnvnelobaTa SerCeviTi Teoriuli ganawilebis standartul

gadaxras ewodeba saSualoTa Soris sxvaobis standartuli Secdoma an ufro

martivad, sxvaobis standartuli Secdoma. is aRiniSneba rogorc σ 21 AXAX − da

ganisazRvreba Semdegnairad:

σ 21 AXAX − =2

22

1

12

A

A

A

A

nnσσ

+

aRsaniSnavia, rom standartuli Secdomis gamoTvla xdeba TiToeuli

populaciis saSualos SerCeviTi ganawilebis standartuli Secdomis SejamebiT.

amitom misi mniSvneloba TiToeuli ganawilebis saSualos standartul

SecdomasTan erTad icvleba. praqtikaSi σ2 ucnobia da σ 21 AXAX − mniSvnelobis

gamoTvla xdeba SerCevis dispersiebidan, s2A1 da s2

A2-dan. saSualoTa Soris

sxvaobis standartuli Secdomis Sefasebis misaRebad gamoiTvleba

kombinirebuli an jamuri dispersia, romelic aRiniSneba s2kombinirebuli

simboloTi da miiReba ori SerCevis dispersiidan Semdegi formulis (tolobis)

gamoyenebiT:

s2kombinirebuli =

2nn)1()1(

A2A1

A22

2A12

1

−+−+− snsn AA

es toloba gviCvenebs, rom s2kombinirebuli –is misaRebad Cven vamravlebT TiToeuli

SerCevis dispersias Tavisuflebis xarisxze anu (nA-1)-ze, vumatebT erTmaneTs

miRebul mniSvnelobebs da Semdeg vyofT am jams ori jgufis Tavisuflebis

xarisxebis jamze. gavixsenoT me-7 Tavidan, rom saSualos standartuli

Secdoma, xs , mocemulia s/√N. es formula miiRebs Semdeg saxes: xs =. )/1(( 2 Ns

saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba, gansazRvruli

rogorc

s 21 AXAX − Semdegnairad gamoisaxeba:

saSualoTa Soris sxvaobis standartuli Secdoma an sxvaobis standartuli Secdoma:

21 AA XX − mniSvnelobaTa SerCeviTi Teoriuli ganawilebis standartuli gadaxra.

18

s 21 AXAX −

+=

A2A1

2

n1

n1ulikombinirebs

s2kombinirebuli CavsvaT formulaSi da miviRebT

saSualoTa Soris sxvaobis standartuli Secdomis Sefasebas:

s 21 AXAX − =

+

−+

−+−

A2A1A2A1

22

212

1

n1

n1

2nn)1(()1( AAAA snsn

sxvaobis standartuli Secdomis Sefaseba t statistikis mniSvnelia. amitom t-s

mniSvneloba gviCvenebs, Tu ramdenad daSorebulia standartuli Secdomebis

Sefaseba 1AX -sa da 2AX -s Soris sxvaobis SerCevis ganawilebis saSualosagan _

realurad miRebuli 21 AA XX − sxvaobisagan. magaliTad, davuSvaT, rom realurad

miRebuli 21 AA XX − =3 da s 21 AXAX − =3. trealuris mniSvneloba maSin iqneba 3/3=1-is

toli, rac imas niSnavs, rom realuri 21 AA XX − sxvaoba aris erTi standartuli

SecdomiT daSorebuli 21 AA XX − SerCeviTi ganawilebis saSualosagan. Tu

realuri 21 AA XX − =6 da maSin 21 AA XX − =3 t tolia 6/3 an 2-is, rac imas niSnavs, rom

realuri 21 AA XX − sxvaoba aris ori standartuli SecdomiT daSorebuli 1AX da

2AX Soris sxvaobis SerCeviTi ganawilebis saSualosagan. aRsaniSnavia, rom Tu

damoukidebeli cvladi ar axdens gavlenas da eqsperimentSi moqmedebs mxolod

SerCevis Secdoma, mosalodnelia, rom trealuris mniSvneloba iqneba mcire, maSin

roca damoukidebel cvlads aqvs gavlena da iwvevs 1AX da 2AX Soris arsebul

gansxvavebas, trealuris mniSvneloba ufro didi iqneba.


1. gansazRvreT:YsaSualoTa Soris sxvaobis SerCevis ganawileba, saSualoTa

Soris sxvaobis standartuli Secdoma, saSualoTa Soris sxvaobis

SerCevis Teoriuli ganawileba.

2. axseniT, ratom ar SeiZleba ubralod SexedoT gansxvavebul

eqsperimentul pirobebSi myofi ori jgufis saSualoTa mniSvnelobebs

saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba (Estimated standard error of the differnce between means): saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba miiReba

s2-is gamoyenebiT σ2-is SefasebisaTvis.

19 Soris gansxvavebas da gadawyvitoT, aris Tu ara es gansxvaveba

ganpirobebuli damoukidebeli cvladis moqmedebiT.

3. gamoTvaleT s 21 AXAX − qulaTa Semdegi erTobliobebisaTvis:

a. b.

d.

statistikuri Semowmeba tdamoukidebelis gamoyenebiT

saSualoTa Soris mniSvnelovani gansxvavebis Sesamowmeblad tdamoukidebelis

gamoyenebisas unda gaviaroT statistikuri hipoTezis Semowmebis qvemoT

mocemuli etapebi, romelic aRwerili iyo me-8 TavSi:

• nulovani hipoTezis, H0-is da alternatiuli hipoTezis H1-is

formulireba.

• H0–is WeSmaritebis daSvebis pirobiT SerCevis ganawilebis miReba. es

ganawileba mocemulia cxilSi #a-2.

jgufi

A1 A2

104 99

111 107

120 105

113 110

108 115

jgufi

A1 A2

77 82

84 86

68 78

71 74

82 69

79

80

jgufi

A1 A2

32 38

37 42

28 29

41 36

35 30

33 37

36

39

34

jgufi

A1 A2

71 85

82 80

63 73

74 76

68 70

76 67

20 • mniSvnelovnobis donis arCeva.

• t kritikuli mniSvnelobis (tkritikulis) dadgena t-s SerCeviTi

ganawilebidan, romelic mocemulia cxrilSi #a-2.

• uaryofis areebi ganlagebulni arian t statistikis SerCeviT Teoriul

ganawilebaze.

• t statistika, romelic ganisazRvreba rogorc trealuri, gamoTvlilia

SerCevis monacemebidan.

• H0–is uaryofis Tu misi ver uaryofis gadawyvetileba keTdeba imis

safuZvelze xvdeba Tu ara trealuri uaryofis areSi.


parametruli kriteriumisaTvis statistikuri hipoTeza aris debuleba

populaciis parametrebis Sesaxeb. parametrebi, romlebic Seesatyvisebian 1AX da

2AX statistikur kriteriumebs, arian populaciis saSualoebi, µ1 da µ2

Sesabamisad. tdamoukidebelisaTvis statistikuri hipoTezebia:

H0: µ A1=µA2. (nulovani hipoTeza)

H1: µ A1≠µA2 (alternatiuli hipoTeza)

monacemTa ori populaciisaTvis am ori hipoTezidan erT-erTi unda iyos

WeSmariti.

nulovani hipoTeza. nulovani hipoTeza ar gansazRvravs populaciis

parametrebis ricxobriv mniSvnelobebs, romlebic, ra Tqma unda, ucnobia.

is adgens mxolod mas, rom populaciis saSualoebi erTmaneTis tolia.

arsebiTad H0: µ A1=µA2 Seesatyviseba im situacias, roca damoukidebeli cvladi

ar moqmedebs damokidebul cvladze. Tu damoukidebeli cvladi ar axdens

gavlenas damokidebul cvladze, eqsperimentSi monawile yvela cdis piri

yofila erTi da imave populaciis warmomadgeneli da nebismieri realuri

gansxvaveba ori SerCevis saSualoebs Soris ganpirobebulia mxolod SerCevis

SecdomiT. amrigad, H0 ayalibebs situacias, roca SeiZleba iyos miRebuli t

statistikis SerCevis Teoriuli ganawileba. t–s SerCeviTi ganawilebis codniT,

SegviZlia ganvsazRvroT rogoria t-s nebismieri mniSvnelobis albaToba, Tu

gaviTvaliswinebT imas, rom SerCevis saSualoTa Soris gansxvaveba miewereba

mxolod SerCevis Secdomas.

21 alternatiuli hipoTeza. alternatiuli hipoTeza asaxavs iseT situacias, roca

1AX da 2AX gansxvavdeba erTmaneTisagan, radgan TiToeuli jgufis cdis pirebi

warmoadgenen sxvadasxva saSualoebis mqone populaciebs. es situacia xdeba

maSin, roca damoukidebel cvlads aqvs gavlena. Tu damoukidebeli cvlads aqvs

gavlena, da H1 aris WeSmariti, maSin mosalodnelia, rom 1AX da 2AX gacilebiT

ufro metad gansxvavdeba erTmaneTisagan, vidre mosalodneli iqneboda mxolod

SerCevis Secdomis gamo. amitom, Tu H1 aris WeSmariti, trealuris mniSvneloba

meti iqneba trealuris mniSvnelobaze H0-is WeSmaritebis pirobebSi.

eqsperimentis kvlevis hipoTeza dasturdeba H0-is uaryofiTa da H1-is

miRebiT. H0-is uaryofa da H1-is miReba aris gadawyvetileba imis Sesaxeb, rom

SerCevis saSualoebs Soris arsebuli realuri gansxvaveba ganpirobebulia ara

SerCevis SecdomiT, aramed damoukidebeli cvladis gavleniT.

am daskvnebis magaliTad SeiZleba ganvixiloT unrig-neervortisa da

sxvaTa eqsperimenti, romlis kvlevis hipoTezis Tanaxmad, diazepamis miReba

cvlida pirovnebis mier drois aRqmas im jgufTan SedarebiT, romelic iRebda

placebos. es hipoTeza dadasturdeba mxolod im SemTxvevaSi, roca orive

jgufSi Sefasebuli drois saSualoebi asaxavs gansxvavebuli saSualoebis

mqone populaciebs, anu situaciaSi, romelic Camoyalibebulia alternatiul

hipoTezaSi. amrigad, H0-is uaryofa da H1-is miReba kvlevis hipoTezis

mxardaWeraa.

mniSvnelovnobis donis SerCeva

mniSvnelovnobis done aris albaTobis is mniSvneloba, romelic iZleva

nulovani hipoTezis uaryofis pirobas statistikur kriteriumSi. rogorc

viciT, bihevioristi mecnierebi Cveulebriv irCeven alfas 0.05 an 0.01

mniSvnelovnobis dones. isini amis saSualebiT iReben gadawyvetilebas, rom t-s

mniSvnelobebis xdomiloba 100 SemTxvevidan mxolod 5-jer Tu ufro naklebad

an 100 SemTxvevidan mxolod erTxel an ufro naklebad aris statistikurad

iSviaTi SemTxveva H0-is WeSmaritebis pirobebSi. magaliTad, Cven virCevT α=0.05.

uaryofis areebis mdebareobis gansazRvra da t-s kritikuli mniSvnelobebis

povna: ormxrivi kriteriumebi

uaryofis are warmoadgens t-s mniSvnelobebs t-s SerCeviT ganawilebaSi,

romelsac aqvs α-s toli an masze naklebi albaToba H0-is WeSmaritebis

22 pirobebSi. aqedan gamomdinare, uaryofis are gansazRvravs trealuris

mniSnvelobebs, romlebic akmayofileben H0-is uaryofis kriteriums.

rogorc me-8 TavSi iyo ganxiluli, ormxriv an aramimarTulebiT

kriteriumSi, t-s uaryofis areebi ganlagebulia SerCevis Teoriuli

ganawilebis erT-erTi kudis mxareze, rogorc naCvenebi iyo suraTze #9.4. roca

H0 WeSmaritia, t-s mniSvnelobaTa umravlesoba dajgufebulia nulis garSemo,

21 AA XX − mniSvnelobaTa umravlesoba odnav iqneba gansxvavebuli nulisagan.

magram, t-s zogierTi mniSvneloba xvdeba ganawilebis romelime boloSi, maSinac

ki, roca H0 WeSmaritia. t-s es SedarebiT maRali mniSvnelobebi gvxvdeba maSin,

roca t-s mricxveli (anu 21 AA XX − ) mniSvnelze metia (anu sxvaobis standartul

Secdomaze). t-s es maRali mniSvnelobebi iSviaTi xdomilobebia, Tumca mainc

gvxvdebian H0: µ A1=µA2 WeSmaritebis pirobebSic.

Tu H1 WeSmaritia da µ A1≠µA2, maSin 21 AA XX − sxvaobis maRali dadebiTi an

uaryofiTi mniSvnelobebis da amrigad, t-s aseTive mniSvnelobebis xdomiloba

ufro xSirad aris mosalodneli. amitom, t-s uaryofis areebi mdebareoben t-s

SerCeviTi Teoriuli ganawilebis TiToeul kudTan. sruli albaToba, romelic

gansazRvrulia α-s saxiT, gayofilia t ganawilebis orive kuds Soris. Tu

trealuri xvdeba romelime uaryofis areSi, maSin H0 iqneba uaryofili da H1

miRebuli. rogorc erTi SerCevis t kriteriumis SemTxvevaSi, ormxrivi

kritikuli mniSvnelobebi, romelic mocemulia cxrilSi #a-2, gansazRvravs

uaryofis areebis zust mdebareobas. am cxrilis gamosayeneblad Cven unda

vicodeT trealuris Tavisuflebis xarisxi.

tdamoukidebelis Tavisuflebis xarisxi. Tavisuflebis xarisxi asaxavs qulaTa

raodenobas, romelic Tavisufalia cvalebadobis dros statistikis

gamoTvlisas. me-8 TavSi Cven vaCveneT, rom s2 aqvs N-1 df, sadac N aris im

qulaTa raodenoba, romelTa dispersiac gamoiTvleba. tdamoukidebeli

statistikisaTvis, s 21 AXAX − gamoiTvleba TiToeuli jgufis dispersiidan, s2A1

da s2A2-dan. Sesabamisad, tdamoukidebelis df efuZneba s2

A1 da s2A2 -is Tavisuflebis

xarisxebis jams. roca nA1 da nA2 asaxavs A1 da A2 jgufebSi qulaTa

raodenobas , s2A1-s aqvs nA1-1 df da s2

A2 aqvs nA2-1 df. amitom df-is jamuri

mniSvneloba tdamoukidebelisaTvis aris (nA1-1)-s mimatebuli (nA2-1). Tuki N

warmoadgens orive jgufis qulaTa saerToO raodenobas, (e.i N= nA1+ nA2), maSin df

23 tdamoukidebelisaTvis aris N-2. Sesabamisad, roca eqsperimentSi, romelSic 12 qulaa

TiToeul jgufSi (e.i. nA1=12 da nA2=12, amrigad, N=24), df tdamoukidebelisaTvis aris 24-

2=22. cxrilidan #a-2 tkritikuli 22 df-isaTvis da α=0.05 donisaTvis aris 2.074.

tkritikulis es mniSvneloba naCvenebia suraTze #9.4, sadac gansazRvrulia

uaryofis are t ganawilebis TiToeul kudTan.

suraTi #9.4

ormxrivi uaryofis aris ilustracia t ganawilebisaTvis, roca df aris 22 da

α=0.05

uaryofis aris mdebareobis gansazRvra: calmxrivi kriteriumebi

roca kvlevis hipoTeza winaswarmetyvelebs saSualoTa Soris mimarTulebis

mqone kavSirs (mag., rom 1AX aris 2AX -ze meti) da ar arsebobs sapirispiro

Sedegis miRebis interesi (mag., 1AX aris 2AX -ze naklebi), maSin mkvlevrebs

SeuZliaT gamoiyenon calmxrivi kriteriumi an mimarTulebis mqone kriteriumi.

calmxriv kriteriumSi, uaryofis are, romelic moicavs α-s srul mniSvnelobas

ganlagebulia t ganawilebis im kudSi, romelic Seesabameba kvlevis hipoTeziT

nawinaswarmetyvelevi Sedegis mimarTulebas. tkritikulis mniSvnelobebi

calmxrivi kriteriumisaTvis 0.01 da 0.05 mniSvnelovnobis doneebze

warmodgenilia cxrilSi #a-2. roca df aris 22, calmxrivi tkritikuli

mniSvnelovnobis 0.05 doneze aris 1.717. calmxrivi uaryofis are tkritikulTan =

+1.717 naCvenebia t-s ganawilebis marjvena kudTan suraTze #9.5., am uaryofis

arisaTvis gamoyenebuli nulovani da alternatiuli hipoTezebi Semdegia:

H0: µ A1≤µA2

H1: µ A1>µA2.

-3 -2 -1 0 +1 +2 +3 t-s mniSvneloba

H0 uaryofa uaryofis zona 0.025

H0 uaryofa uaryofis zona 0.025

H0 miRebis zona

t=-2.074 t=+2.074 t-is kritikuli t-is kritikuli mniSvneloba mniSvneloba

24 suraTi aklia (Reject _ H0 uaryofa, Do not reject _ H0 miRebis zona, Rejection region - uaryofis are, value of t_ t-s mniSvneloba, critical value of t _ t-s kritikuli mniSvneloba) suraTi #9.5 calmxrivi uaryofis aris ilustracia t ganawilebisaTvis, roca df

22-is tolia da α=0.05.

Tumca arsebobs winaaRmdegobebi calmxrivi kriteriumebis gamoyenebasTan

dakavSirebiT. cxrilSi #a-2 warmodgenili t-s SerCeviTi ganawileba Seqmnilia

im varaudis safuZvelze, rom H0: µ A1=µA2 WeSmaritia. SerCevis es ganawileba

nulis mimarT simetriulia da nulidan dadebiTi da uaryofiTi gansxvavebani

Tanabrad moxdeba. calmxrivi kriteriumisaTvis statistikuri hipoTeza aris an

H0: µ A1≤µA2 da H1: µ A1>µA2.

an

H0: µ A1≥µA2 da H1: µ A1<µA2

magram Cven ar SegviZlia SevqmnaT t-s SerCeviTi ganawileba arcerTi nulovani

hipoTezid, radgan arsebobs usasrulo raodenobis saSualebebi, rom H0: µ A1≤µA2

da H0: µ A1≥µA2 gaxdes WeSmariti. amrigad, SerCeviTi ganawileba H0: µ A1=µA2-

saTvis gamoiyeneba t-s kritikul mniSvnelobaTa gansazRvrisaTvis, rogorc

calmxrivi, aseve ormxrivi kriteriumebisaTvis. am da sxva mizezebis gamo

zogierTi bihevioristi mecnieri calmxrivi t kriteriumis gamoyenebis

winaaRmdegia (Gito, 1977). amitom, am Tavis yvela magaliTSi ormxrivi

kriteriumebi gamoiyeneba.

tdamoukidebelis gamoTvla

tdamoukidebelis ganmartebiTi formulaa:

21

2121 )()(

AXAX

AAAA

ssXXt

−−−−

=µµ

elidamoukideb

nulovani hipoTeza tdamoukidebelisaTvis H0: µ A1=µA2, aris H0: µ A1-µA2=0 ekvivalenturi.

amrigad, µ A1-µA2 sxvaoba aris savaraudod nulis toli da µ A1-µA2 amovardeba

formulidan,

21

21 )(

AXAX

AA

ssXXt

−−

=elidamoukideb

sadac s 21 AXAX − aris mocemuli Semdegi saxiT

25

s 21 AXAX − =

+

−+

−+−

A2A1A2A1

22

212

1

n1

n1

2nn)1(()1( AAAA snsn

amitom, gamoTvlebisaTvis gamoiyeneba Semdegi formula

+

−+

−+−

−=

A2A1A2A1

22

212

1

21

n1

n1

2nn)1(()1( AAAA

AA

snsn

XXt elidamoukideb

am formulis gamoyenebisas Cven gvWirdeba vicodeT mxolod ( AX ) cdis pirTa

raodenoba (nA) da populaciis dispersia (s2A) TiToeuli jgufisaTvis. am

formulis gamoyeneba naCvenebia magaliTSi #9-1.


statistikurad mniSvnelovani gansxvaveba. Tu trealuri xvdeba uaryofis areSi,

maSin

H0: µ A1=µA uaryofilia. H0-is uaryofa niSnavs H1:µ A1≠µA-s miRebas. SerCevis

saSualoebs Soris gansxvaveba statistikurad mniSvnelovania. Cven vaskvniT, rom

ori SerCevis saSualo, romlebsac moicavs trealuri, ar warmoadgens erTi da

imave populaciis saSualos Sefasebebs da maT Soris gansxvaveba

ganpirobebulia raRac sxva mizeziT da ara mxolod SerCevis SecdomiT. Tu

eqsperimenti yuradRebiT iqneba dagegmili da Catarebuli, maSin saSualoTa

Soris gansxvaveba ganpirobebulia damoukidebeli cvladis manipulaciiT. es

gadawyvetilebani da daskvnebi Sejamebulia cxrilis #9.1 marcxena svetSi.

cxrili #9.1

ori damoukidebeli SerCevis t kriteriumis gamoyenebiT statistikuri

hipoTezebis Semowmebis Sesaxeb gadawyvetilebebisa da daskvnebis Sejameba.

gamoyenebuli iyo 0.05 mniSvnelovnobis done.

26

aramniSvnelovani gansxvaveba. Tu trealuri ar xvdeba uaryofis areSi, maSin H0-s

ver uarvyofT. Tu H0-s ver uarvyofT, es gulisxmobs, rom viRebT H1-s.

saSualoTa Soris gansxvaveba ar aris statistikurad mniSvnelovani anu,

aramniSvnelovania. ori SerCevis saSualo ganxiluli iqneba, rogorc aRebuli

im populaciebidan, romelTac aqvT identuri saSualoebi da isini

erTmaneTisagan gansxvavdebian mxolod SerCevis Secdomis gamo. eqsperimentis

Sedegad vaskvniT, rom damoukidebel cvlads ara aqvs gavlena da SerCevis

saSualoebs Soris gansxvaveba SemTxveviTi xasiaTisaa. es gadawyvetilebani da

daskvnebi Sejamebulia cxrilis #9.1 marjvena svetSi.

daSvebebi tdamoukidebelisaTvis

erTi SerCevis t kriteriumi Semdeg moTxovnebs ayenebs SerCevis qulebisaTvis:

1. SerCevis qulebi SemTxveviTadaa miRebuli populaciis qulebidan da

TiToeuli qula damoukidebelia nebismieri sxva qulisagan;

2. qulebi populaciaSi normalurad arian ganawilebulni;

Tu trealuri xvdeba


Tu trealuri ar xvdeba


trealuris albaToba naklebia an toli

0.05, an p<0.05

H0 uaryofilia.

H1 miRebulia. TiToeuli SerCevis

saSualo aRebulia gansxvavebuli

populaciidan.

Ggansxvaveba or saSualos Soris aris

statistikurad mniSvnelovani 0.05

doneze.


saSualoebs Soris gansxvavebis mizezi

aris ara SerCevis Secdoma, aramed sxva

ram. Kakuratulad Catarebul

eqsperimentSi gansxvaveba miewereba

damoukidebel cvlads.

trealuris albaToba metia an toli

0.05, an p>0.05


H1 ar aris miRebuli

gansxvaveba or saSualos Soris

aris statistikurad

aramniSvnelovani 0.05 doneze.




saSualoebs Soris. Aar arsebobs

damamtkicebeli sabuTi, rom

damoukidebeli cvladi axdens

gavlenas.

27 3. qulaTa dispersiebi populaciebSi aris toli: σ2A1=σ2A2.

me-8 TavSi Cven aRvniSneT, rom t kriteriumis daSvebebi xSirad srulyofilad

ar akmayofileben realur kvlevas. isini xSirad arRveven kanonzomierebas.

miuxedavad amisa, mkvlevrebi mainc agrZeleben t kriteriumis gamoyenebas. ra

aris aseTi darRvevis Sedegebi?

Ppirveli daSvebis moTxovnis Tanaxmad, TiToeuli qula jgufSi

damoukidebelia amave jgufis yoveli sxva qulisagan, anu jgufSi erTi

pirovnebis qulis codna Tqven ar gaZlevT informacias amave jgufis nebismieri

sxva pirovnebis qulis Sesaxeb. es daSveba moiTxovs imas, rom TiToeul

individs mieces damoukidebeli cvladis mxolod erTi done da man

eqsperimentSi mxolod erTi qula mogvces. QqulaTa damoukideblobis es

moTxovna ar SeiZleba iyos darRveuli tdamoukidebelisaTvis. Tu pirovnebas miecema

damoukidebeli cvladis orive done da is mogvcems or qulas, maSin unda iyos

gamoyenebuli t kriteriumi damokidebuli qulebisaTvis, romelic am TavSi

ufro mogvianebiT ganixileba.

Ppirveli daSveba, agreTve, gviCvenebs, rom cdis pirebi unda iyvnen SerCeulni

SemTxveviTad, magram realurad cdis pirebi eqsperimentSi, rogorc wesi,

miRebuli arian xelmisawvdomi SerCevidan da ara SemTxveviTi SerCevidan.

xelmisawvdomi SerCevis gamoyenebis ZiriTadi nakli isaa, rom Cven garkveul

farglebSi SezRuduli varT miRebuli Sedegebis populaciaze

ganzogadebisaTvis.

G

Mmeore (qulaTa normaluri ganawileba) da mesame (dispersiebis toloba,

xSirad iwodeba, rogorc dispersiaTa erTgvarovneba) daSvebebis darRvevebma

SeiZleba Secvalos I gvaris Secdomis daSvebis realuri albaToba alfas

mniSvnelobiT. Mmagram t kriteriums axasiaTebs mdgradoba am daSvebebis

darRvevis sawinaaRdegod. Gmdgradoba niSnavs, rom daSvebebis darRvevas mcire

gavlena aqvs I gvaris Secdomis daSvebis albaTobaze. Bbredlis (Bradley, 1980,

1984) mier Catarebuli kvleva ewinaaRmdegeba mdgradobis aseT aRniSvnas.

Bbredli amtkicebs, rom ar arsebobs pirobebis arc erTi simravle, romlis

drosac es daSvebebi SeiZleba dairRves da am dros t kriteriumis mdgradoba

iyos garantirebuli. magram roca qulebi ar arian normalurad ganawilebulni

da darRveulia dispersiebis tolobis daSvebebi, amas savaraudod eqneba

mdgradoba (Robustness): es termini gamoiyeneba imis aRsaniSnavad, rom statistikuri

kriteriumis daSvebebis darRvevas aqvs mcire gavlena pirveli gvaris Secdomaze.

28 minimaluri gavlena pirveli gvaris Secdomis daSvebis albaTobaze, Tu

daculia Semdegi pirobebi:

• TiToeul jgufSi cdis pirebis raodenoba iqneba toli;

• QqulaTa or ganawilebas daaxloebiT erTi da igive forma aqvs da

ganawilebebs ara aqvs arc Zalian wawvetebuli da arc Zalian gluvi

forma;

• MmniSvnelovnobis done iqneba 0.05 da ara 0.01.

MmagaliTi #9-1

diazepamis gavlenis dadgena drois Sefasebaze tdamoukidebelis gamoyenebiT

amocana

tdamoukidebelis gamoTvlis ilustrirebisaTvis, roca X da s cnobilia,

gamoviyenebT amocanas, romlis winaSec dadgnen unrug-neervorti da sxvebi,

romelic exeboda diazepamis gavlenas drois Sefasebas aRqmaze. U unrig-

neervortsa da sxvebs hyavdaT ori jgufi _ TiToeulSi 12 cdis piri.

Pplacebos sakontrolo jgufis (A1) mier Sefasebuli saSualo dro iyo 19,8 wm,

roca sA1=7.6 wm, eqsperimentuli jgufis mier Sefasebuli saSualo dro iyo 13.6

wm, roca sA2= 4.8 wm. gansxvavdeba Tu ara Sefasebuli saSualo droebi

erTmaneTisagan mniSvnelovnobis 0.05 doneze? gamoiyeneT ormxrivi kriteriumi.

saWiro statistika

+

−+

−+−

−=

A2A1A2A1

22

212

1

21

n1

n1

2nn)1(()1( AAAA

AA

snsn

XXt elidamoukideb

daSvebebi:

1. cdis pirebi SemTxveviTad arian SerCeuli da TiToeuli cdis piri

damoukidebelia nebismieri sxva cdis pirisagan;

2. drois Sefasebis qulebi normalurad arian populaciaSi ganawilebulni;

3. drois Sefasebis qulaTa dispersiebi tolia orive jgufis populaciebSi:

29 statistikuri hipoTezebi: H0: µ A1=µA2

H0: µ A1≠µA2

mniSnvelovnebis done: α=0.05

df t-saTvis: (nA1-1)+( nA2-1)=(12-1)+(12-1)=22, an N-2=24-2=22

t-s kritikuli mniSvneloba: tkritikuli=2.074 cxrilidan #a-2.

uaryofis areebi: trealuris mniSvneloba tolia an naklebia –2.074, an tolia, an

metia +2.074.

gamoTvla:

387.2597.2

2.6122

228.888

2.6

121

121

21212)8.4)(112()6.7)(112(

6.138.1922

+=+

=

+=

+

−+

−+−

−=t

gadawyvetileba:

trealuri xvdeba uaryofis areSi, romelic tolia an metia tkritikuli= +2.074-is

MCven unda uarvyoT H0 da miviRoT H1.

Ddaskvna: ori SerCevis saSualoebi 19.8wm da 13.6wm mniSvnelovnad gansxvavdebian

0,05 doneze. radgan jgufebi ekvivalenturni iyvnen eqsperimentul Carevamde,

saSualoTa Soris gansxvaveba miewereba damoukidebeli cvladis gavlenas. cdis

pirebis mier Sefasebuli saSualo dro diazepamis miRebis pirobebSi ufro

moklea, vidre is dro, romelic cdis pirebma Seafases placebo kontrolis

pirobebSi.

A

magaliTi #9-2

tdamoukidebelis gamoyeneba nedli qulebisaTvis

amocana

gasulma wlebma aCvenes ganaTlebis gaumjobesebis moTxovnileba momavali

profesiisaTvis saWiro unarebis SesaZenad. Aam moTxovnilebis erT-erTi

gamoxmaureba iyo kompiuterebis gamoyeneba laboratoriuli eqsperimentebis

modelirebisaTvis zust mecnierebebSi. Gebanma, askarma da ozkanma (Geban, Askar,

30 Ozkan, 1992) gamoikvlies, ganapirobebs Tu ara aseTi kompiuterulad

modelirebuli eqsperimentebi ufro did miRwevebs qimiaSi, vidre sayovelTaod

miRebuli midgoma, romelSic studentebi asruleben realur laboratoriul

eqsperimentebs. DdavuSvaT, rom Tqven dainteresdiT am kvlevis gameorebiT 28

umaRlesi saswavleblis student-qimikosze. Tqven SemTxeveviTi wesiT

gadaanawileT es studentebi ori toli moculobis jgufSi, Cveulebrivi

swavlebis jgufSi (A1) da kompiuterulad modelirebuli swavlebis jgufSi

(A2). Aam kursis bolos Tqven qimiis codna miRwevis testiT gazomeT, romlis

qulebi meryeobs 0-dan 100-mde. MmiRebuli iyo Semdegi qulebi:

swavlebis meTodi

sayovelTaod

miRebuli (A1)

Kkompiuterulad

(A2)

74 84

76 75

81 94

90 88

67 69

72 90

84 86

87 97

92 81

70 91

78 83

82 92

75 82

85 93

mniSvnelovnad Ggansxvavdeba Tu ara miRwevis testis es qulebi erTmaneTisagan

0.05 doneze? GgamoiyeneT ormxrivi kriteriumi.

saWiro statistika: tdamoukidebeli

daSvebebi:

1. cdis pirebi SemTxveviTad arian SerCeuli da TiToeuli cdis piri

damoukidebelia nebismieri sxva cdis pirisagan;

31 2. qimiaSi miRwevis testis qulebi normalurad arian populaciaSi

ganawilebulni;

3. qimiaSi miRwevis qulaTa dispersiebi tolia orive pirobaSi myofi jgufis

populaciaSi.

statistikuri hipoTezebi: H0: µ A1=µA2

H0: µ A1≠µA2


df t-saTvis: (nA1-1)+( nA2-1)=(14-1)+(14-1)=26, an N-2=28-2=26

t-s kritikuli mniSvneloba: tkritikuli = 2.056 cxrilidan #a-2.

uaryofis areebi: trealuris mniSvneloba tolia an naklebia –2.056 an tolia an

metia +2.056.

gamoTvla: Tavdapirvelad unda vipovoT TiToeuli jgufis saSualo da

standartuli gadaxra. sayovelTaod miRebuli swavlebis meTodis jgufisaTvis,

1AX =79.5 da sA1=7.59 da kompiuteruliad modelirebuli swavlis jgufisaTvis

2AX =86.07 da sA2=7.74. CavsvaT es mniSvnelobebi trealuris gamoTvliT formulaSi

da miviRebT:

266.2899.2

57.6

142

26708.1527

57.6

141

141

21414)74.7)(114()59.7)(114(

07.868.7922

−=−

=

−=

+

−+

−+−

−=t

gadawyvetileba:

trealuri xvdeba uaryofis areSi, romelic tolia an naklebia tkritikuli= - 2.056-

is.M uarvyofT H0 -s da viRebT H1-s .

daskvna: ori SerCevis saSualoebi 79.5 da 86.07 mniSvnelovnad gansxvavdebian 0,05

doneze. saSualoTa Soris gansxvaveba miewereba damoukidebeli cvladis

32 gavlenas. kompiuterulad modelirebuli eqsperimentebi qimiaSi ganapirobeben

ufro maRal miRwevebs, vidre swavlebis Cveulebrivi meTodebi.


1. gansazRvreT: mimarTulebis mqone kriteriumi, aramimarTulebiTi kriteriumi,

calmxrivi uaryofis are, uaryofis are, tkritikuli, trealuri, tdamoukidebeli,

ormxrivi uaryofis are.

2. gansazRvreT statistikuri Semowmebis etapebi tdamoukidebelisaTvis.

3. dawereT statistikuri hipoTezebi ormxrivi tdamoukidebelisaTvis .

4. eqsperimentis romel situacias Sesabameba H0?

5. eqsperimentis romel situacias Sesabameba H1?

6. davuSvaT Tqven gyavT ori jgufi, romelTa moculobebia nA1 da nA2. SeasruleT

Semdegi savarjiSo SerCevis TiToeuli jgufisaTvis, romelic mocemulia

cxrilSi: (a) ipoveT df t kriteriumisaTvis ori SerCevis safuZvelze, (b) ipoveT

tkritikuli ormxrivi kriteriumisaTvis α=0.05 doneze (Tu cxrilSi ar aris

mocemuli df-is zusti mniSvneloba, maSin isargebleT cxrilSi mocemuli

Semdegi dabali mniSvnelobiT), (g) aRniSneT, xvdeba Tu ara trealuri uaryofis

areSi da (d) aRniSneT Tqveni gadawyvetilebani statistikuri hipoTezebis

Sesaxeb.

SerCevebi nA1 nA2 trealuri,

1 10 10 2.341

2 8 12 1.763

3 15 15 2.12

4 16 14 3.479

5 35 35 2.912

6 19 16 2.007

7 65 64 2.002

8 75 75 2.193

7. gansazRvreT daSvebebi, romelic safuZvlad udevs tdamoukidebelis

gamoyenebas.

8. a. t kriteriums axasiaTebs mdgradoba. ras niSnavs es termini.

b. ra pirobebSi aris t kriteriumi savaraudod mdgradi?

33 9 a. 1 da 2 amocanebisaTvis upasuxeT Semdeg kiTxvebs. gamoiyeneT ormxrivi

kriteriumi da mniSvnelovnobis 0.05 done TiToeuli problemisaTvis.

b. risi tolia trealuris mniSvneloba?

g. risi tolia df t kriteriumisaTvis?

d. risi tolia t kriteriumis mniSvneloba?

e. risi tolia uaryofis areebi trealurisaTvis?

v. xvdeba Tu ara trealuri uaryofis areSi?

z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb?

uaryofT Tu ver uaryofT H0 hipoTezas? miiRebT Tu ara H1 hipoTezas?

T. gamoitanT daskvnas imis Sesaxeb, rom SerCevis saSualoebi erTi da

igivea Tu gansxvavebuli populaciebidanaa?

i. saSualoebs Soris gansxvaveba statistikurad mniSvnelovania?

k. gansxvavdebian jgufebi mniSvnelovnad im drois SefasebaSi, romelsac

isini lodinSi atareben? Tu asea, rogoria am gansxvavebis mimarTuleba?

Aamocana #1. AafasebT Tu ara drois Sualeds rogorc xangrZlivs, roca xarT

raime movlenis molodinSi? DdavuSvaT, am kiTxvaze pasuxis gasacemad Tqven

SemTxveviTi wesiT gaanawileT 36 cdis piri sakontrolo da eqsperimentul

jgufebSi, roca nA1= nA2. orive jgufis cdis pirebs Tqven eubnebiT, rom

CautarebT eqsperiments, romelic gagrZeldeba Zalian cota xani da maT unda

Seavson SfoTvis gasazomi kiTxvari. sakontrolo jgufis cdis pirebs eubnebiT,

rom asistenti mova ramdenime wuTSi, xolo eqsperimentuli jgufis cdis pirebs

atyobinebT, rom asistenti mcire xniT daigvianebs. Oorive jgufis SemTxvevaSi

asistenti brundeba 5 wuTSi. roca asistenti brundeba, cdis pirebs sTxoven

Seafason, ramdeni xani mouwiaT lodini. DdavuSvaT, Tqven miiReT Sefasebuli

droebis Semdegi saSualoebi da standartuli gadaxrebi (wuTebSi) orive

jgufisaTvis:

jgufi

sakontolo eqsperimentuli

X 6.9 8.9

s 1.57 1.71

amocana #2. davuSvaT, rom Tqven dainterebuli xarT igive sakiTxiT, romelic

iyo ganxiluli amocanaSi #1: afasebT Tu ara drois Sualeds, rogorc ufro

34 xangrZlivs, roca raime movlenis molodinSi xarT? Dkvlavac Tqven SemTxveviT

gadaanawileT cdis pirebi or jgufSi. TiToeul jgufSi toli raodenobis

nacvlad Tqven gadaanawileT 13 cdis piri sakontrolo jgufSi da 17

eqsperimentulSi. sxva mxriv, eqsperimenti zemoT aRweril amocanaSi moyvanili

eqsperimentis identuri iyo. davuSvaT, rom Tqven miiReT Semdegi saSualo

Sefasebebi da standartuli gadaxrebi (wuTebSi) Tqveni jgufebisaTvis:

jgufi

sakontolo eqsperimentuli

X 6.6 7.8

s 1.8 1.92

simZlavre da tdamoukidebeli

statistikuri kriteriumis simZlavre gansazRvruli iyo me-8 TavSi rogorc H0-

is uaryofis albaToba roca H0 ar aris samarTliani.

vinaidan H0-is uaryofa niSnavs imas, rom eqsperimentatorma miiRo H1Hda

gadawyvita, rom damoukidebel cvlads gavlena aqvs mis eqsperimentze,

statistikuri kriteriumis simZlavre aris imis albaToba, rom damoukidebeli

cvladis moqmedeba iqneba aRmoCenili eqsperimentSi, roca aseTi moqmedeba

realurad arsebobs. eqsperimentators yovelTvis unda, rom gazardos

statistikuri kriteriumis simZlavre, raTa SeZlos damoukidebeli cvladis

gavlenis SemCneva, roca es gavlena realurad arsebobs.

nulovani hipoTeza aris uaryofili, roca trealuri xvdeba uaryofis areSi

da rac ufro metia trealuri, miT ufro didia albaToba, rom is moxvdeba

uaryofis areSi. Aarsebobs ramdenime saSualeba, raTa gavzardoT trealuris

mniSvneloba da amrigad tdamoukidebelis simZlavre. Cven SegviZlia gaviazroT es

saSualebebi tdamoukidebelis formulis GganxilviT:

21

21

AXAX

AA

sXXt

−

−=elidamoukideb

Aam formulidan Cans, rom arsebobs trealuris gazrdis ori gza: (1) gavzardoT

gansxvaveba 1AX da 2AX Soris da (2) SevamciroT s 21 AXAX − -is mniSvneloba.

35 pirveli midgoma aris damoukidebeli cvladis gavlenis gazrda, anu Cven

gvWirdeba damoukidebeli cvladis imgvari manipulacia, rom 1AX da 2AX Soris

rac SeiZleba didi gansxvaveba miviRoT. am mizeziT, bihevioristi mecnierebi

xSirad iyeneben damoukidebeli cvladis Zalian gansxvavebul doneebs.

s 21 AXAX − mniSvnelobis Sesamcireblad Cven SegviZlia gavakeToT ori ram:

gavzardoT SerCevis moculoba da SevamciroT SerCevis qulebis cvalebadoba.

s 21 AXAX − da SerCevis moculobis kavSiris sailustraciod, davuSvaT, rom orive

s2A1 da s2

A2=36. gaviTvaliswinoT, agreTve, rom s2 es mniSvnelobebi sami

SerCevis moculobidan erT-erTs exeba: nA1=nA2=3, nA1=nA2=6, nA1=nA2=9. amis Semdeg

CavsvaT nA mniSvnelobebi s 21 AXAX − formulaSi:

roca nA1= nA2=3, maSin s 21 AXAX − = 4.9

nA1= nA2=6, maSin s 21 AXAX − = 3.5

nA1= nA2=9, maSin s 21 AXAX − = 2.8

aqedan gamomdinare, vxedavT, rom SerCevis moculobis zrdasTan erTad,

sxvaobis standartuli Secdoma mcirdeba. amrigad, 1AX da 2AX Soris

mocemuli sxvaobisaTvis trealuris mniSvneloba izrdeba SerCevis moculobis

zrdasTan erTad da s 21 AXAX − mcirdeba. xolo, roca trealuri izrdeba, maSin

izrdeba imis albaTobac, rom is moxvdeba uaryofis areSi da HH0 iqneba

uaryofili da H1 _ miRebuli.

zogadad, Carevis nebismieri gavlenis Sesafaseblad ara aqvs mniSvneloba

imas, ramdenad mcirea es gavlena, Tuki SerCevis moculoba sakmaod didia.

amitom, mkvlevars SeuZlia minimumamde daiyvanos meore gvaris Secdoma SerCevis

moculobis zrdiT. magram, Tu gaviTvaliswinebT realobas, SerCevis moculobis

zrda xSirad Zviria da moiTxovs didi drois xarjvas. ufro metic, Tuki

saWiroa TiToeul jgufSi 1000 kaciani SerCeva, raTa davadginoT

damoukidebeli cvladis gavlena, maSin Cven SegviZlia vivaraudoT, rom am

damoukidebel cvlads minimaluri gavlena eqneba qcevaze.

meore nabiji, s 21 AXAX − Sesamcireblad aris TiToeuli jgufis qulebs

Soris cvalebadobis Semcireba. Tu qulebi TiToeul jgufSi naklebad

Seicvleba, maSin s2 Semcirdeba TiToeuli jgufisaTvis da amis Sedegad s 21 AXAX −

, agreTve, Semcirdeba. s 21 AXAX − -is es Semcireba gamoiwvevs trealuris gazrdas da

amrigad imis albaTobis gazrdas, rom trealuri moxvdeba uaryofis areSi. aseTi

kavSirebis gamo eqsperimentatorebi Cveulebriv kargad akontroleben kvlevis

36 Catarebis pirobebs, raTa minimumamde daiyvanon qulaTa cvalebadoba TiToeul

jgufSi.

eqsperimentuli pirobis efeqtis siZlieris gazomva

statistikurad mniSvnelovani gansxvaveba gviCvenebs, rom damoukidebel cvlads

gavlena aqvs eqsperimentis damokidebul cvladze, magram is ar gviCvenebs am

zegavlenis siZlieres an sidides. arsebobs statistikebi, romlebic SeiZleba

gamoviyenoT damoukidebeli cvladis zegavlenis siZlieris sazomad mas Semdeg,

rac eqsperimenti dasrulda. am statistikebs xSirad moixsenieben rogorc

efeqtis siZlieris sazomebs an kavSiris siZlieris sazomebs. statistika,

romelsac SemdegSi ganvixilavT, iwodeba eta kvadratad da is warmodgenilia

η2 saxiT, sadac η aris berZnuli aso eta. eta kvadrati gamoiTvleba trealuris

mniSvnelobidan da Tavisuflebis xarisxidan t-saTvis Semdegi Fformulis

saSualebiT:

dftt

+=

realuri2

realuri2

2η

Tu ganvixilavT magaliTs #9-2, sadac sayovelTaod miRebuli da

kompiuterulad modelirebuli swavlis pirobebis gansxvaveba iyo –2.266, roc

df=29. eta kvadrati trealurisaTvis tolia

16.01348.31

1348.526)266.2(

)266.2(2

2

2

=

==

=+−

−=

=+

=dft

trealuri

2

realuri2

η

eta kvadrati aris imis sazomi, Tu ramdenad amcirebs subieqtis mier

miRebuli qulebis winaswarmetyvelebis Secdomas damoukidebeli cvladis im

donis codna, romelic cdis pirs miewoda. amis sailustraciod, davuSvaT, rom

magaliTSi #9-2 gTxoves iwinaswarmetyveloT cdis piris qula miRwevis testSi,

magram Tqven ar icodiT swavlebis romeli meTodi miewoda mas. am magaliTSi

Tqveni saukeTeso winaswarmetyveleba iqneboda sruli saSualos gamoyeneba 28

cdis pirisaTvis, X G=82.79, rogorc nawinaswarmetyvelebi qulisa. ra Tqma unda,

efeqtis siZliere (Strength of effect): damoukidebeli cvladis siZliere gazomili statistikis gavlenis siZlieriT.

37 es winaswarmetyveleba Seicavs Secdomas da saSualo Secdoma iqneba 28 qulis

standartuli gadaxra sruli saSualos garSemo. aq Cven gamoviyenebT ufro S,

vidre s, radgan η2 aris statistika, romelic aRwers mxolod SerCevas, es ar

aris populaciis mniSvnelobis Sefaseba. S mniSvneloba 28 qulisaTvis sruli

saSualos garSemo aris 8.09. am mniSvnelobis kvadrati [e.i. (8.09)2], tolia 65.448-

is da 28-quliani SerCevis srul dispersiaa.

axla davuSvaT, rom gTxoven iwinaswarmetyveloT cdis piris qula, magram

Tqven amjerad iciT swavlebis is meTodi, romelic cdis pirma gaiara. am

SemTxvevaSi saukeTeso winaswarmetyveleba iqneboda im cdis pirebis qulebis

saSualo, romlebmac es swavlebis meTodi gaiares an 79.5, an 86.07. Tqveni

winaswarmetyveleba kvlav Seicavs Secdomas, magram axla saSualo Secdoma

iqneba im jgufis standartuli gadaxra, romelSic es cdis piri imyofeba. Cven

kvlav viyenebT S; amrigad, SA1=7.32 da SA2=7.46. TiToeuli S-is kvadratSi ayvaniT

miviRebT TiToeuli jgufis SerCevis dispersiebs: S2A1 =53.582 da S2

A2=55.652 am

ori dispersiis saSualo aris 54.617 [mag. (53.582+55.652)/2=109.234/2=54.6170]. es

mniSvneloba aris Secdomis saSualo dispersia, roca swavlebis erTi meTodis

saSualo gamoyenebuli iyo cdis piris qulis winaswarmetyvelebisaTvis.

gansxvaveba im dispersiasa, romelic napovni iyo sruli saSualos gamoyenebiT

(65.448) da im saSualo dispersias Soris, romelic gamoTvlili iyo

eqsperimentuli pirobis saSualoebidan (e.i.54.617) aris 65.448-54.617=10.831.

dispersiis es kleba sruli dispersiidan, roca qulis

winaswarmetyvelebisaTvis gamoiyeneba eqsperimentuli pirobebis saSualoebi

aris 10.831. Tuki am sxvaobas gavyofT srul dispersiaze miviRebT

10.831/65.448=0.165, an Tuki davamrgvalebT aseulebamde, 0.16-s. sxva sityvebiT rom

vTqvaT, roca vici damoukidebeli cvladis done, romelic cdis pirs miewoda

da cdis piris qulis winaswarmetyvelebisaTvis viyeneb sruli saSualos

nacvlad eqsperimentuli jgufis saSualos, maSin winaswarmetyvelebis

dispersia Semcirdeba 0.16 an 16%-iT. es mniSvneloba aris η2. amrigad, η2

gviCvenebs sruli dispersiis Semcirebis proporcias damokidebul cvladSi,

roca vici damoukidebeli cvladis done, romelic miawodes cdis pirs. am

mizeziT, η2 xSirad aris ganmartebuli, rogorc damokidebeli cvladis

dispersiis proporciis maCvenebeli, romelic SeiZleba axsnili iyos

damoukidebeli cvladis mier.

38 η2 mniSvneloba icvleba 0-dan 1-mde. nulis toli η2 gviCvenebs, rom

damoukidebeli cvladis codna ar amcirebs damokidebuli cvladis Secdomis

dispersias. sruli saSualos winaswarmetyveleba iseTive kargi

winaswarmetyvelebaa cdis piris qulisaTvis, rogorc jgufis saSualos

winaswarmetyveleba. ar arsebobs raime damokidebuleba damoukidebeli cvladis

donesa da damoukidebuli cvladis qulas Soris. meore mxriv, roca η2 1-is

tolia, es gviCvenebs, rom damoukidebeli cvladis donis codna gvaZlevs cdis

piris qulis sruli winaswarmetyvelebis saSualebas. jgufis saSualos

saSualebiT cdis piris qulis winaswarmetyvelebisas Secdoma ar iqneba

daSvebuli.

roca praqtikaSi η2 0.1-dan 0.15–mdea, iTvleba, rom eqsperimentuli piroba

Zlier gavlenas axdens. aseT interpretacias mxars uWers bihevioristi

mecnierebis gamoqveynebuli kvlevebic. magaliTad, haasma, uaxterma da

solomonma (Haas, Waechter and Solomon, 1982) gamoiTvales η2 statistikuri

mniSvnelovnobis 11044 SemowmebaSi, romelic moxseniebuli iyo fsiqologiuri

konsultaciis Jurnalis 1970 da 1979 wlebis gamocemebSi. maT aRmoaCines, rom

η2-is mniSvnelobis mediana iyo 0.083. interkvartiluri diapazoni η2

mniSvnelobisaTvis iyo 0.043-dan 0.268-mde. es Sedegi gvaZlevs informacias

bihevioristul mecnierebebSi kavSiris tipur siZliereze da, agreTve, η2

mniSvnelobis Sefasebis safuZvels. amerikis fsiqologiuri asociaciis

publikaciebis saxelmZRvanelos (1994) gvirCevs, rom damoukidebeli cvladis

gavlenis siZlieris sazomebi, iseTi rogoricaa eta kvadrati, unda iyos

CarTuli angariSis warmodgenisas statistikur SemowmebaSi. erT-erTi aseTi

angariSi aRwerilia me-10 TavSi.

magaliTi #9-3

η2 gamoTvla eqsperimentSi

amocana

magaliTSi #9-1, placebo sakontrolo da eqsperimentuli jgufebis mier drois

Sefasebebs Soris gansxvavebis trealuri iyo +2.387 df=22. risi tolia

damoukidebeli cvladis gavlenis siZliere medikamentis miRebis pirobaSi?

amoxsna: gavlenis siZliere mocemulia η2-iT, sadac

39

dftt

+=

realuri

realuri

2

22η

Tu trealuris nacvlad CavsvamT ricxviT mniSvnelobebs da df-s, miviRebT

21.0226978.5

6978.522)387.2(

)387.2(2

22

=

=+

=

=+

=η

daskvna: η2-is mniSvneloba gviCvenebs, rom medikamentis pirobis codna 21%-iT

amcirebs eqsperimentis srul dispersias.

ndobis intervalebis ageba ori damoukidebeli populaciis

saSualoTa sxvaobisaTvis t statistikis gamoyenebiT

populaciis saSualosaTvis ndobis intervali gansazRvruli iyo rogorc

intervali, romelic moicavs populaciis saSualos ndobis garkveuli

xarisxiT. am TavSi Cven davinteresdebiT populaciis saSualoebs Soris

sxvaobiT. am SemTxvevaSi Cven dagvWirdeba ndobis intervali damoukidebeli

populaciis saSualoTa µ A1-µA2 sxvaobisaTvis.. midgoma, romelsac Cven aq

gamoviyenebT msgavsia imisa, romelic gamoyenebuli iyo me-8 TavSi. gansxvaveba

mxolod isaa, rom nacvlad X unda gamoviyenoT 21 AA XX − , xolo nacvlad s X

unda gamoviyenoT s 21 AXAX − . Sesabamisad, 95% ndobis intervali damoukidebeli

populaciis saSualoTa Soris sxvaobisaTvis mocemulia intervaliT:

)()( 215.021 AXAXAA stXX−

−− -dan )()( 215.021 AXAXAA stXX−

+− -mde,

sadac

21 AA XX − = realur SerCevis saSualoTa Soris sxvaobisa

s 21 AXAX − = saSualoTa Soris gansxvavebis Sefasebuli standartuli Secdomisa

t0.05 = tkritikulis ormxrivi mniSvnelobisa mniSvnelobis 0.05 doneze, roca

Tavisuflebis xarisxi tolia N-2.

am formuliT mocemul intervals aqvs 0.95 albaToba.

analogiurad 99%-iani ndobis intervali damoukidebeli populaciebis

saSualoebs Soris sxvaobisaTvis mocemulia Semdegnairad

)()( 211.021 AXAXAA stXX−

−− -dan )()( 211.021 AXAXAA stXX−

−−

40 sadac t0.01 aris tkritikulis ormxrivi mniSvneloba mniSvnelovnobis 0.01 doneze,

roca Tavisuflebis xarisxi tolia N-2-is.

am formulebis gamoyenebis sailustraciod ganvixiloT 95%-iani ndobis

intervalis gamoTvla magaliTisaTvis #9-1, romelSic erTmaneTs udarebdnen

drois Sefasebebs placebo sakontrolo jgufsa da diazepamis miRebis jgufebs

Soris. aq 1AX =19.8wm da 2AX =13.6wm da s 21 AXAX − =2.597wm. am magaliTisaTvis gvaqvs

22 Tavisuflebis xarisxi (ix. magaliTi #9-1), amrigad, t kriteriumis ormxrivi

mniSvneloba 0.05 doneze aris 2.074. 95 procentiani ndobis intervali µ A1-µA2..

aris

(19.8-13.6)-(2.074)(2.597) dan (19.8-13.6)+(2.074)(2.579) wm

an, Tu CavatarebT maTematikur gamoTvlebs,

6.2_5.39 –dan 6.2+5.39 wm-mde

rac tolia

0.8 wm-dan 11.6 wm-mde

albaToba imisa, rom µ1-µA2 mniSvneloba moTavsebulia 0.8-dan 11.6-mde

intervalSi 0.95-a. Cven SegviZlia 95 procentiT viyoT darwmunebuli, rom

placebo sakontrolo da diazepamis jgufebis mier drois Sefasebebis

populaciis

saSualoebs Soris sxvaoba aris 0.8-dan 11.6-mde.

ras amowmebs t kriteriumi sinamdvileSi?

t kriteriumi gvaZlevs informacias, romelic saWiroa im gadawyvetilebis

misaRebad, Tu ramdenadaa ganmtkicebuli kvlevis hipoTeza eqsperimentuli

monacemebiT. Zalian mniSvnelovania imis gacnobiereba, rom kvlevis hipoTeza ar

mowmdeba pirdapiri gziT t kriteriumiT. t kriteriumi aris ubralod procedura

statistikuri hipoTezis H0:µ1=µASesamowmeblad. Tu ar gaviTvaliswinebT

damoukidebel da damokidebul cvladebs, romlebic monawileobs kvlevis

hipoTezaSi, nulovani da alternatiuli hipoTezebi ormxrivi t

kriteriumisaTvis aris yovelTvis H0:µ1=µ da H1:µ1≠µ. meore mxriv, kvlevis

hipoTeza eqsperimentSi aris debuleba, romelic damoukidebel da

damokidebul cvladebs Soris kavSirs winaswarmetyvelebs. rogorc zemoT

moyvanil magaliTSi, debuleba: `mosalodnelia, rom im cdis pirebis mier

Sefasebuli dro, romlebsac miewodaT diazepami ufro naklebi iqneba, vidre

placebo sakontrolo jgufis mier Sefasebuli dro", aris kvlevis hipoTeza.

41 statistikurad mniSvnelovani t realuri aramimarTulebiTi

alternatiuli hipoTezisaTvis (e.i. H1:µ1≠µ) ar warmoadgens aucileblad kvlevis

hipoTezis mxardaWeras. magaliTad, eqsperimentuli hipoTezis mxardaWera

iqneboda im SemTxvevaSi, Tu 1AX , placebo sakontrolo jgufis mier Sefasebuli

drois saSualo iqneboda 2AX -ze, anu eqsperimentuli jgufis mier Sefasebul

saSualoze mniSvnelovnad meti. magram 1AX mniSvnelovnad naklebia 2AX -ze, ar

eTanxmeba kvlvevis am hipoTezas. amitom trealuris statistikuri mniSvnelovnobis

miRebis Semdeg, Cven yovelTvis unda SevamowmoT saSualoTa Soris gansxvavebis

mimarTuleba, raTa davadginoT, eTanxmeba Tu ara is kvlevis hipoTezas.

statistikuri mniSvnelovnobisa da mecnieruli mniSvnelovnebis

mimoxilva

me-8 TavSi Cven mivuTiTeT, rom statistikuri hipoTezis testirebaSi sityva

mniSvnelovani niSnavs mxolod imas, rom nulovani hipoTeza uaryofilia. magram

saSualoTa Soris statistikurad mniSvnelovani gansxvaveba eqsperimentSi ar

aris aucileblad mecnierulad mniSvnelovani. eqsperimentis mecnieruli

mniSvnelovnoba ganisazRvreba manam, sanam monacemebi statistikurad

damuSavdeba. kargad gaazrebul da kargad dagegmil eqsperimentSi, mecnierulad

mniSvnelovani SeiZleba iyos statistikurad mniSvnelovani an statistikurad

aramniSvnelovani gansxvaveba. marTlac, Tu damoukidebelsa da damokidebul

cvlads Soris konkretuli kavSiri aris nawinaswarmetyvelebi, maSin am

kavSiris empiriulad povnis warumateblobas SeiZleba hqondes mecnieruli

mniSvneloba. cudad dagegmil eqsperimentSi ki, romelSic damoukidebeli

cvladis Sereva xdeba damatebiT cvladTan, ori SerCevis saSualoebs Soris

statistikurad mniSvnelovani gansxvavebac ki ar SeiZleba naTlad iyos

axsnili.

diskusia statistikuri hipoTezebis Semowmebis Sesaxeb

rogorc pirvel TavSi aRvniSneT, monacemTa statistikuri analizi ar iqneboda

saWiro, rom ar yofiliyo populaciis wevrebs Soris cvalebadoba. magram aseTi

cvalebadoba aris cxovrebiseuli faqti da iwvevs monacemTa statistikuri

analizis saWiroebas. statistikuri hipoTezebis Semowmeba imisaTvis ganviTarda,

42 raTa daexmaros mecnierebs gadawyvetilebebis miRebaSi cvalebadobis im

situaciaSi, romelic yovelTvis aRiniSneba monacemebSi. Tumca statistikuri

hipoTezebis Semowmeba SeiZleba TqvenTvis konceftualurad rTuli Candes, misi

gamoyeneba monacemebisaTvis male xdeba standartuli da advilad

Sesasrulebeli. SesaZloa mogveCvenos, rom es procesi ar Seicavs araviTar

winaaRmdegobas, magram es ase ar aris. piriqiT, gacxovelebuli diskusia

mimdinareobs misi gamoyenebis damcvelebsa da mowinaaRmdegeebs Soris. (Abelson,

1997; Cohen, 1994; Cortina and Dunlap, 1997; Frick, 1996; Hagen, 1997; Harris, 1997; Hunter, 1997;

Wainer, 1999). statistikur hipoTezebs sxvadasxva kuTxidan akritikeben. zogierTi

zedmetad rTulia imisaTvis, rom Sesaval teqstSi ganvixiloT. arsebiTad,

kritikosTa umravlesoba yuradRebas amaxvilebs Semdeg punqtebze:

• statistikuri hipoTezebis Semowmebisas vamowmebT nulovan hipoTezas,

romlis Tanaxmadac H0: µ A1=µA2 . Tumca naklebad mosalodnelia, rom

nulovani hipoTeza odesme iyos WeSmariti;

• statistikuri hipoTezebis Semowmeba Tavs gvaxvevs diqotomiur

gadawyvetilebas: uarvyoT nulovani hipoTeza, an ver uarvyoT is, rac

efuZneba nebismierad SerCeul α-s mniSvnelobas, Cveulebriv 0.05-s.

• II gvaris Secdomis daSvebis albaToba ar SeiZleba zustad

ganvsazRvroT arc erTi kvlevisaTvis. amrigad, roca

eqsperimentatori H0-s ver uaryofs, man SeiZleba dauSvas II gvaris

Secdoma. Tumca bihevioristi mecnierebi fiqroben, rom II gvaris

Secdomis daSvebis albaToba zogadad ufro naklebad mniSvnelovania,

vidre pirveli gvaris Secdomis daSvebis albaToba, es ar exeba yvela

kvleviT problemas.

• statistikuri hipoTezebis Semowmeba ubiZgebs mkvlevrebs guldasmiT

aRar moekidon maT mier miRebul monacemebs da gamoiyenon

kompiuterizebuli statistikuri kriteriumebi, romelic maT

srulyofilad arc esmiT.

ar arsebobs martivi an arawinaaRmdegobrivi pasuxi am kritikaze.

abelsonma (Abelson, 1997) miuTiTa, rom monacemTa analizi ar aris zusti

procesi, radgan is moicavs adamianis mier gakeTebul Sefasebas.

gadawyvetilebis miRebis yvela procesi, bihevioristul mecnierebebSi

kvlevis Sedegebs exeba, mimdinareobs gazomvisa da SerCevis Secdomis

43 cvalebadobis fonze da imis Sefasebisas gansxvavdeba, Tu ara ori

saSualo erTmaneTisagan sakmarisad, raTa es iyos Carevis efeqtis

maniSnebeli, adamianebi ar arian Secdomis cvalebadobis kargi

Semfaseblebi statistikuri kriteriumis gamoyenebis gareSe. Semdegi

Tavis ori amocana, #10.2 da #10.3 amis naTeli dadasturebaa. cxrilSi

#10.13 mocemuli saSualoTa Soris gansxvavebebi amocanaSi #10.2 erTi

SexedviT didia, ramac SeiZleba migviyvanos mcdar gadawyvetilebamde,

rom damoukidebel cvlads aqvs gavlena. magram am eqsperimentSi

Secdomis cvalebadobis didi odenoba gviCvenebs, rom saSualoTa Soris

gansxvavebis yvelaze Sesatyvis interpretacias warmoadgens misi

ganpirobebuloba SerCevis SecdomiT da ara damoukidebeli cvladis

gavleniT. amis sapirispirod, amocanis #10.3 cxrilSi #10.14

warmodgenil saSualoTa mxolod garegnuli dakvirvebiT Cans, rom isini

ar gansxvavdebian didad erTmaneTisagan, ramac SesaZloa dagvarwmunos

damoukidebeli cvladis araefeqturobaSi. magram aseTi martivi

gadaxedviT SesaZloa gamogvrCes monacemebSi Secdomis cvalebadobis

mcire odenoba. statistikuri kriteriumi ki iTvaliswinebs am

cvalebadobas da gviCvenebs, rom ori jgufis saSualoTa Soris

gasxvaveba saukeTesod SeiZleba aixsnas damoukidebeli cvladis

gavleniT. es ar aris SemTxveviT gansxvaveba. yuradRebiT ganixileT es

amocana, roca mas Semdeg TavSi SexvdebiT.

statistikur hipoTezaTa Semowmebis kritikosebis sapasuxod

amerikis fsiqologTa asociaciam daavala statistikur daskvnaTa

Semswavlel specialur komisias, gaekeTebinaT rekomendaciebi kvlevaSi

statistikuri analizis gamoyenebis Sesaxeb. am komisiam gaakeTa

ramdenime rekomendacia, romelic SesaZlebelia ganvixiloT

statistikuri analizis SesavalSi (Wilkonson&the Task Force on Statistical

Inference, 1999):

• mkvlevrebma dawvrilebiT unda aRweron TavianTi monacemebi

SerCevis moculobebis, centraluri tendenciisa da

cvalebadobis sazomebis CaTvliT;

• ndobis intervalebs unda axldes efeqtis siZlieris sazomi,

rogoricaa η2;

44 • mkvlevrebi kargad unda acnobierebdnen monacemTa analizis

romel programas gamoiyeneben. Sedegebis gadamowmebis mizniT

rekomendebulia monacemTa analizis ori gansxvavebuli

programis gamoyeneba.

am rekomendaciebis gaTvaliswinebiT davubrundeT am TavSi mocemul diskusias

simZlavris Sesaxeb. im diskusiaSi warmodgenili iyo rekomendaciebi kvlevis

simZlavris gasazrdelad da amrigad, II gvaris Secdomis albaTobis

Sesamcireblad.

ueWvelia, rom statistikuri hipoTezebis Semowmebisadmi kiTxvebi da

kritika momavalSic iqneba. savaraudoa, agreTve, rom statistikur hipoTezaTa

Semowmebis gamoyeneba, SesaZloa Secvlili formiT, kvlavac gagrZeldes

bihevioristuli mecnierebebis kvlevebSi, radgan misi umkacresi kritikosebic ki

miuTiTeben, rom amJamad statistikuri hipoTezebis Semowmebis Canacvleba raime

sxva xerxiT SeuZlebelia (Cohen, 1994).

t kriteriumis Sedegebis prezentacia amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvanelos (1994)

stilisa da statistikur daskvnaTa Semswavleli specialuri komisiis

(Wilkonson&the Task Force on Statistical Inference, 1999) rekomendaciebis mixedviT

magaliTis #9-1 Sedegebi gamosacem samecniero SromaSi Semdegi saxiT unda

warmovadginoT:

0.05 mniSvnelovnobis alfa donisa da ormxrivi kriteriumisaTvis placebo

sakontrolo jgufis mier Sefasebuli drois saSualo (M=19.8wm, SD=7.6wm)

mniSvnelovnad aRemateboda eqsperimentuli jgufis mier Sefasebuli drois

saSualos (M=13.6 wm, SD =4.8 wm), t(22)= +2.39, p< 0.05, η2=0.21. ori populaciis

saSualoTa Soris gansxvavebisaTvis 95-procentiani ndobis intervali 0.8-dan 11.6

wm-mde Sualedis tolia.


α=0.05 gviCvenebs am kriteriumisaTvis arCeul mniSvnelovnobis dones.

t(22) gansazRvravs kriteriumis statistikas, rogorc t-s. es t aris

trealuri, Tumca indeqsi `realuri” SeiZleba ar axldes. radgan

am magaliTSi aRwerili iyo ori jgufi, Cven viciT, rom t

ganisazRvreba, rogorc tdamoukidebeli. Tavisuflebis xarisxi

trealurisaTvis naCvenebia frCxilebSi. amrigad, Sedegebis

45 analizisaTvis gamoyenebuli iyo 24 cdis piris monacemi

(radgan df=N-2)

=+2.39 trealuris mniSvneloba (ara tkritikulisa, romelsac #a-2 cxrilSi

vpoulobT).


a) trealuris albaToba H0-is WeSmaritebis pirobebSi aris

naklebi an toli 0.05-is. es mniSvneloba aris trealuris

albaToba, an ufro Sors mdebare trealuris mniSvnelobis

albaToba, roca H0 aris WeSmariti. ar aris aucilebeli igive

procedurebis warmoeba, rac α-s arCevis dros;

b) H0: µ A1=µA2 uaryofilia;

H0: µ A1≠µA2 miRebulia;

g) gansxvaveba SerCevis saSualoebs Soris statistikurad

mniSvnelovania;

d) saSualoTa Soris realurad miRebuli gansxvavebis mizezi

aris ara SerCevis Secdoma, aramed sxva ram. radgan es

eqsperimenti Catarda kontrolirebad pirobebSi, Cven vaskvniT,

rom gansxvaveba ganpirobebuli iyo cdis pirebis mier

medikamentis miRebis pirobiT;

e) Tu SedegebSi warmodgenili iqneba p>0.05, maSin metobis

niSani aRniSnavs imas, rom H0 ar aris uaryofili da ori

SerCevis saSualo ar gansxvavdeba mniSvnelovnad

erTamaneTisagan mniSvnelovnobis 0.05 doneze. Aam SemTxvevaSi

Cven davaskvniT, rom medikamentis miRebis piroba ar axdens

gavlenas drois Sefasebaze;

η2=0.21 dispersiis wili, romelic ganpirobebulia damoukidebeli

cvladiT;

95%-iani ndobis es intervali aris ori populaciis saSualoTa Soris

ndobis inter- sxvaobisaTvis. amis nacvlad, SesaZloa warmodgenili iyos

95

vali=0.8-dan procentiani ndobis intervali TiToeuli saSualosaTvis.

11.6 wm-mde ufro iSviaTad ki, avtorebi ndobis intervalis nacvlad

warmoadgenen saSualos standartul Secdomas.

46


1. gansazRvreT: efeqtis zoma, simZlavre, kavSiris siZliere, efeqtis siZliere.

2. ori SerCevisaTvis miRebuli iyo qulaTa ori simravle. pirveli

simravlisaTvis sA1 = 6.3 da sA2 =7.6. meore simravlisaTvis sA1 = 11.7 da sA2 = 10.3.

a) qulaTa romel simravles aqvs Secdomis meti cvalebadoba?

b) Tuki H1 WeSmaritia, qulaTa romelma simravlem SeiZleba gamoiwvios

statistikur SemowmebaSi II gvaris Secdoma?

g) qulaTa romeli simravle gamoiwvevs statistikuri kriteriumis ufro

met simZlavres?

3. ori SerCevidan ra moculobis SerCeva gamoiwvevs t kriteriumSi s 21 AXAX − -is

ufro mcire sidides: nA1= nA2=12 Tu nA1= nA2=20? ratom? ra moculobis SerCevas

mivyavarT ufro meti simZlavris statistikuri kriteriumisaken?

4. ipoveT η2 trealuris Semdegi mniSvnelobebisaTvis. TiToeuli t-s Tavisuflebis

xarisxi frCxilebSia mocemuli. TiToeuli trealuri statistikurad

mniSvnelovania 0.05 doneze ormxrivi kriteriumisaTvis.

a) t(8)=2.413

b) t(14)=2.2.84

g) t(20)=2.174

d) t(29)=2.939

e) t(60)=2.306

5. erTmaneTisagan ganasxvaveT kvlevisa da statistikuri hipoTeza.

6. niSnavs TuU ara statistikurad mniSvnelovani ganxvaveba imas, rom

aucileblad mecnieruladac mniSvnelovani Sedegebi iqneba miRebuli? axseniT

Tqveni pasuxi.

7. tdamoukidebelis gamoyenebis Semdegi angariSebi aRebulia gamoqveynebuli

naSromebidan. UupasuxeT Semdeg kiTxvebs TiToeuli angariSisaTvis.

a. damoukidebeli cvladis ramdeni done iyo gamoyenebuli am eqsperimentSi?

b. ramdeni cdis piri monawileobda am eqsperimentSi?

g. risi tolia SerCevis saSualo TiToeuli eqsperimentuli pirobisaTvis?

d. risi tolia populaciis standartuli gadaxris Sefasebis mniSvneloba

Carevis TiToeuli pirobisaTvis?

e. risi tolia trealuris mniSvneloba?

47 v. risi tolia tkritikulis mniSvnelovnebis 0.05 doneze ormxrivi uaryofis

arisaTvis?

z. uaryofT Tu ver uaryofT H0-s ?

T. miiRebT Tu ara H1-s?

i. saSualoTa Soris arsebuli gansxvaveba statistikurad mniSvnelovania, Tu

aramniSvnelovania?

k. ras miawerT saSualoTa Soris realurad miRebul gansxvavebas?

AangariSi #1. 0.05 alfas da ormxrivi kriteriumisaTvis, t kriteriumma gviCvena,

rom cdis pirebi sworad amCneven davalebaTa ufro did procents, roca usmenen

nacnob leqss (M=79.7, SD=8.57), vidre maSin, roca usmenen ucnob leqss (M=73.,

SD=5.85), t(60)=3.6, p<0.01.

AangariSi #2. 0.05 alfasa da ormxrivi kriteriumisaTvis, t kriteriumma gviCvena,

rom 5-wuTiani intervalis Semdeg aRdgenili sityvebis raodenoba (M=22.78,

SD=5.23) ar gansxvavdeba 0.5-wuTiani intervalis Semdeg aRdgenili sityvebis

raodenobisagan (M=24.33, SD=5.14), t(28)= -0.82, p>0.05

cdis pirTa Sida sqema da t kriteriumi damokidebuli

qulebisaTvis

Cveni interesi am sakiTxisadmi maSin gaCnda, roca vixilavdiT cdis pirTa

Soris sqemas, romelSic damoukidebeli cvladis TiToeuli done miewodeboda

gansxvavebuli jgufis cdis pirebs. meore tipis kvlevis sqema aris cdis pirTa

Sida sqema, romelSic cdis pirTa erTi jgufi aris warmodgenili da xdeba

misi gazomva damoukidebeli cvladis TiToeuli donis mixedviT. amrigad, cdis

pirTa Sida sqemaSi TiToeul cdis pirs miewodeba eqsperimentis orive piroba.

ganvixiloT erTfaqtoriani cdis pirTa Sida sqemis Semdegi magaliTi.

adamianebi, romlebic mowevas Tavs anebeben, xSirad aRniSnaven, rom imateben

wonaSi. am hipoTezis Sesawmeblad, bekonam da vazqezma (Becona &Vazquez, 1998)

erTmaneTs mwevelTa wonebi Seadares im momentisaTvis, roca maT daiwyes

sigaretisaTvis Tavis danebebis programaSi monawileoba da 36 Tvis Semdeg maT

aRmoaCines, rom programis im monawileebma, romlebmac SesZles Tavi

cdis pirTa Sida sqema (Within-subject design): kvlevis sqema, romelSic cdis pirTa erTi jgufia da izomeba damoukidebeli cvladis TiTeuli donis mixedviT. cdis pirTa Sida sqemaSi TiToeul cdis pirs miewodeba eqsperimentis orive piroba.

48 daenebebinaT sigaretisaTvis, 2 kg-iT moimates wonaSi da es mateba

statistikurad mniSvnelovani aRmoCnda. davuSvaT, Tqven gsurT gaimeoroT es

kvleva 8 mweveli qalisagan Semdgar jgufze, romelTa asaki meryeobs 20-dan 24

wlamde da romlebic 1 wlis ganmavlobaSi aRar eweodnen sigarets. TiToeuli

pirovneba awones am programis dawyebisas (A1) da mowevisaTvis Tavis danebebidan

12 Tvis Semdeg (A2); amrigad, gamoyenebuli iyo cdis pirTa Sida sqema. davuSvaT,

Tqven daadgineT, individTa wonebi programis dawyebis momentisaTvis da 1

wlis Semdeg, rac naCvenebia cxrilSi #9.2. statistikurad mniSvnelovnad

gansxvavdeba Tu ara am or periodSi gazomili wonebi erTmaneTisagan? am

kiTxvaze pasuxis gasacemad da saSualoTa Sesadareblad gamoviyeneT t

kriteriumi damokidebuli qulebisaTvis.

cxrili #9.2

20-24 wlis qalebis wonebi (funtebSi) mowevisaTvis Tavis danebebis programaSi

dasawyisisaTvis (A1) da mowevisaTvis Tavis danebebidan 1 wlis Semdeg (A2):

cdis piri awonvis momenti (A)

programaSi Sesvlisas (A1) mowevis danebebidan 1 wlis Semdeg

(A2)

1 122.4 127.9

2 117.1 119.4

3 127.8 127.1

4 123.2 126.5

5 112.5 114.6

6 118.7 125.3

7 121.4 119.8

8 131.7 134.2

X A 121.85 124.35

As 6.03 6.13

t kriteriumi damokidebuli qulebisaTvis

cdis pirTa Sida sqemaSi xdeba erTi da igive cdis pirebis maxasiaTeblebis

gazomva orive eqsperimentul pirobaSi; amrigad TiToeul pirobaSi miRebuli

qulebi erTmaneTTan dakavSirebuli iqneba. faqtori, romelic moqmedebs cdis

49 piris qulaze erT pirobaSi, imoqmedebs am cdis pirTa qulaze sxva pirobaSi.

tdamoukidebeli ar aris Sesatyvisi sazomi am sqemisaTvis; aq viyenebT t kriteriums

damokidebuli qulebisaTvis, romelic ganisazRvreba, rogorc

D

AA

sXXt 21 −

=lidamokidebu

sadac sD aris saSualos standartuli Secdoma TiToeul 1AX da 2AX qulaTa

Soris gansxvavebisaTvis. sD-s gamoTvliTi formulaa

)1(/)( 22

−Σ−Σ

=wyvilebiNN

NDDsi

Dwyvileb

wyvilebi

sadac D asaxavs gansxvavebas cdis piris or qulas Soris da Nwyvilebi aris

qulaTa wyvilebis raodenoba, an zustad cdis pirTa raodenoba.

CavsvaT formulaD sD-saTvis, tdamokidebulis gamoTvliT formulaSi, miviRebT

)1(/)( 22

21

−

Σ−Σ

−=

ii

i

AA

NNNDD

XXt

wyvilebwyvileb

wyvileb

lidamokidebu

df aris toli Nwyvilebi_1-is, romelic asaxavs qulaTa wyvilebis im raodenobas,

romelic Tavusufalia misi cvalebadobisas.

am formulis gamoyeneba cxrilSi #9.2 mocemuli qulebisaTvis naCvenebia

cxrilSi #9.3. gamoTvlebisaTvis saWiro safuxurebi Semdegia:

1. vipovoT 1AX da 2AX . magaliTad, es mniSvnelobebi tolia 121.85 da 124.35

mili/sT, Sesabamisad.

2. ipoveT D-s mniSvneloba, amisaTvis TiToeuli cdis piris mier A1

pirobaSi miRebul qulebs gamoakeliT cdis pirTa mier A2 pirobaSi miRebuli

qulebi. gaiTvaliswineT sxvaobis algebruli niSnebi. es nabiji naCvenebia im

svetSi, romelic cxrilSi dasaTaurebulia D asoTi.

3. aiyvaneT D-s TiToeuli mniSvneloba kvadratSi, rogorc naCvenebia D2

svetSi.

4. ipoveT ΣD (D svetis jami; magaliTad, _20) da ΣD2 (D2 svetis jami;

magaliTad, 103.7)

5. gansazRvreT Nwyvilebi magaliTad, Nwyvilebi=8

cxrili #9.3

50 cxrilis #9.2 qulebis mixedviT tdamokidebulis gamoTvla

awonvis momenti (A)

cdis

piri

programaSi

Sesvlisas (A1)

mowevis danebebidan

1 wlis Semdeg (A2)

D D2

1 122.4 127.9 -5.5 30.25

2 117.1 119.4 -2.3 5.29

3 127.8 127.1 +0.7 0.49

4 123.2 126.5 -3.3 10.89

5 112.5 114.6 -2.1 4.41

6 118.7 125.3 -6.6 43.56

7 121.4 119.8 +1.6 2.56

8 131.7 134.2 -2.5 6.25

AX 121.85 124.35 ΣD= -20 ΣD2=103.7

sA 6.03 6.13

)1(/)( 22

21

−

Σ−Σ

−=

ii

i

AA

NNNDD

XXt

wyvilebwyvileb

wyvileb

lidamokidebu =

554.2979.0

5.2)7)(8(

8/)20(7.10335.12485.1212

−=

−=

−−

−

roca df aris 7

6. CasviT ricxobrivi mniSvnelobebi tdamokidebulis gamoTvliT formulaSi.

7. ipoveT df, sadac df= Nwyvilebi -1

trealuri cxrilSi #9.2 mocemuli qulebisaTvis aris -2.2554. trealuris

gamosaTvleli danarCeni procedura tdamokidebulis identuria.. unda avirCioT α

mniSvneloba, vipovoT tkritikulis mniSvneloba cxrilidan #a-2, da uaryofis

areebi, romlebic mdebareoben t SerCeviT ganawilebaze. Tu trealuris

mniSvneloba xvdeba uaryofis areSi, maSin

H0: µ A1=µA2 uaryofilia

da

H1: µ A1≠µA2 miRebulia

51 roca df=7 da mniSvnelovnobis done ki 0.05, tkritikuli ormxrivi kriteriumisaTvis

aris 2.365. uaryofis are aris Seqmnili trealuris mniSvnelobidan, romelic

naklebia an toli –2.365-is, an toli an meti, +2.365-is. trealuri –2.554 naklebia –

2.365-ze; amitom is xvdeba uaryofis areSi; H0 unda uarvyoT da H1 miviRoT.

SerCevis saSualoTa Soris sxvaoba statistikurad mniSvnelovania

mniSvnelovnobis 0.05 doneze. programaSi monawile pirebi ufro mets iwonidnen

mowevis danebebidan erTi wlis Semdeg, vidre programis dawyebamde.

am tdamokidebulisaTvis eta kvadrati gamoiTvleba Semdegnairad:

dftt

+=

realuri

realuri2η

CavsvaT trealuris da df-is ricxobrivi mniSvnelobebi:

48.07)554.2(

)554.2(2

22

=+−

−=η

es eta kvadrati gviCvenebs, rom damokidebuli cvladis cvalebadobis 48

procenti aris ganpirobebuli damoukidebeli cvladiT _ siCqaris sazomi

iaraRis pirobiT am eqsperimentSi.

erT-erTi mniSvnelovani sakiTxi am sqemis ganxilvisas isaa, rom Tumca wonaSi

mateba 1 wlis Semdeg statistikurad mniSvnelovani aRmoCnda, Cven mainc ar

SegviZlia vamtkicoT, rom es iyo gamowveuli sigaretis mowevis danebebis

programiT. adamianebi xSirad imateben wonaSi wlebis matebasTan erTad da

SesaZlebelia cdis pirebs isedac moematebinaT wonaSi, Tuki isini ar

miatovebdnen mowevas. imisaTvis, raTa davrwmundeT, rom wonis mateba marTlac

ganpirobebuli iyo sigaretis danebebiT, 1 wlis Semdeg unda SevamowmoT msgavsi

cdis pirebisagan Semdgari meore jgufis wonebi, romlebmac ar daanebes Tavis

mowevas. aseTi jgufis CarTva am magaliTSi gaarTulebda ganxilvas, magram

bekonsa da vazqezs hyavdaT aseTi jgufi da aRmoCnda, rom maTi wona samwlian

periodSi mniSvnelovnad ar gazrdila.

ndobis intervalebis ageba ori damokidebuli populaciis

saSualoTa Soris gansxvavebisaTvis

95-procentiani ndobis intervali ori damokidebuli populaciis saSualoTa

Soris gansxvavebisaTvis mocemulia Semdegi intervaliT:

52 )()( 05.021 DAA stXX −− dan )()( 05.021 DAA stXX +− -mde,

sadac 21 AA XX − = ori realuri SerCevis saSualoTa Soris gansxvavebas

sD = TiToeuli 1AX da 2AX qulaTa Soris sxvaobis saSualos

standartul Secdomas

t0.05 = tkritikulis ormxriv mniSvnelobas mniSvnelovnobis 0.05

doneze

N-1 Tavisuflebis xarisxisiT

albaToba imisa, rom µ A1-µA2 mniSvneloba am intervalSiaA moTavsebuli, aris

0.95.

am magaliTisaTvis, 1AX =121.85 da 2AX =124.35, Nwyvilebi=8, sD=0.979 (cxrilidan

#9.3) da t(0.05) 7-is toli Tavisuflebis xarisxisaTvis tolia 2.365-is. SerCevis

saSualoTa Soris uaryofiTi sxvaobis miRebis Tavidan asacileblad, Cven 1AX -s

vaklebT 2AX -s. Tu CavsvamT ricxobriv mniSvnelobebs, 95-procentiani ndobis

intervali µ A1 da µA2 Soris sxvaobisaTvis iqneba:

(124.35-121.85)_(2.365)(0.979)-dan (124.35-121.85)+(2.365)(0.979)-mde

romelic tolia 0.18-dan 4.82 funtamde Sualedis. 95-procentiani ndobis

intervali populaciis saSualoebs Soris gansxvavebisaTvis programis

dawyebidan gazomil wonasa da mowevis danebebidan 1 wlis Semdeg gazomil

wonas Soris aris moTavsebuli 0.18-sa da 4.82 funts Soris.

\


1. axseniT gansxvaveba cdis pirTa Soris kvlevis sqemasa da cdis pirTa Sida

kvlevis sqemas Soris.

2. sul ufro meti adamiani Tvlis sakuTar avtomanqanas Tavisi ofisis nawilad

da transportis moZraobis usafrTxoebis mravali eqsperti miiCnevs, rom

yuradRebis gamfantveli stimulebis zrda iwvevs avariebis ricxvis zrdas.

davuSvaT, gsurT empiriulad gadaamowmoT es problema da 11 adamians sTxoveT

etarebinaT manqana or gansxvavebul xelovnurad Seqmnil situaciaSi:

sakontrolo pirobaSi, roca ar iqneboda yuradRebis gamfantveli stimulebi

53 da eqsperimentul pirobaSi, roca mZRoli isaubrebda mobiluri telefoniT.

izomeboda cdis pirebis reaqciis dro xelovnurad Seqmnil avariul

situaciaze, romelic moiTxovda damuxruWebas. vTqvaT, miiReT Semdegi reaqciis

droebi (miliwamebSi):

gansazRvreT saSualo da standartuli gadaxra TiToeuli pirobisaTvis;

Semdeg gamoiTvaleT tdamokidebeli am qulebisaTvis da upasuxeT Semdeg kiTxvebze:

a. CamoayalibeTY statistikuri hipoTeza tdamokidebelisaTvis am monacemebis

safuZvelze.

b. risi tolia trealuris mniSvneloba?

g. risi tolia df t kriteriumisaTvis?

d. risi tolia trealuris mniSvneloba?

e. risi tolia uaryofis areebi trealurisaTvis?

v. xvdeba Tu ara trealuri uaryofis areSi?

z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb? uaryofT Tu

ver uaryofT H0 hipoTezas? miiRebT Tu ara H1 hipoTezas?

T. gamoitanT daskvnas imis Sesaxeb, rom SerCevis saSualoebi erTi da imave Tu

gansxvavebuli populaciebidanaa?

i. saSualoebs Soris gansxvaveba statistikurad mniSvnelovania?

k. gansxvavdebian pirobebi mniSvnelovnad reaqciis drois mixedviT? Tu asea,

rogoria am gansxvavebis mimarTuleba?

l. risi tolia eta kvadrati am kvlevaSi?

cdis piri piroba

sakontrolo eqsperimentuli

1 426 448

2 501 552

3 525 506

4 478 507

5 537 574

6 452 491

7 394 438

8 443 477

9 491 523

10 382 413

11 572 541

54 m. risi tolia 95-procentiani ndobis intervali populaciis saSualoTa Soris

sxvaobisaTvis?

Sejameba

• eqsperimenti moiTxovs:

o mecnieruli kiTxvis dasmas da kvlevis hipoTezis formulirebas;

o cdis pirTa SerCevas;

o ekvivalenturi jgufebis SerCevas SemTxveviTi ganawilebiT;

o damatebiTi cvladebis kontrolirebas;

o damoukidebeli cvladis manipulirebas;

o damokidebuli cvladis gazomvas.

• SeiZleba tdamoukidebuli gamoyenebuli iyos imis gansazRvrisaTvis, ori

SerCevis saSualoebi gansxvavdeba Tu ara erTmaneTisagan

eqsperimentSi mniSvnelovnad?

• tdamoukidebelis formula aris:

21

21

AA XX

AA

sXXt

−

−=elidamoukideb

sadac, s 21 AXAX − aris saSualoTa Soris sxvaobis standartuli Secdoma;

• statistikuri hipoTezis Semowmeba tdamoukidebelis gamoyenebiT yvela

statistikuri SemowmebisaTvis saerTo safexurebs imeorebs;

• statistikuri kriteriumis simZlavre aris H0-is uaryofis albaToba,

roca H0 ar aris samarTliani;

• t kriteriumis simZlavreze moqmed faqtorebs miekuTvneba

damoukidebeli cvladis gavlenis sidide, cvalebadobis sidide

damokidebuli cvladis gazomvisas da SerCevis moculoba;

• eta kvadrati aris kavSiris siZlieris sazomi SerCevaSi, romelic

trealuridan gaizomeba da gamoiTvleba;

• eta kvadrati gviCvenebs damokidebuli cvladis cvalebadobis im

wils sruli cvalebadobidan, romelic damoukidebeli cvladiT

aixsneba;

• ori damoukidebeli populaciis saSualoTa Soris sxvaobis 95-

procentiani ndobis intervali mocemulia intervaliT:

55 )()( 2105.021 AXAXAA stXX −−− dan )()( 2105.021 AXAXAA stXX −+− -mde

• cdis pirTa Sida sqemaSi cdis pirTa erT jgufs miewodeba TiToeuli

damoukidebuli cvladis yvela done;

• tdamokidebuli SeiZleba gamoyenebuli iyos, roca gvaqvs damoukidebuli

cvladis ori done erTfaqtorian cdis pirTa Sida sqemaSi;

• tdamokidebulis formula aris:

D

AA

sXXt 21 −

=lidamokidebu

sadac sD aris TiToeul XA1 da XA2 qulas Soris sxvaobis saSualos

standartuli Secdoma;

• ori damokidebuli populaciis saSualoTa Soris sxvaobis 99-

procentiani ndobis intervali mocemulia intervaliT:

)()( 2101.021 AXAXAA stXX −−− -dan )()( 2101.021 AXAXAA stXX −+− -mde.


cdis pirTa Soris sqema

SemTxveviTi gansxvaveba

ndobis intervali ori populaciis saSualoTa Soris gansxvavebisaTvis

Sereuli eqsperimenti

xelmisawvdomi (SesaZleblobis) SerCeva

damokidebuli cvladi

mimarTulebis mqone kriteriumi

ekvivalenturi jgufebi

saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba, (s 21 AXAX − )

eta kvadrati (η2)

eqsperimenti

damatebiTi cvladi

faqtori

damoukidebeli cvladi

damoukidebeli cvladis done

95-procentiani ndobis intervali damoukidebeli populaciebis saSualoebs

Soris sxvaobisaTvis

56 99-procentiani ndobis intervali ori damoukidebeli populaciebis saSualoebs

Soris sxvaobisaTvis

aramimarTulebiTi kriteriumi

cdis pirTa Soris erTfaqtoriani sqema

erTmxrivi kriteriumi

placebo kontroli

simZlavre

SemTxveviTi ganawileba

SemTxveviTi SerCeva

mdgradoba

sD

s2kombinirebuli

saSualoTa Soris gansxvavebis SerCeviTi ganawileba

saSualoTa Soris sxvaobis standartuli Secdoma

efeqtis siZliere

tkritikuli t kriteriumi damokidebuli qulebisaTvis (tdamoukidebeli)

t kriteriumi ori damoukidebuli jgufisaTvis (tdamoukidebuli)

trealuri

ormxrivi kriteriumi

cdis pirTa Sida sqema

kiTxvebi

1. davuSvaT, gainteresebT daadginoT, aqvs Tu ara xmaurs raime gavlena

wnevaze. Tqven SemTxveviTad gaanawileT 22 individi sakontrolo da

eqsperimentul jgufebSi (nA1=nA2). uxmauro jgufis cdis pirebi (A1) isvenebdnen

komfortul savarZlebSi wynar oTaxSi 30 wuTis ganmavlobaSi. 30 wuTis Semdeg

maT uzomavdnen sistoluri sisxlis wnevas (anu sisxlis wnevas gulis kunTis

SekumSvebs Soris). xmauriani jgufis (A2) cdis pirebi agreTve isxdnen

savarZlebSi 30 wuTi. drois am Sualedis ganmavlobaSi isini usmendnen didi

qalaqis transportis xmis Canawers pikis saaTebSi. amis Semdeg maT uzomavdnen

57 sisxlis sistolur wnevas. davuSvaT, Tqven miiReT sisxlis wnevis Semdegi

monacemebi vecxliswylis svetis milimetrebSi.

jgufi

uxmauro xmauriani

106 141

117 136

124 124

129 139

115 121

131 119

121 147

115 128

128 115

136 134

127 140

a. ipoveT TiToeuli jgufisaTvis saSualo da standartuli gadaxra.

Semdeg gamoiyeneT t kriteriumi Semdeg kiTxvebze pasuxis gasacemad:

moqmedebs Tu ara xmauri sisxlis sistoluri wnevaze? gamoiyeneT

ormxrivi kriteriumi da mniSvnelovnobis 0.05 done.

b. risi tolia η2-is mniSvneloba am eqsperimentSi? damokidebuli cvladis

cvalebadobis ra wili SeiZleba iyos axsnili damoukidebeli cvladis

donis codniT?

g. ipoveT 95-procentiani ndobis intervali ori populaciis saSualoTa

Soris sxvaobisaTvis.

d. aRwereT Tqveni Sedegebi am Tavis erT-erTi nawilSi mocemuli magaliTis Sesabamisad, romelic exeboda t kriteriumis Sedegebis prezentacias.

2. davuSvaT, rom Tqven miiReT sisxlis wnevis monacemebi, romlebic

mocemulia pirveli kiTxvis gaswvriv. moqmedebs Tu ara sistolur sisxlis

wnevaze xmauri? gamoiyeneT 0.05 mniSvnelovnobis done da ormxrivi

kriteriumi.

jgufi

uxmauro xmauriani

110 116

114 111

58 106 120

117 119

108 123

116 108

120 122

111 115

115 117

109 101

118 116

3. pedersenma (Pedersen, 1987) ivarauda, rom arsebobs sqesTa Soris

gansxvavebebi kolejis studentebSi pirad cxovrebaSi Caurevlobis

moTxovnilebis mixedviT. hipoTeza Semowmebuli iyo pirad

cxovrebaSi Caurevlobis moTxovnilebis sakvlevi kiTxvaris

saSualebiT (Pedersen, 1979) kolejis mdedrobiTi da mamrobiTi sqesis

studentebze. am kiTxvaris erTi aspeqti zomavs ganmartoebis

midrekilebas. am skalis qulebi SeiZleba icvlebodes 6-dan

(arasodes aqvs ganmartoebis moTxovnileba) 30-mde (Cveulebriv

sWirdebaT ganmartoeba). davuSvaT, rom miRebuli iyo Semdegi qulebi

13 mamrobiTi da 17 mdedrobiTi sqesis cdis pirisaTvis.

sqesi

mamrobiTi(A1) mdedrobiTi(A2)

23 16

16 29

13 11

25 14

17 17

21 19

10 18

19 21

24 9

59 16 15

11 17

22 20

14 14

24

23

25

24

a. ipoveT TiToeuli jgufis saSualo da standartuli gadaxra. gamoiyeneT t

kriteriumi Semdeg kiTxvaze pasuxis gasacemad: gansxvavdebian Tu ara

qalebi da mamakacebi ganmartoebis moTxovnilebiT? gamoiyeneT ormxrivi

kriteriumi da mniSvnelovnobis 0.05 done.

b. aRwereT Tqveni Sedegebi am Tavis erT-erT qveTavSi mocemuli magaliTis

Sesabamisad, romelic exeboda t kriteriumis Sedegebis prezentacias.

4. fsiqologebi swavloben iluziebs, raTa ukeTesad axsnan, Tu rogor

aRviqvamT samyaros. erT-erTi magaliTia morinagas gadaxrilobis

iluzia, romelic naCvenebia suraTze #10.3. suraTze gamosaxuli

figuris iluzoruloba isaa, rom mocemuli kuTxeebis wveroebs erT

vertikalur xazze mdebared ar aRviqvamT, Tumca realurad isini

erT vertikalur xazze mdebareoben. ise Cans, TiTqos Sua kuTxis

wertili mdebareobs sxva vertikalur xazze, xolo zeda da qveda

kuTxeebis wertilebi _ sxvaze. deim da kasperCikma (Day and Kasperczyk,

1984) Caatares kvleva, imis dasadgenad, Tu ramdenad SeiZleba am

iluziis ganzogadeba, amisaTvis isini cdis pirebs sTxovdnen

gaesworebinaT Sua kuTxe iseTnairad, rom samive kuTxis wvero erT

vertikalur xazze aRmoCeniliyo. am gasworebis dros daSvebuli

Secdoma izomeboda milimetrebSi (mm). Tu cdis piri samive kuTxis

wverobs sworad ganalagebda, maSin misi Secdoma iqneboda 0 mm da

iluziac ar iqneboda. sxva SemTxvevaSi, Tu cdis piri Siga kuTxes

ise ganalagebda, rom is gamoCndeboda erT vertikalur xazze

mdebared, magram realurad es ase ar iqneboda, eqsperimentatori

iniSnavda Secdomis sidides. davuSvaT, eqsperimentatori

dainteresda imiT, Tu ra gavlenas moaxdens cdis pirebis aRqmaze

iluziis xasiaTis Sesaxeb informaciis miwodeba. eqsperimentators

60 pirvelad hyavda 12 cdis piri, romlebmac gaaswores kuTxis

wertilebi da gazomes TiToeuli maTganis mier gasworebisas

daSvebuli Secdoma. mas Semdeg, rac cdis pirebma gaaswores

wertilebi, maT acnobdnen iluziis xasiaTs da sTxovdnen Tavidan

gaesworebinaT es wertilebi. davuSvaT, rom Secdomebis am ori

gazomvis (milimetrebSi) Sedegad TiToeuli cdis pirisaTvis miiRes

Semdegi Sedegebi.

cdis pirebi

instruqciis piroba (A)

informaciis miwodebamde (A1) informaciis miwodebis

Semdeg (A2)

1 1.3 1.4

2 2.1 2

3 2.5 2.7

4 1.9 2

5 3 3.4

6 2.5 2.8

7 1.8 1.6

8 2.4 2.5

9 3.1 3.3

10 3.4 3.4

11 2.5 2.3

12 2.9 2.8

a. ipoveT saSualo da standartuli gadaxra TiToeuli jgufisaTvis.

Semdeg gamoiyeneT t kriteriumi, raTa upasuxoT Semdeg kiTxvebs:

axdens Tu ara gavlenas iluziis xasiaTis Sesaxeb informaciis

miwodeba aRqmuli iluziis sidideze? gamoiyeneT ormxrivi


b. aRwereT Tqveni Sedegebi am Tavis erT-erT qveTavSi _ “t kriteriumis

Sedegebis prezentacia” _ aRwerili stilis mixedviT.

61 5. kontrolis lokusi exeba pirovnebis aRqmas, Tu ramdenad akontrolebs is im

SemTxvevebs, romlebic moqmedeben mis cxovrebaze. is pirovneba, visac aqvs

kontrolis eqsternaluri lokusi, gareSe movlenebs miiCnevs gadamwyvetad.

xolo pirovneba internaluri lokusiT aRiqvams movlenebs, rogorc misi

Zlieri kontolis qveS myofs. viem (Wiehe, 1986) ivarauda, rom bavSvebi, romlebic

TavianT mSoblebTan ar izrdebodnen, xasiaTdebodnen ufro meti eqsternaluri

kontroliT, vidre bavSvebi, romlebic izrdebodnen TavianT biologiur

mSoblebTan. am hipoTezis dasamtkiceblad, viem gamoyo mozardebis ori jgufi

adgilobriv samSobiaro saxlSi: 45 orsuli mozardi, vinc ar izrdeboda Tavis

mSoblebTan da dafexmZimda, roca mas zrdidnen mamobili da dedobili da 45

orsuli mozardi, romelic dafexmZimda Tavis mSoblebTan cxovrebis periodSi.

TiToeul mozards misces kontrolis lokusis gasazomi sareitingo skala. vies

angariSSi kontrolis lokusis skalis saSualo dedobilTan da mamobilTan

aRzrdili mozardebis jgufSi iyo 15.76(s=4.65), xolo biologiuri mSoblebis

jgufSi iyo 12.27(s=5.74). am sazomiT maRali qulebi gviCvenebs kontrolis ufro

maRal eqsternalur lokuss.

a. gamoiyeneT t kriteriumi am kiTxvaze pasuxis gasacemad: aris lokusis

kontroli dakavSirebuli mozardebis mSoblebis tipTan? gamoiyeneT

ormxrivi kriteriumi da mniSvnelovnobis 0.05 done.

b. risi tolia η2-is mniSvneloba am eqsperimentSi? damokidebuli

cvladis cvalebadobis ra wili SeiZleba iyos axsnili damoukidebeli

cvladis donis codniT?

g. ipoveT 95-procentiani ndobis intervali ori populaciis

saSualoTa Soris sxvaobisaTvis.

d. aRwereT Tqveni Sedegebi am Tavis erT-erT qveTavSi mocemuli

magaliTis Sesabamisad, romelic exeboda t kriteriumis Sedegebis

prezentacias.

6. hiugi (1985) adarebs wesrigis damrRvevad cnobili kolejis studentebis

akademiuri moswrebis qulebs, romlebic im studentebs, romlebsac ar hqondaT

aseTi SeniSvnebi miRebuli. akademiuri moswrebis qulis saSualo wesrigis

damrRvevi 1794 studentisaTvis iyo 1.68 da s=0.34. 3606 studentisaTvis ki,

romlebic ar arRvevdnen wesrigs, akademiuri moswrebis saSualo iyo 2.09 da

s=0.83.

62 a. gamoiyeneT t kriteriumi, raTa upasuxoT am SekiTxvebs: mniSvnelovnad

gansxvavdeba Tu ara akademiuri moswrebis qulebi? gamoiyeneT ormxrivi


b. risi tolia η2-is mniSvneloba am eqsperimentSi? damokidebuli cvladis

cvalebadobis ra wili SeiZleba iyos axsnili damoukidebeli cvladis

donis codniT?

g. ipoveT 95-procentiani ndobis intervali ori populaciis saSualoTa

Soris sxvaobisaTvis.

d. aRwereT Tqveni Sedegebi am Tavis erT-erT qveTavSi mocemuli

magaliTis Sesabamisad, romelic exeboda t kriteriumis Sedegebis

warmodgenas.

7. mravali studenti gamocdis dros ganicdis SfoTvas, romelic Zalian uSlis

xels maT akademiur moswrebas. kolejis konsultantma wamoayena hipoTeza, rom

damamSvidebeli musikis mosmena gamocdisas Seamcirebs am SfoTvas. am hipoTezis

Sesamowmeblad, Terapevtma airCia 34 studenti damwyeb studentTa didi

jgufidan, vinc xasiaTdeboda Zlieri SfoTviT gamocdis dros. studentebi

SemTxveviTad iyvnen gadanawilebulni sakontrolo an eqsperimentul jgufSi,

sadac TiToeul jgufSi cdis pirTa toli raodenoba iyo warmodgenili.

gamocdis win sakontrolo jgufis cdis pirebi atarebdnen erT saaTs wynar

dasasvenebel oTaxSi, sadac isini kiTxulobdnen Jurnalebs. eqsperimentuli

jgufis cdis pirebi ki erT saaTs gamocdis win usmendnen damamSvidebel

musikas. gamocdis qulis saSualo sakontrolo jgufisaTvis iyo 73.2(sA1=8.4) da

eqsperimentuli jgufisaTvis ki 75.6(sA2=8.7). gamoiyeneT t kriteriumi, raTa

upasuxoT Semdeg kiTxvas: gansxvavdebian Tu ara jgufebi mniSvnelovnad

erTmaneTisagan?

a.gamoiyeneT orxmrivi kriteriumi da mniSvnelovnobis 0.05 done.

b. aRwereT Sedegebi am Tavis erT-erT qveTavSi mocemuli magaliTis


8. maT, vinc cdilobs sxvebis darwmunebas, surT, rom ndoba moipovon. erT-erTi

hipoTezis Tanaxmad mosaubreebi, romlebic uSualod iqcevian, mimarTaven

Tvalebis kontaqts, iRimebian da simpaTias gamoxataven auditoriis mimarT

aRiqmebian rogorc ufro sando, vidre isini, vinc ar mimarTavs qcevis aseT

formebs. davuSvaT, am hipoTezis testirebisaTvis fsiqologma Seqmna 13 cdis

pirisagan Semdgari ori ekvivalenturi jgufi da TiToeuli jgufi ismenda

63 damarwmunebel mokle saubars. im pirebisaTvis, romlebic ar imyofebodnen

uSualo qcevis jgufSi (A1), mosaubre ar mimarTavda raime uSualo qcevas

saubrisas. xolo uSualo qcevis jgufSi (A2) mosaubre mimarTavda Tvalebis

kontaqts, iRimeboda da gamoxatavda auditoriis mimarT simpaTias. saubris

dasrulebis Semdeg, cdis pirebi afasebdnen mosaubres sandoobis skalaze,

romlis qulebic icvleboda 1-dan (sruliad arasando) 9-mde (Zalian sando).

davuSvaT, rom miiRes Semdegi qulebi:

uSualobis piroba (A)

uSualo qcevis gareSe (A1) uSualo qceviT (A2 )

4 7

5 6

3 9

4 8

6 8

7 7

7 6

4 4

3 5

5 8

6 7

3 6

5 8

a. gamoiyeneT t kriteriumi, raTa pasuxi gasceT Semdeg kiTxvebs: moaxdina Tu

ara gavlena sandoobis Sefasebaze mosaubris uSualobam? gamoiyeneT

ornapiriani kriteriumi da 0.05 mniSvnelovnobis done.

b. risi tolia η2 am kvlevaSi? damokidebuli cvladis dispersiis ra wili

SeiZleba mieweros damoukidebeli cvladis donis codnas?

g. ipoveT 95%-iani ndobis intervali ori populaciis saSualoebs Soris

gansxvavebisaTvis.

d. aRwereT Sedegebi am Tavis erT-erT qveTavSi mocemuli magaliTis


1

X Tavi. erTfaqtoriani dispersiuli analizi cdis pirTa Soris sqemisaTvis

erTfaqtoriani mravaldoniani sqemis magaliTi jgufebs Soris da jgufSida cvalebadoba

F statistikis miReba

monacemTa dayofa

saSualos kvadratis miReba dayofili monacemebidan

F statistikis gamoTvla

statistikuri hipoTezebis Semowmeba FFrealuriT

MSA da MSSecdomis mniSvnelobaze moqmedi faqtorebi

F statistikiT miRebuli statistikuri gadawyvetilebani

daSvebebi cdis pirTa Soris erTfaqtoriani dispersiuli

analizisaTvis

F-is statistikuri mniSvnelovnobis interpretacia

mravaldonian sqemaSi mravaljeradi Sedarebis kriteriumebi

tukis HSD kriteriumi saboloo SedarebebisaTvis

Carevis efeqtis siZlieris gazomva

eta kvadratis gamoTvla dispersiuli analizisaTvis

tdamoukidebelsa da F Soris damokidebuleba

dispersiuli analizis Sedegebis prezentacia

2

t kriteriumma gagvacno statistikuri hipoTezis SemowmebaSi mravali

mniSvnelovani cneba bihevioristuli mecnierebis kvlevebisaTvis. Tumca, t

kriteriumi mniSvnelovani SezRudvebiT xasiaTdeba, is SeiZleba saSualoTa

Sesadareblad gamoviyenoT mxolod ori jgufisaTvis. magram bihevioristuli

mecnierebis mraval eqsperiments mravaldoniani sqema aqvs. erTfaqtoriani

mravaldoniani eqsperimenti (romelsac, agreTve, uwodeben calmxriv

mravaldonian sqemas) aris eqsperimenti erTi damoukidebeli cvladiT da

damoukidebeli cvladis sami an meti doniT. am sqemebisaTvis statistikuri

hipoTezis Semowmebis mizniT gamoiyeneba statistikuri meTodi, cnobili

rogorc dispersiuli analizi (romlis abreviaturaa ANOVA).

dispersiuli analizi yvelaze farTod gavrcelebuli statistikuri

kriteriumia fsiqologiur kvlevebSi. is SeiZleba gamoyenebuli iyos rogorc

cdis pirTa Soris, ise cdis pirTa Sida sqemebisaTvis da im sqemebisaTvis,

romelSic ori an meti damoukidebeli cvladi icvleba. erTfaqtoriani

sqemisaTvis dispersiuli analizi SeiZleba gamoyenebuli iyos tdamoukidebelis

nacvlad, roca damoukidebeli cvladis mxolod ori donis manipulireba xdeba.

am TavSi warmodgenilia dispersiuli analizi cdis pirTa Soris

erTfaqtoriani mravaldoniani sqemisaTvis, romelSic erTi damoukidebeli

cvladis manipulireba xdeba da am damoukidebeli cvladis TiToeuli done

miwodebulia cdis pirTa gansxvavebuli jgufisaTvis. misi sxva sqemebSi

gamoyeneba warmodgenilia me-11 da me-12 TavebSi.

erTfaqtoriani mravaldoniani sqemis magaliTi

stimulis ra aspeqtebi gansazRvravs, ramdenad kargad vimaxsovrebT Cven mas?

kvlevis erTi hipoTezis, damuSavebis hipoTezis mixedviT, rac ufro

srulyofilia mexsierebaSi warmodgenili stimulis xati, miT ufro ukeTesad

gvamaxsovrdeba is (Craik&Tulving, 1975) kvlevis meore hipoTezis, Zalisxmevis

erTfaqtoriani mravaldoniani sqema (One-factor multilevel design): eqsperimenti erTi damoukidebeli cvladiT da am damoukidebeli cvladis sami an meti doniT.

dispersiuli analizi (Analysis of variance ANOVA): statistikuri kriteriumi, romelic gamoiyeneba mravaldoniani sqemebis analizisaTvis.

3 hipoTezis Tanaxmad, stimulebi, romelTac ufro meti Zalisxmeva dasWirdaT

gadamuSavebisa da damaxsovrebisaTvis, ufro kargad gvaxsendeba, vidre

stimulebi, romelnic nakleb Zalisxmevas moiTxovdnen (Coulter, Coulter&Glover, 1984).

davuSvaT, Cven SevqmeniT Cveulebrivi sagnebis sami gansxvavebuli tipis naxati:

detaluri, eskizi da daumTavrebeli eskizi. damuSavebis hipoTezis mixedviT

detaluri naxatebi yvelaze metad damuSavebas moiTxoven mexsierebaSi da

amrigad, yvelaze kargad unda gvaxsendebodes, xolo eskizebi da dausrulebeli

eskizebi unda iTxovdnen ufro nakleb damuSavebas mexsierebaSi da amitom ufro

cudad unda gvaxsendebodes. Zalisxmevis hipoTeza ki sapirispiro

winaswarmetyvelebas akeTebs. dausrulebeli eskizebi moiTxoven yvelaze met

Zalisxmevas gadamuSavebisa da damaxsovrebis procesSi da amitom ufro kargad

gvaxsendeba. detaluri naxatebi ki moiTxoven yvelaze nakleb Zalisxmevas,

aqedan gamomdinare, isini yvelaze cudad gvaxsendeba.

imis gansazRvrisaTvis, vizualuri stimulebis damaxsovrebis romeli

hipoTezis winaswarmetyvelebaa ukeTesi, davuSvaT, rom Cven gamoviyeneT cdis

pirTa Soris erTfaqtoriani sqema da ganvaxorcieleT damoukidebeli cvladis

_ naxatis tipis manipulacia sam doneze: dausrulebeli eskizi(A1), eskizi(A2) da

detaluri naxatebi(A3). am SemTxvevaSi gavimeoroT me-9 TavSi ganxiluli

procedurebi, romlebic exeboda eqsperimentis sqemasa da Catarebas. Seqmnili

iyo sami ekvivalenturi jgufi, TiToeulSi 5 cdis piri Tanabrad iyo

gadanawilebuli. TiToeuli jgufis cdis pirebs uCvenes naxatebi, romlebzec

gamosaxuli iyo 30 Cveulebrivi sagani. erT jgufs miawodes detaluri naxati,

meore jgufs _ eskizi, xolo mesame jgufs _ dausrulebeli eskizi. TiToeuli

naxatis 4 wm-is ganmavlobaSi daTvalierebis Semdeg, cdis pirebs sTxoves

zepirad gaexsenebinaT rac SeiZleba meti naxati. cxrilSi #10.1 naCvenebia

TiToeuli piris mier gaxsenebuli sagnebis raodenoba, TiToeuli jgufis

saSualo da standartuli gadaxra, qulebis sruli saSualo. sruli saSualo,

romelic aRiniSneba X G-iT, aris 115 qulis saSualo am eqsperimentSi. moqmedebs

Tu ara gaxsenebuli naxatebis raodenobaze naxatis tipi? am kiTxvaze pasuxis

gasacemad, gamoviyenoT cdis pirTa Soris erTfaqtoriani dispersiuli analizi.

sruli saSualo (Grand mean): yvela qulis saSualo eqsperimentSi.

4 cxrili #10.1 gaxsenebuli sagnebis raodenobis hipoTeturi qulebi, rogorc naxatis tipis

funqcia. TiToeuli pirobisaTvis gamoTvlilia saSualo ( X A ) da standartuli

gadaxra (sA ). X G aris 15 qulis sruli saSualo.

naxatis tipi (faqtori A)

dausrulebeli

eskizi(A1)

eskizi(A2) detaluri(A3)

19 14 15

13 13 10

16 16 11

17 12 12

20 10 12

X A 17 13 12 X G=14

sA 2.74 2.24 1.87

jgufebs Soris da jgufSida cvalebadoba

erTfaqtoriani dispersiuli analizi eqsperimentul monacemTa srul

dispersias cdis pirTa Soris sqemisaTvis or nawilad yofs, romlebsac xSirad

uwodeben wyaroebs: (1) dispersia, romelic icvleba damoukidebeli cvladis

sistematur efeqtTan da jgufebis saSualoebs Soris SerCevis SecdomasTan

erTad da (2) dispersia, romelic icvleba mxolod jgufSida Secdomis

cvalebadobiT. miuxedavad amisa, dispersiul analizSi dispersias ewodeba

kvadratebis saSualo. kvadratebis saSualo (abreviatura MS) aris dispersiis

meore saxelwodeba da is dispersiul analizSi gamoiyeneba.

suraTi #10.1 gvexmareba am dispersiebis gaazrebaSi. am suraTis nawili(a)

gviCvenebs yvela 15 qulis sixSireTa ganawilebas cxrilidan #10.1. am qulebis

sruli cvalebadoba aris qulaTa dispersia sruli saSualos anu 14-is garSemo.

nawili (b), (g) da (d) warmogvidgens qulaTa sixSireTa ganawilebebs jgufis

SigniT daumTavrebeli eskizisaTvis, eskizisaTvis da detaluri naxatisaTvis.

kvadratebis saSualo (mean square): dispersiis saxelwodeba, romelic gamoiyeneba dispersiul analizSi.

jgufSida Secdomis dispersia (Within-groups error variance): monacemTa dispersia jgufSi, romelic jgufis saSualos garSemoa gamoTvlili.

5 miaqcieT yuradReba, rom Tumca cdis pirebi TiToeul jgufSi erTnair

pirobebSi imyofebodnen, qulebi jgufis SigniT jgufis saSualos garSemo

icvleba. jgufSida cvalebadoba mocemulia qulaTa dispersiiT maTi Sesatyvisi

jgufis saSualos garSemo. dispersiul analizSi am cvalebadobas ewodeba

jgufSida Secdomis dispersia an ubralod, Secdomis dispersia da MSSecdomis

saSualebiT izomeba. termini Secdoma gamoiyeneba imis aRsaniSnavad, rom jgufis

SigniT yvela qula erTmaneTis toli ar aris, maSinac ki, roca jgufSi cdis

pirs erTi da igive piroba miewodeba.

jgufTa Soris cvalebadoba aisaxeba jgufebis saSualoebis (e.i. 1AX , 2AX , da

3AX ) sruli saSualosagan _ X G-sagan gadaxraSi. jgufebis saSualoebis am

cvalebadobas uwodeben jgufTa Soris dispersias da is MSA-s saSualebiT

izomeba.

suraTi #10.1

(a) 15 qulis sixSireTa ganawileba cxrilidan #10.1. (b) dausrulebeli eskizis

jgufis(A1) qulebis sixSireTa ganawileba (g) eskizis jgufis (A2) qulebis

sixSireTa ganawileba. detaluri naxatis jgufis (A3) qulebis sixSireTa

ganawileba.

suraTi aklia ( a)Total variation in scores: Variation around grand mean - monacemTa

sruli cvalebadoba: monacemTa cvalebadoba sruli saSualos garSemo; frequency

- sixSire; number of items recalled, all groups - gaxsenebuli sagnebis raodenoba, yvela

jgufi;

b) Within groups error variation; variation of scores around group mean - jgufSida Secdomis

cvalebadoba: monacemTa cvalebadoba jgufis saSualos garSemo; frequency-

sixSire, number of items recalled, groupA1 - gaxsenebuli sagnebis raodenoba, A1 jgufi,

Bbetween-groups variation: variation of group mean from grand mean - jgufTa Soris

cvalebadoba: sruli saSualos garSemo jgufis saSualos cvalebadoba;

g) Within groups error variation; variation of scores around group mean - jgufSida Secdomis

cvalebadoba: monacemTa cvalebadoba jgufis saSualos garSemo; frequency-


jgufTa Soris dispersia (Between-groups variance): dispersia, romelic gamoiTvleba sruli

saSualos garSemo jgufebis saSualoebis cvalebadobis gamoyenebiT.

6


cvalebadoba: sruli saSualos garSemo jgufis saSualos cvalebadoba;

Within groups error variation; variation of scores around group mean - jgufSida Secdomis

cvalebadoba: monacemTa cvladebadoba jgusi saSualos garSemo; frequency-



cvalebadoba: sruli saSualos garSemo jgufis saSualos cvalebadoba.

jgufebis saSualoebi eqsperimentSi erTmaneTisagan SeiZleba ori mizeziT

gansxvavdebodes: (1) damoukidebeli cvladis gavleniT da (2) SerCevis Secdomis

gamo. amrigad, MSA asaxavs sistematur cvalebadobas jgufebis saSualoebs

Soris, romelic gamowveulia damoukidebeli cvladis gavleniTa da SerCevis

Secdomis cvalebadobiT. amrigad, ganxilul eqsperimentSi jgufebis

saSualoebi 1AX , 2AX , 3AX gansxvavdeba erTmaneTisagan, Tuki naxatis tipi

gavlenas axdens mis gaxsenebaze, magram isini gansxvavdebian erTmaneTisagan

ubralod SerCevis Secdomis gamoc.

sruli cvalebadobis dayofa jgufTa Soris dispersiad da Secdomis

dispersiad xdeba kriteriumis statistikis saSualebiT, romelsac ronald a.

fiSeris pativsacemad F-s uwodeben (1890-1962). is britaneli statistikosia,

romelmac SeimuSava dispersiuli analizis cnebebi. F statistika aris MSA

Sefardeba MSSecdomasTan da gamoisaxeba, rogorc F

SecdomaMSMSF A=

F statistikis mricxveli aris MSA, romelic zomavs rogorc damoukidebeli

cvladis gavlenas, aseve jgufebis saSualoebs Soris SerCevis Secdomas.

mniSvneli MSSecdoma zomavs mxolod jgufSida qulebis Secdomis cvalebadobas.

Tu damoukidebel cvlads ara aqvs gavlena, maSin jgufebis saSualoebs Soris

cvalebadoba ganpirobebuli iqneba mxolod SerCevis SecdomiT. amitom MSA da

MSSecdoma TiTqmis toli iqneba da F gaxdeba 1.00-is toli. magram Tu

damoukidebel cvlads aqvs gavlena, is gazrdis gansxvavebas SerCevis

saSualoebs Soris imaze metad, rac mosalodneli iqneboda mxolod SerCevis

Secdomis gamo. am SemTxvevaSi, mricxvelis jgufTa Soris dispersia _ MSA, meti

iqneba, vidre mniSvnelis jgufSida dispersia _ MSSecdoma da F Sefardeba iqneba

erTze meti. amrigad, F-is mniSvneloba izrdeba, roca Carevas aqvs gavlena.

7

Sesabamisad, F statistika uzrunvelyofs statistikuri hipoTezis Semowmebis

safuZvels.

F statistikis miReba Semdeg qveTavSi Cven ganvixilavT dispersiebs, romlebic Sedian F SefardebaSi,

raTa uzrunelvyoT ANOVA konceptualuri gaazreba. dispersiuli analizis

Sesruleba realur monacemebze rom mogesurvebinaT, gamosaTvlelad ufro

advil formulas gamoiyenebdiT, romelic mocemulia am Tavis damatebaSi an

kompiuteruli monacemebis analizis programaSi. es gamoTvliTi formula

axals arafers SegmatebT dispersiuli analizis gaazrebaSi, magram

gamoTvlisaTvis advilia.

F statistika moiTxovs monacemebis cvalebadobis or gazomvas, MSA da

MSSecdomis gazomvas. mexuTe TavSi Cven ganvsazRvreT dispersia, anu MS, rogorc

SS/n-1, sadac SS niSnavs kvadratebis jams. mniSvneli, n-1, Seesatyviseba

Tavisuflebis xarisxs, romelic monawileobs dispersiis Sefasebis miRebaSi.

amrigad, dispersiis formula, romelic uzrunvelyofs populaciis dispersiis

Caunacvlebel Sefasebas, aris MS= SS/df. MSA da MSSecdomis misaRebad Cven

Sesabamisi monacemebidan vpoulobT SSA-s da SSSecdomas da Semdeg kvadratebis

TiToeul jams Tavis df-ze vyofT. pirveli nabiji, anu kvadratebis jamebis

miReba iwyeba monacemTa dayofiT.

monacemTa dayofa

dispersiuli analizi yofs eqsperimentis monacemTa srul cvalebadobas

jgufebs Soris da jgufSida dispersiebad cdis pirTa monacemebis martivi

modelis pirobebSi. Tumca zemoT moyvanili magaliTi damoukidebeli cvladis

mxolod sam dones iyenebs, aq moyvanili modeli gamoiyeneba nebismieri

raodenobis doneebis cdis pirTa Soris erTfaqtoriani sqemis dispersiul

analizisaTvis.

cxrilis #10.1 monacemebi simboluradaa warmodgenili cxrilSi #10.2. es

simboloebi msgavsia im simboloebisa, romelic gamoiyeneboda wina paragrafis

magaliTis ganxilvisas. damoukidebeli cvladi ganisazRvreba, rogorc A

faqtori sami doniT A1, A2 da A3. aso a asaxavs A faqtoris doneebis raodenobas.

cxrilebisaTvis #10.1 da #10.2 a=3. cdis pirebis qulebi mocemulia Xij-is

saSulebiT, sadac qveSmiwerili indeqsi i warmoadgens cdis piris ganmsazRvrel

8

A faqtorisa da SerCevis SecdomiT ganpirobebuli cvalebadoba

jgufSida SecdomiT

ganpirobebuli cvalebadoba

ricxvs jgufis SigniT da qveSmiwerili indeqsi j warmoadgens jgufis

ganmsazRvrel ricxvs, A1, A2 da A3. magaliTad, X52 aris A2-jgufis mexuTe cdis

piri. cxrilSi #10.1, am cdis piris qula aris 10, anu X52=10. roca ar aris

aucilebeli ganisazRvros konkretuli cdis piri, ij indeqss vaSorebT da

viyenebT ubralod X-s. SerCevis saSualoebi damoukidebeli cvladis TiToeuli

donisaTvis aRiniSneba ubralod 1AX , 2AX da 3AX -iT an, zogadad, AX -iT. sruli

saSualo aRiniSneba X G-iT. qulaTa raodenoba damoukidebeli cvladis erT

doneze aRiniSneba nA-iT, xolo qulaTa sruli raodenoba eqsperimentSi

warmodgenilia N-iT. cxrilisaTvis #10-1, nA1=5, nA2=5, nA3=5 , N=15, 1AX =17, 2AX =13,

3AX =12 da X G=14. Tumca dispersiuli analizi ar moiTxovs qulaTa tol

raodenobas TiToeul pirobaSi, toli raodenoba ubralod amartivebs saWiro

gamoTvlebs. am qveTavis yvela magaliTi iyenebs qulaTa tol raodenobas

TiToeul pirobaSi.

dispersiuli analizi iRebs qulaTa srul cvalebadobas, anu srul raodenobas

imisa, riTac qula (anu Xij) gansxvavdeba qulaTa sruli saSualosagan (anu X G)

da yofs am srul sxvaobas or nawilad. erTi nawili _ jgufTa Soris

dispersia (e.i. MSA), icvleba damoukidebeli cvladisa da SerCevis Secdomis

gavleniT. meore nawili, romelic zomavs individualuri qulebis Secdomas

maTi jgufis saSualos garSemo, xdeba jgufSida Secdomis dispersia (e.i.

MSSecdoma). es modeli masSi qulaTa asaxvis mizniT SeiZleba aseTnairad

warmovidginoT.

sruli cvalebadoba

10.1

= +

qulaTa terminebiT es toloba SeiZleba Semdegnairad iyos dawerili:

(X- X G) = ( X A- X G) + (X- X A) 10.2

cdis piris eqsperimentuli jgufis cdis piris qulis

qulis gansxvaveba saSualos gansxvaveba gansxvaveba misi jgufis

sruli saSualosagan sruli saSualosagan saSualosagan

sruli cvalebadoba TiToeuli cdis piris qulaSi

a: A faqtoris doneebis raodenoba

nA: qulaTa raodenoba erTfaqtoriani sqemis erT doneze

9

SevamowmoT 10.2 tolobis TiToeuli nawili.

cxrili #10.2

qulaTa asoiTi aRniSvnebi cxrilSi #10.1 moyvanili hipoTeturi eqsperimentidan

damoukidebeli cvladis sami doniT, A faqtoriT. A faqtoris TiToeuli

donisaTvis saSualoebi warmodgenili arian X A da X G.

A faqtori

A1 A2 A3

X11 X12 X13

X21 X22 X23

X31 X32 X33

X41 X42 X43

X51 X52 X53

1AX 2AX 3AX X G

sruli cvalebadoba

qulis gansxvaveba sruli saSualosagan (e.i. X- X G) aris am qulis sruli

cvalebadoba. rogorc 10.2 toloba gviCvenebs, am cvalebadobis nawili

ganpirobebulia damoukidebeli cvladis gavleniT da nawili ki _ Secdomis

cvalebadobiT.

damoukidebeli cvladis gavleniTa da SerCevis SecdomiT ganpirobebuli

cvalebadoba

gansxvaveba jgufis saSualosa da srul saSualos Soris (e.i. X A- X G) icvleba

damoukidebeli cvladis gavleniTa da SerCevis SecdomiT. Sesabamisad, is Seva

MSA-is gamoTvlaSi.

damoukidebeli cvladis efeqti. dispersiuli analizi iyenebs gansxvavebas

TiToeuli jgufis saSualosa da srul saSualos Soris, raTa gaizomos

damoukidebeli cvladis gavlena. es gansxvaveba izrdeba damoukidebeli

cvladis gavlenis gazrdiT. am sakiTxis kargad gaazrebaSi ricxviTi

ilustracia dagvexmareba.

cxrili #10.3 warmoadgens eqsperimentis sam SesaZlo Sedegs, romelic

moicavs mxolod sam qulas TiToeuli sakvlevi jgufisaTvis. am cxrilis (a)

10 nawilSi ar aris warmodgenili Carevis efeqti. TiToeuli jgufis saSualo

sruli saSualos tolia. ufro metic, ori ( X A- X G) sxvaobidan TiToeuli

nulis tolia. Carevis efeqti, +2 emateba A2 qulebs cxrilis (b) nawilSi;

amrigad, arc erTi jgufis saSualo ar aris toli sruli saSualosi. Carevis

es efeqti, romelic +2-is tolia ( X A1- X G) sxvaobisaTvis warmodgenilia

mniSvnelobiT _1 da X A2- X G sxvaobisaTvis ki mniSvnelobiT +1. cxrilis (g)

nawili warmoadgens Carevis efeqts +6 , romelic emateba A2-is qulebs.

dispersiul analizSi es efeqti iyofa -3-is tol mniSvnelobad ( X A1- X G)

sxvaobisaTvis da +3-is tol mniSvnelobad ( X A2- X G) sxvaobisaTvis. miaqcieT

yuradReba, imas, rom es nimuSi gviCvenebs, aqvs Tu ara Carevas efeqti da es

iwvevs jgufis saSualoebis gansxvavebas erTmaneTisagan, is SeiZleba aisaxos

( X A- X G) sxvaobis mniSvnelobaSi, romelic SeiZleba gamoTvlili iyos

eqsperimentSi.

SerCevis Secdoma. garda Carevis efeqtisa, jgufebis saSualoebi, agreTve,

erTmaneTisagan gansxvavdebian SerCevis Secdomis gamo. maSinac ki, Tu

damoukidebel cvlads ara aqvs gavlena, mosalodnelia, rom jgufis saSualo

gansxvavebuli iqneba sruli saSualosagan mxolod SerCevis Secdomis gamo.

amrigad, damoukidebeli cvladis nebismieri efeqti eqsperimentSi warmoiqmneba

SerCevis Secdomis moqmedebis fonze.

jgufSida SecdomebiT gamowveuli cvalebadoba

individualuri qulis gansxvaveba Carevis saSualosagan (e.i. X - X A) Sedis

MSSecdomis gamoTvliT formulaSi. eqsperimentSi is mxolod jgufSida Secdomis

cvalebadobiT icvleba. faqtiurad, es gansxvaveba, romelic me-5 TavSi iyo

gamoyenebuli, rogorc standartuli gadaxrisa da dispersiis mricxveli,

jgufSida qulebs Soris Secdomis cvalebadobas zomavs. am gansxvavebis

ricxobrivi mniSvneloba ar icvleba damoukidebeli cvladis gamo. radgan

damoukidebeli cvladis nebismieri efeqti gulisxmobs TiToeuli cdis piris

qulebis toli sididiT gazrdas an Semcirebas, amitom eqsperimentuli jgufis

saSualo toli sididiT Seicvleba. amrigad, X - X A ucvleli darCeba

damoukidebeli cvladis efeqtisagan damoukideblad. magaliTad, meore cdis

pirisaTvis cxrilSi #10.3 A2-Si (e.i. qula, romelic warmodgenilia X22=13 (a)

svetSi), X - X A gansxvaveba rCeba (a) +1 cxrilis yvela svetisaTvis Carevis

11

efeqtis rogorc arsebobis, aseve ararsebobis SemTxvevaSi. amrigad, X - X A

mniSvneloba asaxavs mxolod qulaTa Soris Secdomis cvalebadobas Carevis

pirobaSi.

cxrili #10.3

cxrilSi naCvenebia, rom eqsperimentSi X A- X G Seesatyviseba damoukidebeli

cvladis efeqts. Carevis saSualoebi warmodgenilia X A-s saSualebiT da srul

saSualo qulaTa TiToeuli simravlisaTvis X G–iT. TiToeuli eqsperimentuli

jgufis saSualo da sruli saSualo naCvenebia qulaTa TiToeuli simravlis

qveviT.

(a) svetis qulebi; ar aRiniSneba Carevis efeqti

(b) svetis qulebi; aRiniSneba Carevis efeqti +2 toli

(c) svetis qulebi; aRiniSneba Carevis efeqti +6 toli

(aA) B(b) (g)

A1 A2 A1 A2 A1 A2

11 12 11 14 11 18

12 13 12 15 12 19

13 11 13 13 13 17

X A

12

12 12 14 12 18

X G 12 13 15

X A1- X G=0 X A1- X G= -1 X A1- X G= _3

X A2- X G=0 X A2- X G= +1 X A2- X G= +3

kvadratebis saSualos miReba monacemTa dayofis saSualebiT

monacemTa nawilebad dayofa, romelic zomavs jgufTa Soris da jgufSida

cvalebadobas, adgens MSA da MSSecdomis miRebis safuZvels, rac saWiroa F

statistikis misaRebad. gaixseneT, rom populaciis dispersiis Sefaseba

gamoiTvleba Semdegi formulis saSualebiT:

1)( 2

2

−−Σ

=n

XXs

12 miaqcieT yuradReba, rom TiToeuli termini 10-2 tolobaSi dispersiis

mricxvels hgavs. Tu s2-is nacvlad MS-s gamoviyenebT, xolo 2)( XX −Σ -is

nacvlad kvadratebis jams (SS), da (n-1)-s ki CavanacvlebT df-iT, es dispersia ase

gamoisaxeba

dfSSMS =

dispersiis es formula amtkicebs, rom Tu 10-2 tolobaSi warmodgenil

TiToeul sxvaobas kvadratSi aviyvanT TiToeuli cdis pirisaTvis da Semdeg

davajamebT yvela cdis pirisaTvis, Sedegi iqneba sxvaobebis kvadratebis jami

anu SS, kvadratebis saSualos mricxveli. 10-2 tolobidan sami aseTi SS termini

SeiZleba iyos miRebuli. termins X- X G-s mivyavarT SSsrulisaken, romelic asaxavs

qulaTa jamur gansxvavebas sruli saSualosagan. X A- X G gansxvavebas mivyavarT

SSA–sken, romelic jgufTaSoris cvalebadobis sazomia, im cvalebadobisa,

romelic ganpirobebulia damoukidebeli cvladisa da SerCevis SecdomiT.

sabolood, X - X A-s mivyavarT SS Secdomisaken, romelic aris jgufSida Secdomis

variaciis sazomi eqsperimentSi.

sami kvadratis jami

ganvixiloT, Tu rogor SeiZleba gamoviyenoT 10-2 toloba kvadratebis jamis

sapovnelad cxrilis #10.1 A1 jgufis pirveli cdis piris qulaTa dayofiT

(e.i.X11), romelic tolia 19-is. am qulisaTvis toloba 10-2 xdeba:

X11 - X G=( X A1- X G)+(X11- X A1).

Tu CavsvamT Sesatyvis ricxviT mniSvnelobebs X11, X A1 da X G , miviRebT

19-14=(17-14)+(19-17),

(+5) = (+3) + (+2).

dayofa gviCvenebs, rom es qula gansxvavdeba sruli saSualosagan (e.i. X11 _ X G=

+5), gansazRvruli sididiT da es gansxvaveba miRebulia im Carevis pirobis gamo,

romelSic imyofeba cdis piri (asaxulia X A1- X G = +3 tolobaSi) da miRebulia

Secdomis arasistematuri gavlenisa da SemTxveviTi faqtorebis gamo (asaxulia

X11 _ X A1= +2 tolobaSi).

SS-is misaRebad, TiToeuli cdis piris qula analogiurad aris

dayofili, rac naCvenebia cxrilSi #10.4 15 qulisaTvis, romelic aRebuli iyo

cxrilidan #10.1. cxrilis #10.4 garegnulma gadatvirTulobam SeiZleba

dagafrTxoT, magram gamoTvlebi aq mxolod mimatebas, gamoklebas, gamravlebas

13 moiTxovs da monacemebi nabij-nabij muSavdeba. dakvirvebiT SeasruleT

TiToeuli nabiji.

cxrili #10.4

kvadratebis jamis miReba cdis pirTa Soris dispersiuli erTfaqtoriani

analizisaTvis. gamoyenebulia qulebi, romelic mocemulia cxrilSi #10.1.

nabiji 1: qulaTa ganawileba 15 cdis pirisaTvis Semdegi tolobis gamoyenebiT

X- X G = ( X A- X G) + ( X - X A)

qulebi

A1-saTvis

X11 19-14 = (17-14) + (19-17)

X21 13-14 = (17-14) + (13-17)

X31 16-14 = (17-14) + (16-17)

X41 17-14 = (17-14) + (17-17)

X51 20-14 = (17-14) + (20-17)

qulebi

A2-saTvis

X12 14-14 = (13-14) + (14-13)

X22 13-14 = (13-14) + (13-13)

X32 16-14 = (13-14) + (16-13)

X42 12-14 = (13-14) + (12-13)

X52 10-14 = (13-14) + (10-13)

qulebi

A3-saTvis

X13 15-14 = (12-14) + (15-12)

X23 10-14 = (12-14) + (10-12)

X33 11-14 = (12-14) + (11-12)

X43 12-14 = (12-14) + (12-12)

X53 12-14 = (12-14) + (12-12)

cxrili #10.4

gagrZeleba:

nabiji 2: pirvel nabijSi mocemul sxvaobaTa povna TiToeuli cdis pirisaTvis ricxviTi

gansxvavebebis misaRebad

X- X G = ( X A- X G) + ( X - X A)

qulebi

A1-saTvis

X11 +5 = +3 + +2

X21 _1 = +3 + _4

X31 +2 = +3 + _1

X41 +3 = +3 + 0

14 X51 +6 = +3 + +3

qulebi

A2-saTvis

X12 0 = _1 + +1

X22 _1 = _1 + 0

X32 +2 = _1 + +3

X42 _2 = _1 + _1

X52 _4 = _1 + _3

qulebi

A3-saTvis

X13 +1 = _2 + +3

X23 _4 = _2 + _2

X33 _3 = _2 + _

X43 _2 = _2 + 0

X53 _2 = _2 + 0

cxrili #10.4

gagrZeleba:

nabiji 3: TiToeuli sxvaobis kvadrati

(X- X G)2 ( X A- X G)2 ( X - X A)2

qulebi

A1-saTvis

X11 25 9 4

X21 1 9 16

X31 4 9 1

X41 9 9 0

X51 36 9 9

qulebi

A2-saTvis

X12 0 1 1

X22 1 1 0

X32 4 1 9

X42 4 1 1

X52 16 1 9

qulebi

A3-saTvis

X13 1 4 9

X23 16 4 4

X33 9 4 1

X43 4 4 0

X53 4 4 0

15 nabiji 4. SeajameT yvela cdis piris sxvaobebis kvadratebis jami TiToeuli

nawilisaTvis

SecdomaSSSSSS

XXXXXX

A

A

n

i

a

jGA

n

i

a

jG

n

i

a

j

AAA

+=+=

−+−=− ΣΣΣΣΣΣ=

=

=

=

=

=

=

=

=

=

=

=

6470134

)()()( 25

1

3

1

25

1

3

1

25

1

3

1

nabiji 1. 15 cdis piris qulebi eqsperimentSi dayofilia 10-2 tolobis

mixedviT.

nabiji 2. TiToeuli qulisaTvis napovnia ricxviTi sxvaobebi.

nabiji 3. TiToeuli dadebiTi da uaryofiTi sxvaoba, romelic napovnia

nabijSi 2 aris kvadratSi ayvanili.

nabiji 4. TiToeuli svetis sxvaobebis kvadratebis mniSvnelobebi

dajamebulia, sul 15 qulis jami. Sedegi aris sami kvadratis jamis miReba:

SSsruli= 134,

SSA=70,

SSSecdoma= 64.

rogorc naCvenebia me-4 nabijSi, SSsruli maTematikurad Semdegi formuliTaa

warmodgenili.

25

1

3

1)( G

n

i

a

jXX

A

−ΣΣ=

=

=

=

ormagi jamis niSani aRniSnavs, rom kvadratebSi ayvanili sxvaobebi [e.i. (X- X G)2]

unda Sejamdes TiToeuli eqsperimentuli pirobis yvela cdis pirisaTvis

(pirveli cdis piridan, i=1-dan, ukanasknel cdis piramde, nA-mde, konkretul

jgufSi) da yvela jgufisaTvis(damoukidebeli cvladis pirveli donidan, j=1-

dan, ukanasknel donemde, a-mde). am magaliTSi TiToeul jgufSi aris 5 qula

(nA=5) da sami jgufi (a=3); amrigad, dajamebis zRvari aris i=1-dan nA=5-mde da

j=1-dan a=3-mde, an

ΣΣ==

5

1

3

1 ij

radgan yovelTvis yvela cdis piris da yvela jgufis Sedegebs vajamebT,

Cveulebriv gamoiyeneba ormagi jamis niSani sazRvrebis gareSe.

16

analogiurad, ASS -is misaRebad, eqsperimentuli jgufis saSualosa da

srul saSualos Soris sxvaobebis kvadratebi yvela cdis pirisaTvis jamdeba

eqsperimentSi. Sesabamisad,

ASS = 2)( GA XX −ΣΣ

sabolood, SSSecdoma miiReba TiToeuli cdis piris qulisa da misi jgufis

saSualos sxvaobebis kvadratebis SejamebiT, anu

SSSecdoma = 2)( AXX −ΣΣ .

mniSvnelovani kavSiri arsebobs kvadratebis jamebs Soris, rac naCvenebia

cxrilis #10.4 qveda nawilSi.

SSsruli = SSA+ SSSecdoma (10-3)

amrigad, eqsperimentis qulebis sruli cvalebadoba, SSsruli, aris sistematuri

cvalebadobis Sedegi, romelic xdeba gansxvavebul eqsperimentul pirobebSi

myof jgufebs Soris, SSA (zogjer aRiniSneba, rogorc SSjgufTaSorisi an SSCarevis)

da im Secdomis cvalebadobis Sedegi, romelic xdeba jgufebs SigniT, SSSecdoma

(zogjer aRiniSneba, rogorc SSjgufSida).

Tavisuflebis xarisxebis povna

saboloo nabiji MSA da MSSecdomis povnaSi aris TiToeuli SS-is gayofa mis df-

ze.

gaixseneT, rom Tavisuflebis xarisxi aris qulaTa raodenoba, romelic

Tavisufalia cvalebadobaSi statistikis gamoTvlisas. am ganmartebidan

advilad SeiZleba ganvsazRvroT Tavisuflebis xarisxi 10-3 tolobis

kvadratebis TiToeuli jamisaTvis.

sruli Tavisuflebis xarisxi. imisaTvis, rom vipovoT SSsruli, unda iyos

cnobili sruli saSualo, X G. radgan kvadratebis sruli jami efuZneba

eqsperimentis TiToeuli qulis gansxvavebas sruli saSualosagan (e.i. GXX − ),

maSin Tavisuflebis xarisxi unda iyos erTiT naklebi qulaTa srul

raodenobaze. magaliTad, cxrilSi #10.1 sruli saSualo 15 qulisaTvis 14-ia.

SSsrulis misaRebad 14 unda gamoakldes am 15 qulidan TiToeuls. am 15

sxvaobis jami unda iyos 0-is toli. amrigad, nebismieri 14 qula 15-dan

Tavisufalia Tavisi cvalebadobisas, magram me-15 qula xdeba fiqsirebuli, Tu

17

X G cnobilia. Sesabamisad SSsrulisaTvis df tolia 14-is. saerTod, sruli

Tavisuflebis xarisxi anu df sruli gasaanalizebel qulaTa raodenobaze erTiT

naklebis ricxvis tolia. is ase Caiwereba:

dfsruli = N-1

sadac N aris qulaTa sruli raodenoba eqsperimentSi.

Tavisuflebis xarisxi ASS -saTvis. ASS gamoiTvleba eqsperimentuli jgufebis (mag.

1AX , 2AX da 3AX ) saSualosa da srul saSualos Soris sxvaobebidan. sami

damoukidebeli cvladis donis mqone kvlevis sqemisaTvis da qulaTa toli

raodenobisaTvis TiToeul eqsperimentul jgufSi, Tu X G cnobilia, maSin

mxolod ori jgufis saSualo SeiZleba Seicvalos. magaliTad, Tu vici, rom

X G=14, 1AX =17, da 3AX =12, rogorc cxrilSi #10.1, maSin 2AX cnobilia. is unda

iyos 13-is toli. 2AX -is nebismieri sxva mniSvneloba ar iqneba X G=14-is

Sesabamisi. Sesabamisad, am magaliTSi SSA-saTvis 2 df arsebobs.

saerTod, df SSA-saTvis an dfA damoukidebeli cvladis doneebis

raodenobaze erTiT naklebi ricxvis tolia, rac Semdegnairad Caiwereba:

df SSA-saTvis = dfA=a-1

sadac a aris A damoukidebeli cvladis doneebis raodenoba.

Tavisuflebis xarisxi SSSecdomisaTvis. SSSecdoma efuZneba TiToeuli jgufis

saSualos, AX -sa da eqsperimentuli jgufebis SigniT yvela cdis piris qulebs

Soris sxvaobas. amitom, damoukidebeli cvladis TiToeuli donisaTvis AX -s

gansazRvris Semdeg, mxolod nA-1 qula aris Tavisufali cvalebadobaSi am

donis SigniT. cxrilSi #10.1, Tu AX cnobilia, am xuTi qulidan mxolod oTxi

aris Tavisufali Tavis cvalebadobaSi TiToeuli eqsperimentuli pirobis A1, A2

da A3-is SigniT. radgan damoukidebeli cvladis sami done gvaqvs da 4 qula

Tavisufalia cvalebadobisas TiToeuli donis SigniT, df SSSecdomisaTvis 12-is

tolia. (4 df A1-saTvis, plus 4 df A2-saTvis , plus 4 df A3-saTvis, rac tolia 12-

is).

saerTod, sadac a aris damoukidebeli cvladis doneebis raodenoba da

arsebobs nA qula damoukidebeli cvladis TiToeul doneze, df SSSecdomisaTvis

18

an dfSecdoma=a(nA-1). radgan (a)(n) qulaTa sruli raodenobis (N) tolia, dfSecdoma

SeiZleba, agreTve gamoisaxos, rogorc N-s gamoklebuli a, anu,

df SSSecdomisaTvis = dfSecdoma = N-a.

Tavisuflebis xarisxis aditiuroba. Tavisuflebis xarisxi garkveuli azriT

aditiuria, iseve rogorc misi Sesatyvisi kvadratebis jamis mniSvnelobebi.

amrigad,

dfsruli=dfA+ dfSecdoma 10-4

es damokidebuleba SenarCunebulia magaliTSic, dfsruli=14, dfA=2, dfSecdoma=12.

kvadratebis saSualos povna SS da df saSualebiT

F statistika aris ori MSA da MSSecdomis kvadratebis saSualos Sefardeba,. ori

saWiro MS mniSvneloba miRebulia SS -dan, romelic, Tavis mxriv, miRebulia 10-

3 tolobaSi da df mniSvnelobidan, romelic dakavSirebulia TiToeul SS -Tan.

konkretulad, kvadratebis saSualo aris

MSAA

A

dfSS

=

da

MSSecdoma=Secdoma

Secdoma

dfΣεχδχμα

SS

Tumca SesaZlebelia MSsrulis miReba SS srulis gayofiT dfsrulze, es mniSvneloba

ar uzrunvelyofs sasargeblo informacias dispersiul analizSi da

Cveulebriv ar gamoiTvleba. cxrilis #10.1 qulebisaTvis, Cven vipoveT SS A=70

da SS Secdoma=64 (ix. cxrili #10.4), agreTve, dfA=2 da dfSecdoma=12, amitom,

MSA 352

70===

A

A

dfSS

da

MSSecdoma= 333.51264

df==

Secdoma

SecdomaSS

F statistikis gamoTvla

am Tavis dasawyisSi F statistika gansazRvruli iyo, rogorc

F=SecdomaMS

MS A

19

cxrilSi #10.1 mocemuli qulebisaTvis: MSA=35 da MSSecdoma=5.333, amrigad

56.6

563.6

=

==

realuri

realuri

FF

an5.333

35

F

F

sadac Frealuri gamoyenebulia F-is im mniSvnelobis aRsaniSnavad, romelic

realurad aris miRebuli gaanalizebuli qulebidan. miaqcieT yuradReba, rom

MSA da MSSecdoma efuZneba kvadratebis jamis mniSvnelobebs, romelic

yovelTvis unda iyos dadebiTi, F agreTve, unda iyos dadebiTi. F-is uaryofiTi

mniSvneloba gviCvenebs, rom gamoTvlebSi Secdomaa daSvebuli.

dispersiuli analizis ricxviTi mniSvneloba xSirad warmodgenilia

cxrilis saxiT, romelic gansazRvravs cvalebadobis wyaroebs; SS , df da MS

mniSvnelobebs da Frealuris mniSvnelobas. cxrili #10.5 gviCvenebs, Tu rogor

aris organizebuli aseTi sistematizebuli cxrili im ricxviTi mniSvnelobebis

gamoyenebiT, romelic miRebulia cxrilis #10.1 monacemebidan. F-is mniSvneloba

ori aseuliTaa gadaweuli sistematizebul cxrilSi, agreTve SS da MS

mniSvnelobebic gadaweulia ori an meti aseuliT, rac saWiroa damrgvalebis

Secdomis minimumamde miyvanisaTvis. cxrili #10.6 ajamebs SS , df da MS-isaTvis

ganmartebiT formulas.

F statistikis miReba da misi mniSvnelobis gamoTvla xSirad iwvevs

mniSvnelovan kamaTs. ufro xSirad praqtikaSi misi mniSvneloba gamoiTvleba

kompiuteruli programiT an gamoTvliTi formuliT. Cveni mizani ar aris

gamoTvla gavamartivoT, aramed unda vuCvenoT rogor gamoiyvaneba F

statistika qulaTa erTobliobidan. Cven ganvixilavT, rogor unda avxsnaT da

rogor unda gamoviyenoT es informacia momavalSi.

cxrili #10.5

dispersiuli analizis ricxviTi Sejameba cxrilSi #10.1 mocemuli

qulebisaTvis

wyaro SS df MS Fa

naxatis tipi (A) 70 2 35 6.56

Secdoma 64 12 5.333

sul 134 14

*p< 0.05 a F-is statistikurad mniSvnelovani gansxvaveba, romelic naCvenebia albaTobis

20

doniT sqolioSi sistematizebul cxrilSi, e.i., *p< 0.05

cxrili #10.6

cdis pirTa Soris erTfaqtoriani dispersiuli analizis ganmartebiTi

formulis Sejameba

wyaro SS dfa MS F

faqtori A 2)( GA XX −ΣΣ a-1 SS A/ dfA

MSA /MSSecdoma

Secdoma 2)( AXX −ΣΣ N-a SS Secdoma/

dfSecdoma

sul 2)( GXX −ΣΣ N-1 ar gamoiTvleba

aa= A faqtoris doneebis raodenoba, N= qulaTa srul raodenobas.

SeamowmeT Tqveni codna X-1

1. gansazRvreT: a, jgufTaSoris dispersia, dfA, dfSecdoma, dfsruli, faqtori A, MSA

MSSecdoma, N, nA, erTfaqtoriani mravaldoniani sqema, SS A, SS Secdoma, SS sruli,

jgufSida dispersia, GXX − , GA XX − , AXX − .

2. rogoria t kriteriumis SezRuduloba eqsperimentuli monacemebis

analizisaTvis?

3. romeli kriteriumis statistika gamoiyeneba dispersiuli analizis dros?

4. dawereT F statistikis formula.

5. romel or wyarod yofs erTfaqtoriani dispersiuli analizi qulaTa srul

cvalebadobas eqsperimentSi?

6. eqsperimentatori iyenebs cdis pirTa Soris erTfaqtorian eqsperiments A A

faqtoris oTxi donisaTvis: A1, A2, A3 da A4 . aq iyo 11 gansxvavebuli cdis piri

TiToeul doneze. risi toli iqneba a, nA1 da N mniSvnelobebi?

7. dawereT sruli toloba qulaTa dayofisaTvis, romelic miRebulia cdis

pirTa Soris erTfaqtorian eqsperimentSi.

8. axseniT, ratom icvleba GA XX − mniSvneloba damoukidebeli cvladis

efeqtTan erTad eqsperimentSi.

9. axseniT, ratom icvleba AXX − mniSvneloba Secdomis cvalebadobasTan erTad

eqsperimentSi.

21

10. daasruleT Semdegi tolobebi cdis pirTa Soris erTfaqtoriani

dispersiuli analizisaTvis.

a. 2)( GXX −ΣΣ =

b. SS sruli=

g. dfA=

d. dfSecdoma=

e. MSA=

v. MSSecdoma=

z. F=

11. risi tolia SS srulis mniSvneloba, Tu SS A=50 da SS Secdoma=100?

12. risi tolia SS A mniSvneloba, Tu SS sruli=75 da SS Secdoma=50?

13. risi tolia dfsruli, Tu dfA=4 da dfSecdoma=75?

14. risi tolia MSA, TuU SS A=100 da dfA=4?

15. risi tolia MSSecdoma, Tu SS Secdoma=760 da dfSecdoma=76?

16. bihevioristi mecnierebi iyeneben cdis pirTa Soris erTfaqtorian A sqemas A

faqtoris oTxi doniT. TiToeul doneze aris 11 cdis piri. risi tolia dfA,

dfSecdoma, dfsruli ?

17. CaatareT eqsperimenti damoukidebeli cvladis ori doniT, sami cdis piriT

TiToeul jgufSi da miiReT Semdegi qulebi:

A faqtori

A1 A2

30 34

32 36

28 32

dayaviT es qulebi cxrilSi #10.4 naCvenebi midgomis Sesabamisad.

ipoveTY SS sruli, SS A, SS Secdoma da df TiToeuli SS -saTvis. Semdeg daasruleT

Semajamebeli cxrilis analizi.

18. CaatareT eqsperimenti damoukidebeli cvladis sami doniT, sami cdis piriT

TiToeul jgufSi da miiReT Semdegi qulebi:

A faqtori

A1 A2 A3

30 34 37

22 32 36 35

28 32 33

dayaviT es qulebi cxrilSi #10.4 naCvenebi midgomis Sesabamisad.

ipoveTY SS sruli, SS A, SS Secdoma da df TiToeuli SS -saTvis. Semdeg daasruleT

Semajamebeli ricxviTi cxrilis analizi.

19. Semdegi ori cxrili aris dausrulebeli Semajamebeli cxrili cdis pirTa

Soris erTfaqtoriani dispersiuli analizisaTvis. davuSvaT, cdis pirTa

raodenoba damoukidebeli cvladis TiToeul doneze tolia. SeavseT

gamotovebuli mniSvnelobebi TiToeuli cxrilSi SS da df Soris kavSiris

gamoyenebiT, romelic mocemulia 10-3 da 10-4 tolobebSi, da cxrilSi #10.6

warmodgenil formulaSi. Semdeg upasuxeT am SekiTxvebs TiToeuli

cxrilisaTvis.

a. damoukidebeli cvladis ramdeni done unda iyos gamoyenebuli?

b. ramdeni cdis piris monacemi unda iyos gazomili TiToeul eqsperimentul

jgufSi?

g. ramdeni cdis piri monawileobs kvlevaSi?

cxrili #1

wyaro SS df MS F

faqtori

A

50 1 _ _

Secdoma 260 26 _

sul _ _

cxrili #2

wyaro SS df MS F

faqtori

A

_ 3 6 _

Secdoma _ 76 _

sul 170 79

23

statistikuri hipoTezis Semowmeba F-iT

Frealuris mniSvneloba gvaZlevs SesaZleblobas gadavwyvitoT mniSvnelovnad

gansxvavdeba Tu ara eqsperimentuli pirobebis saSualoebi erTmaneTisagan. imis

gasagebad, rogor xdeba am daskvnis gamotana, aucilebelia mokled

mimovixiloT is faqtorebi, romlebic gavlenas axdenen MSA da MSSecdomaze da

kavSirze MSA da MSSecdomas Soris.

MSA da MSSecdomis mniSvnelobaze moqmedi faqtorebi

MSA

MSA-s mniSvneloba mocemulia formuliT

,MSAA

A

dfSS

=

sadac

ASS = 2)( GA XX −ΣΣ

rogorc qulis dayofis Sesaxeb ganxilvaSia naCvenebi, nebismier eqsperimentSi

cvalebadobis ori wyaro moqmedebs, jgufis saSualosa da srul saSualos

Soris sxvaobis xarisxze da amrigad, SSA-s mniSvnelobaze nebismier

eqsperimentSi: (1) damoukidebeli cvladis gavlena da (2) SerCevis Secdoma.

aqedan MSA, agreTve, icvleba damoukidebeli cvladis gavleniT da SerCevis

SecdomiT. radgan MSA izrdeba damoukidebeli cvladis gavlenis zrdasTan

erTad, is gamoiyeneba rogorc sistematuri dispersiis sazomi, romelic

damoukidebeli cvladis mieraa Seqmnili.

MSSecdoma

MSSecdoma miRebulia Semdegi formulidan

SecdomadfXX A

2)( −ΣΣ

da zomavs mxolod jgufSida Secdomis cvalebadobas eqsperimentSi. is ar aris

damoukidebeli cvladis gavlenis qveS. rogorc ufro adre iyo aRniSnuli, es

24 imis gamo xdeba, rom qulebSi nebismier sistematur cvlilebas Tan axlavs

Carevis efeqtiT ganpirobebuli jgufis saSualos Sesatyvisi zrda an kleba.

amrigad, es gansxvaveba cdis piris qulasa da jgufis saSualos Soris,

)( AXX − ar icvleba Carevis efeqtiT. amitom, MSSecdoma zomavs eqsperimentSi

mxolod jgufSida Secdomis cvalebadobas qulebSi. faqtorebi, romlebic

moqmedeben TiToeul kvadratebis saSualoze cxrilSi #10.7 Sejamebuli.

cxrili #10.7

faqtorebi, romlebis moqmedeben MSA da MSSecdomaze eqsperimentSi.

kvadratebis saSualo ganicdis gavlenas Semdegi mizezebiT

MSA damoukidebeli cvladis gavlena

SerCevis Secdoma

MSSecdoma jgufSida Secdomis cvalebadoba

MSA da MSSecdomis kavSiri

Caunacvlebeli Sefasebebi. orive MSA da MSSecdomebi populaciis qulaTa

dispersiis, σ2-is Caunacvlebeli Sefasebebia. isini Caunacvlebelia, radgan

TiToeuli MS-is misaRebad kvadratebis jami iyofa Tavisuflebis xarisxebze da

ara qulaTa realur raodenobaze, rogorc es xdeba dispersiis gamoTvlisas.

rogorc Caunacvlebeli Sefaseba, MS ar unda iyos sistematurad mcire da arc

meti σ2-ze, roca mxolod Secdomis cvalebadobaa pasuxismgebeli monacemTa

cvalebadobaze, romlis safuZvelze xdeba Sefaseba.

damoukidebeli Sefasebebi. MSA da MSSecdomebi aris, agreTve, populaciis

dispersiis damoukidebeli Sefasebebi. damoukidebeli Sefaseba niSnavs, rom

nebismieri saSualos kvadratis mniSvneloba SeiZleba Seicvalos iseTnairad,

rom masze ar imoqmedebs meores mniSvneloba. es sakiTxi warmodgenili iyo im

ganxilvaSi, romelic exeboda SS mniSvnelobaze moqmed faqtorebs. MSA da

MSSecdomebis am maxasiaTeblebs mivyavarT F mniSvnelobis Semdeg mosalodnel

Sedegebamde.

• F-is mosalodneli mniSvneloba, roca damoukidebel cvlads ara aqvs

gavlena: Tu damoukidebel cvlads ara aqvs gavlena, maSin

eqsperimentSi mxolod Secdomis cvalebadobaa mosalodneli. am dros

MSA da MSSecdoma Seafaseben mxolod Secdomis cvalebadobas da

25

amrigad, iqneba TiTqmis toli. Sesabamisad, F-is mniSvneloba iqneba

TiTqmis 1-is toli.

• F-is mosalodneli mniSvneloba, roca damoukidebel cvlads aqvs

gavlena: roca damoukidebeli cvlads aqvs gavlena, maSin MSA

Seesatyviseba sistematur cvalebadobas, romelSic wvlili Seaqvs

rogorc damoukidebel cvlads, ise SerCevis Secdomas. Sedegad, MSA

ufro didi iqneba, vidre MSSecdoma da Frealuri 1-ze meti iqneba. es

damokidebuleba statistikuri hipoTezis SemowmebaSi F-is gamoyenebis

saSualebas gvaZlevs.

F statistikiT miRebuli statistikuri gadawyvetilebebi

F-is statistikur Semowmebas mosdevs CvenTvis cnobili Semdegi nabijebi:

• nulovani hipoTezis, H0-isa da alternatiuli hipoTezis H1-is

formulireba;

• H0-is WeSmaritebis daSvebis pirobiT F SerCeviTi ganawilebis miReba.

es ganawileba mocemulia cxrilSi #a-3;

• mniSvnelovnobis donis arCeva;

• F kritikuli mniSvnelobis (Fkritikulis) dadgena F-is SerCeviTi

ganawilebidan, romelic mocemulia cxrilSi #a-3;

• uaryofis areebi ganlagebulni arian F-is SerCeviT ganawilebaze;

• F Sefardeba, romelic ganisazRvreba rogorc Frealuri, gamoTvlilia

SerCevis monacemebidan;

• H0–is uaryofis Tu miRebis gadawyvetileba keTdeba imis safuZvelze,

xvdeba Tu ara Frealuri uaryofis areSi.


nulovani hipoTeza. eqsperimentisaTvis damoukidebeli cvladis sami doniT,

nulovani hipoTezis dispersiuli analiziT Semowmeba niSnavs imas, rom im

populaciebs, romlebidanac iyo aRebuli sami SerCeva, aqvs erTi da igive

saSualoebi da iwereba Semdegnairad:

H0: µ A1=µA2=µA3

es nulovani hipoTeza warmoadgens situacias, romelic im SemTxvevaSi Cndeba,

Tu damoukidebel cvlads ara aqvs gavlena. Tu H0 WeSmaritia eqsperimentSi,

26 maSin jgufebis saSuloebs Soris nebismieri gansxvaveba mxolod SerCevis

SecdomiTaa ganpirobebuli.

populaciis saSualoebis raodenoba, romelic gansazRvrulia nulovan

hipoTezaSi, yovelTvis Seesatyviseba damoukidebeli cvladis doneebis

raodenobas. Tu eqsperimenti moicavs damoukidebeli cvladis 5 dones, maSin

Semdegnairad iwereba:

H 0: µ A1=µA2=µA3=µA4=µA5

alternatiuli hipoTeza. alternatiuli hipoTeza H1, aris

H1: yvela µ A ar aris erTmaneTis toli,

populaciis saSualoebis im raodenobis miuxedavad, romelic CarTulia

nulovan hipoTezaSi. alternatiuli hipoTeza ayalibebs iseT situacias,

romelic maSin warmoiqmneba, roca damoukidebel cvlads aqvs gavlena.

F SerCeviTi ganawileba

H0-is uaryofis Tu miRebis gadawyvetileba damokidebulia imaze, Tu ramdenad

iSviaTi an naklebad mosalodneli iqneba Frealuris mniSvneloba, Tu H0 ar aris

WeSmariti. statistikis SerCeviTi ganawileba gvawvdis statistikis

mniSvnelobis albaTobas, roca H0 WeSmaritia. amrigad, Frealuris garkveuli

mniSvnelobis miRebis albaToba, roca H0 WeSmaritia, gansazRvrulia F

statistikis SerCeviTi ganawilebiT, magram ara mxolod F-is SerCeviTi

ganawilebiT, Tavad es ganawileba damokidebulia damoukidebeli cvladis

doneebis raodenobaze da qulaTa raodenobaze TiToeul eqsperimentul jgufSi.

kerZod, F SerCeviTi ganawileba icvleba Tavisuflebis xarisxis cvlilebasTan

erTad F statistikis mricxvelisaTvis (e.i. dfA) da mniSvnelisaTvis (e.i.

dfSecdoma).

rogorc nimuSi, F SerCeviTi ganawileba magaliTis saxiT moyvanil

eqsperimentSi 2 df-is toli mricxveliTa (e.i. 2 df dfA-saTvis) da 12-is toli

Tavisuflebis xarisxis mniSvneliT (e.i. 12 df dfSecdomisaTvis) naCvenebia suraTze

#10.2. es ganawileba dadebiTad asimetriulia da misi umciresi mniSvneloba

nulia; F iqneba nulis toli, Tu SerCevis saSualoebi erTmaneTis toli iqneba

da MSA mniSvneloba _ nulis toli. F-is yvelaze mosalodneli mniSvneloba

daaxloebiT 1-ia, Tu H0 WeSmaritia, maSin MSA unda iyos daaxloebiT

MSSecdomis toli. ar arsebobs zeda zRvari im mniSvnelobebisaTvis, romelsac F-

27

ma SeuZlia miaRwios. F-ma SeiZleba miaRwios nebismier mniSvnelobas 0-s da

dadebiT usasrulobas Soris. radgan F mniSvnelobis miRebis albaToba 0-sa da

dadebiT usasrulobas Soris 1-is tolia, ganawilebis qveS mdebare arec erTis

tolia.

mniSvnelovnobis donis arCeva

mniSvnelovnobis done aris albaTobis mniSvneloba, romelic iZleva nulovani

hipoTezis uaryofis pirobas statistikur kriteriumSi. rogorc ganvixileT,

mniSvnelovnobis doned bihevioristuli mecnierebis kvlevaSi Cveulebriv

aRebulia an α=0.05 an α=0.01. uaryofis aris mdebareoba 0.05 mniSvnelovnobis

donisaTvis naCvenebia suraTze #10.2.

suraTi #10.2

SerCeviTi Teoriuli ganawileba 2 da 12 Tavisuflebis xarisxisaTvis. naCvenebia

uaryofis are α=0.05-saTvis. Fkritikuli cxrili #a-3-dan aris 3.89.

suraTi aklia (relative frequency - SefardebiTi sixSire Rejection region - uaryofis are, value of F_ F-is mniSvneloba, critical value _ kritikuli mniSvneloba)

uaryofis aris mdebareobis gansazRvra F-isaTvis

uaryofis are warmoadgens F-is iseT mniSvnelobebs, romelTa albaToba tolia

an naklebia α-ze, Tu H0 WeSmaritia. amrigad, uaryofis are gansazRvravs

Frealuris iseT mniSvnelobebs, romlebic akmayofileben H0-is uaryofis pirobas.

uaryofis are FrealurisaTvis yovelTvis F SerCeviTi ganawilebis marjvena kudSia,

rogorc suraTzea #10.2 naCvenebi. SegviZlia vnaxoT, ratom aris uaryofis are

ganlagebuli aq, Tu gavixsenebT, rom Frealuris mniSvneloba damoukidebeli

cvladis gavleniT icvleba.

Tu damoukidebel cvlads ara aqvs gavlena da aqedan gamomdinare, H0

WeSmaritia, maSin mosalodnelia, rom MSA da MSSecdoma TiTqmis erTmaneTis

toli iqneba; amis Sedegad, Frealuri TiTqmis erTis toli unda iyos. Tu

damoukidebel cvlads aqvs gavlena da amitom H0 ar aris WeSmariti, maSin MSA

iqneba MSSecdomaze meti da Frealuri gaxdeba erTze meti. imisaTvis, rom uarvyoT

nulovani hipoTeza, Frealuri unda iyos sakmarisad meti erTze, ase rom aseTi

Frealuris albaToba nulovani hipoTezis WeSmaritebis pirobebSi arCeul alfa

28

doneze toli an naklebi iqneba. amitom F-is uaryofis are yovelTvis mdebareobs

F-is ufro did mniSvnelobebs Soris SerCevis ganawilebis marjvena napirze.

F-is kritikuli mniSvnelobebi, romlebic (aRiniSneba, rogorc Fkritikuli)

gansazRvraven uaryofis aris qveda sazRvrebs α=0.05-saTvis da 0.01-saTvis

warmodgenilni aris cxril #a-3-is danarTSi. cxrili #a-3a nawili 0.05

mniSvnelovnobis donisaTvis warmodgenilia cxrilSi #10.8. am cxrilSi F

Sefardebis mricxvelis Tavisuflebis xarisxebi (e.i. dfA) mocemulia cxrilis

zeda striqonSi. mniSvnelis Tavisuflebis xarisxebi (e.i. dfSecdoma) ki mocemulia

cxrilis marcxena svetSi.

Fkritikulis mniSvnelobis sapovnelad zemoT xsenebuli magaliTisaTvis,

romlis mricxvelis Tavisuflebis xarisxi 2-ia da mniSvnelis _ ki 12, unda

vipovoT sveti, romlis Tavisuflebis xarisxi 2-ia da Semdeg vpoulobT

striqons 12 df-isaTvis. es sveti da striqoni gadaikveTeba 3.88-is tol

mniSvnelobasTan (cxrilSi gamuqebulia). es mniSvneloba aris Fkritikuli

da gansazRvravs uaryofis ares FrealurisTvis, Tavisuflebis xarisxebiT 2 da 12,

rogorc naCvenebia suraTze #10.2. Fkritikulis nebismieri mniSvneloba

2 da 12 Tavisuflebis xarisxiT da mniSvnelobiT, romelic 3.88-is tolia an

metia, uaryofis areSi mdebareobs. F-is mniSvnelobebi, romlebic 3.88-is toli an

metia, xdeba 100 eqsperimentidan 5 an ufro naklebjer, roca H0 WeSmaritia.


Frealuris mniSvnelobebi uzrunvelyofs statistikuri hipoTezebis Sesaxeb

gadawyvetilebebis gamotanis safuZvels. Tu Frealuri xvdeba uaryofis areSi,

maSin H0 iqneba uaryofili da Tu H0 iqneba uaryofili, maSin H1 unda iyos

miRebuli. SerCevis saSualoebs Soris statistikurad mniSvnelovani

gansxvavebaa.

cxrili #10.8

Fkritikulis mniSvnelobebi α=0.05-saTvis.U

mniSvnelis

Tavisuflebis

xarisxi

mricxvelis Tavisuflebis xarisxi

1 2 3 4 5

1 161.4 199.5 215.7 224.6 230.2

2 18.51 19 19.16 19.25 19.3

3 10.13 9.55 9.28 9.12 9.01

29 4 7.71 6.94 6.59 6.39 6.26

5 6.61 5.79 5.41 5.19 5.05

6 5.99 5.14 4.76 4.53 4.39

7 5.59 4.74 4.35 4.12 3.97

8 5.32 4.46 4.07 3.84 3.69

9 5.12 4.26 3.86 3.63 3.48

10 4.96 4.1 3.71 3.48 3.33

11 4.84 3.98 3.59 3.36 3.2

12 4.75 3.89 3.49 3.26 3.11

es cxrili aris mxolod mTliani cxrilis nawili, romelic warmodgenilia danarTSi

cxrilSi# A a-3.

Tu Frealuri ar xvdeba uaryofis areSi, maSin H0-s ver uarvyofT da ver miviRebT

H1-s. SerCevis saSualoebs Soris gansxvavebebi ar aris statistikurad

mniSvnelovani da ubralod aramniSvnelovania.

statistikurad mniSvnelovani gansxvaveba. statistikurad mniSvnelovani

gansxvavebis ilustraciisaTvis ganvixiloT zemoT moyvanili eqsperimentis

magaliTi, romelic naxatebis aRdgenas exeboda. am eqsperimentSi xdeboda sami

tipis naxatis manipulireba imisaTvis, rom dagvedgina, moqmedebs Tu ara naxatis

tipi masSi mocemuli elementebis gaxsenebaze. qvemoT mocemulia gaxsenebuli

elementebis raodenobis saSualo, rogorc naxatis tipis funqcia.

naxatis tipi (A)

dausrulebeli

eskizi (A1 )

eskizi ( A2) detaluri

( A3)

AX 17 13 12

cxrilSi #10.1 mocemul qulaTa dispersiuli analizis Sedegad miRebulia

Frealuri (2.12)=6.56 (ix.cxrili #10.5). roca df tolia 2 da 12, Fkritikuli 0.05

doneze aris 3.88. Frealuri aris meti Fkritikulze da amrigad, uaryofis areSi

xvdeba. Sesabamisad, nulovani hipoTeza unda uarvyoT

H0: µ A1=µA2=µA3

da miviRoT alternatiuli

H1: yvela µ A ar aris erTmaneTis toli

30 davaskveniT, rom sami SerCevis saSualo ar aris erTi da imave populaciidan.

damoukidebeli cvladi moqmedebs gaxsenebuli elementebis raodenobaze.

miaqcieT yuradReba, rom arsebobs mravali sxvadasxva gza, imisaTvis, rom

hipoTeza

yvela H1: µ A ar aris erTmaneTis toli iyos WeSmariti. alternatiuli hipoTeza,

magaliTad, WeSmariti iqneba, Tu µ A1 iqneba meti µA2-ze an µA3-ze, µA2 da µA3

iqnebian tolebi, an Tu µA1 iqneba meti µA2-ze da µA2 ki meti µA3-ze. romeli

damokidebuleba gvaCvenebs saSualoTa Soris mimarTulebas am SemTxvevaSi?

mxolod dispersiuli analizi ver gagvcems am kiTxvaze pasuxs garkveviT, H0-is

uaryofisaTvis, roca arsebobs sami an meti eqsperimentuli jgufi

eqsperimentSi. ubralod is gvaZlevs saSualebas davaskvnaT, rom arsebobs sul

cota erTi statistikurad mniSvnelovani gansxvaveba saSualoTa Soris.

(saSualoTa Soris konkretuli mniSvnelovani gansxvavebebis sapovnelad

aucilebelia gamoviyenoT mravaljeradi SedarebisKkriteriumi). am kriteriums

mogvianebiT ganvixilavT am Tavis Semdgom qveTavSi. gadawyvetilebis miRebis

procedura da Sesatyvisi daskvnebi statistikurad mniSvnelovani

gansxvavebisaTvis Sejamebulia cxrilSi #10.9, marcxena svetSi.

aramniSvnelovani gansxvaveba. Tu ver uarvyofT H0-s da ver miviRebT H1-s, maSin

Cven ar gveqneba imis mtkicebis saSualeba, rom nulovani hipoTeza ar aris

WeSmariti. aseTi eqsperimentis mniSvneloba isaa, rom damoukidebel cvlads ara

aqvs gavlena. saSualoebs Soris nebismieri raodenobrivi gansxvaveba miewereba

SerCevis Secdomas. es gadawyvetilebani da daskvnebi Sejamebulia cxril #10.9,

marjvena svetSi.

cxrili #10.9

cdis pirTa Soris erTfaqtoriani sqemis dispersiuli analizis gamoyenebiT

statistikuri hipoTezebis Semowmebis Sesaxeb gadawyvetilebebisa da daskvnebis

Sejameba. gamoyenebuli iyo 0.05 mniSvnelovnobis done.

31

daSvebebi cdis pirTa Soris erTfaqtoriani dispersiuli

analizisaTvis

t kriteriumis analogiurad, cdis pirTa Soris dispersiuli analizi efuZneba

sam daSvebas eqsperimentSi miRebul monacemTa Sesaxeb:

1. monacemebi SerCevaSi SemTxveviTadaa miRebuli populaciis monacemebidan

da TiToeuli monacemi damoukidebelia nebismieri sxva monacemisagan.

2. monacemebi populaciis SerCevaSi normalurad arian ganawilebulni.

3. qulaTa dispersiebi populaciebSi tolia: σ2A1=σ2A2

es daSvebebi mniSvnelovania, radgan F SerCeviTi ganawileba da amitom

Fkritikulis mniSvnelobebi, romelic mocemulia cxrilSi #a-3, im

Tu Frealuri xvdeba


Tu Frealuri ar xvdeba

uaryofis areSi, roca α=0.05, maSin:

Frealuris miRebis albaToba naklebia an

toli 0.05, an p≤0.05

H0 uaryofilia

H1 miRebulia. SerCevis yvela saSualo

ar aris aRebuli erTi da imave

populaciidan.

sul cota, erTi gansxvaveba mainc aris

saSualoebs Soris statistikurad

mniSvnelovani 0.05 doneze.


SerCevis saSualoebs Soris gansxvaveba

SeiZleba ganpirobebuli iyos sxva

mizeziTac, garda SerCevis Secdomisa.

kargad Catarebul eqsperimentSi

gansxvaveba miewereba damoukidebel

cvlads.

Tu damoukidebel cvlads sami an meti

done aqvs, saWiroa mravaljeradi

Sedarebis kriteriumebi.

Frealuris miRebis albaToba metia an

toli 0.05, an p>0.05


H1 ar aris miRebuli

ar aris statistikurad

mniSvnelovani gansxvavebani SerCevis

saSualoebs Soris 0.05 doneze.




SerCevis saSualoebs Soris. Aar

arsebobs damamtkicebeli sabuTi,

rom damoukidebeli cvladi axdens

gavlenas.

ar aris saWiro mravaljeradi

Sedarebis kriteriumebi.

32 populaciebidanaa Tavmoyrili, romlebic akmayofileben am daSvebebs. imisaTvis,

rom cdis pirTa Soris dispersiuli analizi gamoviyenoT, TiToeuli monacemi

eqsperimentSi unda iyos damoukidebeli nebismieri sxva monacemisagan. amis

Sedegad, yovel cdis pirs SeiZleba miewodos damoukidebeli cvladis mxolod

erTi done da yovelma cdis pirma SeiZleba miiRos monawileoba eqsperimentSi

mxolod erTi monacemiT. Tu cdis pirs miewodeba damoukidebeli cvladis

erTze meti done eqsperimentSi da ori an meti monacemi iqneba gaanalizebuli

TiToeuli cdis pirisaTvis, maSin cdis pirTa Sida dispersiuli analizi unda

gamoviyenoT. es dispersiuli analizi ganxilulia

me-12 TavSi.

zogierTma eqsperimentma SeiZleba ver daakmayofilos meore da mesame

moTxovna. meore (monacemTa normaluri ganawileba) da mesame (dispersiaTa

toloba) daSvebis darRvevam SeiZleba Secvalos Fkritikulis mniSvnelobis

miRebis albaToba da amrigad, imoqmedos pirveli gvaris Secdomis daSvebis

albaTobaze. rogorc mosalodnelia, am debulebebis darRvevebs minimaluri

gavlena eqnebaT pirveli gvaris Secdomis daSvebis albaTobaze, roca Semdegi

pirobebi iqneba dakmayofilebuli:

• TiToeul jgufSi cdis pirebis raodenoba erTi da igivea;

• monacemTa ganawilebis forma TiToeuli jgufisaTvis TiTqmis erTi

da igivea da ganawilebebi arc Zalian wawvetebulia da arc Zalian

gaSlili;

• mniSvnelovnobis doned miRebulia 0.05.


1. gansazRvreT: Fkritikuli, Frealuri, uaryofis are.

2. ra faqtorebi moqmedeben MSA-is mniSvnelobaze eqsperimentSi?

3. ganmarteT, ratom izrdeba MSA-is mniSvneloba, Tuki damoukidebel cvlads

aqvs gavlena eqsperimentSi?

4. ra faqtorebi moqmedeben MSSecdomis mniSvnelobaze eqsperimentSi?

5. axseniT, ratom ar icvleba MSSecdomis mniSvneloba, Tu damoukidebel

cvlads gavlena aqvs eqsperimentSi?

6. Frealuris ra mniSvnelobebia mosalodneli, Tu damoukidebel cvlads ara

aqvs gavlena eqsperimentSi?

33

7. ra mouva Frealuris mniSvnelobas, Tu damoukidebel cvlads eqneba gavlena

eqsperimentSi?

8. dawereT statistikuri hipoTezebi F kriteriumisaTvis:

a. ori damoukidebeli jgufisaTvis

b. oTxi damoukidebeli jgufisaTvis

g. eqvsi damoukidebeli jgufisaTvis

9. romel situacias Seesabameba eqsperimentSi dispersiuli analizis H0?

10. romel situacias Seesabameba eqsperimentSi dispersiuli analizis H1?

11. risi tolia F mniSvnelobis qveda zRvari, Tu H0 WeSmaritia?

12. risi tolia F mniSvnelobis zeda zRvari, Tu H0 WeSmaritia?

13. savarjiSoSi mocemulia dfA, dfSecdomisa da Frealuri cdis pirTa Soris

erTfaqtoriani ramdenime hipoTeturi eqsperimentisaTvis. mniSvnelobaTa

TiToeuli simravlisaTvis miiReT Fkritikulis mniSvneloba 0.05 mniSvnelovnobis

donisaTvis cxrilidan #a-3a. Semdeg aRniSneT, xvdeba Tu ara Frealuri uaryofis

areSi da ra gadawyvetilebas miiRebdiT H0 -isa da H1-is mimarTebaSi. Tu ar aris

mocemuli zusti df cxrilSi dfSecdomisaTvis, maSin gamoiyeneT misi Mmomdevno

mniSvneloba.

eqsperimenti dfA dfSecdoma Frealuri

1 1 10 5.12

2 2 24 3.29

3 1 24 4.26

4 2 90 3.24

5 2 12 3.94

6 2 15 3.6

7 4 45 1.93

8 3 20 1.46

9 6 63 2.3

10 4 95 2.13

14. CamoTvaleT daSvebebi, romlebic safuZvlad udevs cdis pirTa Soris

erTfaqtoriani dispersiuli analizis gamoyenebas.

15. ra pirobebSi eqneba populaciis monacemebis normaluri ganawilebidan

gadaxras minimaluri gavlena pirveli gvaris Secdomaze cdis pirTa Soris

erTfaqtoriani dispersiuli analizisaTvis?

34

F-is statistikuri mniSvnelovnobis interpretacia mravaldonian

eqpsperimentSi

mravaljeradi Sedarebis kriteriumebi

adre aRvniSneT, rom roca eqsperimentSi gamoiyeneba damoukidebeli cvladis

sami da meti done, H0-is uaryofisas, vaskvniT, rom arsebobs sul cota erTi

statistikurad mniSvnelovani gansxvaveba saSualoebs Soris. magram analizi ar

gansazRvravs, konkretulad romeli saSualoebi gansxvavdebian erTmaneTisagan.

imis gadasawyvetad, saSualoTa romeli wyvilebi gansxvavdebian erTmaneTisagan

mniSvnelovnad, unda gamoviyenoT mravaljeradi Sedarebis kriteriumebi.

mravaljeradi Sedarebis kriteriumebi statistikuri kriteriumebia, romlebic

gamoiyeneba wyvilTaSoris SedarebebSi imis dasadgenad, romeli saSualoebi

gansxvavdeba mniSvnelovnad erTmaneTisagan erTfaqtorian mravaldonian sqemaSi.

roca mocemuli gvaqvs sami SerCevis saSualo eqsperimentidan, romelic

exeboda naxatis tipis gavlenas detalebis gaxsenebaze, SegviZlia gavakeToT

sami Sedareba or saSualos Soris:

1AX SevadaroT 2AX ,

1AX SevadaroT 3AX ,

2AX SevadaroT 3AX .

radgan Sedarebebidan TiToeuli moicavs or saSualos, maT wyvilTaSoris

Sedarebebs uwodeben. mas Semdeg, rac statistikur Semowmebas am sami wyvilis

SedarebiT SevasrulebT, SegveZleba davadginoT, Tu saSualoTa romeli

wyvilebi gansxvavdeba erTmaneTisagan mniSvnelovnad.

am wyvilTa Sedarebebis ganxorcielebis erT-erTi gza iqneboda

tdamoukidebelis gamoyeneba sami t kriteriumis dasadgenad, TiTo t kriteriumisa

TiToeuli SedarebisaTvis. am sami Semowmebis Sedegidan Cven SegviZlia

ganvsazRvroT, romeli saSualoebi gansxvavdebian erTmaneTisagan mniSvnelovnad.

es sakiTxi ar aris ase advilad gadasawyveti SedarebebSi pirveli gvaris

mravaljeradi Sedarebis kriteriumebi (Multiple comparison tests): statistikuri kriteriumebi, romlebic gamoiyeneba wyvilTaSoris SedarebebSi imis dasadgenad, Tu romeli saSuloebi gansxvavdeba mniSvnelovnad erTmaneTisagan erTfaqtorian mravaldonian sqemaSi.

wyvilTaSoris Sedarebebi (Pairwise comparisons): statistikuri Sedarebebi, romlebic erTmaneTs adareben or saSualos.

35 Secdomis daSvebis albaTobis gamo. gavixsenoT, rom pirveli gvaris Secdoma

xdeba maSin, roca H0 aris realurad WeSmariti, magram uaryofilia


Semowmebisas. imisaTvis, raTa SeinarCunon pirveli gvaris Secdomis daSvebis

albaTobis dabali done, mkvlevrebi Cveulebriv adgenen 0.05 an 0.01

mniSvnelovnobis doneebs. mTliani dispersiuli analizisaTvis ganxilul

magaliTSi (romelic moxseniebulia cxrilSi #10.5), pirveli gvaris Secdomis

albaToba α-s, an 0.05-is tolia. magram, roca mravaljerad Sedarebebs vatarebT,

mkvlevari atarebs mraval statistikur Semowmebas, romelTa raodenoba

damokidebulia Sedarebebis raodenobaze. ganvixiloT, magaliTad, tdamoukidebelis

gamoTvla TiToeuli wyvilTa Soris SedarebebisaTvis ganxilul eqsperimentSi.

TiToeuli t kriteriumisaTvis pirveli gvaris Secdomis albaToba aris α-s

toli. aqedan gamomdinare, sul cota, erTi pirveli gvaris Secdomis daSvebis

albaToba sam Sedarebas Soris daaxloebiT 0.14-ia, rac miuReblad maRali

mniSvnelobaa. Secdomis xarisxi eqsperimentSi aris sul cota erTi pirveli

gvaris Secdomis daSvebis albaToba mainc Sesrulebul SedarebebSi. Secdomis

xarisxi eqsperimentSi Zalian swrafad izrdeba Sedarebebis ricxvis zrdasTan

erTad. magaliTad, ganvixiloT t kriteriumis gamoyeneba 5 wyvilTaSorisi

SedarebisaTvis α=0.05 TiToeuli SedarebisaTvis. Tu H0 aris WeSmariti, maSin

sul cota erTi pirveli gvaris Secdomis albaToba 5 SedarebaSi 0.23-is tolia.

Secdomis es xarisxi aSkarad Zalian maRalia mkvlevarTa umetesobisaTvis.

Secdomis xarisxi gasakontroleblad Seqmnilia t kriteriumis alternativa

mravaljeradi SedarebebisaTvis. Cven ganvixilavT erT konkretul kriteriums

mravaljeradi Sedarebebis erTi tipisaTvis, saboloo SedarebebisaTvis.

saboloo Sedarebebi axorcieleben yvela SesaZlo wyvilTa Sedarebebs, mas

Semdeg rac statistikurad mniSvnelovani Frealuri dadasturdeba sruli

dispersiuli analizisaTvis. termini Post hoc (saboloo Sedareba) laTinuri

terminia da niSnavs amis Semdeg an am faqtis Semdeg. misi gamoyeneba aq niSnavs,

saboloo Sedarebebi (Post hoc comparisons): statistikuri kriteriumebi, romlebic asruleben yvela SesaZlo wyvilTa Sedarebebs, mas Semdeg, rac statistikurad

mniSvnelovani Frealuri dadasturdeba sruli dispersiuli analizisaTvis.

Secdomis xarisxi eqsperimentSi (Error rate in an experiment) aris sul cota erTi pirveli gvaris Secdomis daSvebis albaToba mainc statistikur SedarebebSi, romelic Catarebulia eqsperimentSi.

36

rom Sedareba Sesrulebuli iqneba mas Semdeg, rac dadasturdeba, rom Frealuri

aris statistikurad mniSvnelovani. Tu sruli dispersiuli analizis Frealuri

aramniSvnelovania da H0 ar aris uaryofili, monacemTa Semdgom analizs azri

ara aqvs da saboloo Sedarebebi ar Catardeba. sxvadasxva miznebisaTvis

Sedarebis mravali kriteriumi arsebobs. amjerad, ganvixilavT mxolod tukis

(Tukey) HSD kriteriums.

tukis namdvilad mniSvnelovani sxvaobis (Tukey HSD) kriteriumi

saboloo SedarebebisaTvis

tukis namdvilad mniSvnelovani sxvaobis kriteriumi (Tukey HSD – honestly significant

difference), uzrunvelyofs kritikul sxvaobas (abreviatura CD), rac warmoadgens

minimalur sxvaobas ori Carevis saSualos Soris, romelic statistikurad

mniSvnelovania arCeul α doneze. realurad miRebuli gansxvavebis absoluturi

mniSvneloba saSualoebis wyvilTaSoris SedarebebSi tukis CD-iTaa

Sedarebuli. Tu saSualoTa Soris realurad miRebuli sxvaobis absoluturi

mniSvneloba CD-s toli an metia, maSin SerCevis saSualoebi mniSvnelovnad

gansxvavdeba da ganixileba rogorc sxvadasxva populaciebis warmomadgenlebi.

CD tukis namdvilad mniSvnelovani sxvaobis kriteriumi moiZebneba

Semdegi formulidan

AnMS

qCD Secdoma= ,

sadac

q aris studentizebuli rangobrivi statistikis ricxviTi mniSvneloba,

romelic damokidebulia (1) arCeul α doneze, (2) damoukidebeli

cvladis doneebis raodenobaze da (3) df-ze MSSecdomisaTvis

dispersiul analizSi. q-s mniSvnelobebi mocemulia cxrilSi #a-

4, roca α=0.01 da α=0.05 damoukidebeli cvladis 10-ze meti

donisaTvis.

MSSecdoma aris Secdomis aRmniSvneli termini, romelic aRebulia sruli

dispersiuli analizidan.

kritikuli gansxvaveba (CD Critical difference): aris or saSualos Soris raodenobrivi gansxvaveba, romelic statistikurad mniSvnelovania.

37

nA aris monacemTa raodenoba TiToeul eqsperimentul pirobaSi.

imisaTvis, rom gamoviyenoT es formula, nA mniSvnelobebi

TiToeuli eqsperimentuli pirobisaTvis toli unda iyos.

tukis kritikuli sxvaobis miReba

ganvixiloT tukis namdvilad mniSvnelovani CD sxvaobis miReba im magaliTSi

moyvanili qulebis mixedviT, romelic mocemulia cxrilSi #10.1 da im qulebis

dispersiuli analizis mixedviT, romelic Sejamebulia cxrilSi #10.5.

Frealuri, 6.56 statistikurad mniSvnelovania 0.05 doneze; amrigad, unda uarvyoT

H0:µA1=µA2=µA3 da miviRoT H1: yvela µA ar aris erTmaneTis toli. . sruli

dispersiuli analizi gviCvenebs, rom arsebobs sul cota erTi mniSvnelovani

gansxvaveba im sam saSualos Soris, romlebsac erTmaneTs adareben. imisaTvis,

rom vipovoT konkretulad romeli saSualoebi gansxvavdeba erTmaneTisagan,

unda gamoviTvaloT tukis namdvilad mniSvnelovani kritikuli sxvaoba (HSD

CD).

dispersiuli analizidan, romelic Sejamebulia cxrilSi #10.5,

MSSecdoma=5.333. monacemTa raodenoba TiToeul eqsperimentul jgufSi 5-ia.

amrigad, nA=5. q-s mniSvneloba miRebulia cxrilidan #a-4. es cxrili moiTxovs

(1) damoukidebeli cvladis, raodenobis, a-s, codnas (am magaliTSi a=3) da (2)

dfSecdomis codnas dispersiuli analizidan (am SemTxvevaSi, 12). 0.05 doneze, a=3

da 12 dfSecdomisaTvis, q=3.77. CavsvaT es mniSvnelobebi formulaSi CD-saTvis da

miviRebT:

.9.35333.577.3

MS

==

=An

qCD Secdoma

tukis kritikuli sxvaobis interpretacia

tukis kriteriumisaTvis am magaliTSi CD=3.9-s. gansxvavebis absoluturi

mniSvneloba (roca + an _ niSani ar miiReba mxedvelobaSi) or eqsperimentul

saSualos Soris cxrilSi #10.1, romelic 3.9-is tolia an metia statistikurad

mniSvnelovania 0.05 doneze. sami wyvilTaSorisi Sedareba, TiToeuli Sedarebis

absoluturi mniSvneloba, statistikuri hipoTeza da gadawyvetileba TiToeuli

hipoTezisaTvis naCvenebia cxrilSi #10.10.

38 miaqcieT yuradReba, rom ori wyvilTaSoris Sedareba aris meti 3.9-ze

absoluturi mniSvnelobiT: 1AX da 2AX Soris (17-13=4) da 1AX da 3AX Soris (17-

12=5). TiToeuli es Sedareba avlens statistikurad mniSvnelovan gansxvavebas

Sesadarebel saSualoebs Soris. 2AX -sa da 3AX -s Soris Sedarebis absoluturi

mniSvneloba (13-12=1) naklebia 3.9-ze. amrigad, gansxvaveba eskizsa da detalur

naxats Soris aramniSvnelovania. aqedan gamomdinare, tukis kriteriums

mivyavarT daskvnamde, rom dausrulebeli eskizis gaxseneba( 1AX ) mniSvnelovnad

ufro kargia, vidre eskizis ( 2AX ) an detaluri naxatis( 3AX ), magram eskizis ( 2AX )

gaxseneba ar gansxvavdeba detaluri naxatisagan ( 3AX ).

es Sedegi mexsierebaSi informaciis gadamuSavebis Zalisxmevis hipoTezis

dadasturebaa, romelic warmodgenili iyo am Tavis dasawyisSi. dausrulebeli

eskizebi, romlebic yvelaze met Zalisxmevas moiTxoven gadamuSavebis procesSi,

yvelaze kargad gaaxsendebaT. es Sedegi ar adasturebs gadamuSavebis hipoTezas

romlis mixedviTac detaluri naxati gaaxsendebodaT.

cxrili #10.10

tukis HSD kritikuli sxvaobis gamoyeneba wyvilTa Soris Sedarebebisas

ganxilul eqsperimentSi

Sedarebebi Sedarebis

absoluturi

mniSvneloba

statistikuri

hipoTezebi

gadawyvetileba

1AX da 2AX

Soris (17-13)

4 H0:µ A1=µA2

H0:µ A1≠µA2

uaryofilia

miRebulia

1AX da 3AX

Soris (17-12)

5 H0:µ A1=µA3

H0:µ A1≠µA3

uaryofilia

miRebulia

2AX da 3AX

Soris (13-12)

1 H0:µ A2=µA3

H0:µ A2≠µA3

ver uarvyofT

ar aris miRebuli

A1 = dausrulebeli eskizi, A2 = eskizi, A3= detaluri naxati

ori SemTxveva, roca saboloo Sedarebis kriteriumi ar aris aucilebeli

arsebobs ori magaliTi, roca erTfaqtoriani dispersiuli analizis

gamoyenebisas saboloo kriteriumi ar aris saWiro. pirveli, maSin, roca

39

Frealuri dispersiuli analizisaTvis aramniSvnelovania. aramniSvnelovani

Frealuri gviCvenebs, rom ar arsebobs mniSvnelovani gansxvavebebi saSualoTa

Soris arc erTi wyvilisaTvis analizSi. amitom ar aris saWiro saboloo

Sedareba am gansxvavebebis dasadgenad.

meore magaliTi, roca arsebobs damoukidebeli cvladis mxolod ori

done. Tu Frealuri aris mniSvnelovani am analizisaTvis, maSin H0:µA1=µA2 aris

uaryofili da yvela : µ A ar aris erTmaneTis toli iqneba miRebuli. aSkaraa,

rom erTaderTi saSualoebi, romlebic erTmaneTisagan gansxvavdebian 1AX da 2AX -

ia. amrigad, am SemTxvevaSi ar aris saWiro am saSualoebis Semdgomi Semowmeba.

Carevis efeqtis siZlieris gazomva

t kriteriumis analogiurad dispersiuli analizi gviCvenebs, aqvs Tu ara

damoukidebel cvlads gavlena damokidebul cvladze eqsperimentSi, magram is

ar gviCvenebs gavlenis sidides. Frealuri statistikuad mniSvnelovani

gansxvavebisas gavlenis siZlieris sazomad SeiZleba kvlavac iyos gamoyenebuli

η2.

eta kvadratis gamoTvla dispersiuli analizisaTvis

cdis pirTa Soris erTfaqtoriani dispersiuli analizisaTvis η2 SeiZleba

miRebuli iyos ori alternatiuli formulidan. Tu kvadratebis jamebis

mniSvnelobebi cnobilia, η2 mocemuli iqneba Semdegi saxiT

η2sruliSSSS A=

roca dispersiuli analizisaTvis Frealuri da df cnobilia, η2 SeiZleba miRebuli

iyos Semdegi formulidan:

Secdomarealuri

realuri

df))(())((2

+=

FdfFdf

A

Aη

η2 gamoTvlis magaliTi cdis pirTa Soris erTfaqtoriani dispersiuli

analizisaTvis

40

η2 SeiZleba gamoviTvaloT nimuSis saxiT moyvanili amocanisaTvis an SS

mniSvnelobidan, an dispersiuli analizis Frealurisa da df mniSvnelobebidan,

romelic Sejamebulia cxrilSi #10.5.

η2 SS mniSvnelobidan

η2sruliSSSS A=

52.013470

=

=

η2 Frealurisa da df mniSvnelobebidan

Secdomarealuri

realuri

df))(())((2

+=

FdfFdf

A

Aη

52.012.2512.13

12)56.6)(2()56.6)(2(

=

=

+=

es η2 gviCvenebs, rom monacemTa srul dispersiaSi xdeba 52-procentiani kleba,

roca damoukidebeli cvladis doneebis saSualoebi gamoiyeneba qulaTa

winaswarmetyvelebisaTvis, sruli saSualos rogorc prognozirebuli qulis

mixedviT. es damoukidebeli cvladi axsnis gaxsenebuli qulebis dispersiis did

nawils.

tdamoukidebelsa da F Soris damokidebuleba

tdamoukidebelsa da cdis pirTa Soris erTfaqtorian dispersiul analizs

mivyavarT igive statistikuri gadawyvetilebebisaken, roca gasaanalizebelia

eqsperimentuli monacemebi damoukidebeli cvladis ori doniT. Tu uarvyofT

H0-s ormxrivi SedarebisaTvis 0.05 doneze, maSin unda uarvyoT, agreTve, H0

dispersiuli analizisaTvis, romelic igive monacemebiTaa Sesrulebuli. amis

mizezi isaa, rom t2 =F an t=√F es damokidebuleba, agreTve, SenarCunebulia

41

cxrilSi mocemuli t-sa da F kritikuli mniSvnelobebisaTvis, roca xdeba ori

SerCevis saSualoebis analizi erTi da imave mniSvnelovnobis doneze.

magaliTad, α=0.05-saTvis, Fkritikulis 1, 8 df 5.32-ia. tkritikulis mniSvneloba 8 df da

α=0.05 ormxriv kriteriumSi aris 2.306 da is 5.32-is kvadratuli fesvis tolia.

amrigad, (2.306)2=5.32 da t2=F. t-sa da F-s Soris am damokidebulebis gamo, orive

kriteriumi SeiZleba iyos gamoyenebuli ori damoukidebeli SerCevis

saSualoebis SedarebisaTvis. kriteriumebi uzrunvelyofen identur Sedegebs da

arc erTs ar SeiZleba mivaniWoT upiratesoba. dispersiuli analizis

upiratesoba maSin Cans, roca sam an met eqsperimentul pirobas vadarebT

erTmaneTs.

amocana #10-1

cdis pirTa Soris erTfaqtoriani dispersiuli analizis gamoyeneba tukis

HSD kriteriumiT

amocana

ganwyobilebaze damokidebuli mexsierebis fenomeni niSnavs, rom gaxseneba

umjobesdeba, roca subieqts masalis gaxsenebisas igive ganwyobileba aqvs, rac

misi Seswavlis dros (Bower, 1981). davuSvaT, rom ganwyobilebaze damokidebuli

mexsierebis fenomenis Seswavlisas Tqven SeqmeniT cdis pirTa xuTi

ekvivalenturi jgufi, TiToeulSi 10-10 cdis piriT. TiToeul jgufs Tqven

umuSavebT ganwyobilebas gansxvavebuli emociebis gamomxatveli debulebebis

wakiTxvis saSualebiT, romlebic qmnian an mxiarul, an sevdian, an neitralur

ganwyobilebas. mas Semdeg, rac ganwyobileba gaCndeba, cdis pirebi 15

Cveulebrivi arsebiTi saxelis sias swavloben. 24 saaTis Semdeg cdis pirebi

brundebian, Tqven kvlav umuSavebT ganwyobilebas da sTxovT cdis pirebs

gaixsenon adre naswavli arsebiTi saxelebis sia. gaxsenebis am seriaSi

ganwyobileba igive an gansxvavebuli unda iyos im ganwyobilebisagan, romlis

pirobebSic xdeboda swavla. ganwyobileba, romelic Seqmnili iyo swavlis

dros da gaxsenebis dros 5 jgufisaTvis mocemulia Semdeg cxrilSi:

jgufis ganwyobileba

A1 A2 A3 A4 A5

swavlis dros mxiaruli sevdiani neitraluri mxiaruli sevdiani

42 Seqmnili

ganwyobileba

gaxsenebis

dros

Seqmnili

ganwyobileba

mxiaruli sevdiani neitraluri sevdiani mxiaruli

ganwyobilebaze damokidebuli mexsierebis Teoria winaswarmetyvelebs, rom

jgufebi, romelTac aqvT swavlisa da gaxsenebis kongruentuli mdgomareoba

(e.i. jgufebi A1, A2, A3) unda ixsenebdnen ufro met arsebiT saxels, vidre

jgufebi, romelTac hqondaT gansxvavebuli ganwyobilebebi swavlisa da

gaxsenebis dros (e.i. jgufebi A4 da A5). davuSvaT, rom Tqvens mier miRebuli

Sedegebi _ gaxsenebuli punqtebis raodenoba naCvenebia cxrilSi #10.11. romeli

jgufis saSualoebi gansxvavdebian mniSvnelovnad erTmaneTisagan 0.05 doneze?

amoxsna

gamoyenebulia cdis pirTa Soris erTfaqtoriani sqema damoukidebeli cvladis,

ganwyobilebis jgufis 5 doniT. monacemTa analizSi pirveli nabiji aris

monacemebis cdis pirTa Sorisi erTfaqtoriani dispersiuli analizis Catareba.

Tu Frealuri statistikurad mniSvnelovania, maSin SesaZlebelia gamoviyenoT

tukis namdvilad mniSvnelovani sxvaobis kriteriumi imisaTvis, raTa

davadginoT statistikurad mniSvnelovani wyvilTaSorisi gansxvavebebi

eqsperimentul saSualoebs Soris.

saWiro statistika: F= MSA / MSSecdoma

cxrili #10.11

sworad gaxsenebuli arsebiTi saxelebis raodenoba rogorc ganwyobilebis

jgufis funqcia

ganwyobilebis jgufi

A1 A2 A3 A4 A5

8 14 9 5 6

12 13 12 9 9

12 6 7 6 8

7 9 7 8 8

10 12 11 4 5

43 9 10 6 6 4

13 8 10 10 5

8 7 8 3 7

14 11 13 7 7

11 9 10 3 3

AX 10.4 9.9 9.3 6.1 6.2

sA 2.37 2.6 2.31 2.42 1.93

daSvebebi:

1. cdis pirebi namdvilad SemTxveviTad arian SerCeuli;

2. gaxsenebuli qulebi normaluradaa populaciaSi ganawilebuli;

3. obieqtebis gaxsenebis dispersiebi populaciebSi erTmaneTis tolia.

statistikuri hipoTezebi:

H0: µ A1=µA2=µA3=µA4=µA5

H1: yvela µ A ar aris erTmaneTis toli.


df: dfA= a-1=5-1=4

dfSecdoma= N_a=50-5=45

dfsruli = N-1=50-1=49

F kritikuli mniSvnelobebi: Fkritikuli (4.45)=2.61. 45 df mniSvneloba

mricxvelisaTvis ar aris warmodgenili cxrilSi #a-3a. maSin unda gamoviyenoT

df-is momdevno dabali mniSvneloba, romelic mocemulia cxrilSi dfSecdomisaTvis,

40.

uaryofis are: Frealuris mniSvneloba tolia an metia 2.61-ze.

gamoTvla: dispersiuli analizi SeiZleba gamoTvlili iyos im gamoTvliTi

formulis gamoyenebiT, romelic mocemulia Tavis damatebaSi, an kompiuteruli

programis gamoyenebiT. am Tavis damatebis formula gamoyenebuli iyo

dispersiuli analizis Semajamebeli Semdegi cxrilis misaRebad.

wyaro SS df MS F ganwyobilebis 171.88 4 42.97 7.86*

44 jgufi(A)

Secdoma 245.9 45 5.464

sul 417.78 49

*p<0.05

gadawyvetileba: radgan Frealuri=7.86 xvdeba uaryofis areSi, unda uarvyoT

H0: µ A1=µA2=µA3=µA4=µA5 da miviRoT H1: yvela µ A ar aris erTmaneTis toli. H0-is

uaryofa gviCvenebs, rom arsebobs statistikurad mniSvnelovani erTi

gansxvaveba mainc SerCevis saSualoebs Soris. statistikurad mniSvnelovani

wyvilTaSoris gansxvavebebis dasadgenad, unda gamoviyenoT tukis namdvilad

mniSvnelovani sxvaobis kriteriumi.

tukis kriteriumi: ricxviTi mniSvnelobebi, romelic saWiroa am

kriteriumisaTvis:

• q miRebulia cxrilidan #a-4. gvaqvs damoukidebeli cvladis 5 done

da dispersiuli analizis Semajamebeli cxrilSi mocemulia, rom df

MSSecdomisaTvis tolia 45-is. radgan cxrilSi #a-4 ar aris mocemuli 45

df, unda gamoviyenoT 40 df _ cxrilSi mocemuli momdevno dabali df

MSSecdomisaTvis. amrigad, q=4.04 mniSvnelovnobis 0.05 donisaTvis.

• dispersiuli analizidan miRebuli MSSecdoma 5.464-ia.

• nA, qulebis raodenoba TiToeuli Sesadarebeli saSualosaTvis aris

10.

CavsvaT es ricxviTi mniSvnelobebi formulaSi:

An

MSqCD Secdoma=

=

sityva310464.504.4

=

=

gansxvaveba or saSualos Soris Tavisi absoluturi mniSvnelobiT tolia an

metia 3 sityvaze statistikurad sarwmunoa mniSvnelovnobis 0.05 doniT.

wyvilTaSorisi Sedarebebi da maTi absoluturi mniSvnelobebi, Semowmebuli

statistikuri hipoTezebi da gadawyvetilebebi, romelic miiReba statistikuri

hipoTezebis Sesaxeb TiToeuli SedarebisaTvis, mocemulia cxrilSi #10.12.

45

Sedarebebis absoluturi mniSvnelobebi 1AX -sa da 4AX -s Soris; 1AX -sa da

5AX -s Soris; 2AX -sa da 4AX -s Soris, 2AX -sa da 5AX -s Soris, 3AX -sa da 4AX -s,

3AX -sa da 5AX -s Soris aRemateba CD-s, romelic 3 sityvis tolia. TiToeuli

gansxvaveba statistikurad mniSvnelovania. darCenili Sedarebebis absoluturi

mniSvnelobebi 1AX -sa da 2AX -s Soris, 1AX -sa da 3AX -s Soris, 2AX -sa da 3AX -s

Soris, 4AX -sa da 5AX -s Soris kritikul sxvaobaze naklebia. es Sedarebebi

aramniSvnelovania da SerCevis saSualoebs Soris miRebuli gansxvavebebi

saukeTesod aixsneba SerCevis SecdomiT.

cxrili #10.12

tukis namdvilad mniSvnelovani sxvaobis kritikuli sxvaobis gamoyeneba

wyvilTaSori SedarebebisaTvis ganwyobilebis 5 jgufisaTvis.

Sedarebebi Sedarebis

absoluturi

mniSvneloba

statistikuri

hipoTezebi

gadawyvetileba

1AX da 2AX

Soris (10.4-9.9)

0.5 H0:µ A1=µA2

H0:µ A1≠µA2

ver uarvyofT

ar aris miRebuli

1AX da 3AX

Soris (10.4-9.3)

1.1 H0:µ A1=µA3

H0:µ A1≠µA3

ver uarvyofT

ar aris miRebuli

1AX da 4AX

Soris (10.4-6.1)

4.3 H0:µ A1=µA4

H0:µ A1≠µA4

uaryofilia

miRebulia

1AX da 5AX

Soris(10.4-6.2)

4.2 H0:µ A1=µA5

H0:µ A1≠µA5

uaryofilia

miRebulia

2AX da 3AX

Soris(9.9-9.3)

0.6 H0:µ A2=µA3

H0:µ A2≠µA3

ver uarvyofT

ar aris miRebuli

2AX da 4AX

Soris (9.9-6.1)

3.8 H0:µ A2=µA4

H0:µ A2≠µA4

uaryofilia

miRebulia

2AX da 5AX

Soris (9.9-6.2)

3.7 H0:µ A2=µA5

H0:µ A2≠µA5

uaryofilia

miRebulia

3AX da 4AX

Soris (9.3-6.1)

3.2 H0:µ A3=µA4

H0:µ A3≠µA4

uaryofilia

miRebulia

3AX da 5AX

Soris(9.3-6.2)

3.1 H0:µ3=µA5

H0:µ A3≠µA5

uaryofilia

miRebulia

46

4AX da 5AX

Soris (6.1-6.2)

0.1 H0:µ A4=µA5

H0:µ A4≠µA5

ver uarvyofT

ar aris miRebuli

A1 = mxiaruli / mxiaruli jgufi, A A2 = sevdiani/sevdiani jgufi , A3 =neitraluri

/ neitraluri jgufi, A4 =mxiaruli/ sevdiani jgufi, da A5 =sevdiani/ mxiaruli

jgufi.

efeqtis Zala: η2=171.88/417.78=0.41

daskvna: tukis kriteriums mivyavarT daskvnamde, rom ganwyobilebis jgufebi,

romlebsac hqondaT daswavlisa da gaxsenebis kongruentuli pirobebi (e.i.

mxiaruli / mxiaruli jgufi, A A2 = sevdiani/sevdiani jgufi, A3 =neitraluri /

neitraluri jgufi) gaxsenebis mxriv erTmaneTisagan mniSvnelovnad ar

gansxvavdebian, Tumca TiToeul am jgufSi cdis pirebi gacilebiT met sityvas

ixseneben, vidre romelime arakongruentuli ganwyobilebis jgufSi

(mxiaruli/sevdiani, da sevdiani/ mxiaruli jgufebi). garda amisa,

mxiaruli/sevdiani da sevdiani/mxiaruli jgufebi erTmaneTisagan mniSvnelovnad

ar gansxvavdebodnen. Sesabamisad, dadgenili kavSiri populaciis saSualoebs

Soris aris

H0: µ A1=µA2=µA3 > (µA4==µA5)

η2 gviCvenebs, rom ganwyobilebis jgufis codna winaswarmetyvelebis srul

dispersias 41 procentiT amcirebs.

magaliTi #10-2

cdis pirTa Soris erTfaqtoriani dispersiuli analizis gamoyeneba:

aramniSvnelovani Frealuri

amocana

moqmedebs Tu ara alkoholis miReba pirovnebis gaxsnilobis xarisxze

socialuri gavlenis efeqtebis mimarT? gustafsonma (Gustafson, 1987) ivarauda,

rom moqmedebs. am hipoTezis Sesamowmeblad, man cdis pirebs Sesafaseblad misca

sxvadasxva sigrZis monakveTebi, romelTa sigrZe icvleboda 5-dan 100

santimetramde. cdis pirebs werilobiT atyobinebdnen TavianT Sedegebs da

sTxovdnen meored SeefasebinaT imave monakveTis sigrZeebi. TiToeuli cdis

piri afasebda 75 monakveTis sigrZes. 75-dan 60 Sefasebis SemTxvevaSi maT

47 uwerdnen, rom pirveli Sefaseba ar iyo swori. iTvleboda im SesworebaTa

raodenoba, romelsac TiToeuli cdis piri akeTebda, roca meorejer afasebda

monakveTebs. eqsperimentSi cdis pirTa sami jgufi iyo gamoyenebuli.

sakontrolo jgufs asmevdnen forToxlis wvens eqsperimentis dawyebidan 15

wuTiT adre. meore jgufi iyo placebo sakontrolo jgufi, romelic

elodeboda, rom alkohols miiRebda. am jgufis cdis pirebs, agreTve,

awodebdnen forToxlis wvens sigrZeTa Sefasebamde, magram maT arwmunebdnen,

rom sasmeli Seicavs alkohols. mesame jgufis cdis pirebs miewodaT davalebis

dawyebamde 15 wuTiT adre forToxlis wvenSi gaxsnili 0.8 mililitri

alkoholi sxeulis erT kilogram wonaze. gustanfsonma ivarauda, rom cdis

pirebi alkoholisa da placebo jgufebSi ufro advilad daeqvemdebarebian

werilobiT miwodebul ukukavSirs da amitom TavianT meore Sefasebebs ufro

xSirad Secvlian, vidre sakontrolo jgufis cdis pirebi. davuSvaT Sefasebebis

raodenoba icvleba jgufis TiToeuli 13 cdis pirisaTvis, romelTa monacemebi

mocemulia cxrilSi #10.13. cvlis Tu ara alkoholi Sefasebebis raodenobas?

amoxsna: gamoyenebulia cdis pirTa Soris erTfaqtoriani sqema damoukidebeli

cvladis sami doniT, Carevis jgufiT. cdis pirTa Soris erTfaqtoriani

dispersiuli analizi am monacemebis damuSavebis Sesatyvisi statistikuri

kriteriumia. gamoviyenoT 0.05 mniSvnelovnobis done.

saWiro statistika: FF= MSA / MSSecdoma

daSvebebi:

1. cdis pirebi namdvilad SemTxveviTad arian SerCeulebi;

2. monakveTis sigrZeTa SefasebebSi daSvebuli cvlilebebi normalurad aris

populaciaSi ganawilebuli;

3. monakveTis sigrZeTa SefasebebSi daSvebuli cvlilebebis dispersiebi

tolia SerCeul populaciebSi.


H0: µ A1=µA2=µA3



df: dfA=a-1=3-1=2

48


dfsruli = N-1=39-1=38

F-is kritikuli mniSvnelobebi: Fkritikuli (2.36)=3.32. Fkritikulis mniSvneloba 2, 36

df-saTvis ar aris warmodgenili cxrilSi #a-3a. maSin unda gamoviyenoT F

kritikuli 2, 30 df-isaTvis.

uaryofis are: Frealuris mniSvneloba tolia an metia 3.32-is.

gamoTvla: dispersiuli analizi SeiZleba gamoTvlili iyos im gamoTvliTi

formulis gamoyenebiT, romelic mocemulia Tavis damatebaSi. am analizis

Sejameba Semdegi saxiTaa mocemuli:

wyaro SS df MS F ganwyobilebis

jgufi(A)

389.898 2 194.949 2.24

Secdoma 3137.846 36 87.162

sul 3527.744 38

gadawyvetileba: Frealuri=2.24 Fkritikuli=3.32-ze naklebia da ar xvdeba uaryofis

areSi; amrigad, H0-s ver uarvyofT da H1 miRebuli ar iqneba.

cxrili #10.13

Secvlili Sefasebebis raodenoba, rogorc Carevis jgufis funqcia.

Carevis jgufi(A)

sakontrolo

(A1)

placebo

(A2)

alkoholi

(A3)

18 17 31

2 19 27

11 26 16

3 4 24

26 18 41

18 23 17

9 31 12

54 35 32

17 11 16

49 21 8 19

14 29 35

19 25 26

33 38 17

AX 16.54 21.85 24.08

sA 8.83 10.29 8.81

daskvna: realuri gansxvavebebi saSualoebs Soris aramniSvnelovania; SerCevis

saSualoebi 16.5, 21.8 da 24.1 miekuTvneba erTi da imave populaciis SerCevebs. ar

arsebobs damoukidebeli cvladis gavlenis mtkicebis safuZveli. cdis pirebi,

romelTac miawodes alkoholuri sasmeli, monakveTebis sigrZis TavianT

Sefasebebs ar cvlian ufro xSirad, vidre cdis pirebi, romelTac miawodes

placebo da forToxlis wveni. miRebuli gansxvaveba SerCevis saSualoebs Soris

gamowveulia mxolod SerCevis SecdomiT. vinaidan Frealuri iyo

aramniSvnelovani, ar aris mravaljeradi Sedarebis kriteriumebis an kavSiris

siZlieris gazomvis saWiroeba.

am magaliTis Sedegebi gviCvenebs statistikuri hipoTezis Semowmebis

aucileblobas, raTa ganisazRvros, gansxvavdeba Tu ara jgufebis saSualoebi

ufro metad, vidre mxolod SemTxveviTobis gamo iqneboda. mxolod

daTvalierebiT Cans, rom saSualoebi mniSvnelovnad gansxvavdebian

erTmaneTisagan. magram jgufebSi qulaTa Soris gansxvavebac didia, rac iwvevs

Secdomis cvalebadobis mniSvnelovnad gazrdas. es gamoixateba standartuli

gadaxrebis SedarebiT did mniSvnelobebSi. roca Secdomis cvalebadobis

mniSvneloba didia, maSin Cven SeiZleba velodeT did gansxvavebas jgufis

saSualoebs Soris, rac ganpirobebuli iqneba ubralod SerCevis SecdomiT,

rogorc zemoT ganxilul magaliTSi.

amocana #10-3

cdis pirTa Soris erTfaqtoriani dispersiuli analizi damoukidebeli

cvladis ori doniT

amocana

ganvixiloT me-9 Tavis me-4 kiTxva, romelSic gavecaniT morinagas gadaxrilobis

iluzias. is naCvenebia (a) svetSi suraTze #10.3. rogorc naCvenebi iyo me-4

50 kiTxvaSi, figura warmoadgens iluzias, radgan misi kuTxeebis wveroebi erT

vertikalur xazze ganlagebulad ar aRiqmeba. SenarCundeba Tu ara iluzia, Tu

figura rkaluri iqneba, ise rogorc naCvenebia suraTze #10.3(b)? am kiTxvaze

pasuxis gasacemad, davuSvaT, rom fsiqologebma Seqmnes ori ekvivalenturi

jgufi - TiToeulSi 10 cdis piri. cdis pirebs A1 jgufidan uCvenebdnen

kuTxeebian figurebs da A2 jgufidan _ naxevarrkalovan figurebs, rogorc

naCvenebia suraTze #10.3. damokidebul cvlads warmoadgenda stimulebis aRqmis

dros daSvebuli Secdomebi. is gazomes orive jgufisaTvis da davuSvaT, miiRes

cxrilSi #10.14 mocemuli Secdomis qulebi. romelic mocemuli iyo cxrilSi

damokidebulia Tu ara aRqmuli Secdomis sidide figuris tipze?

suraTi #10.3

morinagas gadaxrilobis iluzia.

(a) figura uCvenes A1 jgufs;

(b) figura uCvenes A2 jgufs.

cxrili #10.14

perceptuli Secdoma, (milimetrebSi), rogorc aRqmuli figuris funqcia

figuris tipi (A)

kuTxovani wriuli

2 1.6

1.8 1.7

2 1.8

2.1 1.6

2.2 1.5

1.8 1.8

1.9 1.7

1.8 1.8

1.9 1.7

1.7 1.6

AX 1.92 1.68

sA 0.15 0.10

51 amoxsna: imis sapovnelad, mniSvnelovnad gansxvavdebian Tu ara saSualoebi, Cven

gamoviyenebT cdis pirTa Soris erTfaqtorian dispersiul analizs. (radgan

mxolod ori jgufi monawileobda, tdamoukidebeli, agreTve, Sesatyvisi

statistikuri kriteriumia). gamoviyenoT 0.05 mniSvnelovnobis done.

saWiro statistika: FF= MSA / MSSecdoma

daSvebebi:

1. cdis pirebi namdvilad SemTxveviTad arian SerCeulebi;

2. iluziis aRqmisas daSvebuli Secdomebi normalurad aris populaciaSi

ganawilebuli;

3. SerCeul populaciebSi aRqmis Secdomebis qulebis dispersiebi tolia.


H0: µ A1=µA2



df: dfA=a-1=2-1=1


dfsruli = N-1=20-1=19

F-is kritikuli mniSvnelobebi: Fkritikuli (1.18)=4.41

uaryofis are: Frealuris mniSvneloba tolia an metia 4.41-ze

gamoTvla: gamoiyeneT gamoTvliTi formula, romelic mocemulia am Tavis

damatebaSi da saboloo analizi Sejamebuli saxiT mocemulia:

wyaro SS df MS F ganwyobilebis

jgufi(A)

0.288 1 0.288 16.62

Secdoma 0.312 18 0.01733

sruli 0.6 19

gadawyvetileba: Frealuri xvdeba uaryofis areSi; amrigad H0 unda uarvyoT da

H1 miviRoT.

52

efeqtis Zala: η2=0.288/0.6=0.48

daskvna: saSualoebi 1.92mm da 1.68 mm mniSvnelovnad gansxvavdebian

erTmaneTisagan 0.05 mniSvnelovnobis doneze. 1.92 mm-is toli saSualo

mniSvnelovnad aRemateba saSualos, romelic tolia 1.68 mm-is. radgan ori

jgufi iyo ekvivalenturi, sanam damoukidebeli cvladi Semovidoda da

damatebiTi cvladebic iyo gakontrolebuli, jgufebs Soris realurad

miRebuli gansxvaveba miewereboda damoukidebeli cvladis gavlenas. cdis

pirebi ufro did iluzias xedaven kuTxovani stimulebis aRqmisas, vidre

wriuli stimulebis aRqmisas. tukis namdvilad mniSvnelovani sxvaobis

kriteriumi ar aris saWiro, radgan saSualos mxolod ori mniSvnelobaa

Sedarebuli. η2 =0.48 gviCvenebs damoukidebeli cvladis did gavlenas, rac xsnis

damokidebuli cvladis dispersiis 48%-s. Tu tdamoukidebels gamoviyenebT

dispersiuli analizis nacvlad, trealuri(18)=4.077. tdamoukidebelis es mniSvneloba

Frealurisagan kvadratuli fesvis tolia (1.18)=16.62. es t statistikurad

mniSvnelovania 0.05 doneze da igive daskvnamde mivyavarT, rasac dispersiul

analizs. es Sedegi, agreTve, gviCvenebs statistikuri hipoTezis Semowmebis

mniSvnelobas monacemTa analizisas. erTi SexedviT, jgufis saSualoebi

erTmaneTisagan mniSvnelovnad ar gansxvavdeba, magram yuradReba miaqcieT imas,

rom jgufis SigniT cdis pirebs Soris cvalebadobac SedarebiT mcirea.

jgufSida Secdomis mcire cvalebadobisas, jgufebs Soris mcire gansxvavebac

ki SeiZleba statistikurad mniSvnelovani iyos.


amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvanelos (1994)

stili moiTxovs dispersiuli analizis Sedegebis prezentacia mocemuli iyos

jgufebis saSualoebi, standartuli gadaxrebi, arCeuli alfas mniSvneloba, F-

is realurad miRebuli mniSvneloba da misi Tavisuflebis xarisxi, MSSecdoma,

da realurad miRebuli F-is albaToba, Tu H0 WeSmaritia. qvemoT mocemulia

Sedegebis prezentaciis es stili erT-erT magaliTze, romelic exeboda naxatis

tipis gavlenas gaxsenebuli detalebis raodenobaze. dispersiuli analizi am

amocanisaTvis warmodgenilia cxrilSi #10.5. Tumca, Cven viyenebT Semajamebel

cxrils dispersiuli analizisaTvis, aseTi cxrili ar SeiZleba moTavsebuli

iyos Jurnalis statiaSi beWdvis maRali fasis gamo. Tu Semajamebeli cxrili

53 ar aris mocemuli, dispersiuli analizi teqstSi Sejamebulia ise, rogorc

qvemoT aris naCvenebi.

sworad gaxsenebuuli detalebis raodenoba iyo 17 dausrulebeli

eskizisaTvis(SD=2.74), 13 eskizisaTvis(SD=2.24) da 12 detaluri

naxatisaTvis(SD=1.87). alfa tolia 0.05-sas cdis pirTa Soris

erTfaqtorianma dispersiulma analizma gviCvena, naxatis tipis

mniSvnelovani gavlena: F(2.12)=6.56, MSE=5.333, p<0.05. saboloo Sedarebam

tukis namdvilad mniSvnelovani sxvaobis kriteriumis gamoyenebiT(CD=3.9,

α=0.05) gviCvena, rom mniSvnelovnad meti iyo gaxsenebuli obieqtebis

raodenoba dausrulebel naxatSi, vidre eskizSi an detalur naxatSi,

magram ar arsebobda mniSvnelovani gansxvaveba gaxsenebuli obieqtebis

raodenobis TvalsazrisiT eskizsa da detalur naxatSi. eta kvadrati am

qulebisaTvis 0.52 iyo.


α=0.05 gviCvenebs am kriteriumisaTvis arCeul mniSvnelovnobis dones.

F(2, 12) gansazRvravs testis statistikas rogorc F-s. amitom

monacemTa analizisaTvis gamoyenebulia dispersiuli analizi.

es F aris Frealuri, Tumca indeqsi(qveSminaweri) `realuri”

SeiZleba ar axldes. Frealuri df mricxvelisaTvis (e.i. 2) da

mniSvnelisaTvis, (e.i. 12) naCvenebia frCxilebSi. am df-idan Tqven

SegiZliaT gansazRvroT eqsperimentSi monawile jgufebisa da

cdis pirebis raodenoba Semdegnairad: df mricxvelisaTvis

tolia a-1-s. am moxsnebaSi dfA=2, amrigad, a=3. df

mniSvnelisaTvis tolia df Secdomisa, romelic tolia N-

a. am magaliTSi dfSecdoma= 12 da a=3; amitom 12=N-3 da cdis

pirebis sruli raodenoba eqsperimentSi tolia 15-s.

=6.56 gvaZlevs Frealuris mniSvnelobas (ara Fkritikulisa, romelsac

vpoulobT cxrilSi #a-3). Tu Frealuri naklebia 1-ze, es

SeiZleba warmodgenili iyos martivad, rogorc F<1.


a) Frealuris albaToba H0 WeSmaritebis pirobebSi aris naklebi

an toli 0.05. es mniSvneloba aris Frealuris albaToba an ufro

54

eqstremaluri Frealuris mniSvnelobis albaToba, roca H0 aris

WeSmariti.

b) H0: µ A1=µA2 ==µA3 uaryofilia

H1: yvela µ A ar aris erTmaneTis toli miRebulia

g) sul cota, erTi gansxvaveba mainc arsebobs jgufebis

saSualoebs Soris, romelic statistikurad mniSvnelovania.

d) saSualoTa Soris realurad miRebuli gansxvaveba

ganpirobebulia ara SerCevis SecdomiT, aramed sxva mizeziT.

e) Tu p>0.05 iqneba SedegebSi warmodgenili, maSin metobis

niSani aRniSnavs imas, rom H0 ar aris uaryofili da ori

SerCevis saSualo erTamaneTisagan mniSvnelovnad ar

gansxvavdeba mniSvnelovnobis 0.05 doneze. Aam SemTxvevaSi

davaskvniT, rom naxatis tipi ar axdens gavlenas gaxsenebuli

detalebis raodenobaze.

MSE=5.333 gvaZlevs MSSecdomis mniSvnelobas FrealurisaTvis. es

mniSvneloba aris jgufSida Secdomis cvalebadobis sazomi

monacemebSi.

CD=3.9 gvaZlevs kritikul gansxvavebas wyvilTaSoris SedarebebSi

tukis

namdvilad mniSvnelovani sxvaobis kriteriumis gamoyenebiT. am

SemTxvevaSi statistikurad mniSvnelovani wyvilTaSoris

Sedarebebia aRwerili.

eta kvadrati

=0.52 η2 mniSvneloba gviCvenebs, rom am eqsperimentSi gaxsenebuli

masalis qulebis dispersiis 52 procenti aixsneba naxatis tipiT.

zogjer gamoiyeneba kavSiris siZlieris sazomi, ω2 (omega

kvadrati).

ω2-is mniSvneloba iqneba odnav mcire, vidre η2.


1. paragrafSi SeamowmeT Tqveni codna IX-3, pirveli amocanis me-9 kiTxva

iyo eqsperimenti, romelic molodinis gavlenas. zomavda drois Sualedis

Sefasebaze. gaixseneT, rom sakontrolo jgufis cdis pirebs ar mouwiaT

55 lodini eqsperimentis dawyebamde, maSin roca eqsperimentuli jgufis cdis

pirebs uTxres, rom eqsperimentatoris asistenti daigvianebda. miuxedavad amisa,

Sualedi orive jgufisaTvis 5 wuTi iyo. mas Semdeg, rac iwyeboda eqsperimenti,

cdis pirebs sTxovdnen SeefasebinaT drois Sualedis xangrZlivoba. davuSvaT,

Tqven gaimeoreT es eqsperimenti 22 cdis pirze da miiReT 5-wuTiani Sualedis

Semdegi Sefasebebi (wuTebSi):

jgufi

sakontrolo eqsperimentuli

6 8

7 9

9 11

7 10

6 9

8 8

5 7

9 6

6 10

8 10

7 9

ipoveT saSualo da standartuli gadaxra TiToeuli jgufisaTvis, Semdeg

gamoiTvaleT dispersiuli analizi da upasuxeT Semdeg kiTxvebs:

a. CamoayalibeT statistikuri hipoTezebi F kriteriumisaTvis am

monacemebze dayrdnobiT.

b. risi tolia Frealuris mniSvneloba?

g. risi tolia df FrealurisaTvis?

d. risi tolia Fkritikuli 0.05 doneze?

e. risi tolia uaryofis are FrealurisaTvis?

v xvdeba Tu ara Frealuri uaryofis areSi?

z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb?

uaryofT, Tu ver uaryofT H0-s? miiRebT Tu ar miiRebT H1-s?

T. jgufis saSualoebi erTi da igive Tu gansxvavebuli populaciebidan

arian?

56 i. aris Tu ara saSualoTa Soris gansxvaveba statistikurad

mniSvnelovani?

k. gansxvavdebian Tu ara jgufebi im drois Sefasebebis mixedviT,

romelsac isini lodinSi atarebdnen? Tu gansxvavdebian, ra

mimarTulebiT gansxvavdebian?

l. saWiroa Tu ara mravaljeradi Sedarebis kriteriumebi am

eqsperimentSi? axseniT Tqveni pasuxi.

m. risi tolia η2 am eqsperimentSi?

n. ramdenad kargad xsnis damoukidebeli cvladi am eqsperimentSi

dispersias?

2. sadehma, ravivma da gruberma (Sadeh, Raviv, Gruber, 2000) ikvlies Zilis

Taviseburebebi me-2, me-4 da me-6 klasel bavSvebSi. davuSvaT, rom Tqvenc

ikvlieT es sakiTxi da meore, meoTxe da meeqvse klaseli gogonebis

jgufebisaTvis aRmoaCineT Zilis Semdegi periodebi wuTebSi. TiToeul jgufSi

iyo 14 bavSvi. Zilis periodi ganisazRvreboda, rogorc drois raodenoba

daZinebidan diliT gaRviZebamde.

klasi

me-2 me-4 me-6

580 497 525

525 515 506

562 543 475

590 478 493

575 567 537

603 532 532

594 517 480

521 510 501

536 534 472

612 556 463

514 511 521

57

544 523 517

573 572 477

592 574 515

ipoveT TiToeuli jgufisaTvis saSualo da standartuli gadaxra, Semdeg

gamoiTvaleT dispersiuli analizi am qulebis gamoyenebiT da upasuxeT Semdeg

kiTxvebs:

a. CamoayalibeT statistikuri hipoTezebi F kriteriumisaTvis am monacemebze

dayrdnobiT.

b. risi tolia Frealuris mniSvneloba?

g. risi tolia df FrealurisaTvis?

d. risi tolia Fkritikuli 0.05 doneze?

e. risi tolia uaryofis are FrealurisaTvis?

v. xvdeba Tu ara Frealuri uaryofis areSi?

z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb? uaryofT

Tu ver uaryofT H0-s? miiRebT Tu ar miiRebT H1-s?

T. ra SegiZlia daaskvnaT am qulebis safuZvelze sruli dispersiuli

analizidan?

i. saWiroa Tu ara mravaljeradi Sedarebis kriteriumiebi am eqsperimentSi?

daasabuTeT Tqveni pasuxi.

k. risi tolia tukis CD mniSvneloba?

l. romeli wyvilTaSoris gansxvaveba aris statistikurad mniSvnelovani?

m. aRwereT, rogor gansxvavdebian jgufebi Zilis periodSi?

n. risi tolia η2 am eqsperimentSi?

o. ramdenad kargad xsnis damoukidebeli cvladi am eqsperimentSi

dispersias?

3. Semdegi savarjiSoebi warmoadgens cdis pirTa Soris erTfaqtoriani

dispersiuli analizis Sedegebis prezentaciis nimuSs. upasuxeT Semdeg

SekiTxvebs TiToeuli angariSisaTvis.

a. damoukidebeli cvladis ramdeni done iyo gamoyenebuli am eqsperimentSi?

b. ramdeni cdis piri monawileobda am eqsperimentSi?

g. risi tolia Frealuris mniSvneloba?

d. risi tolia Fkritikulis mniSvneloba mniSvnelovnobis 0.05 doneze?

58

e. uaryofT Tu ver uaryofT H0-s?

v. miiRebT Tu ar miiRebT H1-s?

z. arsebobs Tu ara, sul cota, erTi statistikurad mniSvnelovani gansxvaveba

saSualoTa Soris?

T. Tu arsebobs, romeli saSualoebi gansxvavdebian mniSvnelovnad

erTmaneTisagan?

3.1. nacnobi leqsis mosmenisas SemCneuli intonaciebis saSualo raodenoba

aris 79.9. xolo ucnobi leqsis mosmenisas SemCneulisa _ 73. alfa tolia 0.05-

saTvis cdispirTa Soris erTfaqtoriani dispersiuli analizi gviCvenebs, rom

saSualoebi mniSvnelovnad gansxvavdebian erTmaneTisagan, F(1.6)=14.15, MSE=5.74,

p<0.01

3.2. roca alfa 0.05-is tolia, ar aris aRmoCenili gansxvaveba gaxsenebuli

sityvebis, rogorc mexsierebaSi Senaxvis intervalis funqciis raodenobaSi,

F(3,36)=1.13 MSE=14.342, p>0.05.

3.3 beWdvis Secdomebis saSualo raodenoba 18.5 iyo im jgufisaTvis, vinc

usmenda klasikur musikas, 29.9 _ vinc usmenda mZime roks da 17.5 _ vinc saerTod

ar usmenda musikas _ sakontrolo jgufisaTvis. alfa tolia 0.05-isa cdis

pirTa Soris erTfaqtoriani dispersiuli analizi gviCvenebs musikis tipis

mniSvnelovan gavlenas, F(2.36)=33.94, MSE=18.261, p<0.001. saboloo Sedarebebi tukis

namdvilad mniSvnelovani sxvaobis kriteriumis gamoyenebiT gviCvenebs

mniSvnelovnad ufro met beWdviT Secdomebs mZime rokis musikis pirobebSi,

vidre musikis gareSe beWdvis an klasikuri musikis mosmenis SemTxvevaSi.

musikis gareSe beWdvisa da klasikuri musikis pirobebi erTmaneTisagan

mniSvnelovnad ar gansxvavdeba.

Sejameba

• erTfaqtoriani mravaldoniani sqema aris eqsperimenti erTi

damoukidebeli cvladiT da am damoukidebeli cvladis sami an meti

doniT.

• cdis pirTa Soris erTfaqtoriani dispersiuli analizi

gamoyenebulia erTi faqtoris mravaldoniani sqemis monacemTa

gasaanalizeblad.

• F statistika mocemulia F= MSA / MSSecdomis saxiT.

• MS aris dispersia da is napovnia SS/df.

59

• MSA icvleba damoukidebeli cvladisa da SerCevis Secdomis gavleniT.

• MSSecdoma icvleba jgufSida Secdomis cvalebadobiT.

• Tu damokidebel cvlads ara aqvs gavlena, maSin MSA da MSSecdoma

afasebs mxolod Secdomis cvalebadobas monacemebSi, da F unda iyos

daaxloebiT 1-is toli.

• roca damoukidebel cvlads aqvs gavlena, maSin MSA mniSvneloba

izrdeba, magram MSSecdomis mniSvnelobaze es ar axdens gavlenas. F-is

mniSvneloba xdeba 1-ze meti.

• F statistikis gamoyenebas statistikur kriteriumSi mohyveba

statistikuri hipoTezebis formulirebis Cveulebrivi nabijebi,

mniSvnelovnobis donis dadgena, uaryofis areebis gansazRvra,

Frealuris gamoTvla da gadawyvetilebebi, romlebic statistikur

hipoTezebs exeba.

• statistikurad mniSvnelovani Frealuri, roca damoukidebeli cvladis

sami an meti donis manipulireba xdeba, moiTxovs mravaljeradi

Sedarebis kriteriums, raTa dispersiuli analizi gagrZeldes.

• sruli dispersiuli analizisaTvis Frealuris statistikuri

mniSvnelovnobis dadasturebis Semdeg mravaljeradi Sedarebis

kriteriumebi gamoiyeneba imis dasadgenad, TuU romeli saSualoebi

gansxvavdebian mniSvnelovnad erTmaneTisagan erTfaqtorian

mravaldonian sqemaSi.

• tukis kriteriumi gamoiyeneba saSualoTa simravlis yvela SesaZlo

wyvilTa SedarebisaTvis saboloo SedarebebSi. tukis kriteriumi

pirveli gvaris Secdomis daSvebis albaTobas inarCunebs α-s tolad

an masze naklebad yvela wyvilTaSoris SedarebebSi.

• η2 SeiZleba gamoviyenoT Carevis efeqtis sazomad.


a

dispersiuli analizi (ANOVA)

jgufTa Soris dispersia (MSA )

kritikuli gansxvaveba (CD)

df SS A-saTvis (dfA)

60

df SS Secdom-isaTvis (dfSecdoma)

Secdomis xarisxi eqsperimentSi

eta kvadrati (η2)

Fkritikuli

Frealuri

A faqtori

sruli saSualo (XG)

saSualos kvadrati (MS)

mravaljeradi Sedarebis kriteriumebi

N

nA

erTfaqtoriani mravaldoniani sqema

wyvilTaSoris Sedareba

saboloo Sedareba

q

SS A

SS Secdoma

SS sruli

sruli Tavisuflebis xarisxi (dfsruli)

tukis HSD kriteriumi

jgufSida Secdomis dispersia (MSSecdoma)

AXX −

GXX −

GA XX −

Tavis danarTi: cdis pirTa Soris erTfaqtoriani dispersiuli

analizis gamoTvla specialuri gamoTvliTi formulis

saSualebiT

dispersiuli analizis gamoTvliTi formula efuZneba qulaTa jams

eqsperimentul jgufSi (da ara jgufis saSualos) da qulaTa srul jams

61 eqsperimentSi ( da ara srul saSualos). Tu gavixsenebT im aRniSvnebs,

romelTac cxrilSi #10.2 gavecaniT, eqsperimenti damoukidebeli cvladis A-s

sami doniTa da 5 cdis piriT TiToeul doneze SeiZleba Semdegnairad iyos

warmodgenili:

A faqtori

A1 A2 A3

X11 X12 X13

X21 X22 X23

X31 X32 X33

X41 X42 X43

X51 X52 X53

TA1 TA2 TA3 G(sruli jami)

sadac Xij=cdis pirTa qulebi

TA=qulaTa jami damoukidebeli cvladis donisaTvis

G=qulaTa sruli jami

nA=qulaTa raodenoba damoukidebeli cvladis doneze

a=damoukidebeli cvladis doneebis raodenoba

N=qulaTa sruli raodenoba

am terminebis gamoyenebiT sami ricxviTi mniSvnelobis dadgena xdeba:

[1] =∑∑ 2ijX yvela qulaTa kvadratebis jami

[2] =A

A

nT∑ 2

TiToeuli jgufis qulaTa jamis kvadratebis jami, gayofili

jgufSi qulaTa raodenobaze

[3] =N

G 2

sruli jamis kvadrati, gayofili qulaTa srul raodenobaze

am ricxviTi mniSvnelobebis gamoyenebiT miRebuli dispersiuli analizi

Semdegnairad gamoiyureba:

62 wyaro SS df MS F faqtori (A) [2] _ [3] a-1

A

A

dfSS

SecdomaMS

MS A

Secdoma [1] _ [2] N_a

Secdoma

Secdoma

dfSS

sul [1] _ [3] N-1 ar

gamoiTvleba

am gamoTvlebis ilustraciisaTvis gamoviyenebT cxrilSi #10.1 magaliTis

saxiT mocemul qulebs, romlis dispersiuli analizi Sejamebulia cxrilSi

#10.5.

faqtori A

A1 A2 A3

19 14 15

13 13 10

16 16 11

17 12 12

20 10 12

TA 85 65 60

G=85+65+60=210

nA=5, a=3, da N=15

ricxviTi gamoTvlebis Sedegad miiReba Semdegi mniSvnelobebi

[1] = 192 + . . . + 202 + 142 + . . . + 102 + 152 + . . . + 122= 3074

[2] = 3010505.15

5606585 222

==++

[3] = 294015

1.4415

2102

==

maSin

SSA= [2]-[3]=3010-2940=70

dfA= a-1=3-1=2

MSA= 352

70=

SS Secdoma= [1]_[2]=3074_3010=64

dfSecdoma = N-a=15-3=12

63

MSSecdoma 333.51264

==

SS sruli = [1] _ [3] = 3074 _ 2940 = 134

dfsruli = N -1=15-1=14

Frealuri = 56.6333.535

==SecdomaMS

MSA

dispersiuli analizis Sejameba Semdegnairad xdeba:

wyaro SS df MS F

faqtori (A) 70 2 35 6.56*

Secdoma 64 12 5.333

sul 134 14

*p< 0.05

am gamoTvliTi midgomiT miRebuli mniSvnelobebi, cxrilSi #10.5 asaxuli im

mniSvnelobebis identuria, romlebic miRebulia dispersiuli analizis

ganmartebiTi formulis saSualebiT.

kiTxvebi

1. Tanamedrove sazogadoebaSi musika yvelgan ismis da fsiqologebi swavloben

mis gavlenas adadmianis mier sxvadasxva amocanis Sesrulebaze. magaliTad,

davuSvaT, eqsperimentatorma Seqmna 8 cdis pirisagan Semdgari sami

ekvivalenturi jgufi da TiToeul pirovnebas sTxova waekiTxaT gazeTis mokle

statiis koreqtura. A A1 jgufis monawileebi am amocanas musikis Tanxlebis

gareSe asrulebdnen, A2 jgufis cdis pirebi irCevdnen Zvel popularul

melodias, xolo A3 jgufis cdis pirebi asrulebdnen amocanas mZime rokis

TanxlebiT. damokidebuli cvladi iyo aRmoCenili Secdomebis raodenoba 50

SesaZleblidan. davuSvaT, Semdegi qulebi iyo miRebuli:

Carevis jgufi

musikis

gareSe

A1

Zveli

popularuli

melodia

mZime roki

A3

64 A2

40 34 26

41 39 24

39 38 19

36 40 23

35 34 18

32 35 21

31 29 23

34 36 29

a. TiToeuli jgufisaTvis ipoveT saSualo da standartuli gadaxra. Semdeg

gaaanalizeT qulebi cdis pirTa Soris erTfaqtoriani analizis safuZvelze da

upasuxeT Semdeg kiTxvas: axdens Tu ara musika gavlenas cdis piris mier

aRmoCenil Secdomebze? gamoiyeneT 0.05 mniSvnelovnebis done. Tu saWiroa,

gamoiyeneT tukis kriteriumi mravaljeradi arCevanisaTvis.

b. risi tolia η2 am esqperimentSi? ra xarisxiT SeiZleba viwinaswarmetyveloT

damokidebuli cvladi, Tu gaviTvaliswineT, rom viciT damoukidebeli cvladis

done?

g. am eqsperimentis Sedegebi im saxiT aRwereT, rogoric naCvenebia qveTavSi


2. skolis maswavlebeli mravalmxrivi niWiT unda xasiaTdebodes, maT Soris

niWiT, rom gamoumuSavos bavSvebs sxvadasxva fizikuri unari, romelic saWiroa

sportis garkveuli saxeobebisaTvis da sxva aqtivobebisaTvis. imisaTvis, rom

ganvsazRvroT, Tu ramdenad kargad SeuZliaT daeuflon maswavleblebi am

xerxebs videofilmebidan, morisonma da rivma (Morrison and Reeve, 1986) Seqmnes

ekvivalenturi jgufi da uCvenes videofilmebi gansxvavebuli instruqciebiT.

pirveli jgufi sakontrolo iyo da firs ar uyurebda. meore jgufma naxa

firi, sadac fexburTelebi asrulebdnen dartymebs da mesame jgufma naxa firi

srolis, daWeris, dartymis unarebze. amis Semdeg cdis pirebs uCvenebdnen meore

videofirs, romelSic iyo demonstrirebuli srolis, daWeris da dartymis

unarebi da sTxovdnen ganesazRvraT, Tu romeli iyo am moqmedebebidan sworad

Sesrulebeuli. davuSvaT Semdegi qulebi miiRes 9 cdis pirisagan TiToeuli

instruqciis pirobebSi:

65

instruqciis piroba

videofiris gareSe (A1) videofiri fexburTelze

(A2)

videofiri srolis,

daWeris da dartymis

demonstrirebiT (A3)

50 53 61

46 48 58

39 37 49

52 49 56

44 46 45

51 48 66

41 37 58

36 41 64

48 40 57

a. ipoveT saSualo da standartuli gadaxra TiToeuli jgufisaTvis, Semdeg

gaaanalizeT qulebi cdis pirTa Soris erTfaqtoriani dispersiuli analiziT,

raTa upasuxoT Semdeg kiTxvebs: SeiZleba instruqciis pirobam moaxdinos

gavlena cdis piris unarze sworad Seafasos srolis, daWeris da dartymis

moqmedebebis Sesruleba? gamoiyeneT 0.05 mniSvnelovnobis done. Tu saWiroa,

gamoiyeneT mravaljeradi Sedarebis tukis namdvilad mniSvnelovani sxvaobis

kriteriumi.

b. risi tolia η2 am eqsperimentSi? damokidebuli cvladis dispersiis ra wili

SeiZleba aixsnas damoukidebeli cvladis donis codniT?

g. aRwereT am eqsperimentis Sedegebi am Tavis erT-erT qveTavSi _

“dispersiuli analizis Sedegebis prezentacia” _ aRwerili stilis mixedviT.

3. mecxre Tavis kiTxvebis amocana #1 exeboda xmauris fiziologiur gavlenas.

aRweril eqsperimentSi, cdis pirebi isxdnen wynar oTaxSi 30 wuTis

ganmavlobaSi, an isxdenen oTaxSi da usmendnen transportis xmaurs pikis

saaTSi 30 wuTis ganmavlobaSi. 30 wuTis gasvlis Semdeg cdis pirebs gauzomes

sistoluri sisxlis wneva da miiRes Semdegi monacemebi( milimetrebSi)

jgufi

66 uxmauro xmauriani

106 141

117 136

124 124

129 139

115 121

131 119

121 147

115 128

128 115

136 134

127 140

a. ipoveT saSualo da standartuli gadaxra TiToeuli jgufisaTvis, Semdeg

gaaanalizeT qulebi cdis pirTa Soris erTfaqtoriani dispersiuli analiziT,

raTa upasuxoT, moqmedebs Tu ara xmauri sistolur sisxlis wnevaze?

gamoiyeneT 0.05 mniSvnelovnobis done.

b. SeadareT Frealuris mniSvneloba trealuris mniSvnelobas, romelic miRebuli

iyo mecxre TavSi mocemul kiTxvebSi (amocana #1). miiReT Tu ara mosalodneli

kavSiri t-sa da F Soris?

4. eqsperimentatorma gamoiyena cdis pirTa Soris erTfaqtoriani sqema

damoukidebeli cvladis 6 doniT, A1, A2, A3, A4, A5 da A6 da 7 cdis piriT

TiToeul doneze.

a. risi tolia a, nA, da N mniSvnelobebi am eqsperimentSi?

b. risi tolia dfA, dfSecdomisa da dfsrulis mniSvnelobebi am eqsperimentis

dispersiuli analizisaTvis?

5. Semdegi cxrili aris dispersiuli analizis dausrulebeli cxrili.

davuSvaT cdis pirTa toli raodenobaa damoukidebeli cvladis TiToeul

doneze. SS , df da MS Soris arsebuli kavSirebis gamoyenebiT SeavseT

TiToeuli cxrilis carieli ujrebi saWiro mniSvnelobebiT.

a. damoukidebeli cvladis ramdeni donis manipulireba xdeba?

b. ramdeni cdis piri iyo Semowmebuli TiToeul pirobaSi?

g. risi tolia cdis pirTa sruli raodenoba am eqsperimentSi?

d. aris Tu ara Frealuris mniSvneloba statistikurad mniSvnelovani 0.05 doneze?

67

e. ra gadawyvetilebas iRebT H0-sa da H1-isaTvis TiToeuli analizis

SemTxvevaSi?

cxrili #1

wyaro SS df MS F

faqtori (A) 50 2 _ _

Secdoma 270 27 _

sul _ _

cxrili #2

wyaro SS df MS F

faqtori (A) 60 _ 12 _

Secdoma _ 72 _

sul 276 77

cxrili #3

wyaro SS df MS F

faqtori (A) _ _ _ 3

Secdoma 320 _ 5

sul 365 67

6. aris Tu ara H0-is uaryofa da H1-is miReba aucilebeli dispersiul

analizSi, raTa uzrunvelyoT kvlevis hipoTezis mxardaWera? axseniT Tqveni

pasuxi.

7. Tqven gamoiTvaleT tdamoukidebeli or jgufze da trealuri(18)=2.000. Tu qulebs

gaaanalizebT dispersiuli analizis Frealuris ra mniSvneloba iqneba napovni?

aRniSneT df FrealurisaTvis.

8. Tqven SeasruleT dispersiuli analizi or damoukidebel jgufze da

Frealurze(1, 22)=9. Tu monacemebs gaaanalizebdiT tdamoukidebeliT, trealuris ra

mniSvnelobas ipovidiT? aRniSneT df trealurisaTvis.

68 9. axseniT, ratom ar SeuZlia statistikur kriteriums daadginos ori jgufis

erTmaneTisagan mniSvnelovnad gansxvavebis mizezi?

10. fsiqologi dainteresebuli iyo 4 pirobis erTmaneTTan SedarebiT

mowevisaTvis Tavis danebebis programaSi. man Seqmna 4 jgufi 17 cdis

pirisaTvis, TiToeuli SemTxveviTi wesiT iyo ganawilebuli da Seuqmna

TiToeul jgufs gansxavebuli mkurnalobis piroba. damoukidebuli cvladi

izomeboda sigaretebis mowevaTa Soris drois saSualoTi mkurnalobidan 5

kviris Semdeg. mkurnalobis jgufebi da saSualo dro sigaretebis mowevas

Soris(wuTebSi) mocemulia cxrilSi #1, xolo dispersiuli analizi am

monacemebisaTvis warmodgenilia cxrilSi #2.

cxrili #1

Carevis pirobebi da saSualo dro sigaretebis mowevas Soris (wuTebSi)

Carevis piroba saSualo dro

A1 sakontrolo jgufi, ar aris Careva 24

A2 qcevis modifikacia 38

A3 sigaretis anonimuri klubebi 34

A4 sigaretis Cvevis sawinaaRmdego saReWi rezina 28

cxrili #2

dispersiuli analizis Semajamebeli cxrili #1 sigaretis mowevebs Soris

saSualo droisaTvis

wyaro SS df MS F

faqtori (A) 1972 3 657.333 21.64*

Secdoma 1944 64 30.375

sul 3916 67

*p < 0.01

tukis namdvilad mniSvnelovani sxvaobis Sedegad gamoTvlili CD=5 wuTi 0.05

mniSvnelovnobis doneze.

a. risi tolia Carevis pirobebis gavlena im drois raodenobaze, romelic

arsebobs sigaretis moweva Soris?

b. aRwereT am eqsperimentis Sedegebi am Tavis erT-erT qveTavSi _

“dispersiuli analizis Sedegebis prezentacia” _ aRwerili stilis mixedviT.

Tavi XI. cdis pirTa Soris orfaqtoriani sqema da dispersiuli analizi cdis pirTa Soris 2x2 sqemis magaliTi faqtoruli sqemis Sedegad miRebuli informacia ujredis saSualoebi mTavari efeqtis saSualoebi damoukidebeli cvladebis urTierTqmedeba cdis pirTa Soris orfaqtoriani sqemis dispersiuli analizi monacemTa dayofa dayofili monacemebisagan gadaxrebis kvadratebis saSualos miReba F statistikis gamoTvla statistikuri hipoTezis Semowmeba cdis pirTa Soris orfaqtorian dispersiul analizSi

faqtorebi, romlebic gavlenas axdenen gadaxrebis kvadratebis saSualoebze dispersiul faqtorul analizSi F statistikaze damyarebuli statistikuri gadawyvetilebebis miReba cdis pirTa Soris orfaqtoriani dispersiuli analizis daSvebebi

dispersiis 2x2 faqtoriani dispersiuli analizis axsna damoukidebeli cvladis martivi efeqti Sejameba: dispersiuli analizis axsna eqsperimentis magaliTze

Carevis efeqtis siZlieris gazomva dispersiul faqtorul analizSi

dispersiuli faqtoruli analizis Sedegebis prezentacia

`mravali qceva ganisazRvreba ori an meti erTdroulad moqmedi damoukidebeli cvladis mier. magaliTad, xmauris (damoukidebeli cvladi) gavlena wakiTxulis gaazrebaze (damokidebuli cvladi) gansxvavdeba pirovnebis introvertuli da eqstravertuli tipebisaTvis (meore damoukidebeli cvladi) (Standing, Lynn, & Moxness, 1990). an, Tu sxva magaliTs ganvixilavT, arsebiTi saxelebis wyvilebis (damokidebuli cvladi) damaxsovreba damokidebulia im adamianis warmosaxvis unarze (erTi damoukidebeli cvladi), vinc imaxsovrebs sias, suraTebis tipebze, romelic miewoda adamians (meore damoukidebeli cvladi), da damaxsovrebis drois intervalis xangrZlivobaze (mesame damoukidebeli cvladi) (O’Brien & Wolford, 1982). erTfaqtoriani sqema ar iZleva SesaZleblobas, erTdroulad moxdes erTze meti damoukidebeli cvladiT manipulireba; amitom, imisaTvis, rom Seiswavlon ori an meti damoukidebeli cvladis kombinaciis gavlena, gamoiyeneba faqtoruli analizi. faqtoruli sqema: kvlevis iseTi sqema, romelSic erTdroulad icvleba ori an meti damoukidebeli cvladi.

faqtoruli sqema aris kvlevis iseTi sqema, romelSic erTdroulad icvleba ori an meti damoukidebeli cvladi. yvelaze martiv faqtorul sqemaSi cvalebadobs ori damoukidebeli cvladi da TiToeul damoukidebel cvlads ori mniSvnelobis (donis) miReba SeuZlia. am sqemas xSirad 2x2 (“ori orze”) sqemas uwodeben. pirveli ‘2’ aRniSnavs, rom arsebobs pirveli damoukidebeli cvladis (romelic ganisazRvreba, rogorc A faqtori) ori mniSvneloba (ori done), xolo meore ‘2’ ki aRniSnavs meore damoukidebeli cvladis (romelic ganisazRvreba, rogorc B faqtori) mniSvnelobebis (doneebis) raodenobas. 2x2x2 sqema warmoadgens samfaqtorian sqemas, sadac faqtorebia A, B da C, da TiToeuli maTgani or mniSvnelobas (dones) iRebs. arsebobs faqtoruli analizis mravali SesaZlo varianti, rogoricaa 2x4 sqema (A faqtoris ori done, B faqtoris 4 done), 3x3x2 sqema (A da B faqtorebis ori done, C faqtoris ori done) da 2x2x2x2 sqema (oTxi faqtori, A, B, C da D, TiToeuls aqvs ori done). faqtoruli analizi aseve SesaZlebelia iyos srulad cdis pirebs Soris, srulad cdis pirebs SigniT an am oris kombinacia, rac Sedegad Sereul sqemas gvaZlevs. am TavSi Cven ganvixilavT mxolod cdis pirebs Soris orfaqtorian sqemas (romelsac aseve uwodeben ormxriv sqemas). ujredi an Carevis piroba: faqtorul eqsperimentSi TiToeuli damoukidebeli cvladis erTi donisagan warmoqmnili kombinacia.

cdis pirebs Soris 2x2 sqema mosaxerxebelia faqtoruli sqemebis statistikuri monacemebis analizis wardgenisaTvis. cxrili # 11.1 warmoadgens or damoukidebel cvlads an faqtors, romlebic aRniSnulia, rogorc A da B. maTgan TiToeuli iRebs or mniSvnelobas (dones): faqtori A iRebsAA1 da A2 mniSvnelobebs da faqtori B iRebs B1 da B2 mniSvnelobebs. zogadad, arsebobs A faqtoris a mniSvneloba da B faqtoris b mniSvneloba. am magaliTSi a=2 da b=2. ori damoukidebeli cvladis kombinacia cxrilSi qmnis oTx pirobas, romlebsac ujredebs uwodeben. TiToeuli ujredi anu piroba aris kombinacia, romelic iqmneba

TiToeuli damoukidebeli cvladis erTi mniSvnelobiT, magaliTad, A1B1, A1B2, A2B1 da A2B2 ujredebi. Cveulebriv, TiToeul ujredSi SemTxveviTad nawildeba cdis pirebis toli raodenoba. amdenad, Tu cdis pirebs Soris 2x2 sqemaSi gamoyenebulia 20 cdis piri da TiToeul ujredSi iqneba xuTi cdis piri. yoveli cdis piri mxolod erT ujredSi xvdeba. cxrili # 11.1 cdis pirebs Soris 2x2 sqemis gegma. A faqtoris doneebis kombinacia B faqtoris doneebTan qmnis pirobas anu ujreds.

faqtori A

A1 A2

B1 faqtori B B2

A1B1 piroba (A1B1 ujredi)




cdis pirebs Soris 2x2 sqemis magaliTi zogadad adamianebs ar uyvarT sxva adamianebisTvis cudi siaxleebis Setyobineba. am fenomens MUM efeqts uwodeben. am efeqtis asaxsnelad SemoTavazebuli iyo ramdenime hipoTeza. pirovnuli diskomfortis hipoTezis mixedviT, cudi siaxlis momtani adamiani pirovnul diskomforts grZnobs am siaxlis Setyobinebisas da Tavs aridebs am diskomforts imiT, rom ar yveba am siaxles. meore hipoTeza – TviTprezentaciis hipoTeza – amtkicebs, rom individebi ar grZnoben diskomforts cudi siaxlis Setyobinebisas. piriqiT, cudi siaxlis Setyobinebis Seusrulebloba warmoadgens publikaze gaTvlil mcdelobas, SeinarCunon dadebiTi socialuri imiji. bolos da bolos, vis surs, cudi siaxleebis mimtanad iyos cnobili? am hipoTezebis Sesamowmeblad bondma da andersonma (Bond & Anderson, 1987) gamoiyenes cdis pirebs Soris 2x2 sqema. ori damoukidebeli cvladi iyo cdis piris arseboba im adamianis mxedvelobis areSi, visTvisac unda eambnaT cudi ambebi (cdis piris yofna mxedvelobis areSi, faqtori A) da axali ambebis tipi (siaxlis tipi, faqtori B). TiToeuli damoukidebeli cvladi iRebda or mniSvnelobas. cdis piris mxedvelobis areSi yofnis mxriv, igi an ar imyofeboda (mniSvneloba A1) an imyofeboda (mniSvneloba A2) im piris mxedvelobis areSi, visTvisac unda emcnoT axali ambebi. siaxlis tipi iyo an kargi (mniSvneloba B1), an cudi (mniSvneloba B2). eqsmerimentis procedura Semdegi iyo: cdis pirs unda Caetarebina mravlobiTi pasuxis mqone inteleqtis testi sxva pirisaTvis. es piri eqsperimentatoris TanaSemwe iyo, magram cdis pirma ar icoda amis Sesaxeb. cdis piri da TanaSemwe imyofebodnen sxvadasxva oTaxebSi, romelTa Soris calmxrivi SuSa iyo CamontaJebuli. im pirobaSi, romelSic cdis piri ar Canda (mniSvneloba A1), mas eubnebodnen, rom TanaSemwes ar SeeZlo misi danaxva SuSaSi. im pirobaSi, romelSic cdis piri Canda (mniSvneloba A2), mas eubnebodnen, rom TanaSemwes SeeZlo misi

danaxva SuSaSi. siaxlis tipis cvla xdeboda TanaSemwes mier testis warmatebuli Tu warumatebeli SesrulebiT. kargi siaxlis dros (mniSvneloba B1), cdis piri TanaSemwes awvdida informacias, rom misi Sedegi testis Sedegebis saukeTeso 20 procentSi xvdeboda. cudi siaxlis pirobaSi (mniSvneloba B1), cdis piri TanaSemwes awvdida informacias, rom misi Sedegi testis Sedegebis yvelaze dabal 20 procentSi xvdeboda. damokidebuli cvladi iyo cdis piris meryeobis periodi (latenturi dro) testis Sedegis Setyobinebisas. es periodi warmoadgenda dros testis dasrulebidan im momentamde, rodesac cdis piri atyobinebda TanaSemwes Tavis Sedegs. MUM efeqtis Sesaxeb arsebuli ori hipoTeza eqsperimentis gansxvavebul Sedegebs winaswarmetyvelebda. Tu MUM efeqts safuZvlad udevs cdis pirebis pirovnuli diskomforti, maSin cudi siaxlis Setyobinebas ufro didi dro dasWirdeboda, vidre kargi siaxlisa, imisda miuxedavad, aris Tu ara cdis piri TanaSemwes mxedvelobis areSi. sxva sityvebiT, kargi (mniSvneloba B1) da cudi (mniSvneloba B2) siaxleebis pirobebs Soris gansxvaveba ar unda iyos damokidebuli imaze, xedavs Tu ara TanaSemwe cdis pirs. meore mxriv, Tu MUM efeqti ubralod publikaze gaTvlili qmedebaa, maSin latenturi dro cudi siaxlis miwodebisas damokidebuli unda iyos imaze, xedavs Tu ara TanaSemwe cdis pirs. es niSnavs, rom unda moxdes damoukidebeli cvladebis urTierTqmedeba. rodesac xdeba kargi siaxlis Setyobineba, ar unda iyos gansxvaveba latentur droSi im pirobis miuxedavad, xedavs Tu ara TanaSemwe cdis pirs. xolo rodesac cudi siaxlis Setyobineba xdeba, latenturi dro im pirobaSi, rodesac cdis piri ar imyofeba mxedvelobis areSi, unda mniSvnelovnad maRali iyos, vidre latenturi dro im pirobaSi, rodesac cdis piri imyofeba mxedvelobis areSi. Sesabamisad, rodesac cdis piri ar imyofeba mxedvelobis areSi, ar unda iyos gansxvaveba latentur droSi kargi da cudi siaxleebis Setyobinebisas. xolo rodesac cdis piri imyofeba mxedvelobis areSi, cudi ambebis Setyobinebis latenturi dro unda ufro didi iyos, vidre kargi ambebisa. davuSvaT, xuTi mamrobiTi sqesis cdis piri SemTxveviTad gadanawilda TiToeul ujredSi da miviReT latenturi drois monacemebi (wamebSi), romlebic warmodgenilia cxrilSi # 11.2. im SemTxvevaSi, Tu Sedegebi saerTod romelime winaswarmetyvelebas Seesabameba, romel hipoTezas miekuTvneba es winaswarmetyveleba - pirovnuli diskomfortis hipoTezas, Tu publikaze gaTvlili qmedebis hipoTezas? cxrili # 11.2 pasuxis hipoTeturi latenturi dro (wamebSi), rogorc cdis piris mxedvelobis areSi yofnisa da siaxlis tipis funqcia 20 cdis pirisaTvis sanimuSo cdis pirTa Soris 2x2 eqsperimentSi.

cdis piris mxedvelobis areSi

yofnis piroba (faqtori A) ar Cans (A1) Cans (A1)

siaxlis tipi (faqtori B)

kargi (B1)

97.0 90.0 80.0 107.0 80.0

70.0 87.0 81.0 95.0 90.0

cudi (B2)

68.0 87.0 92.0 80.0 84.0

114.0 96.0 127.0 110.0 115.0

faqtoruli analizis Sedegad miRebuli informacia cxrilis # 11.2 monacemebi (qulebi) sqematuradaa warmodgenili cxrilSi # 11.3. cdis piris monacemi aRniSnulia Xijk, sadac qveda indeqsebi migvaniSneben Semdeg informaciaze: i = ujredSi cdis piris nomeri j = A cvladis done k = B cvladis done cxrili # 11.3 cdis pirebs Soris 2x2 sqemaSi monacemTa sqematuri ganawileba, sadac TiToeul damoukidebel cvlads ori done aqvs da yovel ujredSi xuTi cdis piria.

faqtori A A1 A2

B1 faqtori B B2

X111 X211 X311 X411 X511

X121 X221 X321 X421 X521

X112 X212 X312 X412 X512

X122 X222 X322 X422 X522

magaliTad, X311 aris mesame cdis piris monacemi A1B1 pirobaSi (80.0 wami cxrilSi # 11.2), X421 aris meoTxe cdis piris monacemi A2B1 pirobaSi (95.0 wami), X211 aris meore cdis piris monacemi A1B2 pirobaSi (87 wami), X522 aris mexuTe cdis piris monacemi A2B2 pirobaSi (115 wami) da a.S. rodesac ar

aris aucilebeli konkretuli cdis piris aRniSvna, Cven gamovtovebT ijk indeqsebs da ubralod viyenebT X-s. TiToeuli ujredisaTvis arsebobs nAB cdis pirebis raodenoba, xolo N aris cdis pirebis sruli raodenoba eqsperimentSi. Cvens magaliTSi nAB = 5 da N = 20. monacemebis analizis dros pirveli nabiji centraluri tendenciis sazomis, SerCevis saSualos gamoTvla. cxrilSi # 11.4 naCvenebia ujredisa da mTavari efeqtis saSualoebi, romelTa gamoTvlac SesaZlebelia magaliTis monacemebidan. cxrili # 11.4 ujredisa da mTavari efeqtis saSualoebi cdis pirebs Soris 2x2 sqemaSi. _______________________________________________________________________________

a. ujredis saSualo faqtori A

B1 faqtori B B2

A1 A2

XA1B1 X A2B1 X A1B2 X A2B2

(b) mTavari efeqtis saSualo A faqtorisaTvis,X A1 daX A2

(g) mTavari efeqtis saSualo B faqtorisaTvis, X B1 daX B2

ujredis saSualo: ujredSi nAB qulebis saSualo faqtorul sqemaSi.

faqtori A

B1 faqtori B B1

A1 A2

1AX 2AX

faqtori A A1 A2

B1 faqtori B B2

X B1 X B2

yuradReba ar unda mivaqcioT qulebis klasifikacias B faqtoris mixedviT, raTa miiRoT mTavari efeqtis saSualo A faqtorisaTvis. gansxvavebaX A1-sa daX A2-s Soris aris A faqtoris mTavari efeqti.

yuradReba ar unda mivaqcioT qulebis klasifikacias A faqtoris mixedviT, raTa miiRoT mTavari efeqtis saSualo B faqtorisaTvis. gansxvavebaX B1-sa daX B2-s Soris aris B faqtoris mTavari efeqti.

ujredis saSualo, romelic simbolurad aRiniSneba 11BAX , 21BAX , 12BAX , da

22BAX , an, zogadad, rogorc ABX , warmoadgens ujredSi nAB qulebis saSualos (ixileT cxrili # 11.4a). ujredis saSualo aRniSnavs cdis pirebis tipur Sesrulebas am pirobaSi. ujredebis saSualoebis ricxviTi mniSvnelobebi cxrilisaTvis # 11.2 warmodgenilia cxrilSi # 11.5. standartuli gadaxrebi warmodgenilia frCxilebSi. faqtorul sqemaSi standartuli gadaxra, rogorc wesi, gamoiTvleba mxolod ujredis saSualosaTvis. cxrili # 11.5 ujredisa da mTavari efeqtis saSualoebi qulebisaTvis cxrilidan # 11.2. ricxviTi mniSvnelobebi aris pasuxis latenturi dro wamebSi. ujredis saSualos standartuli gadaxrebi mocemulia frCxilebSi.

cdis piris yofna mxedvelobis areSi (faqtori A)

ar Cans (A1) Cans (A2) mTavari efeqtis saSualo siaxlis

tipisaTvis ( BX )

kargi (B1) siaxlis tipi (faqtori B) cudi (B2) mTavari efeqtis saSualo cdis piris yofnisaTvis mxedvelobis

areSi ( AX )

90.8 84.6 87.7 (11.56) (9.61) 82.2 112.4 97.3 (9.07) (11.15) 86.5 98.5 sruli saSualo 92.5

mTavari efeqtis saSualo: faqtorul analizSi damoukidebeli cvladis erTi donisaTvis yvela cdis piris saSualo, meore damoukidebeli cvladis doneebis mixedviT klasifikaciis ignorirebiT.

mTavari efeqtis saSualo aRniSnavs yvela cdis piris tipur Sesrulebas damoukidebeli cvladis erTi donisaTvis, meore damoukidebeli cvladis doneebis mixedviT klasifikaciis ignorirebiT. amdenad, arsebobs mTavari efeqtis saSualoebis ori erToblioba – erTi A faqtorisaTvis, xolo meore – B faqtorisaTvis.

A faqtoris mTavari efeqti: gansxvaveba 1AX -sa da 2AX -s Soris, gamoxatuli 1AX - 2AX .

A faqtori

mTavari efeqtis saSualo A faqtorisaTvis aRiniSneba 1AX -iTa da 2AX -iT. es saSualoebi, romelTac xandaxan svetebis saSualoebs uwodeben, miiReba yvela cdis pirisaTvis rogorc A1, aseve A2 doneebis Sesabamisad, imisda miuxedavad, Tu ra mniSvnelobebi miiRes maT B faqtoris mixedviT. B faqtoris doneebis mixedviT monacemebis ignorireba, raTa miRebul iqnes A faqtoris mTavari efeqtis saSualo, naCvenebia cxrilSi # 11.4b. magaliTad, mTavari efeqtis saSualo A faqtorisaTvis, romelic aris cdis piris yofna mxedvelobis areSi cxrilSi # 11.2, damyarebulia im cdis pirebis 10 monacemze, romlebic an imyofebian, an ar imyofebian eqsperimentatoris TanaSemwes mxedvelobis areSi. siaxlis tipi (faqtori B) ignorirebulia, rodesac viTvliT A faqtoris mTavari efeqtis

saSualos. 1AX -isa da 2AX -s ricxviTi mniSvnelobebi warmodgenilia

cxrilSi # 11.5. A faqtoris mTavari efeqti aris gansxvaveba 1AX -sa da

2AX -s Soris. Tu A faqtors aqvs gavlena damokidebul cvladze, maSin

1AX da 2AX gansxvavebuli iqneba erTmaneTisagan da 1AX - 2AX mniSvneloba ufro didi gaxdeba.

B faqtoris mTavari efeqti: gansxvaveba 1BX -sa da 2BX -s Soris, gamoxatuli 1BX - 1BX .

B faqtori igive logika gamoiyeneba B faqtorisaTvis mTavari efeqtis saSualos gamoTvlisas. mTavari efeqtis saSualo B faqtorisaTvis (romelsac

xandaxan uwodeben striqonebis saSualos) aRiniSneba 1BX -iTa da 2BX -iT da miiReba A faqtoris monacemebis ignorirebiT, rogorc es naCvenebia cxrilSi # 11.4g. B faqtoris, Cvens magaliTSi siaxlis tipis, mTavari efeqtis saSualos gamoTvlisaTvis xdeba cdis pirebis kategorizaciis ignorireba imis mixedviT, imyofebian Tu ara isini mxedvelobis areSi (A faqtori). 1BX -isa da 2BX -s ricxviTi mniSvnelobebi aseve warmodgenilia

11.5 cxrilSi. B faqtoris mTavari efeqti aris sxvaoba 1BX -sa da 2BX -s

Soris. Tu B faqtors aqvs gavlena damokidebul cvladze, maSin 1BX da

2BX gansxvavebuli iqneba erTmaneTisagan da 1BX - 2BX mniSvneloba ufro didi gaxdeba. sruli saSualo

sruli saSualo, romelic aRiniSneba GX , aris cxrilis yvela monacemis saSualo. cxrilis # 11.2 monacemebisaTvis sruli saSualo – 92.5 wami – aris yvela 20 qulis saSualo. Cveulebriv, sruli saSualo ar aris xolme warmodgenili aRweriT statistikaSi, magram am TavSi saWiroa misi ganxilva, raTa movaxdinoT dispersiuli analizi am sqemisaTvis. damoukidebeli cvladebis urTierTqmedeba urTierTqmedeba: situacia faqtorul sqemaSi, rodesac erTi damoukidebeli cvladis efeqti damokidebulia masTan kombinaciaSi myofi meore damoukidebeli cvladis mniSvnelobaze.

faqtorul sqemaSi urTierTqmedeba xdeba, rodesac erTi damoukidebeli cvladi (mag., faqtori A) damokidebulia meore damoukidebeli cvladis mniSvnelobaze (mag., an B1 an B2), cvladisa, romelTanac aris kombinaciaSi. magaliTad, Tu urTierTqmedeba xdeba 2x2 sqemaSi, maSin A faqtoris efeqti, gansxvaveba qcevaSi A1 da A2 pirobebs Soris, damokidebulia B faqtoris mniSvnelobaze. amis msgavsad, Tu xdeba urTierTqmedeba, B faqtoris efeqti, gansxvaveba qcevaSi B1 da B2 pirobebs Soris, damokidebulia A faqtoris mniSvnelobaze. urTierTqmedeba xSirad aRiniSneba AxB, rac ikiTxeba ase: “A-sa da B-s urTierTqmedeba”. publikaze gaTvlili qcevis hipoTezis winaswarmetyvelebis mixedviT, Cvens sanimuSo eqsperimentSi urTierTqmedeba unda momxdariyo. am hipoTezam ivarauda, rom kargi siaxlis tipisaTvis ar iqneboda gansxvaveba pasuxis latentur droSi mxedvelobis areSi yofnis da mxedvelobis areSi aryofnis pirobebis dros (anu ar iqneboda gansxvaveba A1B1 da A2B1 ujredebis saSualoebs Soris). xolo cudi siaxlis tipisaTvis hipoTezam iwinaswarmetyvela mniSvnelovnad grZeli latenturi dro mxedvelobis areSi yofnis pirobisaTvis (A2B2 ujredis saSualo) mxedvelobis areSi aryofnis pirobasTan (A1B2 ujredis saSualo) SedarebiT. miaqcieT yuradReba, rom urTierTqmedebis arseboba gamoiTvleba ujredebis saSualoebs Soris gansxvavebebis SedarebiT, da ara mTavari efeqtebis saSualoebis gansxvavebebis SedarebiT. SeamowmeT Tqveni codna XI.1

1. gansazRvreT: ujredi, ujredis saSualo, svetis saSualo, faqtori, faqtoruli sqema, sruli saSualo, urTierTqmedeba, damoukidebeli cvladis mniSvneloba (done), A faqtoris mTavari efeqti, B faqtoris mTavari efeqti, mTavari efeqtis

saSualo, striqonis saSualo, piroba,XAB,XA,XB,XG . 2. Semdegi faqtoruli sqemebidan TiToeulisaTvis gansazRvreT, ramdeni

damoukidebeli cvladi cvalebadobs da miuTiTeT doneebis raodenoba TiToeuli faqtorisaTvis. Semdeg, im daSvebiT, rom gvaqvs cdis pirebs Soris sqema, miuTiTeT, ramdeni cdis piri iqneboda saWiro, Tu yovel pirobaSi monawileoba 10 cdis pirma unda miiRos.

3. sqemis tipi: (a) 3x3, (b) 3x2, (g) 2x3, (d) 6x2, (e) 4x4, (v) 2x2x2, (z) 2x4x3, (T) 3x3x2. gamoTvaleT ujredebis saSualoebi, mTavari efeqtis saSualoebi da sruli saSualo qvemoT moyvanili TiToeuli cxrilis monacemebisaTvis:

cxrili 1 faqtori A

A1 A2

B1 faqtori B B2

10 17 13 23 8 17 17 21 12 19 20 24 19 27 13 21 10 16 20 19 22 30

17 24 cxrili 2 faqtori A

A1 A2

B1 faqtori B B2

23 41 13 22 28 27 11 32 19 26 20 24 18 39 29 28 33 31 18 26 29 34 36 39 47 44 27 35

cdis pirebs Soris orfaqtoriani sqemis dispersiuli analizi faqtoruli sqemis statistikuri analizi moicavs hipoTezebis Semowmebas, raTa ganisazRvros, mniSvnelovnad gansxvavdeba Tu ara mTavari efeqtis saSualoebi TiToeuli damoukidebeli cvladisaTvis da dadgindes, statistikurad mniSvnelovania Tu ara damoukidebel cvladebs Soris urTierTqmedeba. es Semowmeba xdeba kvlav dispersiuli analizis meSveobiT, romelic monacemebis srul cvalebadobas dayofs cvalebadobis iseT wyaroebad, romlebic erTmaneTisagan damoukidebelia. cdis pirebs Soris erTfaqtoriani dispersiuli analizi (romelic me-10 TavSi ganvixileT) srul cvalebadobas or damoukidebel wyarod yofs: erTi, romelic damoukidebeli cvladis (anu A faqtoris) efeqtis gamo xdeba, da meore, romelic jgufebs SigniT cvalebadobis gamo. cdis pirebs Soris orfaqtorian sqemaSi qulebis sruli cvalebadoba iyofa jgufebs Soris sam wyarod da jgufebs SigniT erT wyarod. jgufebs Soris cvalebadobis wyaroebi Semdegia: cvalebadoba A faqtoris efeqtis gamo cvalebadoba B faqtoris efeqtis gamo

cvalebadoba A da B faqtorebs Soris urTierTqmedebis gamo, romelic simbolurad aRiniSneba AxB

jgufebs SigniT cvalebadobis wyaro aris ujredebs SigniT SemTxveviTi cvalebadoba. Semdeg dispersiuli analizi iyenebs am danawilebas, raTa warmoqmnas kvadratebis oTxi saSualo:

MSA, romelic icvleba A faqtoris efeqtTan erTad MSB, romelic icvleba B faqtoris efeqtTan erTad MSAxB, romelic icvleba A da B faqtorebis urTierTqmedebasTan erTad MSSecdoma, romelic icvleba ujredebs SigniT monacemebis cvalebadobasTan erTad

am etapis Semdeg gamoiTvleba sami F statistika, ori TiToeuli damoukidebeli cvladisaTvis da erTi – urTierTqmedebisaTvis. es Semdegi gziT xdeba: cvalebadobis wyaro F statistika -----------------------------------------------------------------------------------

faqtori A MSA/ MSSecdoma faqtori B MSB/ MSSecdoma urTierTqmedeba AxB MSAxB/ MSSecdoma

----------------------------------------------------------------------------------- TiToeuli am sami F statistikidan gamoiyeneba statistikuri hipoTezebis SemowmebisaTvis, raTa dadgenil iqnes, statistikurad mniSvnelovania Tu ara cvalebadobis is wyaro, romelsac Seesabameba F statistika. dispersiuli faqtoruli analizis konceptualuri gaazrebisaTvis Semdeg qveTavSi ganvixilavT, rogor miiReba gadaxrebis kvadratebis saSualo, romelic Sedis am F statistikebSi. Tumca, Tu Tqven dagWirdebodaT dispersiuli faqtoruli analizis Catareba monacemTa realur bazaze, Tqven gamoiyenebdiT, gamoTvlis TvalsazrisiT, ufro advil formulebs im gamosaTvleli formulebidan, romlebic moyvanilia am Tavis damatebaSi an mimarTavdiT monacemTa analizis kompiuterul programas. monacemTa dayofa dispersiul analizSi sami F statistikisaTvis gadaxrebis kvadratebis saSualoebi gamoiTvleba eqsperimentSi monacemebis sruli cvalebadobis iseT nawilebad dayofiT, romlebic icvleba A faqtorTan (jgufebs Soris), B faqtorTan (jgufebs Soris), AxB urTierTqmedebasTan (jgufebs Soris) da SemTxveviT cvalebadobasTan (jgufebs SigniT) erTad. qulis wardgenis es modeli cdis pirebs Soris orfaqtorian sqemaSi SeiZleba Semdegnairad gamoisaxos: (11.1) = + + +

sruli cvalebadoba cdis piris monacemebSi

cvalebadoba A faqtoris efeqtis gamo plus SerCevis Secdoma

cvalebadoba B faqtoris efeqtis gamo plus SerCevis Secdoma

cvalebadoba A da B faqtorebis urTierTqme-debis gamo plus SerCevis Secdoma

cvalebadoba ujredebs SigniT Secdomis gamo

monacemebis terminebiT es gantoleba SeiZleba Caiweros Semdegnairad:

(X - GX ) = ( AX - GX ) + ( BX - GX ) + ( ABX - AX - BX + GX ) + (X - ABX ). (11.2) Cven ganvixilavT 11.2 gantolebis TiToeul nawils. sruli cvalebadoba me-10 TavSi warmodgenili dispersiis erTfaqtoriani analizis msgavsad,

monacemis gansxvaveba sruli saSualosagan (anu X - GX ) warmoadgens am qulis srul cvalebadobas. rogorc gviCvenebs 11.2 gantoleba, am cvalebadobis nawili xdeba TiToeuli damoukidebeli cvladis efeqtisa da damoukidebeli cvladebis urTierTqmedebis gamo, xolo nawili ki – SemTxveviTi cvalebadobis gamo. cvalebadoba A faqtoris efeqtisa da SerCevis Secdomis gamo A faqtoris mTavari efeqtis saSualosa da srul saSualos Soris

gansxvaveba (anu AX - GX ) icvleba A faqtoris efeqtsa da A faqtoris mTavari efeqtis saSualoebs Soris SerCevis SecdomasTan erTad. misi gamoyeneba orfaqtorian sqemaSi misi cdis pirebs Soris erTfaqtorian sqemaSi gamoyenebis identuria, raTa gaizomos damoukidebeli cvladis gavlena. mosalodnelia, rom A faqtoris nebismieri efeqti yvela cdis pirze Tanabrad moaxdens gavlenas am faqtoris yvela doneze. amdenad, A faqtoris efeqti cvlis mTavari efeqtis saSualoebis mniSvnelobas A faqtorisaTvis. es mTavari efeqtis saSualoebi Semdeg gansxvavebuli

iqneba sruli saSualosagan. aqedan gamomdinare, gansxvaveba AX - GX asaxavs A faqtoris nebismier efeqts (pirobis efeqts A faqtorisaTvis 11.1 gantolebaSi). nebismieri pirobis efeqtis garda, mTavari efeqtis saSualoebi A faqtorisaTvis aseve gansxvavebuli iqneba erTmaneTisagan

SerCevis Secdomis gamo. amitom, AX - GX sxvaobis mniSvneloba asaxavs rogorc A faqtoris pirobis efeqts, aseve SerCevis Secdomas, romelic

gavlenas axdens AX -s mniSvnelobebze. Sesabamisad, is monawileobs MSA.s gamoTvlaSi. cvalebadoba B faqtoris efeqtisa da SerCevis Secdomis gamo B faqtoris mTavari efeqtis saSualosa da srul saSualos Soris

gansxvaveba (anu BX - GX ) icvleba B faqtoris efeqtsa da B faqtoris

cdis piris qulis gan-sxvaveba sruli saSualosagan

A faqtoris mTavari efeqtis saSualos gansxvaveba sruli saSualosagan

B faqtoris mTavari efeqtis saSualos gansxvaveba sruli saSualosagan

urTierTqmedeba: ujredis saSualos gansxvaveba sruli saSualosagan mas Semdeg, rac A da B faqtorebis mTavari efeqtebi gauqmebulia

cdis piris qulis gansxvaveba misi ujredis saSualosagan

mTavari efeqtis saSualoebs Soris SerCevis SecdomasTan erTad. iseve, rogorc A faqtoris SemTxvevaSi, mosalodnelia, rom B faqtoris nebismieri efeqti gavlenas moaxdens yvela individze am faqtoris yvela doneze da amdenad Secvlis mTavari efeqtis saSualo mniSvnelobas B faqtorisaTvis, romelic gansxvavebuli gaxdeba sruli saSualosagan.

amitom, gansxvaveba BX - GX asaxavs B faqtoris nebismier efeqts (pirobis efeqts B faqtorisaTvis 11.1 gantolebaSi). garda amisa, mTavari efeqtis saSualoebi B faqtorisaTvis aseve gansxvavebuli iqneba erTmaneTisagan

SerCevis Secdomis gamo. amitom, BX - GX gansxvaveba asaxavs rogorc B faqtoris pirobis efeqts, aseve SerCevis Secdomas, romelic gavlenas

axdens BX -s mniSvnelobebze da, Sesabamisad, monawileobs MSB-s gamoTvlaSi. cvalebadoba A da B faqtorebis urTierTqmedebis gamo A da B faqtorebis urTierTqmedeba aris sruli saSualosagan ujredebis saSualoebis darCenili gansxvaveba mas Semdeg, rac TiToeuli damoukidebeli cvladis efeqti gauqmebulia. simbolurad es debuleba SeiZleba Semdegnairad warmovadginoT:

A x B = ( ABX - GX ) - ( AX - GX ) - ( BX - GX ) = - - gantoleba gviCvenebs, rom urTierTqmedeba aris ujredebis saSualoebis cvalebadoba mas Semdeg, rac TiToeuli damoukidebeli cvladis mTavari efeqti gauqmebulia. ujredebis saSualoebis es darCenili cvalebadoba xdeba damoukidebel cvladebs Soris urTierTqmedebisa da ujredebis saSualoebs Soris nebismieri SerCevis Secdomis gamo. gantolebaSi moyvanili gamoklebis Catarebis Sedegad viRebT urTierTqmedebis gamartivebul gamosaxulebas:

ABX - AX - BX + GX

urTierTqmedeba am saxiT aris warmodgenili 11.2 gantolebaSi. gantolebis es nawili monawileobs MSAxB-is gamoTvlaSi. cvalebadoba ujredebs SigniT Secdomis gamo

monacemis gansxvaveba misi ujredis saSualosagan (anu X - ABX ) asaxavs mxolod eqsperimentSi ujredebs SigniT SemTxveviT cvalebadobas. amiT gamowveuli Secdoma aris is, rac rCeba monacemis miRma mas Semdeg, rac orive faqtoris mTavari efeqti da maTi urTierTqmedeba gaTvaliswinebul iqna. rogorc dispersiuli erTfaqtoriani analizis dros, Secdoma asaxavs pirovnebis monacemis unikalurobas individebis jgufSi, romelTaganac yvela erTsa da imave pirobaSi iyo. is monawileobs MSSecdomis gamoTvlaSi.

urTierTqmedeba A.sa da B-s Soris

ujredis saSualos gansxvaveba sruli saSualosagan

A faqtoris mTavari efeqti

B faqtoris mTavari efeqti

dayofili monacemebidan gadaxrebis kvadratebis saSualos miReba rogorc ganvixileT me-10 TavSi, dayofis gantoleba gvaZlevs safuZvels, rom miviRoT SS da df, romlebic qmnian kvadratebis saSualos F SefardebisaTvis. pirveli nabijia SS-is miReba cdis pirebis monacemebidan. kvadratebis jamebis miReba me-10 TavSi ganxiluli midgomis msgavsad, kvadratebis jamebi (SS) miiReba TiToeuli qulis dayofiT 11.2 gantolebis mixedviT. am dayofis ilustraciisaTvis Cven gamoviyenebT pirveli cdis piris monacems pirobaSi “ar aris mxedvelobis areSi/kargi siaxle” (anu X111) 11.2 cxrilidan, romelic aris 97.0. am monacemisaTvis 11.2 gantoleba iqneba

(X111 - GX ) = ( 1AX - GX ) + ( 1BX - GX ) + ( 11BAX - 1AX - 1BX + GX ) + (X111 -

11BAX ) ricxviTi mniSvnelobebis CasmiT viRebT 97.0–92.5 = (86.5 – 92.5) + (87.7 – 92.5) + (90.8 – 86.5 – 87.7 + 92.5) + (97.0 – 90.8) anu, miviRebT +4.5 = (-6.0) + (-4.8) + (+9.1) + (+6.2) es dayofa gvaCvenebs, rom es monacemi gansxvavdeba sruli saSualosagan,

radganac adgili aqvs A faqtoris sistematur efeqts (aRniSnulia 1AX - GX = - 6.0 ), B faqtoris sistematur efeqts (aRniSnulia 1BX - GX = - 4.8), A

da B faqtorebis urTierTqmedebas (aRniSnulia 11BAX - 1AX - 1BX + GX = 9.1) da ujredebs SigniT SemTxveviTi cvalebadobis arasistematur gavlenebs

(aRniSnulia X111 - 11BAX = +6.2). cxrilis # 11.2 yvela sxva monacemis msgavsi dayofa naCvenebia cxrilis # 11.6 pirvel nabijSi. cxrilis danarCen nabijebSi am dayofili monacemebidan gamoTvlilia saSualoebis jamebi. yuradRebiT misdieT TiToeuls am nabijebidan. nabiji 1. eqsperimentSi monawile yvela 20 cdis piris monacemi

dayofilia 11.2 gantolebis mixedviT. nabiji 2. TiToeuli monacemisaTvis miRebulia ricxviTi

sxvaobebi. nabiji 3. me-2 nabijSi miRebuli yoveli dadebiTi da uaryofiTi

sxvaoba ayvanilia kvadratSi. nabiji 4. eqsperimentis yvela monacemis kvadratSi ayvanili

sxvaobebis mniSvnelobebi dajamebulia. es dajameba miiRweva ocive mniSvnelobis mimatebiT TiToeul svetSi. Sedegad viRebT kvadratebis xuT jams: SSsruli = 4567.000,

SSA = 720.000, SSB = 460.800, SSAxB = 1656.200, SSSecdoma = 1730.000.

SSsruli misi komponentebis jamia, amitom SSsruli = SSA + SSB + SSAxB + SSSecdoma. (11.3) cxrili # 11.6 kvadratebis jamis miReba orfaqtoriani dispersiuli analizisaTvis cxrilis # 11.2 monacemebze dayrdnobiT.

nabiji 1: 20 cdis piris qulebis dayofa Semdegi gantolebis gamoyenebiT:

(X - GX ) = ( AX - GX ) + ( BX - GX ) + ( ABX - AX - BX + GX ) + (X - ABX ) __________________________________________________________________ 97.0 - 92.5 = (86.5 - 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (97.0 - 90.8) 90.0 – 92.5 = (86.5 – 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (90.0 - 90.8) monacemebi 80.0 - 92.5 = (86.5 - 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (80.0 - 90.8) A1B1-Si 107.0 - 92.5 =(86.5 - 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (107.0 - 90.8) 80.0 - 92.5 = (86.5 - 92.5) + (87.7 - 92.5) + (90.8 - 86.5 - 87.7 + 92.5) + (80.0 - 90.8) 70.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (70.0 - 84.6) 87.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (87.0 - 84.6) monacemebi 81.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (81.0 - 84.6) A2B1-Si 95.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (95.0 - 84.6) 90.0 - 92.5 = (98.5 - 92.5) + (87.7 - 92.5) + (84.6 - 98.5 - 87.7 + 92.5) + (90.0 - 84.6)

68.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (68.0 - 82.2) 87.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (87.0 - 82.2) monacemebi 92.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (92.0 - 82.2) A1B2-Si 80.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (80.0 - 82.2) 84.0 - 92.5 = (86.5 - 92.5) + (97.3 - 92.5) + (82.2 - 86.5 – 97.3 + 92.5) + (84.0 - 82.2)

114.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (114.0 -112.4) 96.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (96.0 - 112.4) monacemebi 127.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (127.0 -112.4) A2B2-Si 110.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (110.0 -112.4) 115.0 - 92.5 = (98.5 - 92.5) + (97.3 - 92.5) + (112.4 - 98.5 - 97.3 + 92.5) + (115.0 -112.4)

nabiji 2: CavataroT pirvel nabijSi miTiTebuli gamokleba, raTa miviRoT ricxviTi sxvaobebi TiToeuli cdis pirisaTvis. + 4.5 = - 6.0 + - 4.8 + + 9.1 + + 6.2 - 2.5 = - 6.0 + - 4.8 + + 9.1 + - 0.8 monacemebi - 12.5 = - 6.0 + - 4.8 + + 9.1 + - 10.8 A1B1-Si +14.5 = - 6.0 + - 4.8 + + 9.1 + + 16.2 _ 12.5 = - 6.0 + - 4.8 + + 9.1 + - 10.8 - 22.5 = + 6.0 + - 4.8 + - 9.1 + - 14.6 - 5.5 = + 6.0 + - 4.8 + - 9.1 + + 2.4 monacemebi - 11.5 = + 6.0 + - 4.8 + - 9.1 + - 3.6 A2B1-Si + 2.5 = + 6.0 + - 4.8 + - 9.1 + + 10.4 - 2.5 = + 6.0 + - 4.8 + - 9.1 + + 5.4

- 24.5 = _ 6.0 + + 4.8 + - 9.1 + - 14.2

- 5.5 = _ 6.0 + + 4.8 + - 9.1 + + 4.8 monacemebi - 0.5 = _ 6.0 + + 4.8 + - 9.1 + + 9.8 A1B2-Si - 12.5 = _ 6.0 + + 4.8 + - 9.1 + - 2.2 - 8.5 = _ 6.0 + + 4.8 + - 9.1 + + 1.8

+ 21.5 = + 6.0 + + 4.8 + + 9.1 + + 1.6 + 3.5 = + 6.0 + + 4.8 + + 9.1 + - 16.4 monacemebi + 34.5 = + 6.0 + + 4.8 + + 9.1 + + 14.6 A2B2-Si + 17.5 = + 6.0 + + 4.8 + + 9.1 + _ 2.4 + 22.5 = + 6.0 + + 4.8 + + 9.1 + + 2.6

cxrili # 11.6 (gagrZeleba) nabiji 3: kvadratSi aiyvaneT TiToeuli sxvaoba:

(X - GX )2 ( AX - GX )2 ( BX - GX )2 ( ABX - AX - BX + GX )2 (X -

ABX )2 -----________________________________________________________________ 20.25 36.0 23.04 82.81 38.44 6.25 36.0 23.04 82.81 0.64 monacemebi 156..25 36.0 23.04 82.81 116.64 A1B1-Si 210.25 36.0 23.04 82.81 262.44 156..25 36.0 23.04 82.81 116.64 506.25 36.0 23.04 82.81 213.16 30.25 36.0 23.04 82.81 5.76 monacemebi 132.25 36.0 23.04 82.81 12.96 A2B1-Si 6.25 36.0 23.04 82.81 108.16 6.25 36.0 23.04 82.81 29.16

600..25 36.0 23.04 82.81 201.64 30.25 36.0 23.04 82.81 23.04 monacemebi 0.25 36.0 23.04 82.81 96.04 A1B2-Si 156..25 36.0 23.04 82.81 4.84 72.25 36.0 23.04 82.81 3.24

462.25 36.0 23.04 82.81 2.56 12.25 36.0 23.04 82.81 268.96 monacemebi 1190..25 36.0 23.04 82.81 213.16 A2B2-Si 306.25 36.0 23.04 82.81 5.76 506.25 36.0 23.04 82.81 6.76

nabiji 4: daajameT kvadratSi ayvanili sxvaobebi TiToeuli dayofisaTvis yvela cdis piris mixedviT. 4567.000 = 720.000 + 460.800 + 1656.200 + 1730.000

SSsruli SSA SSB SSAxB SSSecdoma

es gantoleba gvaCvenebs, rom eqsperimentSi monacemebis sruli cvalebadoba (SSsruli) im cvalebadobis Sedegia, romelic oTxi damoukidebeli wyaros gamo xdeba: A faqtori, B faqtori, A da B faqtorebis urTierTqmedeba da ujredebs SigniT Secdoma. Tavisuflebis xarisxis gamoTvla

kvadratebis saSualo (MS) miiReba TiToeuli SS-is gayofiT Tavis Tavisuflebis xarisxze. gaixseneT, rom Tavisuflebis xarisxebi ganisazRvreba, rogorc monacemebis (qulebis) raodenoba, romelic Tavisuflad icvleba statistikis gamoTvlisas. gamoviyenoT es gansazRvreba 11.3 gantolebaSi TiToeuli SS-isaTvis df–is gamosaTvlelad. Tavisuflebis sruli xarisxi. SSsruli -s gamosaTvlelad unda vicodeT

sruli saSualo GX . radganac kvadratebis sruli jami damokidebulia

eqsperimentSi yoveli monacemis sxvaobaze sruli saSualosagan (anu X -

GX ), amdenad monacemebis srul raodenobaze erTiT naklebs SeuZlia Tavisufali cvlileba SSsruli -s gamoTvlisas. amitom Tavisuflebis sruli xarisxi = dfsruli = N – 1, sadac N aris eqsperimentSi monacemTa sruli raodenoba. rodesac N = 20, dfsruli = 20 – 1 = 19.

Tavisuflebis xarisxi SSA.-isaTvis. SSA gamoiTvleba GX –sagan A faqtoris mTavari efeqtis saSualoebis gamoklebiT (anu Cvens magaliTSi

1AX da 2AX ). faqtorul sqemaSi, sadac A faqtors ori done aqvs, Tu viciT GX , maSin A faqtoris mxolod erT mTavari efeqtis saSualos SeuZlia Tavisuflad cvlileba. zogadad, df SSA.-isaTvis (dfA) udris erTiT naklebs, vidre A faqtoris doneebi, anu Tavisuflebis xarisxi SSA.-isaTvis = dfA = a – 1, sadac a aris A damoukidebeli cvladis doneebis raodenoba. Cvens magaliTSi, rodesac a = 2, amitom dfA = 2 – 1 = 1.

Tavisuflebis xarisxi SSB-isaTvis. SSB gamoiTvleba GX –sagan B faqtoris mTavari efeqtis saSualoebis gamoklebiT (anu Cvens magaliTSi 1BX da

2BX ). faqtorul sqemaSi, sadac B faqtors ori done aqvs, Tu viciT GX , maSin B faqtoris mxolod erT mTavari efeqtis saSualos SeuZlia Tavisuflad cvlileba. zogadad, df SSB-isaTvis (dfB) udris erTiT naklebs, vidre B faqtoris doneebi, anu Tavisuflebis xarisxi SSB-isaTvis = dfB = b – 1, sadac b aris B damoukidebeli cvladis doneebis raodenoba. Cvens magaliTSi, rodesac b = 2, amitom dfB = 2 – 1 = 1. Tavisuflebis xarisxi SSAxB-isaTvis. SSAxB gamoiTvleba TiToeuli ujredis saSualosagan Sesabamisi mTavari efeqtis saSualoebis gamoklebiT. 2x2 sqemisaTvis, Tu cnobilia TiToeuli mTavari efeqtis saSualo, maSin mxolod erT ujredis saSualos SeuZlia Tavisufali cvlileba. rodesac viciT erTi ujredis saSualo da A da B faqtorebis mTavari efeqtebis saSualoebi, SegviZlia vipovoT danarCeni sami ujredis saSualoebi. zogadad, Tavisuflebis xarisxi SSAxB-isaTvis = dfAxB = (a - 1) (b – 1)

2x2 sqemaSi a = 2 da b = 2, amitom dfAxB = (2 - 1) (2 – 1) = 1

Tavisuflebis xarisxi SSSecdomisaTvis. SSSecdoma gamoiTvleba yoveli ABX -is gamoklebiT ujredis yvela monacemidan, eqsperimentis yvela ujredisaTvis. amdenad, yoveli pirobisaTvis, mas Semdeg, rac napovnia

ABX , mxolod nAB – 1 monacemebs SeuZliaT Tavisufali cvalebadoba am ujredSi. magaliTad, cxrilSi # 11.2 xuTi monacemidan mxolod oTxs SeuZlia Tavisufali calebadoba TiToeul ujredSi, Tu cnobilia

TiTeuli ABX . radganac arsebobs oTxi ujredi da TiToeulSi Tavisufali cvlileba oTx qulas SeuZlia, SSSecdomisaTvis arsebobs 16 df. zogadad, rodesac a aris A faqtoris doneebis raodenoba, b – B faqtoris doneebis raodenoba, xolo nAB ki qulebis raodenoba TiToeuli ujredis SigniT, Tavisuflebis xarisxi SSSecdomisaTvis = dfSecdoma = ab (nAB - 1) Tavisuflebis xarisxis aditiuroba. faqtorul analizSi Tavisuflebis xarisxebi imave gziT jamdeba, rogorc erTfaqtorian dispersiul analizSi. amdenad dfsruli = dfA + dfB + dfAxB + dfSecdoma (11.4) es damokidebuleba sruldeba Cvens magaliTSic, radgan dfsruli =19, dfA =1, dfB =1, dfAxB =1, dfSecdoma =16. MS-is miReba SS-is da df-is saSualebiT. 11.3 gantolebaSi TiToeuli SS-is gayofa mis Sesabamis df-ze warmoqmnis kvadratebis oTx saSualos, romlebic saWiroa cdis pirebs Soris orfaqtoriani dispersiuli analizisaTvis:

MSA = A

A

dfSS -

MSB = B

B

dfSS -

MSAxB = AxB

AxB

dfSS

MSSecdoma = Secdoma

Secdoma

dfSS

Cven ar vangariSobT MSsruls, radganac is ar gamoiyeneba am analizSi romelime F Sefardebis misaRebad. cxrilis # 11.2 monacemebisaTvis SSA=720.000, SSB=460.800, SSAxB=1656.200 da SSSecdoma =1730.000. ufro metic, dfA=1, dfB=1, dfAxB=1 da dfSecdoma =16. Sesabamisad,

MSA = 1000.720 = 720.000

MSB = 1800.460 = 460.800

MSAxB = 1

200.1656 = 1656.200

MSSecdoma = 16

000.1730 = 108.125

F statistikis gamoTvla F Sefardeba iqmneba kvadratebis ori saSualos SefardebiT, erTi MS-isa, romelic icvleba damoukidebeli cvladis efeqtTan erTad da aseve SerCevis SecdomasTan erTad, da meore MS_isa, romelic icvleba mxolod Secdomis cvalebadobasTan erTad eqsperimentSi. orfaqtorian cdis pirebs Soris dispersiul analizSi, MSA icvleba A faqtorTan erTad, MSB – B faqtorTan erTad, MSAxB – A da B faqtorebis urTierTqmedebasTan erTad da MSSecdoma ki – mxolod ujredebs SigniT SemTxveviT cvalebadobasTan erTad. ase rom, viRebT sami saxis F statistikas. maTgan TiToeuli asaxavs cvalebadobis erT sistematur wyaros eqsperimentSi: cvalebadobis wyaro F -------------------------------------------------------------------------------------------

faqtori A SecdomaMS

MS A

faqtori B SecdomaMS

MSB

A da B urTierTqmedeba SecdomaMS

MS AxB

------------------------------------------------------------------------------------ gadaxrebis kvadratebis saSualoebis ricxviTi mniSvnelobebis CasmiT viRebT Semdeg F statistikebs: cvalebadobis wyaro F -------------------------------------------------------------------------------------------------

faqtori A 125.108000.720 = 6.66

faqtori B 125.108800.460

= 4.26

A da B urTierTqmedeba 125.108200.1656 = 15.32

--------------------------------------------------------------------------------------------------- mokled ganvixiloT, Tu rogor gamoviyenoT da rogori axsna mivceT TiToeul F-s. cxrilSi # 11.7 moyvanilia cvalebadobis wyaroebi cdis pirebs Soris orfaqtorian dispersiul analizSi da formulebi TiToeuli SS-isTvis, df-isTvis, MS-isTvis da F statistikisaTvis. cxrilis # 11.2 monacemebis dispersiuli analizis Sejamebis Sedegebi mocemulia cxrilSi # 11.8. cxrili # 11.7 cdis pirebs Soris orfaqtoriani dispersiuli analizSi gamoyenebuli ganmartebiTi formulebis Sejameba. ------------------------------------------------------------------------------------------------------------------------------------------------ wyaro SS dfa MS F -----------------------------------------------------------------------------------------------------------------------------------------------------

A faqtori ‡ ∑ ∑ ∑ ( AX - GX )2 A.1 A

A

dfSS

SecdomaMS

MSA

B faqtori ∑ ∑ ∑ ( BX - GX )2 b-1 B

B

dfSS

SecdomaMS

MSB

A da B

urTierTqmedeba ∑ ∑ ∑ ( ABX - AX - BX + GX )2 (A.1)(b-1) AxB

AxB

dfSS

SecdomaMS

MS AxB

Secdoma ∑ ∑ ∑ (X - ABX )2 ab(nAB-1) Secdoma

Secdoma

dfSS

sul ∑ ∑ ∑ (X - GX )2 N-1 ar gamoiTvleba

-------------------------------------------------------------------------------------------------------------------------------------------------------

aa = A faqtoris doneebis raodenoba; b = B faqtoris doneebis raodenoba; nAB = qulebis raodenoba TiToeul ujredSi; N = qulebis sruli raodenoba. -------------------------------------------------------------------------------------------------------------------------------------------------------

cxrili # 11.8 cxrilis # 11.2 hipoTeturi qulebis dispersiuli analizis gamoTvlis Sedegebi.

wyaro SS df MS F

subieqtis yofna mxedvelobis areSi (A) 720.000 1 720.000 6.66* siaxlis tipi (B) 460.800 1 460.800 4.26 A x B 1656.200 1 1656.200 15.32** Secdoma 1730.000 16 1656.200 sul 4567.000 19

* p < .05 ** p < .01

SeamowmeT Tqveni codna XI.2 1. gansazRvreT monacemebis (qulebis) cvalebadobis wyaroebi cdis pirebs

Soris orfaqtorian sqemaSi. 2. dawereT gantoleba, romlis mixedviTac xdeba qulebis dayofa dispersiis

orfaqtorian cdis pirebs Soris analizSi.

3. axseniT, ratom icvleba AX - GX mniSvneloba A faqtoris efeqtTan erTad faqtorul eqsperimentSi.

4. axseniT, ratom icvleba BX - GX mniSvneloba B faqtoris efeqtTan erTad faqtorul eqsperimentSi.

5. axseniT, ratom icvleba ABX - AX - BX + GX mniSvneloba A da B faqtorebs Soris urTierTqmedebis efeqtTan erTad faqtorul eqsperimentSi.

6. axseniT, ratom icvleba X - ABX mniSvneloba mxolod ujredebs SigniT SemTxveviT cvalebadobasTan erTad faqtorul eqsperimentSi.

7. daasruleT Semdegi tolobebi cdis pirebs Soris orfaqtoriani dispersiuli analizisaTvis: a. dfA = b. dfB = g. dfAxB = d. dfSecdoma = e. dfsruli = v. MSA = z. MSB = T. MSAxB = i. MSSecdoma =

8. ras udris SSsrulis mniSvneloba, Tu SSA =40.00, SSB =70.00, SSAxB =30.00 da SSSecdoma =100.00?

9. ras udris SSA–is mniSvneloba, Tu SSsruli = 224.00, SSB =14.00, SSAxB =64.00 da SSSecdoma =101.00?

10. ras udris SSAxB–is mniSvneloba, Tu SSsruli = 302.00, SSA =75.00, SSB =19.00 da SSSecdoma =153.00?

11. ras udris SSSecdomis mniSvneloba, Tu SSsruli = 98.00, SSA =14.00, SSB =7.00 da SSAxB =17.00?

12. ipoveT dfA, dfB, dfAxB, dfSecdoma da dfsruli 2x3 cdis pirebs Soris dispersiuli analizisaTvis, sadac TiToeul ujredSi eqvsi cdis piria.

13. ipoveT dfA, dfB, dfAxB, dfSecdoma da dfsruli 3x2 cdis pirebs Soris dispersiuli analizisaTvis, sadac TiToeul pirobaSi 13 qulaa.

14. ras udris dfsrulis mniSvneloba, Tu dfA=3, dfB=1, dfAxB=3 da dfSecdoma = 72? 15. gamoTvaleT gadaxrebis kvadratebis saSualoebis ricxviTi mniSvnelobebi

Semdegi SS-isa da df-isaTvis: SSA =14.00, SSB =26.00, SSAxB =44.00, SSSecdoma = 480.00, dfA=1, dfB=2, dfAxB=2 da dfSecdoma = 60.

16. davuSvaT, Tqven CaatareT 2x2 faqtoruli eqsperimenti, sadac TiToeul ujredSi sami cdis piria da miiReT Semdegi qulebi:

faqtori A

A1 A2

B1 faqtori B B2

30 34 32 36 28 32 28 38 30 34 26 36

a. dayaviT es qulebi cxrilSi # 11.6 naCvenebi midgomis mixedviT. gamoTvaleT SSsruli, SSA, SSB, SSAxB, SSSecdoma da Sesabamisi Tavisuflebis xarisxebi yoveli dayofisaTvis. Semdeg SeqmeniT ricxviTi Sejamebis cxrili MS-isa da F-is mniSvnelobebiT. b. ratom aris SSB nulis toli am magaliTSi? 17. cxrilebi # 1 da 2 cdis pirebs Soris orfaqtoriani dispersiuli analizis dausrulebeli cxrilebia. dauSviT, rom TiToeul ujredSi qulebis toli raodenobebia. TiToeul cxrilSi SeavseT gamotovebuli mniSvnelobebi SS-isa da df-is cxrilebSi # 11.3 da 11.4 mocemuli kavSiris gamoyenebiT. Semdeg TiToeuli cxrilisaTvis upasuxeT Semdeg kiTxvebs:

a. A faqtoris ramdeni done iyo gamoyenebuli? b. B faqtoris ramdeni done iyo gamoyenebuli? g. ramdeni cdis piri gaizoma TiToeul ujredSi? d. ramdeni cdis piri monawileobda kvlevaSi?

cxrili # 1

wyaro SS df MS F A faqtori ____ 2 16.00 ------ B faqtori 24.00 1 --------- 3.00 A x B -------- ------ 12.00 ------- Secdoma --------- ------ --------- sul 560.00 65 ------------------------------------------------------------------------------------------------------------------------------------------------------- cxrili # 2

wyaro SS df MS F A faqtori ____ 2 --------- 4.50 B faqtori 24.00 4 --------- 1.50 A x B 160.00 ------ --------- 5.00

Secdoma --------- ------ 4.00 sul --------- 224 -------------------------------------------------------------------------------------------------------------------------------------------------------

statistikuri hipoTezis Semowmeba cdis pirTa Soris orfaqtorian dispersiul analizSi imisaTvis, rom gavigoT, Tu rogor xdeba statistikuri hipoTezebis Semowmeba F statistikis gamoyenebiT, romelic miiReba dispersiuli faqtoruli analizis Sedegad, unda vicodeT, Tu ra axdens gavlenas MSA.ze, MSB-ze, MSAxB-sa da MSSecdomaze. faqtorebi, romlebic gavlenas axdenen gadaxrebis kvadratebis saSualoebze dispersiul faqtorul analizSi faqtorul analizSi yovel MS-ze gavlenas axdens cvalebadobis gansxvavebuli wyaroebi. Tu damokidebul cvladze arc erT damoukidebel cvlads ar aqvs gavlena da aseve ar arsebobs urTierTqmedeba damoukidebel cvladebs Soris, maSin yoveli MS asaxavs mxolod SemTxveviT cvalebadobas. aseT pirobebSi oTxive kvadratebis saSualo daaxloebiT erTi da imave mniSvnelobis unda iyos da aCvenebdes mxolod SerCevis Secdomis gavlenas. magram, Tu romelime an orive damoukidebel cvlads aqvs mTavari efeqti an Tu isini urTierTqmedeben, maSin efeqti zrdis Sesabamisi MS-is mniSvnelobas. magaliTad, Tu A faqtori qmnis mTavar efeqts, maSin MSA gaizrdeba MSSecdomasTan SedarebiT, magram es mTavari efeqti ar moaxdens gavlenas MSB-ze, MSAxB-ze da MSSecdomaze. msgavsi situacia iqneba, Tu B faqtors aqvs mTavari efeqti. Tu B faqtori qmnis mTavar efeqts, maSin MSB gaizrdeba MSSecdoma -sTan SedarebiT, magram es mTavari efeqti ar moaxdens gavlenas MSA.ze, MSAxB-ze da MSSecdoma -ze. Tu damoukidebeli cvladebi urTierTqmedebaSia, maSin izrdeba MSAxB, magram MSA.ze, MSB-ze da MSSecdoma -ze gavlena ar aris. MSSecdoma -ze gavlenas ar axdens arc mTavari efeqtebi da arc damoukidebeli cvladebis urTierTqmedeba. is asaxavs mxolod ujredebs SigniT SemTxveviT cvalebadobas eqsperimentSi. amdenad, TiToeuli MS-is mniSvneloba damoukidebelia sxva kvadratebis saSualoebis mniSvnelobebisagan. kvadratebis oTxi saSualo da cvalebadobis wyaroebi, romlebic maTze axdenen gavlenas, SeiZleba Semdegnairad Sejamdes: kvadratebis saSualo masze gavlenas axdens:

MSA A faqtoris gamo arsebuli sistematuri cvalebadoba SerCevis Secdoma MSB B faqtoris gamo arsebuli sistematuri cvalebadoba SerCevis Secdoma MSAxB A da B faqtorebis interaqciis gamo arsebuli

sistematuri cvalebadoba SerCevis Secdoma MSSecdoma ujredebs SigniT SemTxveviT cvalebadoba

imisaTvis, rom yvelaferi zemoT Tqmuli davukavSiroT sam F Sefardebas cxrilidan # 11.8, miaqcieT yuradReba, rom yoveli F Sefardeba iyenebs MSSecdomas, rogorc mniSvnels da damoukidebeli cvladis an urTierTqmedebis Sesabamis MS-s, rogorc mricxvels. Tu konkretuli F Sefardebis gamoTvlaSi MS-ze gavlenis mqone damoukidebel cvlads ara aqvs efeqti, maSin es F Sefardeba daaxloebiT 1-s unda udrides, radganac rogorc mricxvelis, aseve mniSvnelis MS zomavs mxolod SemTxveviT cvalebadobas eqsperimentSi. magram Tu damoukidebel cvlads aqvs efeqti, maSin MS-is mniSvneloba am cvladisaTvis izrdeba da F-is mniSvneloba am faqtorisaTvis 1.0-ze meti xdeba. es dasabuTeba me-10 TavSi dispersiuli erTfaqtoriani analizisaTvis moyvanili dasabuTebis identuria.

F statistikaze damyarebuli statistikuri gadawyvetilebebis miReba axla, rodesac viciT, Tu rogor axdens gavlenas faqtorebi F Sefardebis TiToeul mniSvnelobaze, Cven mzad varT, gamoviyenoT TiToeuli F hipoTezebis Semowmebisas. saWiro nabijebi igivea, rac me-10 TavSi.

a. TiTeuli F SefardebisaTvis yalibdeba nulovani hipoTeza, H0, da alternatiuli hipoTeza, H1;

b. im daSvebiT, rom H0 sworia, F-is TiTeuli mniSvnelobisaTvis gamoiTvleba F-is SerCeviTi ganawileba. es ganawileba mocemulia cxrilSi # A.3;

g. xdeba mniSvnelovnobis donis SerCeva; d. F-is kritikuli done, romelic aRiniSneba Fkritikuli,

TiToeuli F SefardebisaTvis F–is SerCeviTi ganawilebidan gamoiTvleba, romelic mocemulia cxrilSi # A.3;

e. F–is SerCeviT ganawilebaSi TiToeuli F-isaTvis gamoiyofa uaryofis are; v. SerCevis monacemebidan gamoiTvleba sami F-is sami mniSvneloba,

romlebic aRiniSneba, rogorc Frealuri; z. gadawyvetileba imis Taobaze, uarvyoT Tu ara H0 TiToeuli

FrealurisaTvis, miiReba imaze dayrdnobiT, xvdeba Tu ara Frealuri uaryofis areSi.

statistikuri hipoTezebi cdis pirebs Soris orfaqtoriani dispersiuli analiziT mowmdeba sami gansxvavebuli nulovani hipoTeza, TiTo yoveli Frealuri-saTvis. 2x2 faqtorul dispersiul analizSi nulovani (H0) da alternatiuli (H1) hipoTezebi yoveli F-isaTvis Semdegia: F statistika statistikuri hipoTezebi

A faqtorisaTvis H0: µ A1 = µ A2

H1: µ A.ebi ar aris toli

B faqtorisaTvis H0: µ B1 = µ B2

H1: µ B-ebi ar aris toli

A x B H0: (µAB - µ A - µ B + µ G) = 0 H1: yvela (µAB - µ A - µ B + µ G) ar udris 0 mTavari efeqtebi. A da B faqtorebis mTavari efeqtebis nulovani da alternatiuli hipoTezebi formiT identuria erTfaqtoriani analizis statistikuri hipoTezebisa. damoukidebeli cvladis mTavari efeqtis SemTxvevaSi dispersiuli faqtoruli analizi monacemebs ganixilavs ise, TiTqos isini miRebulia erTfaqtoriani sqemidan, rogorc naCvenebia cxrilSi # 11.4. populaciis saSualoebis raodenoba, romelic ganisazRvreba H0-Si, rogorc A, aseve B faqtorisaTvis, Seesabameba eqsperimentSi am faqtoris doneebis raodenobas. magaliTad, Tu eqsperimenti moicavs A faqtoris sam dones, maSin H0 A faqtorisaTvis ase gamoisaxeba:

H0: µ A1 = µ A2 = µ A3, xolo H1 ki ase:

H1: yvela µ A.-ebi ar aris erTmaneTis toli. nulovani hipoTeza TiToeuli faqtorisaTvis aris situacia, romelic arsebobs, Tu damoukidebel cvlads ar aqvs efeqti. Tu nulovani hipoTeza sworia, maSin nebismieri aRmoCenili gansxvaveba mTavari efeqtebis Sesabamis saSualoebs Soris mxolod SerCevis Secdomis gamoa. alternatiuli hipoTeza amtkicebs, rom populaciis saSualoebi ar aris erTmaneTis toli. es situacia arsebobs maSin, Tu damoukidebel cvlads aqvs efeqti. urTierTqmedeba. urTierTqmedebis nulovani hipoTeza amtkicebs, rom Tu

populaciaSi urTierTqmedeba ar xdeba, ujredebis saSualoebis, µAB, gansxvaveba sruli saSualosagan, µG, nulis toli iqneba, mas Semdeg, rac mas gamovaklebT TiToeuli damoukidebeli cvladis mTavar efeqts (anu

(µ A - µ G) da (µ B - µ G)). sxva sityvebiT, es H0 amtkicebs, rom Tu damoukidebel cvladebs Soris urTierTqmedeba ar arsebobs, damoukidebeli cvladebis mTavari efeqtebidan SesaZlebelia zustad viwinaswarmetyveloT TiToeuli ujredis saSualos mniSvneloba. alternatiuli hipoTeza amtkicebs, rom es ase ar aris da SeuZlebelia ujredebis saSualoebis cvalebadobis winaswarmetyveleba mxolod mTavari efeqtebidan. F–is mniSvnelobaze damyarebuli gadawyvetilebebis miReba hipoTezebis samive tipis Sesaxeb gadawyvetilebebis miReba Frealuris mniSvnelobebze dayrdnobiT msgavsia im procesisa, romelic mimdinareobs

erTfaqtoriani dispersiuli analizis dros. α-s mniSvneloba, romelic gansazRvravs uaryofis ares, arCeulia analizis Catarebamde. H0

hipoTezis pirobebSi F -is SerCeviTi ganawileba ganisazRvreba samive F

SefardebisaTvis da uaryofis are arsebobs samive SerCevis ganawilebaSi. es nabiji moiTxovs Fkritikulis sami mniSvnelobis moZebnas Sesabamisi mricxveli da mniSvneli df-ebiT A.3a an b cxrilSi, imis mixedviT, Tu ra

aris arCeuli α-s mniSvneloba. Semdeg, TiToeuli FrealurisaTvis, Tu Frealuri udris an metia, vidre misi Sesabamisi Fkritikulis mniSvneloba, H0

am FrealurisaTvis uaryofilia da miRebulia H1. Frealuris am mniSvnelobaSi Semaval saSualoebs Soris sxvaoba statistikurad mniSvnelovania. Tu Frealuri naklebia, vidre misi Sesabamisi Fkritikulis mniSvneloba, maSin gadawyvetileba isaa, rom cvalebadobis am wyarosaTvis ver uarvyofT H0–s da ver miviRebT H1–s. Frealuris am mniSvnelobaSi Semaval saSualoebs Soris gansxvaveba ar aris statistikurad mniSvnelovani. orfaqtorian dispersiul analizSi TiToeuli FrealurisaTvis arsebuli gadawyvetilebebis ori jgufi Sejamebulia cxrilSi # 11.9. gadawyvetilebis miRebis ilustracias movaxdenT cxrilSi # 11.2 sanimuSo monacemebis dispersiuli analizis meSveobiT, rac Sejamebulia cxrilSi # 11.8. am cxrilSi, TiToeul Frealurs aqvs 1 df mricxvelad da 16 df mniSvnelad. amitom TiToeul Frealurs aqvs Fkritikulis erTi da igive mniSvneloba da, Sesabamisad, erTi da igive uaryofis are. es mimarTeba aucileblad ar aris aseTi faqtoruli analizis yvela SemTxvevaSi. TiToeuli damoukidebeli cladis doneebis raodenobis mixedviT, SesaZlebelia, rom TiToeul Frealurs mricxvelad hqondes gansxvavebuli df da, Sesabamisad, gansxvavebuli uaryofis are. cxrili # 11.9 dispersiis orfaqtorian analizSi statistikuri gadawyvetilebebis Sejameba -------------------------------------------------------------------------------------------------------------------------------------------------------

F Sefardeba:

Frealuri-s mniSvneloba

statistikuri gadawyvetileba

A faqtorisaTvis

Fkritikulze naklebi ver uarvyofT H0-s ver viRebT H1-s

A faqtoris mTavari efeqti ar aris mniSvnelovani.

Fkritikulis toli an masze meti

uarvyofT H0-s viRebT H1-s

A faqtoris mTavari efeqti statistikurad mniSvnelovania.

B faqtorisaTvis


B faqtoris mTavari efeqti ar aris mniSvnelovani.



B faqtoris mTavari efeqti statistikurad mniSvnelovania.

A x B


A da B faqtorebis urTierTqmedeba ar aris mniSvnelovani.



A da B faqtorebis urTierTqmedeba statistikurad mniSvnelovania.

rodesac α=.05, Fkritikuli 1,16 df-iT aris 4.49 (aRebulia cxrilidan # A.3a). aqedan gamomdinare, cxrilSi # 11.8 TiToeuli FrealurisaTvis uaryofis are Sedgeba Frealuris mniSvnelobebisagan, romlebic udris an metia 4.49-ze. dispersiis am analizidan Semdeg gadawyvetilebebs viRebT: wyaro: Frealuri xvdeba

uaryofis areSi


A faqtorisaTvis 6.66 ki uarvyofT H0-s viRebT H1-s

A faqtoris mTavari efeqti statistikurad mniSvnelovania.

B faqtorisaTvis 4.26 Aara ver uarvyofT H0-s ver viRebT H1-s

B faqtoris mTavari efeqti ar aris mniSvnelovani.

A x B 15.32 ki uarvyofT H0-s viRebT H1-s

A da B faqtorebis urTierTqmedeba statistikurad mniSvnelovania.

cdis pirTa Soris orfaqtoriani dispersiuli analizis daSvebebi cdis pirTa Soris dispersiuli faqtoruli analizi damyarebulia imave sam daSvebaze eqsperimentSi miRebuli monacemebis (qulebis) Sesaxeb, rac cdis pirTa Soris erTfaqtorian analizSi:

1. monacemebi TiToeul SerCevaSi SemTxveviTad aris aRebuli populaciis monacemebidan da TiToeuli qula damoukidebelia yvela sxva qulisagan;

2. im populaciaSi, saidanac aris aRebuli SerCeva, monacemebi normaluradaa ganawilebuli;

3. monacemebis dispersiebi populaciebSi tolia. rogorc aRvniSneT, erTfaqtorian analizTan dakavSirebiT, Tu gvsurs, gamoviyenoT cdis pirTa Soris dispersiuli analizi, unda Sesruldes pirveli daSveba monacemebis damoukideblobis Sesaxeb. TiTeul cdis pirs mxolod erTi qulis mocema SeuZlia eqsperimentSi. Tumca, ar aris aucilebeli, eqsperimentis monacemebSi Sesruldes meore da mesame daSvebebi. mosalodnelia, rom normaluridan gadaxra da dispersiebis

tolobis daSvebis darRveva minimalur gavlenas moaxdens I gvaris Secdomis daSvebis albaTobaze, Tu sruldeba Semdegi pirobebi:

1. cdis pirebis raodenoba yovel ujredSi tolia; 2. TiToeul pirobaSi qulebis ganawilebis forma daaxloebiT

erTnairia da ganawilebebi arc Zalian wamaxvilebulia, arc Zalian brtyeli;

3. mniSvnelovnobis done aris .05.

SeamowmeT Tqveni codna XI.3 a. gansazRvreT faqtorebi, romlebsac aqvs gavlena yovel MS-ze cdis pirTa Soris

dispersiul faqtorul analizSi. b. axseniT, Tu ratom izrdeba MSA.-is mniSvneloba, Tu A faqtors aqvs efeqti

faqtorul eqsperimentSi. g. axseniT, Tu ratom izrdeba MSB-is mniSvneloba, Tu B faqtors aqvs efeqti faqtorul eqsperimentSi. d. axseniT, Tu ratom izrdeba MSAxB-is mniSvneloba, Tu A da B faqtorebi urTierTqmedeben faqtorul eqsperimentSi. e. axseniT, Tu ratom ar aqvs gavlena MSSecdomis mniSvnelobaze arc A da B faqtorebs, arc maT urTierTqmedebas faqtorul eqsperimentSi. v. ra aris mosalodneli, rom daemarTos Frealuris mniSvnelobas A faqtorisaTvis, Tu A faqtors aqvs efeqti eqsperimentSi? z. ra aris mosalodneli, rom daemarTos Frealuris mniSvnelobas AxB-saTvis, Tu damoukidebeli cvladebi urTierTqmedeben eqsperimentSi? T. gansazRvreT statistikuri Semowmebis nabijebi dispersiul faqtorul analizSi. i. dawereT statistikuri hipoTezebi SemdegisaTvis:

c. 2x4 cdis pirTa Soris dispersiuli analizi d. 3x2 cdis pirTa Soris dispersiuli analizi b. 3x3 cdis pirTa Soris dispersiuli analizi

k. am savarjiSoSi mocemulia ramdenime cdis pirTa Soris orfaqtoriani sqemis

dispersiuli analizis Sejamebis FrealurisTvis cxrilebi. TiToeuli

ipoveT Fkritikuli 0.5 mniSvnelovnobis donisaTvis cxrilidan # A.3a da miuTiTeT, xvdeba Tu ara Frealuri uaryofis areSi. Semdeg daafiqsireT Tqveni gadawyvetileba H0-Tan da H1-Tan dakavSirebiT. Tu dfSecdomisaTvis cxrilSi ar aris mocemuli zusti df, gamoiyeneT cxrilSi qvemoT miTiTebuli momdevno mniSvneloba.

cxrili 1 cxrili 2 wyaro df Frealuri

A faqtori 1 4.63

B faqtori 3 2.11

A x B 3 3.97 Secdoma 56 sul 63 cxrili 3

wyaro df Frealuri

A faqtori 3 1.42

B faqtori 2 3.51

A x B 6 2.37 Secdoma 144 sul 155

2x2 faqtoruli dispersiuli analizis interpretacia faqtoruli sqemis Sedegebis axsnis Sesaxeb diskusias daviwyebT im nabijebiT, romlebic saWiroa statistikurad mniSvnelovani urTierTqmedebis axsnisaTvis. Semdeg ki avxsniT statistikurad mniSvnelovan mTavar efeqtebs. damoukidebeli cvladis martivi efeqti damoukidebeli cvladis martivi efeqti: faqtorul sqemaSi erTi damoukidebeli cvladis efeqti meore damoukidebeli cvladis mxolod erTi donis dros.

ori damoukidebeli cvladis urTierTqmedeba xdeba, rodesac erTi damoukidebeli cvladis efeqti damokidebulia masTan kombinaciaSi myofi meore damoukidebeli cvladis doneze. Sesabamisad, dispersiul analizSi statistikurad mniSvnelovani urTierTqmedebis interpretacia xdeba eqsperimentSi ujredebis saSualoebs Soris sxvaobebis SedarebiT. ujredebis saSualoebs Soris sxvaobebi gviCveneben damoukidebeli cvladebis martiv efeqtebs. damoukidebeli cvladis martivi efeqti (romelsac xandaxan martiv mTavar efeqtsac uwodeben) faqtorul sqemaSi aris am damoukidebeli cvladis efeqti meore damoukidebeli cvladis mxolod erTi donis dros. cxrilSi # 11.10 moyvanilia oTxi martivi efeqti, iseve, rogorc ori mTavari efeqti 2x2 sqemaSi. TiToeuli damoukidebeli cvladis martivi efeqti gansazRvrulia marTkuTxedebSi ujredebs Soris. 2x2 sqemaSi mxolod ori martivi efeqti arsebobs TiToeuli damoukidebeli cvladisaTvis.

wyaro df Frealuri

A faqtori 2 4.12

B faqtori 4 2.83

A x B 8 1.87 Secdoma 150 sul 164

cxrili # 11.10 martivi da mTavari efeqtebi 2x2 cdis pirebs Soris dispersiuli analizSi. A faqtori

A1 A2

mTavari efeqtis saSualoebi B faqtorisaTvis

B1 B faqtori B2

11BAX 12BAX

21BAX 22BAX

1BX

2BX

mTavari efeqtis saSualoebi A faqtorisaTvis

1AX 2AX

A faqtoris martivi efeqti A faqtoris martivi efeqti B1 doneze 11BAX da 12BAX ujredebis

saSualoebs Soris gansxvavebiTaa mocemuli (anu 11BAX - 12BAX ). es gansxvaveba asaxavs A faqtoris efeqts B faqtoris B1 doneze. amis

msgavsad, A faqtoris martivi efeqti B faqtoris B2 doneze 21BAX da

22BAX ujredebis saSualoebs Soris gansxvavebiTaa mocemuli (anu 21BAX - 22BAX ). es gansxvaveba asaxavs A faqtoris efeqts B2 doneze. martivi

efeqtebis ricxviTi mniSvnelobebis gamoTvlisas mniSvnelovania, SevinarCunoT Sedarebebis mimarTulebis SeTanxmebuloba da efeqtis niSani (+ an _). B faqtoris martivi efeqti B faqtoris martivi efeqti A1 doneze 11BAX da 21BAX ujredebis

saSualoebs Soris gansxvavebiTaa mocemuli (anu 11BAX - 21BAX ), xolo B faqtoris martivi efeqti A2 doneze 12BAX da 22BAX ujredebis saSualoebs

Soris gansxvavebiTaa mocemuli (anu 12BAX - 22BAX ). TiToeuli martivi efeqti miuTiTebs B faqtoris efeqts A faqtoris mxolod erT doneze. martivi da mTavari efeqtebis ricxviTi magaliTi. cxrilis # 11.5 saSualoebisagan martivi da mTavari efeqtebis ricxviTi mniSvnelobebis sxvaobebi mocemulia cxrilSi # 11.11. saSualoebs Soris mTavari efeqtis da martivi efeqtis gansxvavebebi naCvenebia marTkuTxedebSi. magaliTad, kargi siaxlis mosmenisas cdis piris mxedvelobis areSi yofnis/ar

A-s martivi efeqti B1-

ze: 11BAX - 12BAX

A-s martivi efeqti B2-

ze: 21BAX - 22BAX

A-s martivi efeqti B2-ze:

11BAX - 21BAX

A-s martivi efeqti B2-ze:

12BAX - 22BAX

B faqtoris mTavari efeqti:

1BX - 2BX

A faqtoris mTavari efeqti:

1AX - 2AX

yofnis martivi efeqti (A faqtoris martivi efeqti B1 doneze) aris 11BAX - 12BAX , rac udris 90.8-84.6, anu +6.2 wams. amis msgavsad, siaxlis tipis martivi efeqti im pirobisaTvis, rodesac cdis piri ar imyofeba

mxedvelobis areSi (B faqtoris martivi efeqti A1 doneze) aris 11BAX - 21BAX , rac udris 90.8-82.2, anu +8.6 wams.

cxrili # 11.11 pasuxis latenturi drois (wamebSi) mTavari efeqtisa da martivi efeqtis sxvaobebi cdis piris mxedvelobis areSi yofnisa da siaxlis tipisaTvis sanimuSo eqsperimentSi.

subieqtis yofna mxedvelobis areSi (A)

Cans (A1) ar Cans (A2)

mTavari efeqtis saSualoebi siaxlis tipisaTvis (B)

kargi (B1)

siaxlis tipi (B) cudi (B2)

90.8 84.6

82.2 112.4

87.7

97.3

mTavari efeqtis saSualoebi subieqtis mxedvelobis areSi yofnisaTvis (A)

86.5 98.5

martivi efeqtebis axsna, rodesac vlindeba statistikurad mniSvnelovani urTierTqmedeba Tu dispersiuli analizisas xdeba damoukidebeli cvladebis statistikurad mniSvnelovani urTierTqmedeba, maSin faqtoris ori martivi efeqti ar aris erTmaneTis toli an am faqtoris mTavari efeqtis toli. amitom, Tu xdeba statitikurad mniSvnelovani urTierTqmedeba, A faqtoris martivi efeqti B1 doneze, 11BAX - 12BAX , ar udris A faqtoris martiv efeqts B2 doneze, 21BAX - 22BAX . garda amisa, A faqtoris arc erTi martivi efeqti ar udris A faqtoris mTavar efeqts, romelic

ganisazRvreba 1AX - 2AX . Tu am mimarTebas mivusadagebT magaliTis Sedegebs cxrilSi # 11.11, aRmovaCenT, rom kargi siaxlis mosmenisas cdis piris mxedvelobis areSi yofnis martivi efeqti (efeqtia +6.2 wami) ar udris cudi siaxlis mosmenisaTvis cdis piris mxedvelobis areSi yofnis martiv efeqts (efeqtia –30.2 wami), da arc erTi martivi efeqti ar udris cdis piris mxedvelobaSi yofnis mTavar efeqts (efeqtia –12.0 wami). msgavsi urTierToba axasiaTebs B faqtoris martiv efeqtebsac. Tu xdeba statistikurad mniSvnelovani urTierTqmedeba, maSin B faqtoris martivi

+6.2

+8.6 -27.8

-30.2

-12.0

-9.6

efeqti A1 doneze, 11BAX - 21BAX , ar udris B faqtoris martiv efeqts A2

doneze, 12BAX - 22BAX ; da arc erTi martivi efeqti ar udris B faqtoris mTavar efeqts, romelic ganisazRvreba 1BX - 2BX . Tu am mimarTebas mivusadagebT magaliTis Sedegebs cxrilSi # 11.11, naTelia, rom siaxlis tipis martivi efeqti im pirobaSi, rodesac cdis piri ar imyofeba mxedvelobis areSi (efeqtia +8.6 wami) ar udris siaxlis tipis martiv efeqts im pirobaSi, rodesac cdis piri imyofeba mxedvelobis areSi (efeqtia –27.8 wami), da arc erTi efeqti ar udris siaxlis tipis mTavar efeqts (efeqtia –9.6 wami). mravlobiTi Sedarebis kriteriumebi statistikurad mniSvnelovani urTierTqmedebisaTvis. martivi efeqtebis TiToeuli Sedareba cxrilebSi # 11.10 da 11.11 warmoadgens wyvilTaSoris Sedarebas. erTmaneTs vadarebT ori ujredis saSualos. imis gansasazRvrad, mniSvnelovnad gansxvavdeba Tu ara erTmaneTisagan ori ujredis saSualo, gamoiyeneba mravlobiTi Sedarebis kriteriumi, raTa dadgenil iqnes kritikuli gansxvaveba (CD) martivi efeqtisaTvis. Tu martivi efeqtis realuri mniSvneloba tolia an aRemateba CD–s, maSin ujredebis saSualoebs Soris gansxvaveba statistikurad mniSvnelovania. aqac, martivi efeqtebis saboloo SedarebisaTvis xSirad gamoiyeneba tukis (Tukey) HSD kriteriumi. am kriteriumisaTvis CD ase gamoiTvleba:

CD = qABn

MSSecdoma

q-s mniSvneloba gamoiTvleba cxrilidan # A.4 da damokidebulia martivi efeqtis Casatarebeli Sedarebebis raodenobaze da dfSecdomaze. radgan cxrili # A.4 Sedegil iqna erTfaqtoriani sqemisaTvis, Cven cxrilSi viyenebT Semdeg gardaqmnas:

cxrilidan # A.4 gamoiyeneT q, romelic

moTavsebulia svetSi damoukidebeli sqemis tipi cvladis _____ doneebisaTvis

______________________________________________________ 2 x 2 3 2 x 3 5 3 x 2 5 3 x 3 7 2 x 4 8 4 x 3 8 4 x 4 10

______________________________________________________ MSSecdoma aris MSSecdomis mniSvneloba, gamoTvlili dispersiis faqtorul analizSi da nAB aris qulebis raodenoba TiToeul ujredSi. am gamoTvlis ilustraciisaTvis 2x2 eqsperimentSi cdis piris mxedvelobis areSi yofnisa da siaxlis tipis efeqtebisaTvis, viyenebT svets a=3 cxrilidan # A.4, raTa vipovoT q. radganac MSSecdomis df aris

16 (cxrilidan # 11.8) da α = .05, q-s mniSvneloba (3, 16) aris 3.65. cxrilSi # 11.8 MSSecdoma =108.125 da nAB, qulebis raodenoba, romlebic Sedis TiToeuli ujredSi, aris 5. aqedan gamomdinare, tukis CD-s ricxviTi mniSvneloba aris

CD = 3.655125.108 = 17.0 wami, damrgvalebuli erT meaTedamde.

nebismieri martivi efeqti cxrilidan # 11.11, romelic absoluturi mniSvnelobiT tolia an aRemateba 17.0-s, statistikurad mniSvnelovania .05 doneze. martivi efeqtebi cdis piris mxedvelobis areSi yofnisaTvis cudi siaxlis mosmenis pirobaSi (anu - 30.2 wami) da siaxlis tipisaTvis cdis piris mxedvelobis areSi yofnis pirobaSi (anu -27.8 wami) aRemateba absoluturi mniSvnelobiT CD-s, romelic aris 17.0 wami. es martivi efeqtebi gviCvenebs statistikurad mniSvnelovan gansxvavebebs ujredebis saSualoebs Soris. martivi efeqtebi cdis piris mxedvelobis areSi yofnisTvis kargi siaxlis mosmenis pirobaSi (anu + 6.2 wami) da siaxlis tipisaTvis cdis piris mxedvelobis areSi aryofnis pirobaSi (anu + 8.6 wami) absoluturi mniSvnelobiT naklebia, vidre CD, romelic 17.0 wamia. es martivi efeqtebi ar aris mniSvnelovani da monawile ujredebis saSualoebi gansxvavdeba mxolod SerCevis Secdomis gamo. mTavari efeqtebis axsna, rodesac xdeba urTierTqmedeba rodesac xdeba statistikurad mniSvnelovani urTierTqmedeba, damoukidebeli cvladebis mTavari efeqtebi SeiZleba ver gamoviyenoT raime mniSvnelobis mqone axsnisaTvis. es xdeba imis gamo, rom damoukidebeli cvladis mTavari efeqti aris am cvladis martivi efeqtebis saSualo. magaliTad, yvela ujredSi cdis pirebis toli raodenobis arsebobisas, cdis piris mxedvelobis areSi yofnis mTavari efeqti (-12.0 wami) cxrilSi # 11.11 aris kargi da cudi siaxleebisTvis cdis piris mxedvelobis areSi yofnis martivi efeqtebis saSualo (anu [(+6.2) + (-30.2)]/2 = _12.0 wami). msgavsi damokidebulebaa siaxlis tipis mTavari efeqtisaTvis, -9.6 wami. es aris saSualo martivi efeqtebisa siaxlis tipisaTvis cdis piris mxedvelobis areSi aryofnis pirobaSi, +8.6 wami, da cdis piris mxedvelobis areSi yofnis pirobaSi, -27.8 wami. Sesabamisad, rodesac xdeba urTierTqmedeba, ise, rom damoukidebeli cvladis martivi efeqtebi ar udris erTmaneTs, am damoukidebeli cvladis mTavari efeqti SeZleba zustad ar warmoadgendes cvladis efeqts. aseTi mTavari efeqtebi aris urTierTqmedebis artefaqtebi da ar iZlevian raime mniSvnelobis mqone informacias eqsperimentis Sedegebis Sesaxeb. Sesabamisad, artefaqturi mTavari efeqti aris mTavari efeqti, romelic ar iZleva mniSvnelobis mqone informacias damoukidebeli cvladis efeqtis Sesaxeb, radgan is Tavs iCens mxolod realuri urTierTqmedebis specifikuri paternis gamo. es aris am cvladis martivi efeqtebis saxis xelovnuri Sedegi. aqedan gamomdinare, faqtorul sqemaSi mTavari efeqtebis axsnasTan dakavSirebiT SemoRebulia Tavis dasazRvevi mniSvnelovani SeniSvna:

a. Tu faqtorul sqemaSi xdeba statistikurad mniSvnelovani urTierTqmedeba, maSin rogorc A, aseve B Bfaqtorebis mTavari efeqtebi SeiZleba iyos artefaqturi da SeiZleba ar

warmoadgendnen mniSvnelobis mqone Sedegebs am damoukidebeli cvladis efeqtis Sesaxeb.

b. Tu urTierTqmedeba faqtorul sqemaSi aramniSvnelovania, maSin rogorc A, aseve B faqtorebis mTavari efeqtebi warmoadgenen mniSvnelobis mqone Sedegebs am damoukidebeli cvladis efeqtis Sesaxeb.

artefaqturi mTavari efeqti: mTavari efeqti, romelic ar iZleva mniSvnelobis mqone informacias damoukidebeli cvladis efeqtis Sesaxeb faqtorul sqemaSi.

dispersiuli analizSi statistikurad mniSvnelovani urTierTqmedeba ar niSnavs imas, rom mTavari efeqtebi iyos artefaqturi, magram is aris gafrTxileba, rom Semowmdes nebismieri mniSvnelovani mTavari efeqti, raTa gamovlindes, iZleva Tu ara is mniSvnelobis mqone Sedegebs damoukidebeli cvladis efeqtis Sesaxeb. martivi mTavari efeqtebis axsna, rodesac ar xdeba statistikurad mniSvnelovani urTierTqmedeba Tu damoukidebel cvladebs Soris ar aris statistikurad mniSvnelovani urTierTqmedeba, maSin erTi damoukidebeli cvladis efeqti ar aris damokidebuli meore damoukidebeli cvladis doneze. amitom, Tu urTierTqmedeba ar xdeba, A faqtoris martivi efeqti B1 donisaTvis B2 donisaTvis A faqtoris martivi efeqtis toli iqneba. nebismieri realuri gansxvaveba am martiv efeqtebs Soris mxolod SerCevis SecdomiT aris gamowvuli. ufro metic, Tu A faqtoris martivi efeqtebi erTmaneTis tolia, isini aseve A faqtoris mTavari efeqtis tolia, da A faqtoris mTavari efeqti am damoukidebeli cvladis efeqtis saukeTeso aRweras iZleva. msgavsi urTierToba aRiniSneba B faqtoris martivi efeqtebisaTvisac. Tu urTierTqmedeba ar xdeba, maSin B faqtoris martivi efeqti A1 doneze A2 doneze B faqtoris martivi efeqtis toli iqneba. nebismieri realuri gansxvaveba am martiv efeqtebs Soris mxolod SerCevis SecdomiT aris gamowvuli. B faqtoris martivi efeqtebi aseve B faqtoris mTavari efeqtis tolia, da B faqtoris mTavari efeqti am damoukidebeli cvladis efeqtis saukeTeso aRweras iZleva. es mimarTebebi Sejamebulia cxrilis # 11.12 (b) nawilSi. cxrili # 11.12 urTierTqmedeba 2x2 faqtorul dispersiul analizSi da martivi efeqtebis da mTavari efeqtebis axsna.

(a) Tu Frealuri AxB urTierTqmedebisaTvis xvdeba uaryofis areSi, maSin:

1. urTierTqmedebis F Sefardeba statistikurad mniSvnelovania; 2. erTi damoukidebeli cvladis efeqti damokidebulia meore damoukidebeli

cvladis doneze;

3. A faqtoris martivi efeqti B1 doneze (anu 11BAX – 12BAX ) ar udris A

faqtoris martiv efeqts B2 doneze (anu 21BAX – 22BAX ). A faqtoris martiv efeqtebs Soris realuri sxvaoba gamowveulia damoukidebeli cvladebis urTierTqmedebiT da ara mxolod SerCevis SecdomiT;

4. B faqtoris martivi efeqti A1 doneze (anu 11BAX – 21BAX ) ar udris B

faqtoris martiv efeqts A2 doneze (anu 12BAX – 22BAX ). B faqtoris martiv

efeqtebs Soris realuri sxvaoba gamowveulia damoukidebeli cvladebis urTierTqmedebiT da ara mxolod SerCevis SecdomiT;

5. A faqtoris martivi efeqtebi ar udris A faqtoris mTavar efeqts; 6. B faqtoris martivi efeqtebi ar udris B faqtoris mTavar efeqts; 7. TiToeuli martivi efeqti unda Sedardes tukis CD-s, raTa vipovoT, Tu

ujredis romeli saSualoebi gansxvavdeba erTmaneTisagan mniSvnelovnad; 8. rogorc A, aseve B faqtoris mTavari efeqtebi SeiZleba artefaqturi iyos da

SeiZleba ar warmoadgendes mniSvnelobis mqone Sedegebs damoukidebeli cvladis efeqtis Sesaxeb.

(b) Tu Frealuri AxB urTierTqmedebisaTvis ar xvdeba uaryofis areSi, maSin:

1. urTierTqmedebis F Sefardeba ar aris statistikurad mniSvnelovani; 2. erTi damoukidebeli cvladis efeqti ar aris damokidebuli meore damoukidebeli

cvladis doneze; 3. A da B faqtorebis mTavari efeqtebi warmoadgenen mniSvnelobis mqone Sedegebs

eqsperimentis Sedegebis Sesaxeb;

4. A faqtoris martivi efeqti B1 doneze (anu 11BAX – 12BAX ) mniSvnelovnad ar

gansxvavdeba A faqtoris martivi efeqtisagan B2 doneze (anu 21BAX – 22BAX ). A faqtoris martiv efeqtebs Soris realuri gansxvaveba gamowveulia SerCevis SecdomiT;

5. B faqtoris martivi efeqti A1 doneze (anu 11BAX – 21BAX ) mniSvnelovnad ar

gansxvavdeba B faqtoris martivi efeqtisagan A2 doneze (anu 12BAX – 22BAX ). B faqtoris martiv efeqtebs Soris realuri gansxvaveba gamowveulia SerCevis SecdomiT;

6. A faqtoris martivi efeqtebi ar gansxvavdeba A faqtoris mTavari efeqtisagan. A faqtoris mTavari efeqti warmoadgens A faqtoris efeqtis saukeTeso aRweras;

7. B faqtoris martivi efeqtebi ar gansxvavdeba B faqtoris mTavari efeqtisagan. B faqtoris mTavari efeqti warmoadgens B faqtoris efeqtis saukeTeso aRweras;

8. ar aris saWiro martivi efeqtebis gaanalizeba mravlobiTi Sedarebis kriteriumiT.

Sejameba: dispersiuli analizis axsna sanimuSo eqsperimentSi faqtoruli eqsperimentis Sedegebis gageba moiTxovs TiToeuli mTavari efeqtis da urTierTqmedebis ganxilvas da maT axsnas dispersiul analizSi miRebuli gadawyvetilebebis gaTvaliswinebiT. rodesac xdeba statistikurad mniSvnelovani urTierTqmedeba, rogorc es mocemuli eqsperimentis magaliTSia (ix. cxrili # 11.8), Sedegebis axsna urTierTqmedebiT unda daiwyos. cdis piris mxedvelobis areSi yofnisa da siaxlis tipis urTierTqmedebis axsna Frealuris mniSvneloba urTierTqmedebisaTvis cxrilidan # 11.8 = 15.32 metia, vidre Fkritikulis mniSvneloba 4.49 da amdenad xvdeba uaryofis areSi. aqedan gamomdinare, H0 A da B faqtorebis urTierTqmedebisaTvis, H0: yvela (µAB – µ A – µ B + µ G) = 0 uaryofilia da

H1: yvela (µAB – µ A – µ B + µ G) ar udris 0-s miRebulia. am urTierTqmedebis interpretaciisaTvis Tavdapirvelad viyenebT tukis HSD kritikul sxvaobas cxrilis # 11.11 martivi efeqtebisaTvis. tukis CD aris 17.0 wami; amitom nebismieri martivi efeqti cxrilSi # 11.11, romelic absoluturi mniSvnelobiT tolia an metia 17.0 wamze, statistikurad mniSvnelovania .05 doneze.

Tu davubrundebiT cxrils # 11.11, jer ganvixilavT A faqtoris – cdis piris mxedvelobis areSi yofnis martiv efeqtebs. A faqtoris martivi efeqti B1 doneze, sxvaoba +6.2 wami, naklebia CD-ze, romelic 17.0 wamia; es martivi efeqti aramniSvnelovania. cdis piris mxedvelobis areSi yofnas ar aqvs efeqti, rodesac kargi siaxlis Setyobineba xdeba. sxvaoba “Cans” da “ar Cans” ujredebis saSualoebs Soris, rodesac kargi siaxlis Setyobineba xdeba (Sesabamisad, 90.8 da 84.6 wamebi), mxolod SerCevis Secdomis gamoa miRebuli. A faqtoris martivi efeqti B2 doneze, sxvaoba –30.2 wami, absoluturi mniSvnelobiT metia, vidre CD, romelic 17.0 wamia. es martivi efeqti statistikurad mniSvnelovania. cudi siaxlis Setyobineba mniSvnelovnad ufro nela xdeba, rodesac cdis piri mxedvelobis areSi imyofeba (112.4 wami), vidre rodesac cdis piri mxedvelobis areSi ar imyofeba (82.2 wami). axla SevudgeT B faqtoris – siaxlis tipis – martiv efeqtebs. B faqtoris martivi efeqti A1 doneze, sxvaoba +8.6 wami, absoluturi mniSvnelobiT naklebia CD-ze, romelic 17.0 wamia; es martivi efeqti aramniSvnelovania. rodesac cdis piri ar imyofeba mxedvelobis areSi, kargi da cudi siaxleebis Setyobineba erTnairi siswrafiT xdeba. sxvaoba “kargi siaxlisa” da “cudi siaxlis” ujredebis saSualoebs Soris (Sesabamisad, 90.8 da 82.2. wami) mxolod SerCevis Secdomis gamoa miRebuli. B faqtoris martivi efeqti A2 doneze, sxvaoba –27.8 wami, absoluturi mniSvnelobiT metia CD-ze, romelic 17.0 wamia; amdenad, es martivi efeqti mniSvnelovani sxvaobaa. rodesac cdis piri imyofeba mxedvelobis areSi, cudi siaxlis Setyobineba mniSvnelovnad ufro nela (112.4 wami) xdeba, vidre kargi siaxlis Setyobineba (84.6 wami). am analizs mivyavarT cdis piris mxedvelobis areSi yofnasa da siaxlis tips Soris urTierTqmedebis Sedegebis Semdgom Sejamebamde:

cdis piris mxedvelobis areSi yofnis efeqti damokidebulia Sesatyobinebeli siaxlis tipze;

a. kargi siaxlis SemTxvevaSi sxvaoba latentur droSi mxedvelobis areSi myof da armyof cdis pirebs Soris (Sesabamisad, 90.8 da 84.6 wami) ar iyo mniSvnelovani;

b. cudi siaxlis SemTxvevaSi im cdis pirebis latenturi dro, romlebic iyvnen mxedvelobis areSi (112.4 wami) mniSvnelovnad meti iyo im cdis pirebis latentur droze, romlebic ar imyofebodnen mxedvelobis areSi (82.2 wami). siaxlis tipis efeqti damokidebulia cdis piris mxedvelobis

areSi yofnaze; a. im cdis pirebisTvis, romlebic ar imyofebodnen mxedvelobis areSi,

ar iyo mniSvnelovani sxvaoba kargsa da cud siaxleebs Soris; aseTi cdis pirebi karg da cud siaxleebs ekvivalenturi latenturi drois Semdeg atyobinebdnen (Sesabamisad, 90.8 da 82.2 wami);

b. is cdis pirebi, romlebic imyofebian mxedvelobis areSi, mniSvnelovnad did dros andomeben cudi siaxlis Setyobinebas, vidre kargis Setyobinebas, Sesabamisad, 112.4 da 84.6 wami.

es Sedegebi aSkarad Seesabameba TviTprezentaciis hipoTezis winaswarmetyvelebas, rom MUM efeqti warmoadgens mxolod publikaze

gaTvlil qmedebas, romelic dadebiTi socialuri imijis SenarCunebazea mimarTuli. eqsperimentis Sedegebis grafikuli warmodgena. faqtoruli sqemebis gamoqveynebul SedegebSi ujredebis saSualoebi grafikulad xSirad suraTis saxiTaa warmodgenili, da ara # 11.5 msgavsi cxrilis saxiT, radganac suraTi xSirad urTierTqmedebis ufro naTel suraTs iZleva. ujredebis saSualoebi cxrilidan # 11.5 naCvenebia suraTze # 11.1. am suraTze gansazRvrulia TiToeuli ujredis saSualo, magram gamoqveynebul statiebSi ujredebis saSualoebi warmodgenilia xolme mxolod iseTi simboloTi, rogoricaa Savi an TeTri wreebi. iqve suraTze aRniSnulia, Tu ras gamosaxavs es wreebi. rodesac grafiks vagebT, erTi damoukidebeli cvladi gamoisaxeba abscisaze (anu horizontalur RerZze), xolod meore ki – ordinataze (anu vertikalur RerZze). romeli damoukidebeli cvladi gamoisaxeba abscisaze? zogadad, miRebuli wesis mixedviT, abscisaze aigeba is damoukidebeli cvladi, romelic bunebiT ufro raodenobrivia an romelsac ufro uwyveti ganzomileba udevs safuZvlad. Cvens magaliTSi, aseTi cvladia B faqtori, siaxlis tipi, radganac siaxlis tips SeuZlia uwyvetad cvalebadoba kargi-cudi ganzomilebis gaswvriv; zogi siaxle SeiZleba Zalian kargi iyos, zogi – Zalian cudi, xolo zogi siaxle am or ukiduresobas Soris aris. amitom siaxlis tipis cvladi moTavsda abscisaze. meore damoukidebeli cvladi – cdis piris mxedvelobis areSi yofna – naklebad uwyvetad cvalebadobs; ar arsebobs uwyveti gradacia mxedvelobis areSi yofnasa da aryofnas Soris. Sesabamisad, es damoukidebeli cvladi grafikze ori funqciiT gamosaxeba. uwyveti xaziT SeerTebuli Savi wreebi gamosaxavs cdis piris mxedvelobis areSi aryofnis pirobas (A1), xolo wyvetili xaziT SeerTebuli TeTri wreebi - cdis piris mxedvelobis areSi yofnis pirobas (A2). damokidebuli cvladi gamosaxulia ordinataze. radganac yvelaze mokle latenturi

dro aris 82.2 wami, RerZi dayofilia (=) da pirveli aRniSnuli

mniSvneloba aris 80 wami. suraTi # 11.1 pasuxis latenturi dro (wamebSi) sanimuSo eqsperimentSi, rogorc siaxlis tipis (kargi siaxle - cudi siaxle) da cdis piris mxedvelobis areSi yofnis (Cans – ar Cans) funqcia. am suraTze agebuli cxrilis # 11.5 ujredebis saSualoebi.

80

85

90

95

100

105

110

115

1 2

siaxlis tipi (faqtori B)

Set

yobinebis

lat

enturi dro

(wamebi)

ar Cans Cans

suraTi # 11.1 gvaCvenebs gansxvavebul martiv efeqtebs saTiTaod orive damoukidebeli cvladisaTvis. rodesac xdeba statistikurad mniSvnelovani urTierTqmedeba, erTi damoukidebeli cvladis efeqti damokidebulia meore damoukidebeli cvladis doneze, im cvladis, romelTanac is kombinirebulia. urTierTqmedebis es saxe, romelic gamoiTqmeba “damokidebulia”, naTeli xdeba am suraTidan. Tu Tqven gkiTxavdnen: “rogor moqmedebs cdis piris mxedvelobis areSi yofna pasuxis latentur droze?”, Tqveni pasuxi iqneboda: “es damokidebulia im siaxlis tipze, romlis Setyobinebac xdeba. kargi siaxlisaTvis cdis piris mxedvelobis areSi yofnis da mxedvelobis areSi aryofnis pirobebs Soris aramniSvnelovani sxvaoba arsebobs. magram cudi siaxlisaTvis cdis piris mxedvelobis areSi yofnis dros pasuxis latenturi dro mniSvnelovnad metia, vidre cdis piris mxedvelobis areSi aryofnis dros.” amis msgavsad, Tu Tqven gkiTxavdnen: “axdens Tu ara gavlenas siaxlis tipi pasuxis latenturobaze?”, Tqveni pasuxi aseTi iqneboda: “es damokidebulia imaze, Tu rogoria cdis piris mxevelobis areSi yofnis piroba. rodesac cdis piri ar imyofeba mxedvelobis areSi, kargi da cudi siaxleebis Setyobinebis latenturoba aramniSvnelovnad aris gansxvavebuli. xolo rodesac cdis piri imyofeba mxedvelobis areSi, cudi siaxlis Setyobinebis latenturi dro mniSvnelovnad metia, vidre kargi siaxlisa.” monacemTa aRmniSvnelo wertilebis damakavSirebeli xazebi suraTze # 11.1 ar aris paraleluri, rac statistikurad mniSvnelovani urTierTqmedebis amsaxveli suraTis ganmasxvavebeli maxasiaTebelia. radganac urTierTqmedeba gulisxmobs, rom damoukidebeli cvladis martivi efeqtebi ar udris mis mTavar efeqts, statistikurad mniSvnelovani urTierTqmedebis agebis Sedegad araparaleluri xazebi unda miviRoT. magram ar aris aucilebeli, rom xazebma erTmaneTi

ar Cans/kargi siaxle ujredis saSualo

Cans/cudi siaxle ujredis saSualo

ar Cans/cudi siaxle ujredis saSualo

Cans/kargi siaxle ujredis saSualo

B1 B2

cdis piris mxedvelobaSi yofna (faqtori A)

gadakveTon, rogorc es xdeba suraTze # 11.1; es moxdeba mxolod maSin, rodesac damoukidebeli cvladis martiv efeqtebs urTierTsawinaaRmdego niSnebi aqvT, rogorc es am magaliTSi moxda. praqtikaSi iSviaTad xdeba, rom nebismieri ori xazi suraTze zustad paraleluri iyos. SerCevis Secdomis gamo xazebi ar iqneba zustad paraleluri, maSinac ki, rodesac ar arsebobs statistikurad mniSvnelovani urTierTqmedeba. amdenad, mxolod vizualuri Semowmeba ar aris sakmarisi imisaTvis, rom ganisazRvros, moxda Tu ara urTierTqmedeba. urTierTqmedeba xdeba monacemebSi mxolod maSin, rodesac dispersiul analizSi urTierTqmedebisaTvis aRmoCenilia statistikurad mniSvnelovani Frealuri. A damoukidebeli cvladis – cdis piris mxedvelobis areSi yofnis - mTavari efeqtis axsna. cxrilSi # 11.8 Frealuris mniSvneloba A faqtoris mTavari efeqtisaTvis Frealuri (1.16) = 6.66 metia, vidre Fkritikuli (1.16) = 4.49. es Frealuri xvdeba

uaryofis areSi da “ 210 : AAH µµ = ” cdis piris mxedvelobis areSi

yofnisaTvis uaryofilia, xolo “ AH µ:1 -ebi ar udris erTmaneTs” ki –

miRebuli. es gadawyvetileba gulisxmobs, rom cdis piris mxedvelobis areSi yofnis mTavari efeqti (sxvaoba –12.0 wami mTavari efeqtebis saSualoebs - 86.5 da 98.5 wamebs Soris cdis piris mxedvelobis areSi aryofnis da yofnis dros, Sesabamisad) statistikurad mniSvnelovania. es mTavari efeqti miuTiTebs, rom siaxlis Setyobineba ufro swrafad xdeba, rodesac cdis piri ar Cans, vidre roca cdis piri Cans. magram es mTavari efeqti artefaqturia, da kavSiri, romelsac is gamoxatavs, ar aris rogorc kargi, aseve cudi siaxlisaTvis. rogorc aRmoCnda urTierTqmedebis Seswavlis Sedegad, piroba, rodesac cdis piri Cans, mniSvnelovnad xargrZliv latentur dros iwvevs mxolod maSin, rodesac siaxle cudia. kargi siaxlisaTvis, latenturi droebi cdis piris mxedvelobis areSi yofnis da aryofnis pirobebSi mniSvnelovnad ar gansxvavdeba. amdenad, miuxedavad imisa, rom cdis piris mxedvelobis areSi yofnis mTavari efeqti ufro did latentur dros gviCvenebs im pirobisaTvis, rodesac cdis piri Cans, martivi efeqtebis analizi gviCvenebs, rom es mimarTeba marTebulia mxolod cudi siaxlis Setyobinebis dros; is ar aris marTebuli kargi siaxlisaTvis. amdenad, cdis piris mxedvelobis areSi yofnis mTavari efeqti ar gvawvdis mniSvnelovan informacias eqsperimentis SedegebTan dakavSirebiT; is artefaqturia. B damoukidebeli cvladis – siaxlis tipis - mTavari efeqtis axsna Frealuris mniSvneloba B faqtorisTvis – siaxlis tipisaTvis – aris 4.26. Frealuris es mniSvneloba naklebia, vidre Fkritikuli (4.49) da, aqedan

gamomdinare, “ 210 : AAH µµ = ” ar aris uaryofili da “ AH µ:1 -ebi ar udris

erTmaneTs” ar aris miRebuli. es gadawyvetileba niSnavs, rom siaxlis tipis mTavari efeqti (sxvaoba –9.6 wami 87.7 da 97.3 wamebs Soris kargi da cudi siaxlisTvis, Sesabamisad) aramniSvnelovania. mxolod am aramniSvnelovani mTavari efeqtidan davaskvnidiT, rom siaxlis tipi ar axdens gavlenas cdis piris pasuxis latentur droze; mTavari efeqtis 9.6-wamiani sxvaoba kargi da cudi siaxlis pirobebisaTvis SerCevis

Secdomis gamo iqna miRebuli. magram zustad aRwers Tu ara es daskvna siaxlis tipis efeqts? aqac martivi efeqtebis Seswavla gviCvenebs, rom es ase ar aris; siaxlis tips aqvs efeqti, magram efeqti damokidebulia cdis piris mxedvelobis areSi yofnaze. rodesac cdis piri ar aris eqsperimentatoris TanaSemwes mxedvelobis areSi, kargsa da cud siaxles Soris ar arsebobs mniSvnelovani gansxvaveba pasuxis latenturobaSi (anu sxvaoba 90.8 da 82.2 wamebs Soris SerCevis Secdomis gamo miviReT); magram rodesac cdis piri aris eqsperimentatoris TanaSemwes

mxedvelobis areSi, cudi siaxlis ( X A2B2 = 112.4 wams) Setyobineba

mniSvnelovnad ufro nela xdeba, vidre kargi siaxlisa ( X A2B1 = 84.6 wams). statistikurad mniSvnelovani urTierTqmedebis gamo ar aris saWiro siaxlis tipis aramniSvnelovani efeqtis axsna. pirovnuli diskomfortis hipoTezam iwinaswarmetyvela, rom siaxlis tipisaTvis iarsebebs statistikurad mniSvnelovani da mniSvnelobis mqone mTavari efeqti. cudi siaxlis Setyobinebas ufro didi dro unda sWirdebodes, vidre kargi siaxlis Setyobinebas, miuxedavad imisa, aris Tu ara cdis piri TamaSemwes mxedvelobis areSi. am magaliTSi siaxlis tipis mTavari efeqtis Sedegi aSkarad ar Seesabameba pirovnuli diskomfortis hipoTezis winaswarmetyvelebas. urTierTqmedebis saxeebi kavSiri ujredis saSualoebs Soris sanimuSo eqsperimentSi mxolod erT-erTia im mravali kavSiridan, romlis Sedegad statistikurad mniSvnelovan urTierTqmedebas viRebT. dispersiul analizSi statistikurad mniSvnelovani urTierTqmedeba gviCvenebs mxolod imas, rom damoukidebeli cvladis martiv efeqtsa da mTavar efeqts Soris sxvaobebi ar iqneba erTmaneTis toli. mxolod es ar iZleva raime Semdgom informacias im mravali gzis Sesaxeb, romlebiTac es Sedegi SeiZleba migveRo. urTierTqmedebis zusti buneba SeiZleba ganisazRvros mxolod ujredebis saSualoebs Soris kavSiris frTxili analizis gziT.

Carevis efeqtis siZlieris gazomva dispersiul faqtorul analizSi rogorc es iyo erTfaqtoriani dispersiuli analizis dros, statistikurad mniSvnelovani efeqti faqtorul sqemaSi SeiZleba

gaizomos 2η -iT. radganac arsebobs ori mTavari efeqtis da

urTierTqmedebis kriteriumebi, 2η -is sam gansxvavebul mniSvnelobas

viRebT:

2η = sruliSSSS A A faqtoris statistikurad mniSvnelovani mTavari

efeqtisaTvis.

2η = sruliSSSSB B faqtoris statistikurad mniSvnelovani mTavari

efeqtisaTvis.

2η = sruliSS

SS AxB A da B faqtorebis statistikurad mniSvnelovani

urTierTqmedebisaTvis.

vaCvenoT 2η -is gamoyeneba magaliTSi, roca mxolod urTierTqmedebam

mogvca mniSvnelobis mqone informacia; am SemTxvevaSi 2η

urTierTqmedebisaTvis, damyarebuli SS mniSvnelobebze cxrilidan # 11.8, iqneba:

2η = sruliSS

SS AxB = 000.4567200.1656 = .36

es mniSvneloba gviCvenebs, rom pasuxebis latenturobis dispersiis 36 procenti aixsneba cdis piris mxedvelobis areSi yofnisa da siaxlis tipis urTierTqmedebiT. SeamowmeT Tqveni codna XI.4

1. gansazRvreT: artefaqturi mTavari efeqti, mTavari efeqti, martivi efeqti. 2. Semdegi cxrilebi da suraTebi # 11.2 da 11.3 warmoadgenen eqsperimentSi miRebul

ujredebis saSualoebs. saSualoebis TiToeuli erTobliobisaTvis mocemulia dispersiuli Sejamebuli analizi. Sejamebul analizSi statistikurad

mniSvnelovani Frealuri aRniSnulia p < .05. statistikurad aramniSvnelovani

Frealuri aRniSnulia p > .05. rodesac xdeba mniSvnelovani urTierTqmedeba,

tukis HSD CD martivi efeqtebisaTvis mocemulia dispersiis analizis Sejamebis boloSi. davuSvaT, rom A faqtori aris swavlebis meTodi, romelsac aqvs ori done: kompiuterizebuli swavleba (A1) da arakompiuterizebuli swavleba (A2). faqtori B aris saswavlo masala – maTematikuri (B1) da socialuri (B2) Sinaarsis. damokidebuli cvladi aris testSi miRebuli qula. axseniT saSualoebis TiToeuli erToblioba, ipoveT martivi da mTavari efeqtebi TiToeuli damoukidebeli cvladisaTvis da daukavSireT es efeqtebi Sejamebul dispersiul analizs. Semdeg TiToeuli magaliTisaTvis upasuxeT Semdeg kiTxvebs:

a. mniSvnelovnad gansxvavdebian Tu ara erTmaneTisagan A faqtoris mTavari efeqtis saSualoebi?

b. mniSvnelovnad gansxvavdebian Tu ara erTmaneTisagan B faqtoris mTavari efeqtis saSualoebi?

g. xdeba Tu ara statistikurad mniSvnelovani urTierTqmedeba A da B faqtorebs Soris?

d. rogor moqmedebs swavlebis meTodis (A faqtori) efeqti testSi miRebul qulebze?

e. rogor moqmedebs masalis tipis (B faqtori) efeqti testSi miRebul qulebze?

v. iZlevian Tu ara mTavari efeqtebi gamosadeg informacias eqsperimentis Sedegebis Sesaxeb?

suraTi # 11.2 A da B faqtorebis manipulaciis Semcveli eqsperimentis Sedegi

76

80

84

88

92

96

100

1 2

A faqtori

qula

B1 B2

suraTi # 11.3 A da B faqtorebis manipulaciis Semcveli eqsperimentis Sedegi

667074788286909498

1 2

A faqtori

qula

B1 B2

cxrili 1 A faqtori

A1 A2 dispersiuli analizis Sedegebis Sejameba

} B faqtori

dispersiuli analizis Sedegebis Sejameba A faqtori p > .05 B faqtori p < .05 AxB p < .05

A1 A2

} B faqtori

dispersiuli analizis Sedegebis Sejameba A faqtori p < .05 B faqtori p > .05 AxB p < .05

A1 A2

B1 B faqtori B2

91 85 82 87

A faqtori p < .05. B faqtori p > .05. A x B p > .05.

cxrili 2 A faqtori


B1 B faqtori B2

89 90 81 80

A faqtori p > .05. B faqtori p < .05. A x B p > .05.

cxrili 3 A faqtori


B1 B faqtori B2

85 94 78 86

A faqtori p < .05. B faqtori p < .05. A x B p > .05.

cxrili 4 A faqtori


B1 B faqtori B2

88 75 76 90

A faqtori p > .05. B faqtori p > .05. A x B p < .05.

cxrili 5 A faqtori

A1 A2

dispersiuli analizis Sedegebis Sejameba

B1 B faqtori B2

87 88 74 80

A faqtori p < .05. B faqtori p < .05. A x B p < .05.

cxrili 6 A faqtori


B1 B faqtori B2

90 80 81 82

A faqtori p < .05. B faqtori p < .05. A x B p < .05.

magaliTi # 11.1 2x2 sqemis analizi statistikurad mniSvnelovani urTierTqmedebiT da aramniSvnelovani mTavari efeqtiT amocana mxedveloba da smena ori damoukidebeli SegrZnebaa, da mainc maT Soris SesaZlebelia arsebobdes raRac saerTo gancdis TvalsazrisiT. vizualur stimulebs zogjer axasiaTeben, rogorc JRerads, xolo xmebs - rogorc naTels. amis safuZvelze lindnerma da hainanma (Lindner & Hynan, 1987) wamoayenes hipoTeza, rom musikis tipi, romelsac adamiani usmenda, moaxdenda gavlenas am adamianis mier abstraqtuli naxatis aRqmaze. am hipoTezis Sesamowmeblad maT Caatares cdis pirTa Soris 2x2 sqemis eqsperimenti. isini cvlidnen musikis tips, romelsac cdis piri usmenda: es iyo an arastruqturirebuli avangarduli an struqturirebuli minimalisturi musika (A faqtori). aseve icvleboda cdis piris sqesi: mamrobiTi an mdedrobiTi (B faqtori). musikis mosmenisas yoveli cdis piri aTvalierebda rva abstraqtul naxats. naxatebis daTvalierebis Semdeg igi afasebda maT ramdenime 7-quliani Sefasebis skalaze. erTi skala Sedgenili iyo zedsarTavi saxelebis wyvilebisagan: daZabuli (1) - Tavisufali (7), rigiduli (1) - fleqsiburi (7), Seviwrovebuli (1) - gafarToebuli (7). davuSvaT, rom qulebi miRebulia am zedsarTavi saxelebis samive erTobliobis SejamebiT, ise, rom Sefasebis qulebi SeiZleba icvlebodes 3-dan 21-mde. davuSvaT, rom 24 cdis pirisagan miRebulia Semdegi hipoTeturi qulebi: musikis tipi (A)

avangarduli (A1) minimalisturi (A2)

mamrobiTi (B1) cdis pirTa sqesi (B) mdedrobiTi (B2)

4 16 5 14 7 17 6 19 10 16 4 20 12 5 15 9 18 7 13 6 16 10 16 5

axdens Tu ara musikis tipi gavlenas naxatebis Sefasebaze, gansxvavdebian Tu ara mamakacebi da qalebi micemuli Sefasebebis mixedviT, da arsebobs Tu ara urTierTqmedeba musikis tipisa da cdis piris sqess Soris? amoxsna

gamoyenebuli sqema warmoadgens cdis pirTa Soris 2x2 sqemas. pirveli nabiji aris cdis pirTa Soris orfaqtoriani dispersiuli analizis Catareba imis gansasazRvrad, arsebobs Tu ara statistikurad mniSvnelovani mTavari efeqtebi an statistikurad mniSvnelovani urTierTqmedeba. gamosayenebeli daSvebebi:

1. cdis pirebi populaciisagan SemTxveviTad da damoukideblad iqna SerCeuli;

2. Sefasebebis Secdomebi populaciaSi normaluradaa ganawilebuli;

3. populaciis SerCevebSi Sefasebis qulebis dispersiebi tolia.


A faqtori H0: µ A1 = µ A2


B faqtori H0: µ B1 = µ B2


AxB H0: yvela (µAB - µ A - µ B + µ G) = 0 H1: yvela (µAB - µ A - µ B + µ G) ar udris 0

mniSvnelovnobis done: α = .05. df: dfA = a - 1 = 2 - 1 = 1 dfB = b - 1 = 2 - 1 = 1 dfAxB = (a - 1) (b - 1) = (2 - 1)(2 - 1) = 1 dfSecdoma = ab(nAB - 1) = (2)(2)(6 - 1) = 20 dfsruli = N - 1 = 24 - 1 = 23 F-is kritikuli mniSvneloba: TiToeuli Frealuris aqvs 1 df mricxvelad da 20 df mniSvnelad; amitom Fkritikuli (1.20) = 4.35 [cxrilidan # A.3a]. uaryofis are: 4.35-ze toli an meti Frealuri-s mniSvnelobebi. gamoTvla: dispersiuli analizi SeiZleba gamoTvlil iqnes qulebidan im gamoTvliTi formulebis gamoyenebiT, romlebic mocemulia am Tavis damatebaSi an kompiuteruli programis gamoyenebiT. Cven gamoviyeneT formulebi am Tavis damatebidan, raTa migveRo saboloo cxrili: wyaro df SS MS F musika (A) sqesi (B) AxB Secdoma sul

1 1 1 20 23

13.50 1.50 541.50 96.00 652.50

13.50 1.50 541.50 4.80

2.81 0.31 112.81*

* p < .01 gadawyvetilebebi: am analizis statistikuri hipoTezebisaTvis misaRebia sami gadawyvetileba, erTi yoveli FrealurisTvis.

• musikis tipi (A faqtori): Frealuri = 2.81. es Frealuri ar xvdeba uaryofis areSi; H0-is uaryofis safuZveli ar aris da ar viRebT H1-s.

• sqesi (B faqtori): Frealuri = 0.31. es Frealuri ar xvdeba uaryofis areSi; H0-is uaryofis safuZveli ar aris da ar viRebT H1-s.

• musikis tipisa da sqesis urTierTqmedeba (AxB): Frealuri = 112.81. es Frealuri xvdeba uaryofis areSi; uarvyofT H0-s da viRebT H1-s.

efeqtis siZliere: η2 statistikurad mniSvnelovani urTierTqmedebisaTvis udris SSAxB/SSsruli = 541.50/652.50 = .83. axsna da daskvnebi: statistikurad mniSvnelovani urTierTqmedebis asaxsnelad, viRebT tukis HSD CD-s martivi efeqtebisaTvis da Semdeg vadarebT martiv efeqtebs am CD-s. am kriteriumisTvis saWiro ricxviTi mniSvnelobebi Semdegia:

• q aRebulia cxrilidan # A.4. 2x2 sqemis gardaqmnis matricis gamoyenebiT, romelic TandarTulia am cxrilTan, q-s mniSvneloba miRebulia, rodesac a=3 da MSSecdomis df aris 20. es mniSvnelobaa

3.58, rodesac α = .05. • dispersiuli analizis Sedegad miRebuli MSSecdoma udris 4.80.

• nAB, qulebis raodenoba yovel ujredSi, aris 6. CavsvaT es ricxviTi mniSvnelobebi formulaSi:

CD = qABn

MSSecdoma

= 3.58 680.4

= 3.2 Sefasebis wertili nebismieri martivi efeqti, romlis absoluturi mniSvneloba tolia an metia 3.2 Sefasebis wertilze, statistikurad mniSvnelovania .05 doneze. cxrili # 11.13 gviCvenebs ujredis da mTavari efeqtis saSualoebs da martiv da mTavar efeqtebs. am cxrilSi yoveli martivi efeqti absoluturi mniSvnelobiT metia CD-ze, romelic aris 3.2 da amdenad statistikurad mniSvnelovania. Tumca, aRsaniSnavia, rom TiToeuli damoukidebeli cvladisaTvis martiv efeqtebs sapirispiro niSnebi aqvT. musikis tipis efeqti aSkarad damokidebulia cdis piris sqesze. mamakacebi, rodesac isini avangardul musikas usmendnen, naxatebs Cveulebriv 6.0 qulas aniWebdnen (daZabuli, rigiduli, Seviwrovebuli), xolo rodesac isini minimalistur musikas usmendnen, naxatebs bevrad ufro maRali quliT afasebdnen - 17.0 (Tavisufali, fleqsiburi, gafarTovebuli). qalebis SemTxvevaSi Sefasebis es paterni Sebrunebulia. rodesac isini avangardul musikas usmendnen, qalebi naxatebs 15.0 quliT afasebdnen, maSin, rodesac minimalisturi musikis mosmenisas bevrad nakleb qulas aniWebdnen - 7.0. martivi efeqtebi musikis tipisaTvis (-11.0

mamakacebisaTvis da +8.0 qalebisaTvis) ar aris erTmaneTis toli, aseve ar aris musikis aramniSvnelovani mTavari efeqtis - -1.5 - tolic. cxrili # 11.13 naxatebis SefasebebSi mTavari efeqtebisa da martivi efeqtebis sxvaobebi, rogorc musikis tipisa da cdis piris sqesis funqcia.

musikis tipi (A)

avangarduli (A1) minimalisturi (A2)

mTavari efeqtis saSualoebi sqesisaTvis (B)

mamrobiTi (B1)

cdis piris sqesi (B) mdedrobiTi (B2)

6.0 17.0

15.0 7.0

11.5

11.0

mTavari efeqtis saSualoebi musikis tipisaTvis (A)

10.5 12.0

amis msgavsad, naxatebis Sefasebisas gansxvaveba qalebsa da mamakacebs Soris damokidebulia musikis tipze. qalebi, romlebic avangardul

musikas usmenen ( X 21BA = 15.0), naxatebs afaseben, rogorc mniSvnelovnad

ufro Tavisufals, fleqsibursa da gafarTovebuls, vidre mamakacebi

( X 11BA = 6.0), romlebic usmenen avangardul musikas. magram mamakacebi,

romlebic usmenen minimalistur musikas ( X 12BA = 17.0), naxatebs am

ganzomilebaSi mniSvnelovnad maRali qulebiT afaseben, vidre qalebi

( X 22BA = 7.0). amgvarad, martivi efeqtebi sqesisaTvis (-9.0 avangarduli

musikisaTvis da +10.0 minimalisturi musikisaTvis) ar aris arc erTmaneTis da arc cdis pirTa sqesis aramniSvnelovani mTavari efeqtis - +0.5 - toli. mTavari efeqtebi ar iZleva gamosadeg informacias am eqsperimentis Sedegebis Sesaxeb. mxolod mTavari efeqtebidan Cven davaskvnidiT, rom musikis tipisaTvis efeqti ar arsebobs da rom mamakacebi da qalebi ar gansxvavdebian TavianTi Sefasebebis mxriv. magram urTierTqmedebis analizi gviCvenebs, rom musikis tips aqvs efeqti micemul Sefasebebze, magram efeqtis mimarTuleba damokidebulia sqesze. ufro metic, qalebi da mamakacebi gansxvavdebian Sefasebebis mxriv, magram gansxvaveba

-11.0

-9.0 +10.0

+8.0

-1.5

+0.5

damokidebulia musikis tipze. µ2 gviCvenebs, rom Sefasebebis dispersiis 83 procents ganapirobebs urTierTqmedeba. qulebis winaswarmetyveleba ujredis saSualodan, da ara sruli saSualodan, am SerCevaSi srul dispersias 83 procentiT amcirebs.

magaliTi # 11.2 2x2 sqemis analizi statistikurad mniSvnelovani urTierTqmedebiT da mTavari efeqtiT amocana sityva-gasaRebis meTodi warmoadgens mexsierebis gaZlierebis teqnikas, rodesac adamiani iyenebs vizualur xats, raTa warmoqmnas asociacia or erTeuls Soris. am meTodis gamoyenebis erT-erTi magaliTia meore enis leqsikonis Seswavla. davuSvaT, rom Tqven gTxoves, SeiswavloT ucxo franguli sityvebis sia da am sityvebis inglisuri ekvivalentebi, rogoricaa, magaliTad, franguli sityva boue (talaxi) da misi inglisuri ekvivalenti mud (talaxi). aq sityva-gasaRebi SeiZleba iyos boot (Ceqma), inglisuri sityva, romelic franguli boue-s msgavsia JReradobiT. am teqnikis gamoyenebiT, rodesac Tqven gesmiT sityva boue, Tqven fiqrobT boot-ze da Semdeg warmoidgenT adamians, romelsac acvia Ceqmebi da midis talaxSi. bevrma kvlevam aCvena, rom sityva-gasaRebis teqnika aumjobesebs daswavlas da meore enis sityvis dauyovnebel gaxsenebas (Wang, Thomas, & Ouellette, 1992). magram Riad rCeba kiTxva, Tu ramden xans grZeldeba es gaumjobesebuli gaxseneba. vangma da misma kolegebma (Wang et al. 1992) aRmoaCines, rom miuxedavad imisa, rom sityva-gasaRebis teqnikam, sityvebis martivi gameorebis gziT meqanikur SeswavlasTan SedarebiT, gaaumjobesa mexsierebaSi Senaxva da dauyovnebeli gaxseneba, cdis pirebs didwilad sityva-gasaRebic aviwydebodaT erTkviriani Senaxvis intervalis Semdeg.

davuSvaT, Tqven gainteresebT, marTebulia Tu ara es kavSiri zrdasruli adamianebisaTvis iseve, rogorc kolejis studentebisaTvis, rac Seamowmes vangma da misma kolegebma. Sesabamisad, Tqven gamoiyenebT cdis pirTa Soris 2x2 sqemas, sadac Seswavlis sxvadasxva meTodi - an meqanikuri, an sityva-gasaRebiT - iqneba faqtori A da Senaxvis intervali - dauyovnebeli an erTkviriani aRdgena - iqneba faqtori B. Tqven SemTxveviTad gadaanawileT rva frangulis armcodne zrdasruli adamiani TiToeul pirobaSi. cdis pirebs, romlebzec meqanikuri Seswavlis meTodi gamoiyeneboda sTxoves, ubralod mravaljer gaemeorebinaT franguli sityvebi da maTi inglisuri ekvivalentebi damaxsovrebis mizniT. sityva-gasaRebis meTodis gamoyenebisas cdis pirebs aswavles sityva-gasaRebis teqnika da misces sityva-gasaRebebi yoveli frangul-inglisuri wyvilisaTvis. dauyovnebeli aRdgenis pirobaSi inglisuri ekvivalentebis gaxseneba mowmdeboda 5 wuTSi siis Seswavlis Semdeg. gadavadebuli aRdgenis pirobaSi cdis pirebs sTxovdnen, gaexsenebinaT inglisuri ekvivalentebi Seswavlidan erTi kviris Semdeg. sityvebis sia 25 sityvas Seicavda, amitom cdis piris gaxsenebis qula SeiZleboda Secvliliyo 0-dan (verc erTi sityva ver gaixsena) 25-mde (srulyofilad gaixsena). dauSviT, rom 32 cdis pirisagan miRebulia gaxsenebis hipoTeturi qulebi. gaaanalizeT da aRwereT am eqsperimentis Sedegebi.

daswavlis piroba (A) meqanikuri (A1) gasaRebi sityviT (A2)

dauyovnebeli aRdgena (B1) Senaxvis intervali (B) gadavadebuli aRdgena (B2)

12 9 11 19 16 14 13 15

25 22 19 23 17 24 20 21

6 7 3 11 5 10 5 8

9 4 7 11 5 8 12 6

amoxsna. eqsperimentis statistikuri gaanalizebisaTvis saWiroa cdis pirTa Soris dispersiuli orfaqtoriani analizi. viyenebT mniSvnelovnobis dones .05. daSvebebi:

1. cdis pirebis SerCeva moxda SemTxveviTad da populaciisagan damoukideblad;

2. gaxsenebis qulebi populaciaSi normaluradaa ganawilebuli; 3. gaxsenebis qulebis dispersiebi SerCeul populaciaSi tolia.


A faqtori H0: µ A1 = µ A2


B faqtori H0: µ B1 = µ B2


AxB H0: yvela (µAB - µ A - µ B + µ G) = 0 H1: yvela (µAB - µ A - µ B + µ G) ar udris 0

mniSvnelovnobis done: α = .05. df: dfA = a - 1 = 2 - 1 = 1 dfB = b - 1 = 2 - 1 = 1 dfAxB = (a - 1) (b - 1) = (2 - 1)(2 - 1) = 1 dfSecdoma = ab(nAB - 1) = (2)(2)(8 - 1) = 28 dfsruli = N - 1 = 32 - 1 = 31 F-is kritikuli mniSvneloba: Fkritikuli (1,28) = 4.20 TiToeuli Frealuris mniSvnelobisaTvis.

uaryofis are: Frealuris mniSvnelobebi, romlebic tolia an metia 4.20-ze. gamoTvla: Semdegi Sejamebis cxrilis misaRebad gamoyenebul iqna formulebi Tavis damatebidan: wyaro SS df MS F* Seswavlis piroba (A) Senaxvis intervali (B) AxB Secdoma sul

148.781 830.281 94.531 224.125 1297.718

1 1 1 28 31

148.781 830.281 94.531 8.004

18.59* 103.73* 11.81*

* p < .01 gadawyvetilebebi:

• daswavlis piroba (A faqtori): Frealuri = 18.59. es Frealuri xvdeba uaryofis areSi; uarvyofT H0-s da viRebT H1-s.

• Senaxvis intervali (B faqtori): Frealuri = 103.73. es Frealuri xvdeba uaryofis areSi; uarvyofT H0-s da viRebT H1-s.

• daswavlis pirobisa da Senaxvis intervalis urTierTqmedeba (AxB): Frealuri = 11.81. es Frealuri xvdeba uaryofis areSi; uarvyofT H0-s da viRebT H1-s.

efeqtis siZliere:

Seswavlis piroba: η2 = SSA/SSsruli = 148.781/1297.718 = .11.

Senaxvis intervali: η2 = SSB/SSsruli = 830.281/1297.718 = .64. Seswavlis piroba x Senaxvis intervali:

η2 = SSAxB/SSsruli = 94.531/1297.718 = .07. axsna da daskvnebi: Tavdapirvelad urTierTqmedebas avxsniT tukis HSD CD-s moZebniT martivi efeqtebisaTvis da Semdeg martivi efeqtebis am CD-sTan SedarebiT. am kriteriumisaTvis saWiro ricxviTi mniSvnelobebi Semdegia:

• q aRebulia cxrilidan # A.4, rodesac a=3 da MSSecdomis df aris 28. radgan MSSecdomis 28 df ar aris moyvanili cxrilSi, gamoviyeneT

momdevno df-is, 24-is q. es mniSvnelobaa 3.53, rodesac α = .05. • MSSecdoma aris 8.004.

• nAB aris 8. CavsvaT es ricxviTi mniSvnelobebi formulaSi:

CD = qABn

MSSecdoma

= 3.53 8004.8

= 3.5 sityva, damrgvalebuli erT meaTedamde.

nebismieri martivi efeqti, romelic absoluturi mniSvnelobiT tolia an metia 3.5 sityvisa, statistikurad mniSvnelovania .05 doneze. cxrilSi # 11.14 warmodgenilia ujredisa da mTavari efeqtis saSualoebi da martivi da mTavari efeqtebi am qulebisaTvis. Tavdapirvelad avxsniT urTierTqmedebas TiToeuli damoukidebeli cvladisaTvis da Semdeg am cvladis mTavar efeqts. cxrili # 11.14 sworad aRdgenili sityvebis raodenobebis mTavari efeqtisa da martivi efeqtis gansxvavebebi, rogorc daswavlis pirobisa da Senaxvis intervalis funqcia. Seswavlis piroba (A)

meqnikuri (A1) gasaRebi sityviT (A2)

mTavari efeqtis saSualoebi Senaxvis intervalisaTvis (B)

dauyovnebeli aRdgena (B1)

Senaxvis intervali (B) gadavadebuli aRdgena (B2)

13.6 21.4

6.9 7.8

17.5

7.3

mTavari efeqtis saSualoebi Seswvalis pirobisaTvis (A)

10.2 14.6

Seswavlis piroba: Seswavlis pirobis martivi efeqti dauyovnebeli aRdgenisaTvis, -7.8, absoluturi mniSvnelobiT metia, vidre CD - 3.5 da statistikurad mniSvnelovania. dauyovnebeli aRdgenisas gasaRebi sityvis

pirobas ( X A2B1 = 21.4 sityva) Sedegad moyveba mniSvnelovnad ufro didi

gaxseneba, vidre meqanikur daswavlas ( X A1B1 = 13.6 sityva). xolo

gadavadebuli aRdgenisas Seswavlis pirobis martivi efeqti, -0.9, absoluturi mniSvnelobiT naklebia, vidre CD - 3.5. es martivi efeqti

aramniSvnelovania. amdenad, gadavadebuli aRdgenisas meqanikuri (X A1B2 =

6.9 sityva) da gasaRebi sityvis ( X A2B2 = 7.8 sityva) pirobebi mniSvnelovnad

ar gansxvavdeba. Seswavlis pirobas ar aqvs efeqti gaxsenebaze gadavadebuli aRdgenisas. Frealuri (1.28) = 18.59 Seswavlis pirobis mTavari efeqtisaTvis statistikurad mniSvnelovania da migviTiTebs, rom

-7.8

+6.7 +13.6

-0.9

-4.4

+10.2

Seswavlis pirobis mTavari efeqtis saSualoebi ( X A1 = 10.2 da X A2 = 14.6) mniSvnelovnad gansxvavdebian erTmaneTisagan. magram es mniSvnelovani mTavari efeqti ar gvawvdis mniSvnelobis mqone informacias eqsperimentis Sedegis Sesaxeb. es mTavari efeqti miuTiTebs, rom cdis pirebi ufro met sityvebs ixseneben gasaRebi sityvis pirobaSi, magram Seswavlis pirobis martivi efeqtebis analizma daadgina, rom es kavSiri arsebobs mxolod dauyovnebeli aRdgenis intervalis dros; is ar aris marTebuli gadavadebuli aRdgenis intervalis dros. aqedan gamomdinare, Seswavlis pirobis mTavari efeqti ar iZleva Seswavlis pirobis efeqtis swor aRweras. amis gamo Cven mas ugulvebelvyofT eqsperimentis Sedegebis aRwerisas. Senaxvis intervali: mniSvnelovani urTierTqmedeba aseve migvaniSnebs imaze, rom Senaxvis intervalis efeqti damokidebulia Seswavlis pirobaze. Seswavlis pirobis orive donisaTvis Senaxvis intervalis martivi efeqtebi (+6.7 meqanikuri SeswavlisaTvis da +13.6 gasaRebi sityviT SeswavlisaTvis) absoluturi mniSvnelobiT metia, vidre CD = 3.5. es statistikurad mniSvnelovani martivi efeqtebi migviTiTebs, rom rogorc meqanikuri, aseve gasaRebi sityviT Seswavlis SemTxvevaSi gadavadebuli aRdgena Sedegad mniSvnelovnad nakleb gaxsenebas iZleva, vidre dauyovnebeli aRdgena. magram martivi efeqtebis Seswavla gviCvenebs, rom Senaxvis Semcireba dauyovneblidan gadavadebuli aRdgenisken ufro metia gasaRebi sityviT Seswavlis pirobaSi (13.6 sityva), vidre meqanikuri daswavlisas (6.7 sityva). Senaxvis intervalis statistikurad mniSvnelovani mTavari efeqti [anu Frealuri (1.28) = 103.73, p < .05] gviCvenebs, rom Senaxvis intervalis mTavari efeqtis saSualoebi

( 1BX = 17.5 da 2BX = 7.3) mniSvnelovnad gansxvavdebian. mxolod am statistikurad mniSvnelovani mTavari efeqtidan vaskvniT, rom dauyovnebel aRdgenas ufro karg Senaxvamde mivyavarT, vidre gadavadebul aRdgenas. es daskvna sworia, radgan es kavSiri erTnairad marTebulia rogorc meqanikuri, aseve gasaRebi sityviT Seswavlis pirobebisaTvis. aqedan gamomdinare, Senaxvis intervalis mTavari efeqti mniSvnelobis mqonea da gamosadeg informacias gvawvdis eqsperimentis Sedegebis Sesaxeb. Tumca, am mTavari efeqtis axsnas Tan unda mohyves imis aRiareba, rom urTierTqmedeba migviTiTebs imaze, rom gansxvavebis xarisxi dauyovnebel da gadavadebul aRdgenas Soris Seswavlis pirobazea damokidebuli.

η2 Senaxvis intervalis mTavari efeqtisaTvis .64 migviTiTebs, rom Senaxvis intervali gaxsenebis qulebis dispersiis did nawilzea

pasuxismgebeli. η2 urTierTqmedebisaTvis, .07, gviCvenebs, rom urTierTqmedeba pasuxismgebelia gaxsenebis qulebis dispersiis ufro mcire nawilze. radganac Seswavlis pirobis mTavari efeqti artefaqturia, am faqtorisaTvis ar viyenebT eta kvadrats.

dispersiuli faqtoruli analizis Sedegebis prezentacia faqtoruli eqsperimentis Sedegebis wardgenis stili igivea, rac erTfaqtorian sqemaSi, romelic aRwerilia me-10 TavSi. Sedegebis wardgenis magaliTs moviyvanT sanimuSo amocanis Sedegebis gamoyenebiT,

sadac ganxiluli iyo cdis piris mxedvelobis areSi yofnis (cdis piri imyofeba an ar imyofeba mxedvelobis areSi) da siaxlis tipis (kargi an cudi siaxle) efeqtebi am siaxlis Setyobinebis latenturobaze. am eqsperimentis ujredis saSualoebi warmodgenilia cxrilSi # 11.5, xolo dispersiuli analizi Sejamebulia cxrilSi # 11.8. cxrilebis dabeWdvis siZviris gamo eqsperimentis Sedegebi, ujredebis saSualoebisa da statistikuri analizis CaTvliT, xSirad teqstis formatiT aris xolme warmodgenili, rogoc es aq aris naCvenebi. yvela statistikuri kriteriumisaTvis gamoyenebuli iyo alfas .05 done. cdis piris mxedvelobaSi yofnasa da Sesatyobinebeli siaxlis tips Soris arsebobda mniSvnelovani urTierTqmedeba, F(1,16) = 15.32, MSE = 108.125, p < .01. tukis HSD kriteriumi (kritikuli sxvaoba = 17.0, rodesac α = .05) gamoviyeneT, raTa Segvemowmebina martivi efeqtebis statistikuri mniSvnelovnoba. rodesac kargi siaxlis Setyobineba xdeboda, cdis piris mxedvelobis areSi yofnis efeqti aramniSvnelovani iyo (M = 90.8, SD = 11.56, rodesac cdis piri ar Cans, da M = 84.6, SD = 9.61, rodesac cdis piri Cans). xolo rodesac cudi siaxlis Setyobineba xdeboda, cdis piris mxedvelobis areSi yofnis pirobis latenturoba (M = 112.4, SD = 11.15) mniSvnelovnad ufro xangrZlivi iyo, vidre cdis piris mxedvelobis areSi aryofnis pirobisa (M = 82.2, SD = 9.07). sxvaoba latenturobaSi kargsa da cud siaxleebs Soris cdis piris mxedvelobis areSi aryofnis pirobaSi aramniSvnelovani iyo. xolo cdis piris mxedvelobis areSi yofnis pirobisTvis cudi siaxlis Setyobinebis latenturoba mniSvnelovnad xangrZlivi iyo, vidre kargi siaxlisa. urTierTqmedebis eta kvadrati iyo .36. cdis piris mxedvelobis areSi yofnis mTavari efeqti statistikurad mniSvnelovani iyo, F(1,16) = 6.66, p < .05, magram ar iyo mniSvnelobis mqone. siaxlis tipis mTavari efeqti aramniSvnelovani iyo, F(1,16) = 4.26, p < .05. am prezentaciaSi:

α = .05 aRniSnavs mniSvnelovnobis dones, SerCeuls am statistikuri kriteriumebisaTvis.

F(1,16) aRniSnavs kriteriumis statistikas, rogorc F-s. TiTeuli

warmodgenili Frealuris mniSvnelisa da mricxvelis df naCvenebia frCxilebSi. wamodgenilia Frealuris sami mniSvneloba, ori yoveli damoukidebeli cvladisaTvis da erTi - damoukidebeli cvladebis urTierTqmedebisaTvis. Tanmimdevroba, romliTac warmodgenilia Frealurebi, ar aris aucileblad A faqtori, B faqtori da AxB. xSirad eqsperimentatori ufro metad dainteresebulia urTierTqmedebiT da Sedegebis aRwerisas pirvel rigSi Frealuris am mniSvnelobas gamosaxavs, rogorc es naCvenebia magaliTSi.

= 15.32 gvaZlevs Frealuri-s mniSvnelobas (da ara Fkritikulis

mniSvnelobas cxrilidan # 11.3). zogierT SemTxvevaSi, SesaZlebelia warmodgenili iyos dispersiuli analizis Sedegad miRebuli sami Frealuridan mxolod ori. aseT

SemTxvevebSi SegiZliaT dauSvaT, rom im Frealuris mniSvneloba, romelic ar aris warmodgenili, aramniSvnelovania.

p < .01 aRniSnavs, rom a. Frealuris albaToba, Tu Ho WeSmaritia, naklebia an udros

.01. es mniSvneloba aris Frealuris albaToba an Frealuris ufro ukiduresi mniSvneloba, Tu Ho WeSmaritia; is SeiZleba ar

iyos igive, rac SerCeuli α-s mniSvneloba. b. Ho uaryofilia da H1 miRebulia. g. Frealuri miRebul iqna dispersiuli analizis urTierTqmedebis efeqtisaTvis; amdenad, is miuTiTebs, rom A da B faqtorebis urTierTqmedeba statistikurad mniSvnelovania. d. A da B faqtorebis urTierTqmedeba xdeba ara SemTxveviTi Secdomis, aramed sxva mizezis gamo.

MSE = 108.125 iZleva MSSecdomas FrealurisTvis. es mniSvneloba qulebis

ujredebs Soris cvalebadobis sazomia. CD = 17.0 gvaZlevs kritikul sxvaobebs martivi efeqtebis Sesadareblad

tukis HSD kriteriumis gamoyenebiT. Semdeg aRiwereba statistikurad mniSvnelovani martivi efeqtebi.

η2 = .36 gviCvenebs, rom pasuxebis latenturobebis cvalebadobis 36 procenti am eqsperimentSi miRebulia A da B faqtorebis urTierTqmedebis gamo. am magaliTSi urTierTqmedeba iZleva erTaderT mniSvnelovan informacias eqsperimentis Sedegebis

Sesaxeb; amdenad, η2 mocemulia cvalebadobis mxolod am wyarosaTvis.

SeamowmeT Tqveni codna # 11.5 amoxseniT es amocanebi cdis pirTa Soris orfaqtoriani dispersiuli analizis gamoyenebiT. misdieT sanimuSo amocanebSi gamoyenebul formats. yoveli amocanisaTvis gamoiyeneT .05 mniSvnelovnobis done.

1. erTma fsiqologma wamoayena hipoTeza, rom instruqciis tipis efeqti adamianis mier anagramis amoxsnaze damokidebulia am adamianis kontrolis lokusze. am hipoTezis Sesamowmeblad man eqsperimenti Caatara 12 qalze, romlebmac dabali qulebi miiRes kontrolis lokusis skalaze (rac niSnavs kontrolis Sida mdebareobas) da 12 qalze, romlebmac maRali qulebi miiRes am skalaze (rac niSnavs kontrolis gare mdebareobas). TiToeuli jgufidan qalebis naxevari SemTxveviTad ganawilda oridan erT instruqtirebis pirobaSi - unarebis an SemTxveviTobis instruqtirebaSi. unarebis instruqtirebis pirobaSi cdis pirebs uTxres, rom maT mier anagramis warmatebuli amoxsna damokidebuli iyo maT verbalur unarebze da, Sesabamisad, maT kontrols eqvemdebareboda. SemTxveviTobis instruqcia miuTiTebda, rom maT mier am davalebis Sesruleba ar iyo damokidebuli maT verbalur unarebze da maTi kontrolis miRma iyo. Semdeg cdis pirebi xsnidnen anagramebis seriebs da maT mier am davalebis Sesrulebaze daxarjuli dro aRiniSneboda eqsperimentatoris mier. dauSviT, rom miRebul iqna Semdegi droebi (wamebSi), romlebsac TiToeuli cdis piri xarjavda anagramebis amoxsnaze:

kontrolix lokusi (A) Sida (A1) gare (A2)

unari (B1) instruqciebi (B) SemTxveviToba (B2)

175 202 193 186 150 212

115 106 87 93 99 157

125 100 77 101 131 152

218 202 196 244 237 180

a. gamoTvaleT ujredis da mTavari efeqtis saSualoebi. Semdeg gaaanalizeT qulebi,

raTa gansazRvroT (1) arsebobs Tu ara mTavari efeqti romelime damoukidebeli cvladisaTvis da (2) arsebobs Tu ara urTierTqmedeba damoukidebel cvladebs Soris.

b. aRwereT instruqciis tipis efeqti. Tu saWiroa, gamoiyeneT tukis HSD kriteriumi statistikurad mniSvnelovani martivi efeqtebis gansazRvrisaTvis.

g. gaqvT Tu ara SesaZlebloba, mniSvnelobis mqone axsna misceT romelime statistikurad mniSvnelovan mTavar efeqts, Tu isini artefaqturebia?

d. damokidebuli cvladis dispersiis ra nawili SeiZleba mivaweroT damoukidebeli cvladebis urTierTqmedebas?

2. mraval marTlmsajulebaSi potenciur nafic msajulebs winaswar uCveneben films,

romelic acnobs maT iuridiul process da iuridiuli sistemis arss. davuSvaT, rom iurists ainteresebs, daadginos, cvlis Tu ara filmi nafici msajulis qcevas standartul verbalur instruqciasTan SedarebiT. ufro metic, iurists surs, gaigos, SesaZlebelia Tu ara, filmis efeqti damokidebuli iyos im danaSaulis tipze, romelic unda Seafasos msajulma. am kiTxvebze pasuxis gasacemad iuristma Caatara 2x2 cdis pirTa Soris faqtoruli eqsperimenti, sadac TiToeul pirobaSi cxra cdis piri - fiqtiuri msajuli - monawileobda. A faqtori iyo msajulebisaTvis micemuli instruqciis tipi: A1 - normaluri verbaluri instruqciebi, gacemuli sasamarTlos muSakis mier da A2 - instruqciebi filmis saxiT. B faqtori iyo danaSaulis tipi, romlis Sefasebasac sTxovdnen cdis pirs. am cvlads cvlidnen Semdegnairad: cdis pirs akiTxebdnen avtomanqanis moparvis SemTxvevas (B1) an fizikuri Zaladobis gamoyenebiT Tavdasxmis SemTxvevas (B2). Semdeg TiToeul cdis pirs sTxovdnen, miesadagebina patimrobis vada SemTxvevaSi aRwerili braldebulisaTvis. miCenili patimrobis vadebi (TveebSi) mocemulia momdevno cxrilSi:

instruqciis tipi (A)

normaluri (A1) filmis saxiT (A2) manqanis moparva (B1)

15 10 10 14 17 9 8 12 10

13 16 9 11 14 8 10 10 12

danaSaulis tipi (B) Tavdasxma fizikuri Zaladobis gamoyenebiT (B2)

25 20 23 20 19 17 18 22 20

30 33 26 25 30 35 30 28 32

a. gamoTvaleT ujredisa da mTavari efeqtis saSualoebi. Semdeg gaaanalizeT

qulebi, raTa gansazRvroT (1) arsebobs Tu ara mTavari efeqti romelime damoukidebeli cvladisaTvis da (2) arsebobs Tu ara urTierTqmedeba damoukidebel cvladebs Soris.

b. aRwereT instruqciis tipis efeqti. g. aRwereT danaSaulis tipis efeqti. d. SegviZlia Tu ara mniSvnelobis mqone axsna mivceT romelime

statistikurad mniSvnelovan mTavar efeqts, Tu es efeqtebi artefaqturebia? e. damokidebuli cvladis dispersiis ra nawili SeiZleba mivaweroT danaSaulis

tips da damoukidebeli cvladebis urTierTqmedebas? 3. Semdeg savarjiSoSi warmodgenilia cdis pirTa Soris orfaqtoriani dispersiuli analizi. TiToeuli prezentaciisaTvis upasuxeT Semdeg kiTxvebs:

a. A faqtoris ramdeni done iyo am eqsperimentSi? b. B faqtoris ramdeni done iyo am eqsperimentSi? g. ramdeni cdis piri iRebda monawileobas eqsperimentSi? d. ra aris Fkritikulis mniSvneloba .05 mniSvnelovnobis doneze yoveli Frealuri-

sTvis? e. aris Tu ara Frealuris mniSvneloba A faqtorisaTvis statistikurad

mniSvnelovani? v. aris Tu ara Frealuris mniSvneloba B faqtorisaTvis statistikurad

mniSvnelovani?

z. aris Tu ara Frealuris mniSvneloba AxB-saTvis statistikurad mniSvnelovani?

3.1. alfas done .05 gamoyenebul iqna yvela statistikuri kriteriumisaTvis. medikamentis donis (A faqtori) mTavari efeqti statitikurad mniSvnelovani iyo, F(2,36) = 9.43, p < .01, MSE = 17.814. aseve miRebuli iyo mTavari efeqti Senaxvis intervalisaTvis (B faqtori), F(1,36) = 5.60, p < .05. urTierTqmedeba aramniSvnelovani iyo, F(2,36) = 2.46, p > .05. 3.2. alfas done .05 gamoyenebul iqna yvela statistikuri kriteriumisaTvis. miRebul iqna aTletebis sqesisa (A faqtori) da sportis saxeobis (B faqtori) mniSvnelovani urTierTqmedebis gavlena TviTSefasebis qulebze, F (3,88) = 3.28, p < .05, MSE = 42.163. garda amisa, mamakacebma ufro maRali TviTSefasebis qulebi mogvces, vidre qalebma, F (1,88) = 4.82, p < .05. sportis saxeobebs Soris ar aRmoCnda mTavari efeqtebis sxvaobebi, F (3,88) = 1.65, p > .05. 3.3. alfas done .05 gamoyenebul iqna yvela statistikuri kriteriumisaTvis. mniSvnelovani urTierTqmedeba gamovlinda xmauris pirobasa da pirovnebis tips Soris F (1,56) = 5.33, p < .05, MSE = 26.718. mniSvnelovani ar aRmoCnda mTavari efeqtebi arc xmauris pirobis, F (1,56) < 1.0, arc pirovnebis tipisaTvis, F (1,56) = 1.43, p > .05.

Sejameba

• faqtorulia kvlevis iseTi sqemebi, sadac ori an meti damoukidebeli cvladi erTdroulad icvleba.

• faqtoruli sqemis TiToeuli ujredi anu Carevis piroba warmoadgens yoveli damoukidebeli cvladis erTi donisagan warmoqmnil kombinacias.

• mTavari efeqtis saSualo gviCvenebs yvela individis tipur Sesrulebas damoukidebeli cvladis erTi donisaTvis, meore

damoukidebeli cvladis doneebis mixedviT klasifikaciis ignorirebiT.

• faqtorul sqemaSi urTierTqmedeba xdeba, rodesac erTi damoukidebeli cvladis efeqti damokidebulia masTan kombinaciaSi myofi meore damoukidebeli cvladis doneze.

• cdis pirTa Soris orfaqtorian dispersiul analizSi miiReba sami F statistika:

cvalebadobis wyaro F A faqtori B faqtori AxB urTierTqmedeba

MSA/MSSecdoma

MSB/MSSecdoma

MSAxB/MSSecdoma

• statistikuri Semowmebis nabijebi am F SefardebebiT me-10 TavSi ganxiluli nabijebis identuria.

• faqtorul sqemaSi damoukidebeli cvladis martivi efeqti aris am damoukidebeli cvladis efeqti meore damoukidebeli cvladis mxolod erT doneze.

• Tu xdeba damoukidebeli cvladebis statistikurad mniSvnelovani urTierTqmedeba, maSin erTi faqtoris ori martivi efeqti ar udris erTmaneTs an am faqtoris mTavar efeqts.

• artefaqturi mTavari efeqti aris mTavari efeqti, romelic ar SeiZleba mniSvnelovnad aixsnas; is xdeba mxolod realuri urTierTqmedebis specifikuri saxis gamo.

• Tu xdeba mniSvnelovani urTierTqmedeba, saWiroa Semdgomi Semowmeba, raTa statistikurad gaanalizdes martivi efeqtebi. tukis HSD kriteriumi gamoiyeneba saboloo SedarebisaTvis.

• η2 gamoiyeneba, rogorc Carevis efeqtis siZlieris sazomi statistikurad mniSvnelovani mTavari efeqtebisa da urTierTqmedebisaTvis.

__________________________________________________________________ ZiriTadi terminebi da simboloebi

a artefaqturi mTavari efeqti b ujredi

ujredis saSualo ( ABX ) kritikuli sxvaoba (CD) dfA dfB dfAxB dfSecdoma dfsruli A faqtori B faqtori faqtoruli sqema

sruli saSualo ( GX ) urTierTqmedeba (AxB) A faqtoris mTavari efeqti

MSA MSB MSAxB MSSecdoma nAB damoukidebeli cvladis martivi efeqti SSA SSB SSAxB SSSecdoma SSsruli Carevis piroba tukis HSD kriteriumi

AX

BX

B faqtoris mTavari efeqti mTavari efeqtis saSualo

GX

__________________________________________________________________

Tavis damateba cdis pirTa Soris orfaqtoriani dispersiuli analizis gamoTvla formulebis meSveobiT __________________________________________________________________ am damatebaSi mocemulia gamoTvliTi formulebi cdis pirTa Soris orfaqtoriani dispersiuli analizisaTvis, sadac TiToeul ujredSi cdis pirTa Tanabari raodenobaa. amis simboluri gamosaxuleba TiToeuli faqtoris ori donisaTvis da TiToeul ujredSi xuTi qulisaTvis aseTia:

faqtori A

A1 A2

B1

faqtori B B2

X111 X211 X311 TA1B1 X411 X511

X121 X221 X321 TA2B1 X421 X521

TB1

X112 X212 X312 TA2B1 X412 X512

X122 X222 X322 TA2B2 X422 X522

TB2

TA1 TA2 G sadac Xijk = cdis piris qula TA = qulebis sruli jami A faqtoris donisaTvis TB = qulebis sruli jami B faqtoris donisaTvis TAB = qulebis jami ujredSi G = qulebis saerTo sruli jami nAB = qulebis raodenoba ujredSi a = A faqtoris doneebis raodenoba b = B faqtoris doneebis raodenoba

N = qulebis sruli raodenoba am aRniSvnebis gamoyenebiT gamoiTvleba xuTi ricxviTi mniSvneloba:

[1] = ∑∑∑ 2ijkX yvela qulis kvadratebis jami,

( ∑∑∑ aRniSnavs, rom kvadratSi ayvanili qulebi dajamebulia ujredSi yvela qulis mixedviT da Semdeg A da B faqtorebis yvela donis mixedviT),

[2] = AB

AB

nT 2∑∑

TiToeuli ujredis kvadratebis jami, gayofili

ujredSi qulebis raodenobaze,

[3] = AB

A

bnT 2∑

A faqtoris yoveli donis sruli jamebis kvadratebis

jami, gayofili TiToeul srul jamSi Semavali qulebis raodenobaze.

[4] = AB

B

anT 2∑

B faqtoris yoveli donis sruli jamebis kvadratebis

jami, gayofili TiToeul srul jamSi Semavali qulebis raodenobaze.

[5] = N

G 2

sruli saerTo jamis kvadrati, gayofili qulebis srul

raodenobaze. am ricxviTi mniSvnelobebis gamoyenebiT, dispersiuli analizi ase miiReba: cvalebadobis wyaro

SS df MS F

A faqtori B faqtori AxB Secdoma sul

[3] – [5] [4] – [5] [2] – [3] – [4] + [5] [1] – [2] [1] – [5]

a – 1 b – 1 (a – 1)( b – 1)

ab(nAB - 1) N - 1

A

A

dfSS

B

B

dfSS

AxB

AxB

dfSS

Secdoma

Secdoma

dfSS

ar gamoiTvleba

SecdomaMSMS A

SecdomaMSMSB

SecdomaMSMS AxB

am gamoTvlebis ilustraciisaTvis gamoviyenoT monacemebi magaliTidan. es monacemebi moyvanilia cxrilSi # 11.2, xolo maTi dispersiuli analizi Sejamebulia cxrilSi # 11.8. faqtori A

A1 A2

B1

faqtori B B2

97 90 80 TA1B1 = 454 107 80

70 87 81 TA2B1 = 454

95 90

TB1 = 877

68 87 92 TA2B1 = 411

80 84

114 96 127 TA2B2 = 562

110 115

TB2 = 973

TA1 = 865 TA2 = 985 G = 1850 da nAB = 5, a = 2, b = 2 da N = 20. ricxviTi mniSvnelobebia [1] = 972 + . . . + 802 + 702 . . . + 902 + 682 + . . . + 842 + 1142 + . . . + 1152 =

175,692.000

[2] = 2222

5)562()411()423()454( +++=

500.810,869

= 173,962.000

[3] = )5)(2(

)985()865( 22 + =

1000.450,718,1

= 171,845.000

[4] = )5)(2(

)973()877( 22 + =

1000.858,715,1

= 171,585.800

[5] = 20

18502

= 20

00.500,422,3 = 171,125.000

Semdeg SSA = [3] - [5] = 171,845.000 - 171,125.000 = 720.000 dfA = a - 1 = 2 - 1 = 1

MSA = 1000.720 = 720.000

SSA = [4] - [5] = 171,585.800 - 171,125.000 = 460.800 dfB = b - 1 = 2 - 1 = 1

MSB = 1800.460 = 460.800

SSAxB = [2] - [3] - [4] + [5] = 173,962.000 - 171,845.000 - 171,585.800 + 171,125.000 =

1656.200 dfAxB = {a - 1}(b - 1) = (2 - 1) (2 - 1) = 1

MSAxB = 1

1656.200 = 1656.200

SSSecdoma = [1] - [2] = 175,692.000 - 173,962.000 = 1730.000 dfSecdoma = ab(nAB - 1) = (2)(2)(5-1) = 16

MSSecdoma = 16

1730.000 = 108.125

SSsruli = [1] - [5] = 175,692.000 - 171, 125.000 = 4567.000 dfsruli = N - 1 = 20 - 1 = 19 amdenad, Frealuris mniSvnelobebia

A faqtori: Frealuri = SecdomaMS

MSA = 125.108000.720

= 6.66

B faqtori: Frealuri = SecdomaMS

MSB = 125.108800.460

= 4.26

A da B faqtorebis urTierTqmedeba: Frealuri = SecdomaMS

MSAxB

Frealuri = 125.108200.1656

= 15.32

dispersiuli analizis Sejamebaa wyaro SS df MS F cdis piris mxedvelobis areSi yofna (A) siaxlis tipi (B) AxB Secdoma sul

720.000 460.000 1656.200 1730.000 4567.000

1 1 1 16 19

720.000 460.000 1656.200 108.125

6.66* 4.26 15.32**

* p < .05 ** p < .01 am gamoTvliTi midgomis Sedegad miRebuli mniSvnelobebi cxrilSi # 11.8 warmodgenili im mniSvnelobebis identuria, romlebic miviReT dispersiis analizis ganmartebiTi formulebis gamoyenebis Sedegad.

kiTxvebi

mocemul qulebze cdis pirTa Soris orfaqtoriani dispersiuli analizis Catarebis Sedegad upasuxeT sam kiTxvas.

1. rogor icvleba fiziologiuri agznebis done, rodesac Cven socialuri gacvlis procesSi verTvebiT? am kiTxvaze nawilobrivi pasuxis gasacemad velensma (Wellens, 1987) gamoiyena cdis pirTa Soris 2x2 sqema, sadac eqsperimentatori intervius utarebda cdis pirebs. erTi damoukidebeli cvladi (A) iyo eqsperimentatoris mxedvelobiTi kontaqti cdis pirTan intervius dros. aq icvleboda ori done: mxedvelobiTi kontaqtis ararseboba (A1) da mxedvelobiTi kontaqtis damyareba (A2). meore damoukidebeli cvladi (B faqtori) iyo emociiT manipulireba or doneze: dadebiTi (B1) da uaryofiTi (B2). dadebiTi emociis pirobaSi cdis pirebs werilobiTi saxiT aZlevdnen maTze eqsperimentatoris "pirvel STabeWdilebas", romelic Zalian dadebiTi iyo. uaryofiTi emociis pirobaSi maT werilobiTi saxiT aZlevdnen maTze eqsperimentatoris pirvel STabeWdilebas, romelic Zalian uaryofiTi iyo. damokidebuli cvladi iyo cdis piris guliscema, gamoxatuli wuTSi dartymebis raodenobiT, romelsac afiqsirebdnen intervius Catarebis procesSi. dauSviT, rom Tqven gaimeoreT es eqsperimenti TiToeul ujredSi rva cdis pirze da guliscemis Semdegi monacemebi miiReT:

mxedvelobiTi kontaqtis piroba (A)

araseboba (A1) damyareba (A2) dadebiTi (B1) emociis piroba (B) uaryofiTi (B2)

75 77 68 72 83 78 70 65

61 68 71 59 74 66 67 57

79 82 75 77 84 72 78 69

84 79 88 90 75 75 77 85

a. damokidebulia Tu ara mxedvelobiTi kontaqtis efeqti emociis

doneze? Tu ki, gamoiyeneT tukis HSD kriteriumi martivi efeqtebis SedarebisaTvis. aRwereT Tqvens mier aRmoCenili urTierToba.

b. damokidebulia Tu ara emociis pirobis efeqti mxedvelobiTi kontaqtis pirobaze? Tu ki, gamoiyeneT tukis HSD kriteriumi martivi efeqtebis SedarebisaTvis. aRwereT Tqvens mier aRmoCenili urTierToba.

g. SegiZliaT Tu ara, mniSvnelovnad CavTvaloT romelime mTavari efeqti, Tu isini artefaqturia?

d. damokidebuli cvladis cvalebadobis ra nawili xdeba

damoukidebeli cvladebis urTierTqmedebis gamo? e. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic

ilustrirebuli iyo am Tavis seqciaSi "dispersiuli faqtoruli analizis Sedegebis prezentacia".

2. adamianebis umetesobis mier aRiarebuli pirovnebis erT-erTi

ganzomileba aris introversia-eqstraversia. eqstravertebs axasiaTeben, rogorc Ria, socialur da xalisiT aRsavse adamianebs, xolo introvertebs ki, rogorc Caketil da naklebad socialur tipebs. radganac introvertebi ufro metad sakuTar fiqrebsa da ideebisken mimarTulni Canan, SeiZleba vifiqroT, rom introvertebi da eqstravertebi SeiZleba gansxvavbulad reagirebdnen garegan stimulirebaze, rogoricaa xmauri. am sakiTxis ufro Rrmad gamokvlevisaTvis stendingma, linma da moqsnesma (Standing, Lynn and Moxness, 1990) gamoiyenes cdis pirTa Soris 2x2 sqema, sadac cdis pirebi asrulebdnen wakiTxulis Sinaarsis gagebis davalebas da icvleboda pirovnebis tipi (A faqtori) - introvertebi (A1) da eqstravertebi (A2) da xmauris foni (B faqtori) - xmadabali (B1) da xmamaRali (B2). davuSvaT, Tqven CaatareT msgavsi kvleva 12 cdis piriT yovel ujredSi da yovelma cdis pirma Seasrula 15 debulebisgan Semdgari "sworia/arasworia" tipis wakiTxulis Sinaarsis gagebis davaleba. cdis piris qula warmoadgenda sworad gagebuli debulebebis raodenobas. Tqven miiReT Semdegi qulebi:

pirovnebis tipi (A)

introvertuli (A1) eqstravertuli (A2) xmadabali (B1) xmauris foni (B) xmamaRali (B2)

10 8 12 15 10 9 14 13 14 7 12 11

15 14 12 15 11 8 12 8 13 10 9 11

6 8 11 4 5 8 9 10 9 10 7 11

13 8 9 14 13 12 15 11 9 11 12 10

a. ra damokidebuleba arsebobs pirovnebis tipsa da xmauris fonis efeqts Soris? Tu xdeba urTierTqmedeba, martivi efeqtebisTvis gamoiyeneT tukis HSD kriteriumi da aRwereT Tqvens mier dadgenili urTierTqmedeba.

b. ra aris η2 urTierTqmedebisaTvis? g. SegiZliaT Tu ara, mniSvnelovnad axsnaT romelime mTavari efeqti,

Tu isini artefaqturia? d. SegiZliaT Tu ara, mniSvnelovnad CavTvaloT romelime mTavari

efeqti, Tu isini artefaqturia? e. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic


3. romelia kiTxvis damwyebTaTvis fonetikis swavlebis

saukeTeso meTodi? jozefma (Joseph, 2000) Seamowma pirvelklaselebisaTvis fonetikis swavlebis ori axali meTodi - sityvebis yuTis instruqciiT da sityvebis dajgufebis instruqciiT, da aseve tradiciuli fonetikuri meTodis instruqciiT. sityvebis yuTis instruqciis gamoyenebisas bavSvs eZleva oTxkuTxedi, romelic dayofilia seqciebad. TiToeuli seqcia sityvaSi yovel fonemas Seesabameba (fonema metyvelebis yvelaze mcire nawilakia). am instruqciis sawyis fazebze, rodesac sityva warmoiTqmis, bavSvi Jetons aTavsebs oTxkuTxedis im seqciaze, romelic Seesabameba gaJRerebul fonemas. sityvebis dajgufebis instruqcia isaa, rom bavSvebs sTxoven, moaxdinon sityvebis dajgufeba garkveuli zogadi aspeqtebis mixedviT, rogoricaa marTlwera an JReradoba. fonetikis tradiciuli swavlebis dros moswavleebi gundurad kiTxuloben im sityvebs, romelTac marTlweris saerTo tipi aqvT. dauSviT, rom Tqven dainteresdiT fonetikis swavlebis meTodebis efeqtiT da amas garda moisurveT, gagegiT am meTodebze varjiSis raodenobis efeqtebi. ase rom, Tqven cvlidiT fonetikis swavlebis (A faqtori) sam meTods: tradiciuls (A1), sityvebis yuTis (A2) da sityvebis dajgufebis (A3). aseve varjiSis raodenoba (B faqtori) icvleboda ori donis mixedviT: erTi kvira (B1) da eqvsi kvira (B2). cxra pirvelklaseli bavSvi SemTxveviTad gadanawilda TiToeul ujredSi. varjiSis periodis bolos yoveli bavSvi Semowmda marTlweris 30-debulebiani testiT da miRebuli iqna Semdegi qulebi (yoveli qula sworad dawerili sityvebis raodenobaa):

instruqciis meTodi (A) tradiciuli sityvebis yuTi sityvebis dajgufeba (A1) (A2) (A3)

erTi kvira (B1) varjiSis raodenoba (B) eqvsi kvira (B2)

10 14 16 12 14 13 15 11 13

16 17 23 19 20 15 19 18 21

21 24 23 20 23 25 21 26 22

15 18 22 18 21 17 18 16 16

18 24 19 22 21 20 20 22 23

21 19 20 23 24 24 22 25 20

a. damokidebulia Tu ara instruqciis meTodi varjiSis raodenobaze?

Tu ki, gamoiyeneT tukis HSD kriteriumi martivi efeqtebisaTvis. aRwereT Tqvens mier aRmoCenili urTierToba instruqciis meTodsa da varjiSis raodenobas Soris.

b. ra aris η2 urTierTqmedebisaTvis? g. SegiZliaT Tu ara, mniSvnelovnad CavTvaloT romelime mTavari

efeqti Tu isini artefaqturia? d. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic


4. erTma fsiqologma Caatara cdis pirTa Soris 2x2 eqsperimenti rva cdis piriT TiToeul ujredSi.

a. ra aris am sqemisaTvis A.s, b-s, nAB-s da N-is mniSvnelobebi? b. ra aris am sqemisaTvis dfA.s, dfB-s, dfAxB-s, dfSecdomis da dfsrulis

mniSvnelobebi? 5. Semdegi cxrilebi warmoadgens cdis pirTa Soris dispersiuli faqtoruli analizis arasrul Sejamebis cxrilebs. dauSviT, rom yovel ujredSi cdis pirTa toli raodenobebia. gamoiyeneT SS-s, df-sa da MS-s Soris kavSiri, raTa SeavsoT TiToeul cxrilSi gamotovebuli mniSvnelobebi. Semdeg yoveli cxrilisaTvis upasuxeT Semdeg kiTxvebs:

a. A faqtoris ramdeni done iyo? b. B faqtoris ramdeni done iyo? g. ramdeni cdis piri gamoiyenes TiToeul ujredSi? d. ra iyo cdis pirTa sruli raodenoba eqsperimentSi? e. Frealuris romeli mniSvnelobebia statistikurad mniSvnelovani .05 doneze? v. am analizSi rogor gadawyvetilebebs gaakeTebT yoveli nulovani da alternatiuli hipoTezisaTvis?

cxrili # 1 wyaro SS df MS F davalebis tipi (A) xmauris done (B) AxB Secdoma sul

------- 16.00 36.00 ------- -------

3 ------- 6 60 -------

5.00 ------- ------- 2.00

------- ------- -------

cxrili # 2 wyaro SS df MS F treningis xangrZlivoba (A) unaris done (B) AxB Secdoma sul

40.00 ------- 960.00 3200.00 -------

------- 3 ------- 160 179

10.00 40.00 80.00 -------

------- ------- -------

cxrili # 3 wyaro SS df MS F ukukavSiris tipi (A) davalebis sirTule (B) AxB Secdoma sul

200.00 ------- ------- ------- 3300.00

------- ------- 8 ------- 149

50.00 ------- 30.00 20.00-

------- ------- -------

6. dauSviT, rom Tqven sTxoveT cdis pirebs, SeeswavlaT winadadebebis seriebi da Semdeg gaexsenebinaT arsebiTi saxelebi am winadadebebidan. Tqven iyenebdiT xatebis or dones (A faqtori) – Cveulebriv da ucnaur xatebs, da damuSavebis or dones (B faqtori) – dabal da maRal damuSavebas. yovel cdis pirs 30 winadadeba unda eswavla, Sesabamisad, aRsadgeni arsebiTi saxelebis maqsimumi iyo 30. cxrilSi # 1 warmodgenilia ujredis saSualoebi sworad aRdgenili arsebiTi saxelebis raodenobebisaTvis. am monacemebis dispersiuli analizi warmodgenilia cxrilSi # 2. dauSviT, rom gamoyenebuli iyo cdis pirTa Soris sqema xuTi cdis piriT TiTo ujredSi. cxrili # 1 sworad aRdgenili arsebiTi saxelebis saSualo raodenoba, rogorc xatis tipisa da damuSavebis donis funqcia. xatis tipi (A)

Cveulebrivi (A1) ucnauri (A2) dabali (B1) damuSavebis done (B) maRali (B2)

21.4

4.8

10.4

18.8

cxrili # 2 dispersiuli analizis Sejamebis cxrili aRdgenili arsebiTi saxelebis saSualo raodenobisaTvis cxrilidan # 1. wyaro SS df MS F xatis tipi (A) damuSavebis done (B) AxB Secdoma sul

84.050 11.250 781.250 274.000 1150.550

1 1 1 16 19

84.050 11.250 781.250 17.125

4.91* 0.66 45.62**

* p < .05 ** p < .01

tukis HSD kriteriumma gviCvena, rom CD = 6.8 sityvas martivi efeqtebisaTvis.

a. axdens Tu ara xatebis tipi gavlenas arsebiTi saxelebis aRdgenaze?

b. axdens Tu ara damuSavebis done gavlenas arsebiTi saxelebis aRdgenaze?

7. fsiqologi dainteresebuli iyo pilotebis unariT, waikiTxon TviTmfrinavis xelsawyoebis monacemebi Zilis deprivaciis periodis Semdeg. rogorc es naCvenebia naxatze # 11.4, man gamoiyena ori sxvadasxva tipis xelsawyo, romelic pilotebs unda waekiTxaT – analoguri da cifruli. fsiqologma gamoiyena pilotebis SerCeva. maT naxevars unda waekiTxa analoguri xelsawyos monacemebi, xolo meore naxevars ki – cifruli xelsawyos monacemebi. TiToeul jgufSi pilotebis naxevari Semowmda normaluri Ramis Zilis Semdeg, xolo meore naxevari ki – mas Semdeg, rac 24 saaTis ganmavlobaSi ar eZinaT. ase rom, gamoyenebul iqna cdis pirTa 2x2 sqema, sadac xelsawyos tipi - analoguri an cifruli - iyo erTi damoukidebeli cvladi, da Zilis deprivaciis raodenoba - arc erTi an 24 saaTi - iyo meore damoukidebeli cvladi. damokidebuli cvladi iyo 100 cdidan swori wakiTxvis raodenoba. ujredis saSualoebi swori wakiTxvis raodenobisaTvis mocemulia cxrilSi # 1. dispersiuli analizi am monacemebisaTvis warmodgenilia cxrilSi # 2. cxrili # 1 xelsawyos sworad wakiTxvis saSualo raodenobebi, rogorc xelsawyos tipisa da Zilis deprivaciis raodenobis funqcia.

xelsawyos tipi (A)

analoguri (A1) cifruli (A2) arc erTi (B1) Zilis deprivaciis raodenoba (B) 24 saaTi (B2)

89.0

91.0

72.0

81.0

cxrili # 2 dispersiuli analizis Sejameba xelsawyos sworad wakiTxvis saSualo raodenobisaTvis cxrilidan # 1. wyaro SS df MS F xelsawyos tipi (A) Zilis deprivaciis raodenoba (B) AxB Secdoma sul

211.750 1275.750 85.750 433.920 2007.170

1 1 1 24 27

211.750 1275.750 85.750 18.080

11.71* 70.56 4.74**

* p < .05 ** p < .01

tukis HSD kriteriumis gamoyenebiT gamoTvlili CD martivi efeqtebisaTvis udris 5.7 swor wakiTxvas.

a. rogoria xelsawyos tipis efeqti swori wakiTxvis raodenobaze? b. rogoria Zilis deprivaciis raodenobis efeqti sworad wakiTxvis

raodenobaze?

1

Tavi XII. cdis pirTa Sida erTfaqtoriani sqema da dispersiuli analizi cdis pirTa Sida erTfaqtoriani sqemis magaliTi cdis pirTa Sida erTfaqtoriani sqemis dispersiuli analizi monacemTa dayofa da kvadratebis jamebis gamoTvla

Tavisuflebis xarisxebis gamoTvla gadaxrebis kvadratebis saSualos da F statistikis gamoTvla

statistikuri hipoTezis Semowmeba Frealuris meSveobiT Carevis efeqtis siZlieris gazomva cdis pirTa Sida dispersiuli analizis daSvebebi

dispersiuli analizis Sedegebis prezentacia kvlevis sqemis arCeva: cdis pirebs Soris Tu cdis pirebs Sida cdis pirebis saWiro raodenoba damoukidebeli cvladis efeqturoba mravaljeradi Carevis efeqtebi

2

me-9 TavSi warmodgenili iyo cdis pirTa Sida sqemebi da t kriteriumi damokidebuli qulebisTvis. am TavSi grZeldeba cdis pirTa Sida sqemebis ganxilva da warmodgenilia maTi dispersiuli analizi. cdis pirTa Sida sqema: kvlevis sqema, rodesac cdis pirTa erTi jgufi ganicdis damoukidebeli cvladis zemoqmedebas, rac izomeba am cvladis yoveli donis Sesabamisad.

cdis pirTa Sida erTfaqtorian sqemaSi cdis pirTa erTi jgufi ganicdis damoukidebeli cvladis zemoqmedebas, rac izomeba am cvladis yoveli donis Sesabamisad. radgan cdis pirTa Sida sqemaSi cdis pirebi ganmeorebiT izomebian, am sqemas aseve uwodeben ganmeorebiTi gazomvis sqemas an “Careva cdis pirebze” sqemas, rac imaze migviTiTebs, rom TiToeuli cdis piri damoukidebeli cvladis yvela donis mixedviT mowmdeba. cxrilSi # 12.1 sqematuradaa warmodgenili cdis pirTa Sida erTfaqtoriani sqemis monacemebi damoukidebeli cvladis oTxi doniT da xuTi cdis piris monawileobiT. damoukidebeli cvladi gansazRvrulia, rogorc A faqtori A1, A2, A3 da A4 doneebiT. A faqtoris doneebis raodenoba aRniSnulia a-Ti; cxrilisaTvis # 12.1 a = 4. cdis piris monacemi (qula) aRniSnulia Xij-T, sadac indeqsi i migviTiTebs cdis piris aRmniSvnel ricxvs (mag., X1j, X2j, . . . X5j) da indeqsi j migviTiTebs ricxvs, romelic aRniSnavs damoukidebeli cvladis dones. amdenad, X34 aRniSnavs me-3 cdis piris qulas A faqtoris me-4 doneze. rodesac ar aris saWiro konkretuli cdis piris identificireba, gamovtovebT ij indeqsebs da ubralod X-s viyenebT. damoukidebeli cvladis TiToeuli donis

saSualoebi aRiniSneba 1AX , 2AX , 3AX da 4AX an zogadad AX . qulebis raodenoba damoukidebeli cvladis donisaTvis aRiniSneba nA. miaqcieT yuradReba, rom nA aseve aris cdis pirTa raodenoba cdis pirTa Soris sqemaSic. qulebis sruli raodenoba eqsperimentSi aRniSnulia N-iT. cxrilisaTvis # 12.1, nA = 5 da N = 20. cdis piris saSualo A faqtoris yvela doneze aRiniSneba SX . es saSualo gamosaxavs cdis piris tipur Sesrulebas damoukidebeli cvladis yvela doneze.

cdis pirTa Sida erTfaqtoriani sqemis magaliTi cxrili # 12.1 qulebis sqematuri gamosaxuleba cdis pirTa Sida erTfaqtorian sqemaSi damoukidebeli cvladis oTxi doniTa da xuTi cdis piriT. cdis piri A faqtori

________________________________________________ A1 A2 A3 A4

cdis piris saSualo

S1 S2 S3 S4 S5

X11 X21 X31 X41 X51

X12 X22 X32 X42 X52

X13 X23 X33 X43 X53

X14 X24 X34 X44 X54

1SX

2SX

3SX

4SX

5SX

A faqtoris saSualoebi

1AX 2AX 3AX 4AX GX

3

SexedeT suraTs # 12.1. miuxedavad imisa, rom aq mxolod nawilobriv Sevsebuli wreebia warmodgenili, suraTze isini TiTqos kideebs qmnian, romelTa Sorisac samkuTxedia moTavsebuli. am kideebs subieqturi konturebi ewodeba, radganac isini TavisTavad stimulSi ar aris mocemuli. SesaZlebelia Tu ara subieqturi konturebis gamoyeneba aRqmis iluziis gamosawvevad? davuSvaT, gadavwyviteT am sakiTxis gamorkveva poncos (Ponzo) iluziis ramdenime formis gamoyenebiT, rogorc es maierma gaakeTa (Meyer, 1986). suraTze # 12.2a naCvenebia poncos standartuli iluzia. es suraTi iluziaa, radganac ori horizontaluri xazi fizikurad erTi da imave sigrZisaa, Tumca isini tolad ar gamoiyureba. zeda xazi ufro grZeli Cans, vidre qveda. suraTze # 12.2b warmodgenilia am iluziis versia subieqturi konturebiT. versia suraTze # 12.3g gvawvdis saorientacio informacias, magram mTliani wertilebi ar iZleva subieqtur konturebs. dabolos, suraTze # 12.2d mocemulia sakontrolo piroba, sadac iluzia ar unda iyos. davuSvaT, cdis pirs sTxoven, ise Secvalos qveda horizontaluri xazis sigrZe, rom is zeda xazis tolad aRiqmebodes, Semdeg ki davuSvaT, rom zeda xazis sigrZe 14 santimetria (sm). Tu iluzia ar Cndeba, maSin qveda xazi aseve 14 sm unda gaxdes. xolo Tu iluzia Cndeba, maSin zeda xazi ufro grZeli unda aRmoCndes, vidre es sinamdvileSia (anu is 14 sm-ze grZeli unda aRmoCndes) da qveda xazi ise unda Seicvalos, rom aRqmuli sigrZis Sesabamisi iyos. gamotovebulia suraTi 12.1 cdis pirTa Sida erTfaqtoriani eqsperimenti iluziis tipis (A faqtori) oTxi doniTa da xuTi cdis piriT gamoiyeneba imisaTvis, raTa Semowmdes iluziis pirobis efeqti iluziis aRqmul sidideze. yoveli cdis piri aTvalierebs oTxive stimuls, romlebic naCvenebia suraTze # 12.2 da cvlis qveda horizontalur xazs ise, rom is sigrZeSi zeda xazis tolad gamoiyurebodes. santimetrebSi gamosaxuli qveda xazis Secvlili sigrZe aris damokidebuli cvladi. davuSvaT, miiReT monacemebi, romlebic mocemulia cxrilSi # 12.2. imoqmeda Tu ara aRqmuli iluziis sidideze iluziis pirobam? cxrili # 12.2 im xuTi cdis pirisagan miRebuli xazebis aRqmuli toli sigrZeebi (santimetrebSi), romelTac poncos iluziis oTxi versia naxes. cdis piri iluziis piroba (A)

________________________________________________ standartuli subieqturi mTliani sakontrolo konturebi wertilebi (A1 ) (A2 ) (A3) (A4)

cdis piris saSualo

1 2 3 4 5

16.6 16.9 17.1 17.2 16.8

15.8 15.6 16.1 16.3 15.7

14.7 15.1 14.8 15.2 15.3

14.4 13.8 13.9 14.1 14.3

15.38 15.35 15.48 15.70 15.53

AX sA

16.92 0.24

15.90 0.29

15.02 0.26

14.10 0.25

4

iseve, rogorc cdis pirTa Soris sqemebSi, cxrilSi # 12.2 moyvanili monacemebis analizi aRweriTi statistikebis gamoTvliT iwyeba. amdenad, pirobebis saSualoebi da standartuli gadaxrebi naCvenebia cxrilis yoveli svetis boloSi. Tu damoukidebeli cvladi gavlenas axdens iluziis aRqmaze, maSin pirobebis saSualoebi gansxvavebuli unda iyos. magram, rogorc Cven viciT, pirobebis saSualoebs Soris miRebuli nebismieri sxvaoba yovelTvis unda SevadaroT Secdomis cvalebadobas. ramdenad aris SesaZlebeli, rom saSualoebis gansxvaveba mxolod SerCevis Secdomis gamo momxdariyo? saSualoebis ubralo mxedvelobiTi Semowmeba ar gvaZlevs am kiTxvaze pasuxis gacemis SesaZleblobas. rogorc es cdis pirTa Soris sqemisaTvis gavakeTeT, am gadawyvetilebis miRebis problemas statistikuri kriteriumis gamoyenebiT vwyvetT. es aris cdis pirTa Sida erTfaqtoriani dispersiuli analizi.

cdis pirTa Sida erTfaqtoriani sqemis dispersiuli analizi cdis pirTa Sida erTfaqtoriani dispersiuli analizi gamoiyeneba erTfaqtoriani mravaldoniani cdis pirTa Sida sqemebis gaanalizebisaTvis. Tu damoukidebeli cvladis mxolod ori done icvleba, cdis pirTa Soris erTfaqtoriani dispersiuli analizi SeiZleba tdamokidebulis magivrad gamoviyenoT. es analizi eqsperimentSi qulebis srul cvalebadobas sam wyarod yofs:

1. A faqtoris efeqtiT gamowveuli cvalebadoba; 2. cdis pirTa Soris individualuri gansxvavebebis, S faqtoris

efeqtiT gamowveuli cvalebadoba; 3. pirobebisa da cdis pirebis urTierTqmedebiT (AxS) gamowveuli

cvalebadoba. es sami dayofa gamoiyeneba sami kvadratebis saSualos misaRebad:

1. MSA, romelic A faqtoris efeqtTan da Secdomis cvalebadobasTan erTad icvleba;

2. MSS, romelic cdis pirebs Soris gansxvavebebTan erTad icvleba; 3. MSAxS, romelic pirobebis cdis pirebTan urTierTqmedebasTan

erTad icvleba da eqsperimentSi Secdomis cvalebadobis zomas iZleva.

F statistika, romelic statistikuri hipoTezis Semowmebisas gamoiyeneba, aris

F = AxS

A

MSMS

monacemis dayofa da kvadratebis jamebis miReba am seqciaSi ganvixilavT monacemebis dayofas da kvadratebis jamebis miRebas cdis pirTa Sida sqemebisaTvis. ramdenadac aq gamoyenebuli midgoma Zalian hgavs imas, romelic gamoiyeneba cdis pirTa Soris sqemebSi, dayofis wardgena mokle iqneba.

5

imis gasagebad, Tu rogor unda daiyos cvalebadobis wyaroebi cdis pirTa Sida sqemebSi, cdis pirTa Sida sqema warmovidginoT, rogorc orfaqtoriani sqema, sadac damoukidebeli cvladi erTi faqtoria (A faqtori), da cdis pirebi ki - meore faqtori (S faqtori, sadac S warmoadgens cdis pirs). radganac TiToeuli cdis piri damoukidebeli cvladis yovel doneze mowmdeba, cdis pirTa Sida erTfaqtoriani sqema SeiZleba ganvixiloT, rogorc AxS faqtoruli sqema A faqtoris a doneebiT da S faqtoris nA doneebiT. ase rom, miviRebT a x nA pirobebs anu ujredebs, da TiToeuli ujredi warmoadgens, konkretul cdis pirTan kombinirebul damoukidebeli cvladis erT dones. cxrilSi # 12.2 A faqtoris oTxi done da xuTi cdis piria; amdenad, eqsperimenti SeiZleba ganvixiloT, rogorc 4 (iluziis pirobis doneebi) x 5 (cdis pirebis raodenoba) faqtoruli sqema, sadac sul 20 gansxvavebuli Careva/cdis piri (AxS) pirobaa. magram TiToeuli AxS pirobisaTvis mxolod erTi qula miiReba. cdis pirTa Sida erTfaqtoriani dispersiuli analizis ganxilva, rogorc orfaqtoriani dispersiuli analizisa, A da S faqtorebisaTvis "mTavari efeqtebis" saSualoebs gvaZlevs. cxrilSi # 12.2 miaqcieT yuradReba, rom saSualoebi mocemulia TiToeuli cdis pirisaTvis, iseve, rogorc damoukidebeli cvladis TiToeuli donisaTvis. TiToeuli cdis piris saSualo aris erTi cdis piris qulebis saSualo iluziis yvela pirobisaTvis. es saSualoebi asaxavs iluziis aRqmis individualur gansxvavebebs eqsperimentSi monawile cdis pirebs Soris. miuxedavad imisa, rom cdis pirebis saSualoebi dispersiuli analizis relevanturia, es saSualoebi, Cveulebriv, ar aris xolme warmodgenili im eqsperimentebis aRweriT statistikaSi, romlebSic cdis pirTa Sida sqema gamoiyeneba. cdis pirTa Sida erTfaqtoriani dispersiuli analizis ganxilva, rogorc orfaqtoriani analizisa, ar cvlis eqsperimentis arss: icvleba mxolod erTi damoukidebeli cvladi da xdeba misi efeqtebis analizi. Tumca, analizis orfaqtorian sqemad warmodgena gvexmareba, gavigoT AxS urTierTqmedebis Secdomis nawilad SemoReba F statistikisaTvis. F statistikis misaRebad cdis pirTa Sida dispersiul analizSi iReben srul raodenobas, romliTac qula gansxvavdeba sruli saSualosagan da am gansxvavebas sam komponentad yofen: = + + . (12.1) dispersiuli analizis ZiriTadi midgomis mixedviT, dayofili qulebis komponentebi ayvanilia kvadratSi da jamdeba yvela cdis piris da Carevis yvela pirobis mixedviT, raTa kvadratebis jami miviRoT: SSsruli = SSA + SSS + SSAxS (12.2)

sruli cvalebadoba cdis piris qulaSi

cvalebadoba A faqtoris gamo plius Secdoma A da S faqtorebis urTierTqme-debis gamo

cvalebadoba cdis pirTa Soris indi-vidualuri gansxvavebe-bis gamo (S faqtori)

cvalebadoba A da S faqtorebis urTierTqme-debis gamo

6

es kvadratebis jamebi migviTiTebs, rom eqsperimentSi qulebis sruli cvalebadoba (aRniSnuli SSsruli) aris im cvalebadobis Sedegi, romelic xdeba A faqtoris (aRniSnuli SSA), S faqtoris (aRniSnuli SSS) da A da S faqtorebis urTierTqmedebis gamo. cxrilSi # 12.2 mocemuli qulebisaTvis es procesi Semdeg ricxviT mniSvnelobebs gvaZlevs: SSsruli = 22.926, SSA = 21.830, SSS = 0.314, SSAxS = 0.782. am mniSvnelobebis misaRebad gamyenebulia gamoTvliTi formulebi am Tavis damatebidan. ganvixiloT 12.2 tolobaSi mocemul TiToeuli kvadratebis jami. SSsruli SSsruli warmoadgens eqsperimentSi qulebis srul cvalebadobas. rogorc 12.2 tolobaSia aRniSnuli, am cvalebadobis nawili damoukidebeli cvladis efeqtis, nawili - cdis pirebs Soris individualuri gansxvavebebis, xolo nawili ki - pirobebis da cdis pirTa urTierTqmedebis Sedegia. SSA SSA icvleba damoukidebeli cvladis efeqts da pirobebsa da cdis pirTa urTierTqmedebasTan erTad. rogorc cdis pirTa Soris erTfaqtorian sqemaSi, damoukidebeli cvladis efeqti zrdis SSA -is mniSvnelobas. garda amisa, SSA reagirebs pirobebisa da cdis pirTa nebismier urTierTqmedebaze. cdis pirTa Sida sqemaSi pirobebisa da cdis pirTa urTierTqmedeba ganixileba Secdomis cvalebadobad. SSS SSS icvleba eqsperimentSi cdis pirTa Soris individualur gansxvavebebTan erTad. cdis pirTa Soris am gansxvavebebs aRvniSnavT S faqtoriT. cxrilSi # 12.2 miaqcieT yuradReba, rom me-4 cdis pirs aqvs tendencia, iluziis yvelaze meti raodenoba dainaxos damoukidebeli

cvladis yvela doneze ( 4SX = 15.70 sm), maSin, rodesac me-2 cdis piri

yvelaze nakleb iluzias aRiqvams ( 2SX = 15.35 sm). rac ufro metad gansxvavdeba eqsperimentSi cdis piris saSualoebi erTmaneTisagan, miT ufro didi xdeba SSS. SSAxS pirobebis urTierTqmedeba cdis pirebTan qulis darCenili cvalebadobaa da gavlenas axdens SSAxS-ze. pirobebisa da cdis pirebis es urTierTqmedeba SeiZleba vnaxoT suraTze # 12.3, romelic cxrilis # 12.2 hipoTetur monacemebs gamoxatavs. abscisaze iluziis oTxi tipia warmodgenili. suraTze moTavsebuli funqciebi gamoxataven xuTi cdis pirisagan TiToeulis miRebul oTx qulas. xazis toli aRqmuli sigrZe yvelaze didia yvela cdis pirisaTvis standartuli iluziis pirobaSi. amdenad, aSkaraa iluziis pirobis efeqti xazis aRqmul sigrZeze. Tumca, radganac xuTi funqcia paraleluri ar aris, xdeba pirobebisa da cdis

7

pirebis urTierTqmedebac. xazis aRqmul sigrZes Soris sxvaobebi iluziis pirobebs Soris ar aris erTi da igive yvela cdis pirisaTvis. magaliTad, pirveli cdis piris SemTxvevaSi qulebis sxvaoba standartul pirobasa da subieqturi konturis pirobas Soris aris 16.6 - 15.8 = 0.8 sm, xolo meore cdis piris SemTxvevaSi sxvaoba aris 16.9 - 15.6 = 1.3 sm. ase rom, iluziis pirobis efeqti xazis aRqmul sigrZeze cdis pirebzea damokidebuli. SSAxS zomavs pirobebisa da cdis pirebis urTierTqmedebis xarisxs cdis pirTa Sida eqsperimentSi. magram, radganac Carevis TiToeul pirobaSi erTi cdis pirisagan mxolod erT qulas viRebT, pirobebisa da cdis pirebis urTierTqmedebas ver gamovacalkevebT nebismieri Secdomisagan, romelic SeiZleba gavlenas axdendes qulaze. Sesabamisad, SSAxS ganixileba, rogorc Secdomis cvalebadobis sazomi cdis pirTa Sida erTfaqtorian sqemaSi da warmoadgens Secdomis nawilis safuZvels F statistikisaTvis am analizSi. pirobebisa da cdis pirTa urTierTqmedeba (AxS): situacia cdis pirTa Sida erTfaqtorian sqemaSi, rodesac damoukidebeli cvladis efeqti damokidebulia cdis pirze.

Tavisuflebis xarisxebis gamoTvla F statistikisaTvis saWiro gadaxrebis kvadratebis saSualos misaRebad TiToeuli SS iyofa mis Sesabamis Tavisuflebis xarisxze. Tavisuflebis xarisxebis gamoTvlis procesi Zalian hgavs cdis pirTa Soris sqemaSi gamoyenebul gamoTvlas.

Tavisuflebis sruli xarisxi dfsruli = N - 1, sadac N aris qulebis mTeli raodenoba eqsperimentSi. magaliTad, N = 20; amitom dfsruli = 20 - 1 = 19. Tavisuflebis xarisxi SSA-sTvis dfA = a - 1, sadac a aris A damoukidebeli cvladis doneebis raodenoba. Cvens magaliTSi A faqtoris oTxi donisaTvis a = 4; aqedan gamomdinare, dfA = 4 - 1 = 3.

Tavisuflebis xarisxi SSS-sTvis dfS = nA - 1 Cvens magaliTSi xuTi cdis piria; Sesabamisad, nA = 5 da dfS = 5 - 1 = 4.

Tavisuflebis xarisxi SSAxS-isTvis dfAxS = (a - 1)( nA - 1)

8

Cvens magaliTSi a = 4 da nA = 5; Sesabamisad, dfAxS = (4 - 1)( 5 - 1) = 12.

Tavisuflebis xarisxis aditiuroba Tavisuflebis xarisxebi aditiuria, ase rom dfsruli = dfA + dfS + dfAxS es mimarTeba Cvens magaliTSic sruldeba, radgan dfsruli = 19, dfA = 3, dfS = 4, da dfAxS = 12. gadaxrebis kvadratebis saSualoebisa da F statistikis gamoTvla kvadratebis sami saSualo miiReba SS-idan 12.2 tolobaSi TiToeuli SS-is mis df-ze gayofiT:

MSA = A

A

dfSS

,

MSS = S

S

dfSS

,

MSAxS = AxS

AxS

dfSS

.

MSsruli ar gamoiTvleba, radganac is ar gamoiyeneba dispersiul analizSi. cxrilis # 12.2 qulebisaTvis SSA = 21.830, SSS = 0.314, SSAxS = 0.782. am kvadratebis saSualoebisaTvis Tavisuflebis xarisxebia dfA = 3, dfS = 4, dfAxS = 12. aqedan gamomdinare,

MSA = 3830.21

= 7.277,

MSS = 4314.0

= 0.078,

MSAxS = 12782.0

= 0.065.

9

F statistika cdis pirTa Sida erTfaqtorian dispersiul analizSi aris

F = AxS

A

MSMS .

ase rom, cxrilis # 12.1 qulebisaTvis

Frealuri = 065.0277.7

= 111.95.

es Frealuri aris am analizSi gamosaTvleli erTaderTi F; S faqtorisaTvis F ar gamoiTvleba. cxrilis # 12.2 qulebis dispersiuli analizis ricxviTi Sejameba warmodgenilia cxrilSi # 12.3. gamoTvliTi formulebi am analizisaTvis mocemulia Tavis damatebaSi. cxrili # 12.4 ajamebs cdis pirTa Sida erTfaqtorian dispersiul analizSi cvalebadobis wyaroebs - SS, df, MS da F. cxrili # 12.3 dispersiuli analizis cxrili xazebis hipoTeturi Tanabari aRqmuli sigrZis qulebisaTvis cxrilidan # 12.2. wyaro SS df MS F iluziis tipi (A) cdis pirebi (S) AxS sul

21.830 0.314 0.782 22.926

3 4 12 19

7.277 0.078 0.065

111.95*

* p < .01

cxrili # 12.4 formulebis Sejameba cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis. wyaro SS dfa MS F A faqtori S faqtori AxS sul

SSA SSS SSAxS SSsruli

a - 1 nA - 1 (a - 1)( nA - 1) N - 1

A

A

dfSS

S

S

dfSS

AxS

AxS

dfSS

ar gamoiTvleba

AxS

A

MSMS

aa = A faqtoris doneebis raodenoba; nA = qulebis raodenoba Carevis pirobaSi anu, Sesabamisad, cdis pirebis raodenoba; N - qulebis sruli raodenoba.

SeamowmeT Tqveni codna # 12.1

1. gansazRvreT: AxS urTierTqmedeba, dfA, dfAxS, dfS, dfsruli, A faqtori, S faqtori, MSA, MSAxS, MSS, MSsruli, nA, cdis pirTa Sida erTfaqtoriani sqema, ganmeorebiTi gazomvebis sqema, SSA, SSAxS, SSS, SSsruli, pirobebisa da cdis pirebis urTierTqmedeba.

2. gansazRvreT qulebis cvalebadobis wyaroebi cdis pirTa Sida erTfaqtorian sqemaSi.

10

3. daasruleT Semdegi gantolebebi cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis:

a. SSsruli = b. dfA = g. dfS = d. dfAxS = e. MSA = v. MSS = z. MSAxS = T. F =

4. ra aris SSsrulis mniSvneloba, Tu SSA = 74.00, SSS = 29.00 da SSAxS = 96.00? 5. ra aris SSA-is mniSvneloba, Tu SSsruli = 426.00, SSS = 92.00 da SSAxS = 301.00? 6. ra aris SSS -is mniSvneloba, Tu SSsruli = 96.00, SSA = 15.00 da SSAxS = 56.00? 7. ra aris SSAxS -is mniSvneloba, Tu SSsruli = 173.00, SSA = 41.00 da SSS = 36.00? 8. erTma fsiqologma gamoiyena cdis pirTa Sida erTfaqtoriani dispersiuli

analizi A faqtoris xuTi doniT da 13 cdis piriT. ra aris dfA, dfS, dfAxS da dfsrulis mniSvnelobebi?

9. ra aris dfsrulis mniSvneloba cdis pirTa Sida erTfaqtorian dispersiul analizSi, Tu dfA = 2 da dfS = 19?

10. ipoveT gadaxrebis kvadratebis saSualoebis ricxviTi mniSvnelobebi Semdegi SS-isa da df-isaTvis: SSA = 22.00, SSS = 18.00, SSAxS = 36.00, dfA = 2, dfS = 9 da dfAxS = 18.

11. cxrilebi # 1 da 2 warmoadgens arasrul Sejamebis cxrilebs cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis. TiToeul cxrilSi SeavseT gamotovebuli mniSvnelobebi cxrilSi # 12.4 SS-sa da df-s Soris mocemuli mimarTebis gamoyenebiT. Semdeg upasuxeT am kiTxvebs yoveli cxrilisaTvis.

a. damoukidebeli cvladis ramdeni done icvleba? b. ramdeni qulaa miRebuli TiToeul pirobaSi? g. ramdeni cdis piri iRebda monawileobas kvlevaSi?

cxrili # 1 wyaro SS df MS F A faqtori S faqtori AxS sul

48.00 27.00 72.00 -------

4 9 ------- -------

------- ------- -------

-------

cxrili # 2 wyaro SS df MS F A faqtori S faqtori AxS sul

6.00 36.00 ------- 78.00

------- 12 24 -------

------- 3.00 -------

-------

statistikuri hipoTezis Semowmeba Frealuris meSveobiT Frealuri = MSA/MSAxS gamoiyeneba imis gadasawyvetad, mniSvnelovnad gansxvavdeba Tu ara Carevis pirobebis saSualoebi. rogorc SSAxS-is ganxilvisas vnaxeT, F-is mniSvneli, MSAxS, am analizSi Secdomis cvalebadobas warmoadgens. F-is mricxveli, MSA, icvleba damoukidebeli cvladis sistematur efeqtTan erTad da pirobebisa da cdis pirebis urTierTqmedebasTan erTad. Tu damoukidebel cvlads ar aqvs efeqti da pirobebis saSualoebis cvalebadoba mxolod pirobebisa da cdis pirebis urTierTqmedebis gamo xdeba, maSin MSA-is mniSvneloba daaxloebiT igive

11

unda iyos, rac MSAxS-is mniSvneloba da Frealuri unda iyos daaxloebiT 1.00. xolo, Tu A faqtors aqvs efeqti da pirobebis saSualoebi gansxvavdeba rogorc am efeqtis, aseve pirobebisa da cdis pirebis urTierTqmedebis gamo, maSin MSA-s ufro didi mniSvneloba unda hqondes, vidre MSAxS-s, da Frealuri 1.00-ze meti xdeba. F Sefardeba MSS-is monawileobiT, romelic individualur gansxvavebebis kvadratis saSualos aRniSnavs, ar gamoiTvleba. miuxedavad imisa, rom MSS icvleba eqsperimentSi cdis pirebs Soris gansxvavebebis xarisxTan erTad, am analizSi ar gamoiTvleba MS, romelic iZleva individebs Soris mosalodneli Secdomis cvalebadobis Sefasebas. aqedan gamomdinare, MSS-is mniSvneloba xSirad ar aris warmodgenili saboloo cxrilSi. sami kvadratebis saSualo da maTze gavlenis mqone cvalebadobis wyaroebi SeiZleba Semdegnairad Sejamdes: kvadratebis saSualo masze gavlenas axdens:

MSA A faqtoriT gamowveuli sistematuri cvalebadoba

pirobebisa da cdis pirebis urTierTqmedeba (anu Secdomis cvalebadoba)

MSS S faqtoriT gamowveuli sistematuri cvalebadoba

MSAxS pirobebisa da cdis pirebis urTierTqmedeba (anu Secdomis cvalebadoba)

statistikuri Semowmeba F-is gamoyenebiT ukve nacnob proceduras warmoadgens:

• yalibdeba nulovani hipoTeza, H0, da alternatiuli hipoTeza, H1;

• im daSvebiT, rom H0 sworia, gamoiTvleba F-is SerCevis ganawileba. es ganawileba mocemulia cxrilSi # A-3;

• xdeba mniSvnelovnobis donis SerCeva;

• F-is kritikuli done, romelic aRiniSneba FkritikuliT, gamoiTvleba F–is SerCeviTi ganawilebidan, romelic mocemulia cxrilSi # A-3;

• F–is SerCeviT ganawilebaSi gamoiyofa uaryofis are;

• SerCevis monacemebidan gamoiTvleba F statistika, romelic aRiniSneba, rogorc Frealuri;

• gadawyvetileba imis Taobaze, uarvyoT Tu ara H0, miiReba imaze dayrdnobiT, xvdeba Tu ara Frealuri uaryofis areSi.

statistikuri hipoTezebi nulovani hipoTeza. nulovani hipoTeza cdis pirTa Sida sqemisaTvis A faqtoris oTxi doniT ase gamoixateba:

H0: µ A1 = µ A2 = µ A3 = µ A3

12

es nulovani hipoTeza asaxavs situacias, romelic Cndeba, Tu damoukidebel cvlads ar aqvs efeqti. Tu H0 marTldeba, maSin pirobebis saSualoebs Soris miRebuli sxvaobebi ubralod Secdomis cvalebadobis gamoa. rogorc cdis pirTa Soris sqemaSi, nulovan hipoTezaSi gansazRvruli populaciis saSualoebis raodenoba yovelTvis Seesabameba damoukidebeli cvladis doneebis raodenobas. alternatiuli hipoTeza. alternatiuli hipoTezaa:

H1: µ A-ebi ar aris toli. hipoTezaSi monawile populaciis saSualoebis raodenobis miuxedavad. alternatiuli hipoTeza asaxavs situacias, romelic Cndeba, Tu damoukidebel cvlads aqvs efeqti. F statistikaze damyarebuli gadawyvetilebebis miReba cdis pirTa Sida erTfaqtorian sqemaSi statistikuri gadawyvetilebebis miRebis procesi cdis pirTa Soris erTfaqtorian dispersiul analizSi

gamoyenebuli procesis identuria. α–s mniSvneloba, romelic gansazRvravs uaryofis ares FrealurisaTvis, analizis Catarebamde SeirCeva. Semdeg cxrilSi # A-3a an b moiZebneba Fkritikulis mniSvneloba Sesabamisi mricxveli da mniSvneli df-iT. Tu Frealuri udris an metia Fkritikulze, maSin Frealuri xvdeba uaryofis areSi. nulovani hipoTeza uaryofilia da miRebulia alternatiuli hipoTeza. arsebobs sul mcire erTi statistikurad mniSvnelovani sxvaoba pirobebis saSualoebs Soris. Tu Frealuri naklebia, vidre Fkritikuli, maSin Frealuri ar xvdeba uaryofis areSi; nulovani hipoTeza ar aris uaryofili da alternatiuli hipoTeza ar miiReba. pirobebis saSualoebi mniSvnelovnad ar gansxvavdeba; saSualoebs Soris miRebuli sxvaobebi Secdomis cvalebadobis gamo xdeba. vaskvniT, rom damoukidebel cvlads ar hqonda efeqti. Frealurze damyarebuli gadawyvetilebebi da Semdgomi daskvnebi Sejamebulia cxrilSi # 12.5. cxrili # 12.5 gadawyvetilebebis da daskvnebis Sejameba statitikuri hipoTezebis SemowmebaSi cdis pirTa Sida erTfaqtoriani sqemisaTvis dispersiuli analizis gamoyenebiT. mocemulia mniSvnelovnobis done .05.

Tu Frealuri xvdeba uaryofis areSi α = .05-isaTvis, maSin:

Tu Frealuri ar xvdeba uaryofis areSi α = .05-isaTvis, maSin:

Frealuris albaToba naklebia, an tolia

.05-is, anu p≤ .05.

H0 uaryofilia.

H1 miRebulia. SerCevis yvela saSualo ar aris erTi da imave populaciidan. pirobebis saSuloebs Soris sul mcire erTi sxvaoba statistikurad mniSvnelovania .05 doneze. gadawyvetilia, rom SerCevis Secdomis garda kidev raRac sxva ganapirobebs sxvaobas SerCevis saSualoebs Soris. yvela pirobis dacviT Catarebul eqsperimentSi sxvaoba damoukidebel

Frealuris albaToba metia .05-ze, anu p>.05.

H0 ar aris uaryofili.

H1 ar aris miRebuli. SerCevis saSualoebs Soris ar aris mniSvnelovani sxvaobebi .05 doneze. gadawyvetilia, rom SerCevis Secdoma SerCevis saSualoebs Soris sxvaobis yvelaze metad savaraudo axsnaa. ar arsebobs imis safuZveli, rom damoukidebel

13

cvlads miewereba. Tu damoukidebel cvlads sami an meti done aqvs, saWiroa mravlobiTi Sedarebis kriteriumebi.

cvlads efeqti hqonda. mravlobiTi Sedarebis kriteriumebi ar aris saWiro.

gadawyvetilebis procesis ilustraciisaTvis gamoviyenebT cxrilSi # 12.3 Sejamebul analizs cxrilis # 12.2 sanimuSo qulebisaTvis. am analizSi Frealurs aqvs 3 (mricxveli) da 12 (mniSvneli) df. amdenad,

rodesac α = .05, Fkritikuli = 3.49 (miRebulia cxrilidan A-3a.) Frealuri = 111.95 metia Fkritikulze da xvdeba uaryofis areSi. uarvyofT

H0: µ A1 = µ A2 = µ A3 = µ A3 da viRebT

H1: µ A-ebi ar aris toli, es gadawyvetileba migviTiTebs, rom arsebobs sul mcire erTi statistikurad mniSvnelovani sxvaoba cxrilis # 12.2 oTx saSualos Soris, magram is ar aRniSnavs konkretulad, Tu romeli saSualoebi gansxvavdeba erTmaneTisagan. imis dasadgenad, Tu romeli wyvilTaSorisi Sedarebebi gansxvavdeba mniSvnelovnad, gamoiyeneba mravlobiTi Sedarebis kriteriumebi. mravlobiTi Sedarebis kriteriumebi cdis pirTa Sida dispersiuli analizisaTvis mravlobiTi Sedarebis kriteriumebTan dakavSirebuli problemebi da sakiTxebi cdis pirTa Sida dispersiuli analizisaTvis igivea, rac ganvixileT me-10 TavSi cdis pirTa Soris dispersiuli analizisaTvis. saboloo SedarebisaTvis SeiZleba gamoyenebul iqnes tukis HSD kriteriumi kritikuli sxvaobiT

CD = qA

AxS

nMS .

Tu am kriteriums Cvens eqsperiments mivusadagebT:

• MSAxS = 0.065 (miRebulia cxrilidan # 12.3)

• nA – 5. • q .05 doneze damoukidebeli cvladis oTxi donisaTvis da 12 df –

MSAxS-saTvis aris 4.20 (miRebulia cxrilidan # A-4). am mniSvnelobebis CasmiT viRebT:

CD = 4.205065.0

= 4.20 013.0 = 0.48 sm.

14

TiToeuli wyvilTaSorisi SedarebisaTvis pirobebis saSualoebs Soris sxvaoba, romelic absoluturi mniSvnelobiT 0.48 sm-ze metia, statistikurad mniSvnelovania .05 doneze. eqvsi wyvilTaSorisi Sedareba, TiToeuli Sedarebis absoluturi mniSvnelobebi, statistikuri hipoTeza da gadawyvetileba TiToeuli hipoTezisaTvis naCvenebia cxrilSi # 12.6 (sadac A1 = standartuli iluzia, A2 = subieqturi konturebis iluzia, A2 = mTliani wertilebis iluzia, A4 = sakontrolo piroba). TiToeuli sxvaobis absoluturi mniSvneloba CD – 0.48 sm-ze metia; amdenad, TiToeuli Sedareba statistikurad mniSvnelovania .05 doneze. standartuli iluzia iluziis mniSvnelovnad nakleb Sedegs qmnis, mTliani wertilebi – mniSvnelovnad kidev ufro nakleb iluzias da sakontrolo piroba ki – yvelaze nakleb iluzias. Carevis efeqtis siZlieris gazomva eta kvadrati efeqtis siZlieris sazomad SeiZleba gamoviyenoT cdis pirTa Sida erTfaqtorian sqemaSi. gamoyenebuli formula mciredaa Secvlili cdis pirTa Soris sqemasTan SedarebiT:

η2 = AxSA

A

SSSSSS+

miaqcieT yuradReba, rom am formulis mixedviT individualuri gansxvavebebiT gamowveuli cvalebadoba, romelic mocemulia SSS-iT, ar aris CarTuli eqsperimentis srul cvalebadobaSi. cxrili # 12.6 tukis HSD CD-s gamoyeneba wyvilTaSorisi SedarebebisaTvis sanimuSo eqsperimentSi. Sedareba Sedarebis

absoluturi mniSvneloba


gadawyvetileba

1AX 2AX -Tan

(16.92 – 15.90)

1AX 3AX -Tan

(16.92 – 15.02)

1AX 4AX -Tan

(16.92 – 14.10)

2AX 3AX -Tan

(15.90 – 15.02)

2AX 4AX -Tan

(15.90 – 14.10)

3AX 4AX -Tan

(15.02 – 14.10)

1.02 1.90 2.82 0.88 1.80 0.92

H0: µ A1 = µ A2 H1: µ A1 ≠ µ A2

H0: µ A1 = µ A3 H1: µ A1 ≠ µ A3

H0: µ A1 = µ A4 H1: µ A1 ≠ µ A4

H0: µ A2 = µ A3 H1: µ A2 ≠ µ A3

H0: µ A2 = µ A3 H1: µ A2 ≠ µ A3

H0: µ A3 = µ A4 H1: µ A3 ≠ µ A4

uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia

15

Cveni magaliTisaTvis, SSA = 21.830 da SSAxS = 0.782. ase rom,

η2 = 782.0830.21

830.21+

= .97.

xazis sigrZis gazomvis cvalebadobis 97 procenti xdeba iluziis pirobis gamo, mas Semdeg, rac individualuri gansxvavebebis gamo cvalebadoba gauqmebulia. es damoukidebeli cvladis Zalian Zlieri efeqtia. cdis pirTa Sida dispersiuli analizis daSvebebi daSvebebi, romlebsac emyareba cdis pirTa Sida erTfaqtoriani analizis gamoyeneba, Semdegia:

1. TiToeuli cdis piri mowmdeba damoukidebeli cvladis yovel doneze;

2. monacemebi SerCevaSi SemTxveviTad aris aRebuli populaciidan; 3. monacemTa populaciebi Carevis sxvadasxva pirobisaTvis

normaluradaa ganawilebuli; 4. populaciaSi monacemTa dispersiebi Carevis sxvadasxva pirobisaTvis

erTnairia; 5. cdis piris individualuri gansxvavebebis wili misi qulebisaTvis

erTi da igive rCeba Carevis yvela pirobaSi. pirveli daSveba aucilebelia cdis pirTa Sida dispersiul analizSi monacemebis saTanado statistikuri kriteriumis arsebobisaTvis; am daSvebis ugulvebelyofa dauSvebelia. meore, mesame da meoTxe daSvebebi cdis pirTa Soris dispersiul analizSi arsebuli daSvebebis identuria. xandaxan eqsperimentebSi maTi darRveva xdeba, magram dispersiuli analizi inarCunebs garkveul mdgradobas aseTi darRvevebis mimarT. mexuTe daSvebis mixedviT, adamianis qceva, Carevis efeqtisagan damoukideblad, stabiluria damoukidebeli cvladis yvela doneze. mosalodnelia, rom es daSveba dairRveva bihevioristuli mecnierebis kvlevis mraval SemTxvevaSi. aseTi darRvevis gavlena isaa, rom mniSvnelovnobis donis mixedviT dadgenili mniSvnelobis zeviT izrdeba I gvaris Secdomis daSvebis albaToba. Secdomis am gazrdil SesaZleblobasTan gamklavebis ramdenime midgoma arsebobs. gaiser-grinhausis (Geisser-Greenhouse) koreqcia cvlis Fkritikulis misaRebad gamoyenebul df-s ise, rom Frealuris gamosaTvlelad gamoiyeneba ufro didi Fkritikuli. am koreqciiT, Fkritikuli miiReba mricxvelad 1 df-is da mniSvnelad (nA - 1) df -is gamoyenebiT, Cveulebriv gamoyenebuli a – 1, (a - 1)( nA - 1) df-is nacvlad. gaiser-grinhausis koreqciis Cvens magaliTze misadagebiT, Fkritikuli (1.4) = 7.71. Frealuri (3.12) = 111.95 mainc ufro didia, vidre Fkritikuli da H0 uaryofilia – miRebulia igive gadawyvetileba, rac 3 da 12 df-iani Fkritikulis gamoyenebiT. es da sxva midgomebi, romlebic amcirebs Secdomis SesaZleblobas, Tu darRveulia me-5 daSveba, ganxilulia kepelis naSromSi (Keppel, 1991).

16

alternatiuli midgomaa, gamoviyenoT araparametruli uilkoqsonis niSniani rangebis kriteriumi an fridmanis dispersiuli analizi cdis pirTa Sida dispersiuli analizis magivrad. uilkoqsonis kriteriumi warmodgenilia me-15 TavSi. SeamowmeT Tqveni codna XII.2

1. gansazRvreT TiToeul MS-ze gavlenis mqone faqtorebi cdis pirTa Sida erTfaqtorian dispersiul analizSi.

2. axseniT, Tu ratom izrdeba MSA-s mniSvneloba, Tu A faqtors aqvs efeqti cdis pirTa Sida erTfaqtorian eqsperimentSi.

3. dawereT statistikuri hipoTezebi cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis damoukidebeli cvladis xuTi doniT.

4. gansazRvreT daSvebebi, romlebic safuZvlad udevs cdis pirTa Sida erTfaqtoriani dispersiuli analizis gamoyenebas.

magaliTi # 12.1 cdis pirTa Sida erTfaqtoriani sqemis analizi damoukidebeli cvladis oTxi doniT amocana: mravali davalebis Sesruleba moiTxovs adamianebisagan, fxizlad iyvnen drois xangrZlivi periodis ganmavlobaSi. amis magaliTia sahaero moZraobis kontroliori, romelic radaris ekrans akvirdeba, raTa TviTmfrinavebs Soris saWiro distancia SenarCundes. ase, fsiqologebma Seiswavles sifxizleze moqmedi mravali faqtori. magaliTad, pfendlerma da uidelma (Pfendler & Widdel, 1986) Caatares eqsperimenti, sadac cdis pirebi 2,5 saaTis ganmavlobaSi akvirdebodnen ekrans masze gamosaxuli skaliT. rac ufro meti dro gadioda, cdis pirebs ufro didi dro sWirdebodaT ekranze cvlilebebis dasadgenad. dauSviT, rom Catarda analogiuri kvleva, sadac cdis pirebs 4 saaTis ganmavlobaSi ramdenime skala unda ekontrolebinaT. cdis piris davaleba iyo, mieTiTebina, rodis Seicvleboda monacemi skalaze. damokidebuli cvladi zomavda im drois xangrZlivobas, romelic cdis pirs cvlilebis SesamCnevad sWirdeboda. cvlilebis SemCnevis dro (wamebSi) 13 cdis pirisaTvis oTxsaaTiani periodidan TiToeuli saaTisaTvis mocemulia cxrilSi # 12.7. mniSvnelovnad gaizarda Tu ara SemCnevis dro? amoxsna. sqema warmoadgens cdis pirTa Sida erTfaqtorian sqemas damoukidebeli cvladis – drois periodis - oTxi doniT. monacemTa analizis pirveli nabijia cdis qulebis pirTa Sida erTfaqtoriani dispersiuli analizi. Tu Frealuri statistikurad mniSvnelovania, gamoviyenoT tukis HSD kriteriumi, raTa vipovoT statistikurad mniSvnelovani wyvilTaSorisi sxvaobebi pirobebis saSualoebs Soris. saWiro statistika: F = MSA/MSAxS gamosayenebeli daSvebebi:

1. TiToeuli cdis piri mowmdeba drois yovel periodSi. 2. SerCevis qulebi SemTxveviTad aris aRebuli populaciidan.

17

3. SemCnevis droebis populacia gansxvavebuli drois periodebisaTvis normaluradaa ganawilebuli.

4. SemCnevis droebis populaciis dispersiebi gansxvavebuli drois periodebisaTvis erTnairia.

5. cdis piris individualuri gansxvavebebis wvlili igive rCeba misi qulebisaTvis drois yvela periodisaTvis.

cxrili # 12.7 skalaze cvlilebis SesamCnevad saWiro wamebis saSualo raodenoba oTxsaaTiani drois periodisaTvis 13 cdis pirisaTvis. cdis piri drois periodi (saaTebSi) (A)

________________________________________________ 1 (A1 ) 2 (A2 ) 3 (A3) 4 (A4)

1 2 3 4 5 6 7 8 9 10 11 12 13

6.2 5.9 8.4 7.6 4.1 5.4 6.6 6.1 4.9 8.2 5.7 5.9 6.9

6.7 4.8 8.7 7.8 4.7 5.3 6.7 5.8 5.1 8.6 5.7 6.4 6.6

7.4 6.1 9.9 8.7 5.4 5.9 7.2 6.4 5.2 9.3 6.5 6.9 7.1

7.8 6.9 10.3 8.9 6.6 7.1 7.5 6.7 6.8 10.4 7.2 7.6 7.5

AX sA

6.30 1.24

6.38 1.34

7.08 1.45

7.79 1.28


H0: µ A1 = µ A2 = µ A3 = µ A4 H1: µ A-ebi ar aris toli,

mniSvnelovnobis done: α = .05. df: dfA = a – 1 = 4 – 1 = 3 dfS = nA – 1 = 13 – 1 = 12 dfAxS = (a – 1) (nA – 1) = (4 – 1) (13 – 1) = 36 dfsruli = N – 1 = 52 – 1 = 51 F-is kritikuli mniSvneloba: Fkritikuli (3.36) = 2.92. df-is mniSvneloba 36, rogorc mniSvneli, ar aris warmodgenili cxrilSi # A-3a. dfSecdomisaTvis gamoviyeneT kritikuli df-is uaxloesi Semdgomi mniSvneloba cxrilSi – 30. uaryofis are: Frealuris mniSvnelobebi tolia an meti 2.92-ze. gamoTvla: Tavis damatebaSi mocemuli formulebis gamoyenebiT gamoviTvaleT dispersiuli analizi, raTa migveRo dispersiuli analizis Semdegi cxrili:

18

wyaro SS df MS F drois periodi (A) cdis pirebi (S) AxS sul

18.985 81.188 3.727 103.900

3 12 36 51

6.328 6.766 0.104

60.85*

* p < .05

gadawyvetileba: Frealuri = 60.85 xvdeba uaryofis areSi; uarvyofT H0: µ A1

= µ A2 = µ A3 = µ A3 da viRebT H1: µ A-ebi ar aris toli. H0–is uaryofa gviCvenebs, rom arsebobs sul mcire erTi statistikurad mniSvnelovani sxvaoba Carevis pirobebis saSualoebs Soris. statistikurad mniSvnelovani wyvilTaSorisi sxvaobebis sapovnelad viyenebT tukis HSD kriteriums. tukis kriteriumi: am kriteriumisaTvis saWiro ricxviTi mniSvnelobebi Semdegia:

• q miiReba cxrilidan # A-4. gvaqvs damoukidebeli cvladis oTxi done da 36 df MSAxS-isaTvis. radganac 36 df ar aris mocemuli cxrilSi # A-4, viyenebT 30df-s, df-is uaxloes Semdgom mniSvnelobas cxrilSi. ase rom, q = 3.85 .05 mniSvnelovnobis donisaTvis;

• dispersiuli analizidan miRebuli MSAxS aris 0.104;

• nA, qulebis raodenoba TiToeul Sesadarebel saSualoSi, aris 13. formulaSi am ricxviTi mniSvnelobebis CasmiT

CD = qA

AxS

nMS = 3.85

13104.0 = 0.34 wami.

nebismieri sxvaoba, romelic absoluturi mniSvnelobiT 0.34 wamis tolia an metia, statistikurad mniSvnelovania .05 doneze. wyvilTaSorisi Sedarebebi, maTi absoluturi mniSvnelobebi, Semowmebuli statistikuri hipoTezebi da statistikur hipoTezebze damyarebuli gadawyvetilebebi TiToeuli SedarebisaTvis mocemulia cxrilSi # 12.8 (sadac A1 = 1 saaTs, A2 = 2 saaTs, A3 = 3 saaTs da A4 = 4saaTs). efeqtis siZliere:

η2 = AxSA

A

SSSSSS+

= 727.3985.18

985.18+

= .84.

cxrili # 12.8 tukis HSD CD-s gamoyeneba oTxi drois periodis wyvilTaSoris SedarebaSi. Sedareba Sedarebis

absoluturi mniSvneloba


gadawyvetileba

1AX 2AX -Tan

(6.30 – 6.38)

0.08

H0: µ A1 = µ A2 H1: µ A1 ≠ µ A2

ar aris uaryofili ar aris miRebuli

19

1AX 3AX -Tan

(6.30 – 7.08)

1AX 4AX -Tan

(6.30 – 7.79)

2AX 3AX -Tan

(6.38 – 7.08)

2AX 4AX -Tan

(6.38 – 7.79)

3AX 4AX -Tan

(7.08 – 7.79)

0.78 1.49 0.70 1.41 0.71

H0: µ A1 = µ A3 H1: µ A1 ≠ µ A3

H0: µ A1 = µ A4 H1: µ A1 ≠ µ A4

H0: µ A2 = µ A3 H1: µ A2 ≠ µ A3

H0: µ A2 = µ A3 H1: µ A2 ≠ µ A3

H0: µ A3 = µ A4 H1: µ A3 ≠ µ A4

uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia uaryofilia miRebulia

daskvna: erTaderTi Sedareba, romelic naklebia CD-ze – 0.34 wamze, aris pirvel da meore saaTebs Soris. es sxvaoba aramniSvnelovania. yvela sxva sxvaoba statistikurad mniSvnelovania. ase rom, Sedarebebma gvaCvena, rom SemCnevis droebSi ar aris zrda davalebis Sesrulebis pirvel da meore saaTebs Soris, magram statistikurad mniSvnelovani zrda aRiniSneba me-2

da me-3 saaTebs Soris da me-3 da me-4 saaTebs Soris. efeqtis siZliere, η2 = .84, gviCvenebs, rom drois periodi pasuxs agebs SemCnevis drois cvalebadobis did nawilze mas Semdeg, rac individualuri gansxvavebebis cvalebadoba eqsperimentSi ugulvebelyofilia.

magaliTi # 12.2 cdis pirTa Sida erTfaqtoriani dispersiuli analizis gamoyeneba damoukidebeli cvladis ori doniT amocana: rodesac studenti leqciis dros amTqnarebs, leqtori amas xSirad mowyenis niSnad aRiqvams. magram aris Tu ara mTqnareba namdvilad amis niSani? am hipoTezis gamosakvlevad provinma da hamernikma (Provine & Hamernik, 1986) Caatares eqsperimenti imis Sesamowmeblad, namdvilad zrdis Tu ara mTqnarebis sixSires uintereso masalis swavla. am eqsperimentSi ori sxvadasxva sesiis dros studentebi 30 wuTis ganmavlobaSi uyurebdnen rok-musikis videoklipebs (A1, saintereso stimuli) da 30 wuTis ganmavlobaSi – vertikalur ferad-zolebian ekrans (A2, uintereso stimuli). aRiricxeboda damTqnarebis raodenoba ori sesiis dros. davuSvaT, es eqsperimenti 11 cdis pirze gavimeoreT da aRvricxeT damTqnarebis raodenoba cxrilSi # 12.9. mocemul TiToeul sesiaSi moqmedebs Tu ara videoklipis tipi mTqnarebis raodenobaze? cxrili # 12.9 mTqnarebis miRebuli raodenoba, rogorc nanaxi videoklipis funqcia 11 cdis pirisaTvis.

20

cdis piri videoklipis tipi (A) _______________________________________ saintereso (A1 ) uintereso (A2)

1 2 3 4 5 6 7 8 9 10 11

5 2 4 3 0 4 7 6 3 1 8

7 1 7 8 2 5 6 9 3 4 9

AX sA

3.9 2.47

5.5 2.77

amoxsna. cdis pirTa Sida erTfaqtoriani dispersiuli analizi Sesabamisi statistikuri kriteriumia am monacemebisaTvis. ramdenadac gamoyenebuli iyo mxolod ori Carevis piroba, tdamokidebuli aseve Sesabamisi statistikuri kriteriumi iqneboda. viyenebT .05 mniSvnelovnobis dones. saWiro statistika: F = MSA/MSAxS gamosayenebeli daSvebebi:

1. TiToeuli cdis piri mowmdeba videoklipis orive tipis pirobaSi;

2. SerCevis monacemebi SemTxveviTad aris aRebuli populaciidan; 3. mTqnarebis monacemebis populacia Carevis gansxvavebuli

pirobisaTvis normaluradaa ganawilebuli; 4. mTqnarebis monacemebis dispersiebi populaciaSi Carevis

gansxvavebuli pirobebisaTvis erTnairia; 5. cdis piris individualuri gansxvavebebis wvlili igive rCeba

misi qulebisaTvis videoklipis orive tipis pirobaSi. statistikuri hipoTezebi:

H0: µ A1 = µ A2 H1: µ A-ebi ar aris toli,

mniSvnelovnobis done: α = .05. df: dfA = a – 1 = 2 – 1 = 1 dfS = nA – 1 = 11 – 1 = 10 dfAxS = (a – 1) (nA – 1) = (2 – 1) (11 – 1) = 10 dfsruli = N – 1 = 22 – 1 = 21 F-is kritikuli mniSvneloba: Fkritikuli (1,10) = 4.96. uaryofis are: Frealuris mniSvnelobebi tolia an meti 4.96-ze.

21

gamoTvla: am Tavis damatebaSi mocemuli formulebis gamoyenebiT, am analizis Sejameba aseTia: wyaro SS df MS F videoklipis tipi (A) cdis pirebi (S) AxS sul

14.728 120.364 17,272 152.364

1 10 10 21

14.728 12.036 1.727

8.53*

* p < .05

gadawyvetileba: Frealuri xvdeba uaryofis areSi; amitom uarvyofT H0-s da viRebT H1-s. efeqtis siZliere:

η2 = 272.17728.14

728.14+

= .46.

daskvna: sxvaoba ori Carevis pirobis saSualos Soris – 3.9 da 5.5 damTqnarebas Soris – statistikurad mniSvnelovania. rodesac saintereso videoklipebs uyurebdnen, cdis pirebi naklebad amTqnarebdnen, vidre uintereso videoklipebis yurebisas. radganac mxolod ori saSualoa CarTuli, mravlobiTi Sedarebis kriteriumi ar

aris saWiro. η2 miuTiTebs, rom damokidebuli cvladis dispersiis 46 procentze pasuxs agebs videoklipis tipi, mas Semdeg, rac eqsperimentidan amoRebulia SSS.

dispersiuli analizis Sedegebis prezentacia iluziis pirobasa da xazis aRqmul sigrZeze (ix. cxrilebi # 12.2 da 12.3) moyvanili magaliTis analizis Sedegebis aRwera, romelic misdevs amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils, statiis Sedegebis seqciaSi ase SeiZleba daiweros:

xazis aRqmuli sigrZeebis saSualoebi iluziis standartuli, subieqturi konturebis, mTliani wertilebis da sakontrolo pirobebisaTvis aRmoCnda 16.92 (SD = 0.24), 15.90 (SD = 0.29), 15.02 (SD = 0.26) da 14.10 (SD = 0.25) sm, Sesabamisad. alfas .05 doneze, cdis pirTa Sida erTfaqtorianma dispersiulma analizma gviCvena, rom saSualoebi mniSvnelovnad gansxvavdeba, F (3,12) = 111.95, MSE = 0.065, p < .01. wyvilTaSorisi Sedarebebisas tukis kriteriumma aCvena, rom iluziis TiToeuli piroba mniSvnelovnad gansxvavdeboda iluziis yoveli sxva pirobisagan, p < .05 (CD = 0.48 sm). eta kvadrati qulebisaTvis iyo .97.

am angariSSi informaciis didi nawili cdis pirTa Soris sqemis msgavsia. mokled ganvixiloT es prezentacia.

SerCevis saSualo SerCevis saSualoebis da standartuli gadaxrebis miRebuli mniSvnelobebi warmodgenilia pirvel winadadebaSi.

22

radganac miRebuli iyo mxolod oTxi saSualo, isini warmodgenilia winadadebis da ara cxrilis an suraTis saxiT.

α = .05 miuTiTebs kriteriumisaTvis SerCeul mniSvnelovnobis dones.

F (3,12) aRniSnavs kriteriumis statistikis saxels,

rogorc F-s. frCxilebSi naCvenebi ricxvebi aRniSnavs Frealuris mricxvelis da mniSvnelos df-ebs (Sesabamisad, 3 da 12).

= 111.95 iZleva Frealuris mniSvnelobas. p < .01 aRniSnavs, rom:

a. Frealuris albaToba, Tu H0 marTldeba, naklebia an tolia .01-is. es mniSvneloba aris Frealuris albaToba an Frealuris kidev ufro ukiduresi mniSvneloba, Tu H0marTalia.

b. H0: µ A1 = µ A2 = µ A3 = µ A4 uaryofilia.

H1: µ A-ebi ar aris toli, miRebulia. g. arsebobs sul mcire erTi statistikurad mniSvnelovani sxvaoba pirobebis saSualoebs Soris.

d. SerCevis Secdomis garda kidev sxva mizezi agebs pasuxs pirobebis saSualoebs Soris sxvaobaze.

MSE = 0.065 gvaZlevs MSAxS mniSvnelobas FrealurisaTvis. es

MS aris Secdomis nawili F SefardebisaTvis; agreTve, is SeiZleba ganisazRvros, rogorc MSAxS.

CD = 0.48 sm gvaZlevs kritikul sxvaobas wyvilTaSorisi

SedarebebisaTvis tukis HSD kriteriumis gamoyenebiT. Semdeg aRiwereba statistikurad mniSvnelovani wyvilTaSorisi Sedarebebi.

η2 = .97 gviCvenebs, rom xazis aRqmuli Tanabari sigrZeebis dispersiis 97 procenti am eqsperimentSi iluziis pirobis gamo xdeba, mas Semdeg, rac individualuri gansxvavebebis gamo arsebuli cvalebadoba gauqmebulia.

kvlevis sqemis arCeva: cdis pirebs Soris Tu cdis pirebs Sida rogor unda gadawyvitos bihevioristma mecnierma, romeli kvlevis sqema gamoiyenos kvlevis hipoTezis Sesafaseblad – cdis pirebs Soris Tu

23

cdis pirebs Sida? sqemis TiToeul tips aqvs upiratesobebi da SezRudvebi. zogadad, am gadawyvetilebis miRebisas sami sakiTxi dgeba xolme: (1) cdis pirTa saWiro raodenoba, (2) damoukidebeli cvladis mosalodneli efeqturoba da (3) Carevis mravaljeradi efeqtebis miRebis albaToba. cdis pirTa saWiro raodenoba cdis pirTa Sida sqemaSi cdis pirs Carevis yovel pirobaSi zomaven; ase rom, aq naklebi cdis pirebia saWiro, vidre cdis pirTa Sida sqemaSi. cxrilSi # 12.2 naCveneb magaliTSi iluziis pirobis oTxi donidan TiToeulisaTvis miRebulia xuTi qula, Sesabamisad, sul 20 qulaa. magram mxolod xuTi cdis piria saWiro am 20 qulis misaRebad. cdis pirTa Soris sqemaSi am 20 qulis misaRebad saWiro iqneboda 20 sxvadasxva cdis piri. ase rom, rodesac cdis pirebis moZieba Znelia an rodesac cdis pirebis moZieba da maTTvis instruqciebis micema bevrad met dros moiTxovs, vidre TviTon eqsperimentuli davalebis Sesruleba, cdis pirTa Sida sqema SeiZleba upiratesi iyos cdis pirTa Soris sqemasTan SedarebiT cdis pirebis gamoyenebis TvalsazrisiT. damoukidebeli cvladis efeqturoba is, Tu ramdenad advilad xdeba eqsperimentSi damoukidebeli cvladis efeqtis dadgena, nawilobriv damokidebulia am cvladis efeqtis siZliereze. cdis pirTa Sida sqema xSirad ufro mgrZnobiarea damoukidebeli cvladis efeqtis aRmoCenis mxriv, vidre cdis pirTa Soris sqema. amdenad, cdis pirTa Sida sqemas SeiZleba upiratesoba mieniWos, rodesac damoukidebeli cvladis mosalodneli efeqti sustia. cdis pirTa Sida sqema metad mgrZnobiarea imis gamo, rom TiToeuli cdis piri iRebs damoukidebeli cvladis yovel dones da xvdeba Carevis yvela pirobaSi. cdis pirTa Sida dispersiuli analizi gamoacalkevebs cvalebadobas cdis pirTa Soris individualuri gansxvavebebis gamo (anu S faqtoris gamo cvalebadobas - MSS), ase rom am ukanasknels ar Seaqvs wvlili F-is Secdomis nawilSi (anu MSAxS). cdis pirTa Soris sqemaSi es cvalebadoba Sedis MSSecdomaSi. ase rom, Secdomis veqtori cdis prTa Sida sqemaSi (anu MSAxS) ufro mcire iqneba, vidre Secdomis veqtori cdis pirTa Soris sqemaSi (anu MSSecdoma). ufro mcire Secdomis veqtori iwvevs Frealuris ufro did mniSvnelobas, ris Sedegadac ufro mosalodnelia, rom H0-s uaryofa miviRoT. ase rom, eqsperimentSi, sadac cdis pirTa Soris didi individualuri gansxvavebebia da damoukidebeli cvladis efeqti ar unda iyos Zlieri, cdis pirTa Sida sqema ufro kargad gamoavlens damoukidebeli cvladis efeqts, vidre cdis pirTa Soris sqema. Carevis mravaljeradi efeqtebi cdis pirTa Sida sqemaSi cdis pirs damoukidebeli cvladis yovel doneze zomaven drois garkveuli periodis ganmavlobaSi. Carevebis es Tanmimdevroba cdis pirTa Sida sqemaSi Carevis mravaljeradi efeqtebis moxdenis SesaZleblobas qmnis; damoukidebeli cvladis donis efeqti SeiZleba damokidebuli iyos mis winamorbed doneebze. Carevis mravaljeradi efeqtebi moicavs rogorc gavarjiSebis, aseve Carevis gadatanis efeqtebs.

24

gavarjiSebis efeqtebi aris Sesrulebis cvlilebebi, romlebic imis gamo xdeba, rom eqsperimentuli davaleba ganmeorebiT sruldeba. am cvlilebebma SeiZleba gamoiwvios an gaumjobesebuli, an gauaresebuli Sesruleba eqsperimentis msvlelobisas. magaliTad, eqsperimentis magaliTSi, sadac izomeba iluziis xarisxi iluziis oTx gansxvavebul pirobaSi, cdis pirebs SeiZleba mobezrdeT eqsperimenti misi msvlelobisas an SeiZleba isini, rac ufro met Sefasebebs akeTeben, miT ufro skrupulozurni xdebian TavianT SefasebebSi. orive SemTxvevaSi cdis piris Sesrulebam SeiZleba Secvalos, moqmedebs Tu ara qcevaze iluziis piroba. Carevis gadatanis efeqtebi aris damoukidebeli cvladis erTi donis specifikuri efeqtebi, romlebic SeiZleba ise iqnen gadatanili, rom gavlena moaxdinon Sesrulebaze, damoukidebeli cvladis Semdgom doneze. magaliTad, amocanaSi, sadac iluziebis aRqma ganixileba, rogorc iluziis pirobis funqcia, Tu cdis pirebi sakontrolo iluzias yovelTvis bolo rigSi xedavdnen, maT SeiZleba aReqvaT iluzia, radgan sakontrolo pirobis win isini xedavdnen iluziis Semqmnel pirobebs. maT aqvT iluziis danaxvis molodini, romelic gadaitaneba sakontrolo pirobaze. Tu isini sakontrolo pirobas pirvel rigSi naxavdnen iluziis danaxvis molodinis gareSe, isini mas ar aRiqvamdnen rogorc iluzias. gavarjiSebis efeqtebi da Carevis gadatanis efeqtebi mniSvnelovania, radgan Tu isini Cndebian da ar arian kontrolirebadi, isini arTuleben eqsperiments. am efeqtebis kontroli Cveulebriv xorcieldeba cdis pirebisaTvis Carevis pirobebis sxvadasxva TanmimdevrobiT miwodebis gziT. ganxiluli sakiTxebidan – cdis pirebis saWiro raodenoba, damoukidebeli cvladis mosalodneli efeqturoba da Carevis mravaljeradi efeqtebis miRebis albaToba – TiToeuli unda miviRoT mxedvelobaSi, rodesac kvlevis sqemas virCevT. cdis pirebis efeqturad gamoyeneba da ufro meti mgrZnobiaroba damoukidebeli cvladis efeqtis dasafiqsireblad cdis pirTa Sida sqemis upiratesobebia. xolo Carevis mravaljeradi efeqtebis miRebis SesaZlebloba am sqemis naklovanebaa. Carevis mravaljeradi efeqtebi ar SeiZleba moxdes cdis pirTa Soris sqemaSi, radganac yoveli cdis piri eqvemdebareba damoukidebeli cvladis mxolod erTi dones. ase rom, rodesac eqsperimentSi mosalodnelia Carevis mravaljeradi efeqtebis miReba da maTi gakontroleba ar aris advili Carevebis Tanmimdevrobis gziT, sasurvelia cdis pirTa Soris sqemis gamoyeneba. radganac Carevis mravaljeradi efeqtebi xdeba maSin, rodesac mravali damoukidebeli cvladi gvaqvs da am efeqtebis gakontroleba ar aris advili cdis pirTa Sida sqemaSi, bihevioristi mecnierebi xSirad arCeven cdis pirTa Soris sqemis gamoyenebas. Carevis mravaljeradi efeqti: cdis pirTa Soris sqemaSi Sesrulebis cvlilebebi, romlebic xdeba imis gamo, rom mowmdeba damoukidebeli cvladis yovel doneze. gavarjiSebis efeqti: Carevis mravaljeradi efeqti, romelic xdeba imis gamo, rom cdis pirebi SeiZleba gavarjiSdnen an daiRalnen eqsperimentis davalebis Sesrulebisas. Carevis gadatanis efeqti: Carevis mravaljeradi efeqti, romelic xdeba maSin, rodesac damoukidebeli cvladis erTi donis erTi efeqti gadaitaneba, gavlenas axdens Sesrulebaze damoukidebeli cvladis meore doneze.

25

SeamowmeT Tqveni codna XII.3 1. gansazRvreT: Carevis mravaljeradi efeqtebi, gavarjiSebis efeqti, Carevis gadatanis efeqti. 2. rodesac adamianebis geografiul adgilmdebareobebze fiqroben, adgilis maT mier identifikaciaze xSirad gavlenas axdens Tanadaqvemdebarebis gadaxra. es niSnavs, rom iseTi geografiuli adgilebi, rogoricaa qalaqebi, moTavsebulia ufro did geografiul arealSi, rogoricaa regioni an qveyana. ufro didi geografiuli arealebi (anu Tanadaqvemdebarebuli adgilebi) SeiZleba gavlenas axdendnen imaze, Tu rogor warmovidgenT maTSi moTavsebul adgilebs. magaliTad, kanada amerikis SeerTebuli Statebisagan CrdiloeTiT aris ganlagebuli, Tumca arsebobs zogierTi kanaduri qalaqi, romelic bevrad ufro samxreTiT mdebareobs, vidre zogierTi aSS-s qalaqi. magram adamianebis umetesoba CaTvlida, rom aSS-s qalaqebi ufro samxreTiTaa, ubralod imis gamo, rom SeerTebuli Statebi kanadis samxreTiT mdebareobs. marTlac, fridmanma da braunma (Friedman and Brown, 2000) aRmoaCines, rom adamianebi Tvlidnen, rom vaSingtonSi mdebare sietli (SeerTebuli Statebi) bevrad ufro samxreTiTaa, vidre niufaundlendSi mdebare sent-jonsi (kanada), Tumca orive qalaqi erTsa da imave ganedzea. davuSvaT, Tqven gainteresebT es sakiTxi da sTxoveT cdis pirebs, daesaxelebinaT sami arseboTad erT ganedze ganlagebuli qalaqis ganedebi. es qalaqebia dalasi, texasis Stati (aSS), tihuana (meqsika) da tripoli (libia). yovel cdis pirebs eZleva instruqcia da utardeba ganedis dadgenis mcire savarjiSo, amis Semdeg ki sTxoven, daasaxelos yoveli qalaqis ganedi. miRebuli iyo Semdegi monacemebi: ' cdis piri

qalaqi (A) dalasi (A1) tihuana (A2) tripoli (A3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

33 29 35 28 30 40 32 31 30 25 27 36 26 29

18 27 21 17 18 26 25 19 23 20 20 24 22 23

20 24 26 19 27 23 25 24 17 21 20 23 18 20

ipoveT saSualo da standartuli gadaxra Carevis TiToeuli pirobisaTvis. Semdeg am qulebze CaatareT cdis pirTa Sida erTfaqtoriani dispersiuli analizi imis gansasazRvrad, arsebobs Tu ara Tanadaqvemdebarebis gadaxra ganedebis dasaxelebisas da upasuxeT Semdeg kiTxvebs: a. CamoayalibeT F kritetiumis statistikuri hipoTezebi am qulebisaTvis.

b. ra aris Frealuris mniSvneloba? g. ra aris Frealuris df-ebi? d. ra aris Fkritikuli .05 doneze? e. ra aris uaryofis are FrealurisaTvis? v. xvdeba Tu ara Frealuri uaryofis areSi?

z. ra gadawyvetilebebs miiRebT statistikuri hipoTezebis Sesaxeb? uaryofT Tu ar uaryofT H0-s? miiRebT Tu ar miiRebT H1-s? T. ras daaskvniT, saSualoebi erTi da igive Tu sxvadasxva populaciebidanaa? i. arsebobs Tu ara statistikurad mniSvnelovani sxvaoba sul mcire saSualoebis erT wyvils Soris? k. saWiroa Tu ara am eqsperimentSi mravlobiTi Sedarebis kriteriumebi? daasabuTeT Tqveni pasuxi.

l. ra aris tukis CD-s mniSvneloba? m. romeli wyvilTaSorisi sxvaobebia statistikurad mniSvnelovani?

26

n. ra aris η2 am eqsperimentisaTvis? o. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic ilustrirebuli iyo am Tavis seqciaSi "dispersiuli analizis Sedegebis prezentacia".

savarjiSoebi # 3 da 4 warmoadgens cdis pirTa Sida erTfaqtoriani dispersiuli analizis angariSis nimuSebs. TiToeuli savarjiSosaTvis upasuxeT Semdeg kiTxvebs: a. damoukidebeli cvladis ramdeni done iyo gamoyenebuli eqsperimentSi? b. ramdeni cdis piri iRebda monawileobas eqsperimentSi? g. ra aris Frealuris mniSvneloba? d. ra aris Fkritikulis mniSvneloba .05 mniSvnelovnobis doneze? e. uaryofT Tu ar uaryofT H0-s? v. miiRebT Tu ar miiRebT H1-s?

z. arsebobs Tu ara sul mcire erTi statistikurad mniSvnelovani sxvaoba saSualoebs Soris? T. romeli Carevis pirobebi gansxvavdeba mniSvnelovnad, Tuki aseTebi saerTod arseboben?

1. saSualo reaqciis dro martivi stimulis pirobaSi 396 miliwami (mw) iyo, maSin, rodesac rTuli stimulis pirobaSi - 412 mw. alfas .05 doneze cdis pirTa Sida erTfaqtorianma dispersiulma analizma aCvena saSualoebs Soris mniSvnelovani sxvaoba, F (1,18) = 4.76, MSE = 510.924, p < .05.

2. sworad amocnobili saxeebis saSualo raodenoba 13.9 iyo pirvel cdaSi, 14.6 - meore cdaSi da 14.1 - mesame cdaSi. alfas .05 doneze cdis pirTa Sida erTfaqtorianma dispersiulma analizma ar aCvena sxvaoba cdebSi sworad amocnobis raodenobebs Soris, F (2,22) = 2.81, MSE = 0.555, p > .05.

Sejameba

• cdis pirTa Sida erTfaqtorian sqemaSi cdis pirTa erTi jgufi ganicdis damoukidebeli cvladis yoveli donis zemoqmedebas.

• cdis pirTa Sida erTfaqtoriani dispersiuli analizi gamoiyeneba cdis pirTa Sida erTfaqtoriani mravaldoniani sqemis qulebis gasaanalizeblad.

• cdis pirTa Sida erTfaqtoriani dispersiuli analizidan miiReba sami kvadratebis saSualo: MSA, romelic icvleba damoukidebeli cvladis efeqtTan erTad da Carevis pirobebis da cdis pirTa urTierTqmedeba MSS, romelic icvleba cdis pirTa Soris gansxvavebebTan erTad, da MSAxS, romelic icvleba pirobebis cdis pirebTan urTierTqmedebasTan erTad da iZleva eqsperimentSi Secdomis cvalebadobis sazoms.

• F statistika, romelic gamoiyeneba cdis pirTa Sida erTfaqtorian dispersiul analizSi, aris F = MSA/ MSAxS.

• F statistikis gamoyeneba misdevs statistikuri hipoTezebis Camoyalibebis Cveulebriv nabijebs, rogorebicaa mniSvnelovnobis donis dadgena, uaryofis aris dadgena, Frealuris gamoTvla da statistikur hipoTezebTan dakavSirebuli gadawyvetilebebis miReba.

• rodesac damoukidebel cvlads sami an meti done aqvs, statistikurad mniSvnelovani F moiTxovs mravlobiTi Sedarebis kriteriumis gamoyenebas, raTa dadgindes statistikurad mniSvnelovani wyvilTaSorisi Sedarebebi.

27

• tukis kriteriumi gamoiyeneba yvela SesaZlebeli wyvilTaSorisi saboloo SedarebisaTvis.

• η2 SeiZleba gamoyenebuli iqnes, rogorc Carevis efeqtis siZlieris sazomi.

• cdis pirTa Soris da cdis pirTa Sida sqemebs Soris arCevanis gakeTeba moiTxovs Semdegi sakiTxebis ganxilvas: cdis pirTa saWiro raodenoba, damoukidebeli cvladis mosalodneli efeqturoba, da Carevis mravaljeradi efeqtebis miRebis SesaZlebloba.

------------------------------------------------------------------------------------------------------------------------------ ZiriTadi terminebi da simboloebi

kritikuli sxvaoba (CD) dfA dfS dfAxS dfsruli eta kvadrati (η2) A faqtori S faqtori MSA MSS MSAxS Carevis mravaljeradi efeqtebi

nA gavarjiSebis efeqti ganmeorebiTi gazomvis sqema SSA SSS SSAxS SSsruli Carevis gadatanis efeqti Carevis pirobebis da cdis pirebis urTierTqmedebis sqema Carevis pirobebis da cdis pirebis urTierTqmedeba (AxS) cdis pirTa Sida sqema

__________________________________________________________________ Tavis damateba cdis pirTa Soris orfaqtoriani dispersiuli analizis gamoTvla formulebis meSveobiT __________________________________________________________________ am damatebaSi mocemulia gamoTvliTi formati cdis pirTa Sida erTfaqtoriani dispersiuli analizisaTvis. eqsperimenti A damoukidebeli cvladis oTxi doniT da xuTi cdis piriT yovel doneze qvemoTaa warmodgenili: cdis piri A faqtori

________________________________________________ A1 A2 A3 A4

S1 S2 S3 S4 S5

X11 X21 X31 X41 X51

X12 X22 X32 X42 X52

X13 X23 X33 X43 X53

X14 X24 X34 X44 X54

1ST

2ST

3ST

4ST

5ST

1AT 2AT 3AT 4AT G

28

sadac Xij = cdis piris qula Ta = qulebis jami damoukidebeli cvladis donisaTvis TS = qulebis jami cdis pirisaTvis G = qulebis saerTo sruli jami nA = qulebis jami damoukidebeli cvladis doneze a = damoukidebeli cvladis doneebis raodenoba N = qulebis sruli raodenoba. am aRniSvnebis gamoyenebiT miiReba oTxi ricxviTi mniSvneloba:

[1] = ∑∑ 2ijX yvela qulis kvadratebis jami

[2] = A

A

nT∑ 2

damoukidebeli cvladis TiToeuli donis kvadratebis

jami, gayofili cdis pirebis raodenobaze.

[3] = aTS∑ 2

TiToeuli cdis piris qulebis kvadratebis jami,

gayofili A faqtoris doneebis raodenobaze.

[4] = NG2

saerTo sruli jamis kvadrati, gayofili qulebis srul

raodenobaze. am ricxviTi mniSvnelobebis gamoyenebiT miiReba Semdegi dispersiuli analizi: wyaro SS dfa MS F A faqtori cdis pirebi (S) AxS sul

[2] – [4] [3] – [4] [1] – [2] – [3] + [4] [1] – [4]

a - 1 nA - 1 (a - 1)( nA - 1) N - 1

A

A

dfSS

S

S

dfSS

AxS

AxS

dfSS

ar gamoiTvleba

AxS

A

MSMS

am gamoTvlebis ilustraciisaTvis gamoviyenoT cxrilSi # 12.2 moyvanili monacemebi, romelTa dispersiuli analizi Sejamebulia cxrilSi # 12.3. cdis piri

A faqtori cdis piris jamebi A1 A2 A3 A4

1 16.6 15.8 14.7 14.4 1ST = 61.50

2 16.9 15.6 15.1 13.8 2ST = 61.40

3 17.1 16.1 14.8 13.9 3ST = 61.90

29

4 17.2 16.3 15.2 14.1 4ST = 62.80

5 16.8 15.7 15.3 14.3 5ST = 62.10

AT = 84.60 79.50 75.10 70.50 G = 309.70

nA = 5, a = 4 da N = 20. ricxviTi mniSvnelobebia: [1] = [(16.6)2 + ... + (16.8)2 + (15.8)2 + ... + (15.7)2 + (14.7)2 + ... (15.3)2 + (14.4)2 + ... +

(14.3)2] = 4818.630

[2] = 5

)50.70()10.75()50.79()60.84( 2222 +++ = 4817.534

[3] = 4

)10.62()80.62()90.61()40.61()50.61( 22222 ++++ = 4796.018

[4] = 20

)70.309( 2

= 4795.704

Semdeg SSA = [2] – [4] = 4817.534 – 4795.704 = 21.830 dfA = a – 1 = 4 – 1 = 3

MSA = 3830.21

= 7.277

SSS = [3] – [4] = 4796.018 – 4795.704 = 0.314 dfS = nA – 1 = 5 – 1 = 4

MSS = 4314.0

= 0.078

SSAxS = [1] – [2] – [3] + [4] = 4818.630 - 4817.534 - 4796.018 + 4795.704 = 0.782 dfAxS = (a – 1)(nA – 1) = (4 - 1)(5 – 1) = 12

MSAxS = 12782.0

= 0.065

SSsruli = [1] – [4] = 4817.534 – 4795.704 = 22.926 dfsruli = N – 1 = 20 – 1 = 19

Frealuri = AxS

A

MSMS

= 065.0277.7

= 111.95

dispersiuli analizis Sejamebaa:

wyaro SS df MS F A faqtori cdis pirebi AxS sul

21.830 0.314 0.782 22.926

3 4 12 19

7.277 0.078 0.065

111.95*

30

* p < .01

am gamoTvliTi midgomis Sedegad miRebuli mniSvnelobebi identuria im mniSvnelobebisa, romelic warmodgenilia cxrilSi # 12.3 dispersiuli analizis ganmartebiTi formulebidan. __________________________________________________________________

kiTxvebi ________________________________________________________

1. davuSvaT, gainteresebT, moqmedebs Tu ara sityviT (mxiaruli, neitraluri an sevdiani) gamoxatuli xasiaTis tipi imaze, Tu ramdenad kargad xdeba mocemuli siidan sityvis damaxsovreba. SeadgenT eqvsi samxiarulo sityvisagan (magaliTad, mxiaruli, gaciskrovnebuli), eqvsi neitraluri sityvisagan (magaliTad, gamoyvana, gadacema) da eqvsi sevdiani sityvisagan (magaliTad, daRvremili, martosuli) Semdgar sias. Semdeg rva cdis piridan TiToeuli Seiswavlis sias, sanam orjer sruli sizustiT ar gaimeorebs mas. erTi kviris Semdeg TiToeuli cdis piri cdilobs siis gaxsenebas. sworad aRdgenili erTeulebis raodenobebi, rogorc sityvis tipis funqcia, ase gamoiyureba:

cdis piri

sityvis tipi bednieri neitraluri mowyenili

1 2 3 4 5 6 7 8

5 6 4 5 3 6 2 5

4 3 5 3 1 3 2 3

3 4 2 1 2 4 3 1

a. sityvis pirobis TiToeuli tipisaTvis ipoveT saSualo da

standartuli gadaxra. Semdeg gaaanalizeT qulebi cdis pirTa Sida erTfaqtoriani analizis meSveobiT, raTa upasuxoT Semdeg kiTxvebs: moqmedebs Tu ara sityvis tipi aRdgenaze? gamoiyeneT .05 mniSvnelovnobis done. Tu saWiroa, gamoiyeneT tukis HSD kriteriumi mravlobiTi SedarebisaTvis.

b. ra aris damoukidebeli cvladis - sityvis tipis - efeqtis siZliere?

g. aRwereT Sedegebi im stilis gamoyenebiT, romelic ilustrirebuli iyo am Tavis seqciaSi "dispersiuli analizis Sedegebis prezentacia".

2. sanimuSo magaliTSi 10.3 warmoadgenili iyo cdis pirTa Sida sqemis

mixedviT agebuli morinagas (Morinaga) SeuTavseblobis iluziis kvleva [ix. suraTi # 10.3, paneli (a)]. davuSvaT, eqsperimentators surs, icodes, Secvlis Tu ara cdis pirebis mier iluziis aRqmas maTi instruqtireba am iluziis Sesaxeb. Tavdapirvelad eqsperimentatorma sTxova 12 cdis pirs, SeeTavsebinaT (erT xazze ganelagebinaT) kuTxeebis wveroebi da gazoma Secdomis raodenoba yoveli adamianisTvis. mas Semdeg, rac cdis pirebma SeaTavses

31

wveroebi, maT Seatyobines iluziis Sesaxeb da kidev erTxel sTxoves kuTxeebis SeTavseba. miRebuli iqna Secdomis ori gazomva (milimetrebSi) yoveli cdis pirisaTvis:

cdis piri

instruqciis piroba (A) _______________________________________ instruqciamde (A1 ) instruqciis Semdeg (A2)

1 2 3 4 5 6 7 8 9 10 11 12

1.3 2.1 2.5 1.9 3.0 2.5 1.8 2.4 3.1 3.4 2.5 2.9

1.4 2.0 2.7 2.0 3.4 2.8 1.6 2.5 3.3 3.4 2.3 2.8

a. instruqciis TiToeuli pirobisaTvis ipoveT saSualo da

standartuli gadaxra. Semdeg gaaanalizeT qulebi cdis pirTa Sida erTfaqtoriani analizis meSveobiT, raTa upasuxoT Semdeg kiTxvebs: iqonia Tu ara instruqciam gavlena aRqmuli iluziis raodenobaze? gamoiyeneT .05 mniSvnelovnobis done.

b. SeadareT Frealuri me-9 Tavis "kiTxvebidan" meoTxeSi moyvanil

tdamokidebuls. sworia Tu ara realuriF = tdamokidebuli?

g. dispersiuli analizisa da tdamokidebuli kriteriumis gamoyenebis Sedegad erTsa da imave daskvnebs gaakeTebT damoukidebeli cvladis efeqtis Sesaxeb, Tu gansxvavebulebs?

d. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic ilustrirebuli iyo am Tavis seqciaSi "dispersiuli analizis Sedegebis prezentacia".

3. davuSvaT, gegmavT, CaataroT cdis pirTa Sida erTfaqtoriani

eqsperimenti damoukidebeli cvladis xuTi doniTa da 12 cdis piriT.

a. dawereT dispersiuli analiziT Semowmebuli statistikuri hipoTezebi im qulebisaTvis, romlebsac Tqven miiRebdiT.

b. ipoveT df am sqemis dispersiuli analizisaTvis. g. ra aris Fkritikulis mniSvneloba .05 mniSvnelovnobis doneze am

sqemis dispersiuli analizisaTvis? d. davuSvaT, rom Frealuris mniSvneloba ufro didi iyo, vidre

Fkritikulis mniSvneloba. ra gadawyvetilebebs miiRebdiT H0-Tan da H1-Tan dakavSirebiT?

e. davuSvaT, rom am eqsperimentisaTvis Tqven uaryaviT H0 da miiReT H1. ra daskvnas gaakeTebdiT am gadawyvetilebebidan Carevis saSualoebis Sesaxeb?

4. qvemoT moyvanilia dispersiuli analizis arasruli Sejamebis

cxrilebi. SS-s, df-sa da MS-s Soris arsebuli mimarTebebis

32

gamoyenebiT TiToeul cxrilSi SeitaneT gamotovebuli mniSvnelobebi. Semdeg TiToeuli cxrilisaTvis upasuxeT Semdeg kiTxvebs:

a. ramdeni done hqonda damoukidebel cvlads? b. damoukidebeli cvladis TiToeul doneze ramdeni qula iyo

miRebuli? g. ramdeni cdis piri iRebda monawileobas kvlevaSi? d. aris Tu ara Frealuris mniSvneloba statistikurad

mniSvnelovani .05 doneze? e. ra gadawyvetilebebs miiRebT H0-isa da H1-isaTvis yovel

analizSi? cxrili # 1

wyaro SS df MS F A faqtori S faqtori AxS sul

____ 138.00 253.00 ____

____ 23 46 71

11.00 6.00 ____

____

cxrili # 2


60.00 133.00 ____ 573.00

____ 19 95 ____

____ 7.00 ____

____

5. ramdenime eqsperimentatorma gamoikvlia Cvilis interesi ucnobi saxeebis mimarT nacnob saxeebTan SedarebiT. 14 Cvilidan TiToeuli aTvalierebda nacnobi saxeebis (magaliTad, dedis da mamis fotoebi), ucnobi saxeebis (magaliTad, CvilisTvis ucnobi mamakacebisa da qalebis fotoebi) da saxis zomis carieli ovalebis fotoebs. eqsperimentatorebi afiqsirebdnen dros, romlis ganmavlobaSic TiToeuli Cvili akvirdeboda saxes. cxrilSi # 1 warmodgenilia stimulis piroba da drois saSualo xangrZlivoba, romelic ixarjeboda stimulze TiToeul pirobaSi. cxrilSi # 2 warmodgenilia am monacemebis dispersiuli analizi.

cxrili # 1 dakvirvebis saSualo dro (wamebSi), rogorc stimulis pirobis funqcia.

stimulis piroba

nacnobi saxe (A1) ucnobi saxe (A2) carieli ovali (A3) 34.2 26.8 18.5

cxrili # 2 dispersiuli analizis Sejameba cxrilSi # 1 warmodgenili dakvirvebis saSualo droebisaTvis.

wyaro SS df MS F stimulis piroba (A) 1727.32 2 863.66 15.99*

33

cdis pirebi (S) AxS sul

2047.37 1404.52 5179.21

13 26 41

157.49 54.02

* p < .01

tukis HSD-m mogvca CD = 6.9 wuTi mniSvnelovnobis .05 doneze. a. ra aris stimulis pirobis efeqti dakvirvebis droze? b. aRwereT am eqsperimentis Sedegebi im stilis gamoyenebiT, romelic

ilustrirebuli iyo am Tavis seqciaSi "dispersiuli analizis Sedegebis prezentacia".

Tavi XIII. korelacia _______________________________________________________________________

1

Tavi XIII korelacia __________________________________________________________

wertilTa gabnevis diagrama dadebiTi kavSiri uaryofiTi kavSiri

pirsonis korelaciis koeficienti korelaciis sazomis miReba: jvaredina namravlebi

r : XYCP -is Sedareba qulebis saerTo cvalebadobasTan pirsonis korelaciis koeficientis alternatiuli formulebi

r -is standartuli qulebis formula r -is gamoTvliTi formula

r -is formulebi _ Sejameba

r -is Tvisebebi kavSiris mimarTuleba kavSiris xarisxi diapazonis SezRudva da r -is mniSvneloba determinaciis koeficienti korelacia da mizezobrioba

pirsonis r koeficientis statistikuri mniSvnelovnobis Semowmeba

populaciis korelacia, ρ aranulovani korelaciis Semowmeba populaciaSi

korelaciuri kvlevis Sedegebis warmodgena spirmenis ranguli korelaciis koeficienti

Sr -is gamoyenebis magaliTi

spirmenis ranguli korelaciis koeficientis statistikuri mniSvneloba

Sr -is Tvisebebi


2

korelaciuri kvlevebi (Correlational studies): kvlevebi, romlebSic izomeba ori an meti cvladi maTi korelaciis mimarTulebisa da xarisxis dasadgenad.

mecxredan meTormete Tavebis CaTvliT ganixileboda statistikis

gamoyeneba eqsperimentebSi, romlebSic xdeba damoukidebel cvladze manipulacia da izomeba damokidebuli cvladi. Tumca bevr kvlevaSi mkvlevari mxolod imiT aris dainteresebuli, rom aRmoaCinos, dakavSirebulia Tu ara erTmaneTTan ori cvladi. korelaciuri kvleva gulisxmobs kvlevas, romelSic izomeba ori an ramdenime cvladi, raTa gairkves maTi kavSiris mimarTuleba da xarisxi. am kvlevebSi arc erT cvladze ar xdeba manipulireba da, Sesabamisad, ar arsebobs damoukidebeli da damokidebuli cvladebi. am TavSi gagacnobT korelaciis cnebebs, romlebic saWiroa aseTi kvlevebis gasagebad da statistikas, romelsac korelaciis koeficienti ewodeba. erTmaneTTan korelaciaSi myofi qulebis gamoyeneba imisaTvis, rom erTis mixedviT meore viwinaswarmetyveloT, ganxilulia me-14 TavSi. korelaciuri kvlevis umartivesi SemTxveva gulisxmobs cdis pirTa jgufis TiToeuli wevrisgan ori gansxvavebuli cvladis monacemis miRebas. Semdeg xdeba monacemTa am wyvilebis statistikuri analizi, raTa dadgindes, arsebobs Tu ara raime kavSiri maT Soris. ganvixiloT Semdegi magaliTi:

martooba fsiqologiuri mdgomareobaa, romelsac yvela Cvengani ginicdis odesme. bevrisaTvis martooba xangrZlivi mdgomareobaa da is SeiZleba dakavSirebuli iyos fsiqikuri janmrTelobis problemebTan. (Roberts, Lewinsohn, & Seeley, 1993). davuSvaT, rom gvainteresebs martoobasa da depresias Soris kavSiris kvleva. viRebT 10 adamianisagan Semdgar SerCevas da vzomavT maTi depresiisa da martoobis dones. martoobis gasazomad adamians vTxovT, upasuxos 10 debulebas rogoricaa, magaliTad, ,,Tavs sxvebisgan gariyulad vgrZnob”. SesaZlo pasuxebia: arasodes (0), iSviaTad (1), zogjer (2), xSirad (3). frCxilebSi mocemuli cifrebi gamoxatavs pasuxebisaTvis miniWebul ricxviT mniSvnelobas (wonas). subieqti, romlis pasuxi iqneba ,,arasodes” aTive debulebaze, miiRebs nul qulas martoobis skalaze maSin, roca subieqti, romlis pasuxi iqneba ,,xSirad” aTive debulebaze, miiRebs 30 qulas martoobis skalaze. depresiac aseve 10 debulebiT izomeba, rogoricaa ,,me mowyenili var”. aqac cdis piri upasuxebs: arasodes (1), iSviaTad (2), zogjer (3), xSirad (4). Tumca yuradReba miaqcieT imas, rom am skalaze pasuxebisaTvis miniWebuli ricxviTi mniSvnelobebi gansxvavebulia martoobis skalis pasuxebis ricxviTi mniSvnelobebisagan. yvelaze dabali qulaa 10, romelsac iRebs adamiani, romelic upasuxebs ,,arasodes” aTive debulebaze da umaRlesi qulaa 40, romelsac iRebs adamiani, romelic pasuxobs ,,xSirad” aTive debulebaze. maSasadame, ToToeuli subieqtisagan miviReT ori qula: pirveli - martoobis da meore - depresiis.

davuSvaT, rom miRebuli martoobis qulebi (aRniSnuli rogorc X cvladi) da depresiis qulebi (aRniSnuli roogrc Y cvladi) warmodgenilia cxrilSi #13.1. cvladis X -iT an Y -iT aRniSvna pirobiTia. Tumca zogadad mklevari svams kiTxvas: rogoria Y -is, rogorc X -is funqciis mniSvneloba?

es debuleba gamoxatavs, rom mkvlevari dainteresebulia, erTi cvladis ( X ) mniSvnelobis mixedviT iwinaswarmetyvelos meore cvladis


3

(Y ) mniSvneloba. cvladi, romelsac prediqtorad viyenebT, aRiniSneba X -iT, xolo nawinaswarmetyvelebi cvladi - Y -iT. maSasadame, rodesac mkvlevari svams kiTxvas: `ras udris depresiis qula martoobis garkveuli qulis SemTxvevaSi?~, is cdilobs, depresiis done (Y ) iwinaswarmetyvelos martoobis donis ( X ) mixedviT. erTi qulis winaswarmetyvelebisaTvis meore cvladis gamoyenebis detalebi warmodgenilia me-14 TavSi.

qulaTa wyvilebi cxrilSi # 13.1 warmoadgens qulebis organzomilebian (bivariaciul) ganawilebas.

prefiqsi ,,bi” niSnavs ors (2); es qulebi organzomilebiania, radgan

TiToeuli cdis pirisgan miRebulia 2 monacemi (qula). dakavSirebulia Tu ara es qulebi erTmaneTTan? iqneb martoobis maRali qulebi dakavSirebulia depresiis maRal qulebTan, xolo martoobis dabali qulebi dakavSirebulia depresiis dabal qulebTan? an iqneb kavSiri Sebrunebulia: martoobis maRali qulebi dakavSirebulia depresiis dabal qulebTan da martoobis dabali qulebi dakavSirebulia depresiis maRal qulebTan. aseve SesaZlebelia, rom qulebi ar korelirebdes da martoobisa da depresiis qulebi urTierTdamoukidebeli iyos. imisaTvis, rom davadginoT, dakavSirebulia Tu ara qulebi, Tavdapirvelad vagebT monacemebis grafiks, romelsac wertilTa gabnevis diagrama ewodeba; Semdeg am monacemebisTvis viTvliT pirsonis korelaciis koeficients.

cxrili # 13.1 10 cdis piris hipoTeturi qulebi martoobisa da depresiis skalebze

cdis piri

martoobis qula: X

depresiis qula: Y

1 2 3 4 5 6 7 8 9 10

4 27 18 7 30 12 18 23 19 12

16 37 33 23 34 32 24 29 26 26

wertilTa gabnevis diagramebi

wertilTa gabnevis diagrama (Scatterplot): organzomilebiani ganawilebis grafiki, romelSic X cvladi gadazomilia horizontalur RerZze da Y cvladi gadazomilia vertikalur RerZze.

organzomilebiani ganawileba (Bivariate Distribution): ganawileba, romelSic TiToeuli cdis pirisagan miiReba ori monacemi (qula).


4

cxrilis #13.1 monacemebis wertilTa gabnevis diagrama warmodgenilia suraTze #13.1. wertilTa gabnevis diagramaze X -iT aRniSnuli cvladi (Cvens magaliTSi martoobis qula) gadazomilia horizontalur RerZze (abscisaze), xolo Y cvladi (Cvens magaliTSi depresiis qula) gadazomilia vertikalur RerZze (ordinataze). cdis piris mier TiToeul skalaze miRebuli qula gafantulobis diagramaze erTi wertiliT gamoisaxeba. magaliTad, grafikze #13.1 pirveli cdis piris qulebi warmodgenilia martoobis Sesabamisi 4 qulis da depresiis 16 qulis gadakveTis wertiliT. imis saCveneblad, Tu rogor aigeba wertilTa gabnevis diagrama, grafikze #13.1 TiToeuli cdis piris Sesabamisi wertilis gverdiT miTiTebulia am cdis piris nomeri. Tumca Cveulebriv, wertilTa gabnevis diagramaze cdis pirebis saxeliT an nomriT aRniSvna ar xdeba.

dadebiTi kavSiri

korelacia (covary): ori cvladi korelirebs, rodesac erTi cvladis cvlileba dakavSirebulia meore cvladis SeTanxmebul cvlilebasTan. wertilTa gabnevis diagrama gviCvenebs, rom monacemebs aqvT korelaciis tendencia: depresiis SedarebiT maRali qulebi dakavSirebulia martoobis SedarebiT maRal qulebTan, xolo depresiis SedarebiT dabali qulebi dakavSirebulia martoobis SedarebiT dabal qulebTan. monacemebi korelirebs (msgavsad varirebs) rodesac erTi qulis cvlilebas Tan axlavs meore qulis SeTanxmebuli cvlileba. grafikze #13.1 monacemebs Soris kavSiri dadebiTia. dadebiTi kavSiris SemTxvevaSi (rasac pirdapiri kavSiric ewodeba), X cvladis mniSvnelobis zrdasTan erTad izrdeba Y cvladis mniSvnelobac.

suraTi #13.1 10 cdis piris martoobisa da depresiis qulebis wertilTa gabnevis diagrama. wertilebis gverdiT miTiTebuli ricxvebi aRniSnavs cdis pirebs. qulebi aRebulia 13.1 cxrilidan.

maSasadame, grafikze #13.1 martoobis qulebis zrdasTan erTad xdeba depresiis qulebis zrdac. radgan orive cvlads vzomavT da arcerTiT ar vaxdenT manipulirebas, Tanabrad marTebulia vTqvaT, rom depresiis qulebis zrdasTan erTad izrdeba martoobis qulebic. magram grafikze #13.1 gamosaxuli kavSiri ar aris sruli. magaliTad, me-9 cdis pirs ufro maRali martoobis qula aqvs, vidre me-10 cdis pirs, magram orives depresiis qulaa 26. analogiurad, me-6 da me-10 cdis pirebis martoobis qulaa 12, magram pirvelis depresiis qulaa 32 da meoris - 26.

sruli dadebiTi kavSiri depresiisa da martoobis qulebs Soris gamosaxulia suraTze @13.2. am SemTxvevaSi wetilTa gabnevis diagrama gvaZlevs wrfes. or cvlads Soris kavSirs, romelic SeiZleba aRiweros wrfiT, wrfivi kavSiri ewodeba.

dadebiTi kavSiri (positive relationship): or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad izrdeba meore cvladis mniSvnelobac.


5

wrfivi kavSiri (linear relationship): or cvlads Soris kavSiri, romelic SeiZleba aRiweros wrfiT.

sruli dadebiTi kavSiris SemTxvevaSi, rogoric gamosaxulia suraTze #13.2, X -is TiToeul mniSvnelobas Seesabameba Y -is mxolod erTi mniSvneloba. amgvarad, erTi cvladis mniSvnelobis codna saSualebas gvaZlevs, zustad gavigoT an viwinaswarmetyveloT meore cvladis mniSvneloba. magaliTad, cdis pirs, romelsac aqvs 8 qula martoobis skalaze, eqneba 18 qula depresiis skalaze; an cdis pirma, romelmac miiRo 24 qula depresiis skalaze, miiRo 14 qula martoobis skalaze. sruli kavSiris SemTxvevaSi erTi qulis mixedviT meores winaswarmetyveleba SecdomasTan ar aris dakavSirebuli. Tumca realuri kvlevis monacemebs Soris sruli dadebiTi kavSirebi iSviaTad an arasodes aRiniSneba.

suraTi #13.2 10 cdis piris martoobisa da depresiis qulebs Soris sruli dadebiTi kavSiri.

uaryofiTi kavSiri davuSvaT, rom martoobisa da depresiis miRebul maCveneblebs Soris kavSiri naCvenebia suraTze #13.3. es grafiki gamoxatavs uaryofiT kavSirs am or cvlads Soris: X cvladis mniSvnelobis matebasTan erTad Y cvladis mniSvneloba mcirdeba.

uaryofiTi kavSiri (negative relationship): or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad meore cvladis mniSvneloba mcirdeba.

grafikze #13.3 xedavT, rom martoobis dabali qulebi dakavSirebulia depresiis maRal qulebTan da martoobis maRali qulebi dakavSirebulia depresiis dabal qulebTan. uaryofiT kavSirs zogjer Sebrunebul (inversiul) kavSirsac uwodeben, radgan cvladebi Sebrunebulad aris erTmaneTTan dakavSirebuli: X -is dabali mniSvnelobebi dakavSirebulia Y -is maRal mniSvnelobebTan da X -is maRali mniSvnelobebi dakavSirebulia Y -is dabal mniSvnelobebTan.

suraTze @#13.3 gamosaxuli uaryofiTi kavSiri ar aris sruli kavSiri martoobisa da depresiis qulebs Soris. magaliTad, ori cdis piris martoobis qulaa 20, magram maTi Sesabamisi depresiis qulebia 17 da 24. aseve, ori cdis piris depresiis qulaa 32, magram erTis martoobis qulaa 5 da meores martoobis qulaa 12.

depresiisa da martoobis qulebs Soris sruli uaryofiTi kavSiri naCvenebia cxrilze #13.4. am SemTxvevaSi wertilTa gabnevis diagrama qmnis wrfes da erTi monacemis (qulis) codna meore monacemis (qulis) zusti winaswarmetyvelebis saSualebas iZleva. magaliTad, Tu CemTvis cnobilia, rom cdis piris martoobis qulaa 8, aseve cnobilia, rom misi depresiis qula aris 32. aseve, Tu viciT, rom cdis pirma miiRo 22 qula depresiis skalaze, maSin misi martoobis qula aris 18. sruli dadebiTi kavSiris msgavsad, realuri monacemebis SemTxvevaSi sruli uaryofiTi kavSiric


6

iSviaTad gvxvdeba. yuradReba miaqcieT imas, rom `dadebiTi~ da `uaryofiTi~ wertilTa gabnevis diagramasTan mimarTebaSi gamoxatavs am kavSiris mxolod mimarTulebas da ara siZlieres. erTi qulis mixedviT meores winaswarmetyvelebisaTvis dadebiTi da uaryofiTi kavSirebi Tanabrad gamosadegia. suraTi #13.3 10 cdis piris martoobisa da depresiis qulebs Soris uaryofiTi kavSiri.

suraTi #13.4 10 cdis piris martoobisa da depresiis qulebs Soris sruli uaryofiTi kavSiri.

SeamowmeT Tqveni codna XIII-1

1. ganmarteT: monacemebis organzomilebiani ganawileba, korelaciuri kvleva, korelacia, pirdapiri kavSiri, Sebrunebuli kavSiri, wrfivi kavSiri, uaryofiTi kavSiri, dadebiTi kavSiri, wertilTa gabnevis diagrama.

2. cxrilisaTvis #13.1 SeabruneT da cvladebis aRniSvna (e.i. depresiis qula aRniSneT X -iT da martoobis qula Y -iT) da Semdeg am wertilebisaTvis aageT suraTze #13.1 gamosaxulis msgavsi wertilTa gabnevis diagrama. cvladis X -iT an Y -iT aRniSvna cvlis Tu ara qulebs Soris kavSirs?

3. 1-li da me-2 cxrilebis monacemebisaTvis ujrian furcelze aageT wertilTa gabnevis diagrama da Semdeg TiToeul cxrilTan dakavSirebiT upasuxeT Semdeg kiTxvebs: a. qulebs Soris kavSiri dadebiTia, uaryofiTi Tu kavSiri ar

arsebobs? b. qulebs Soris kavSiri srulia Tu ara? g. martoobis qulebis codna gazrdida Tu ara depresiis

qulebis winaswarmetyvelebis sizustes?

cxrili #1

cdis piri

martooba

depresia

1 4 19 2 12 20 3 28 33 4 18 32 5 14 26 6 6 19 7 22 28 8 23 36 9 15 30 10 18 27

cxrili #2

cdis piri

martooba

depresia


7

1 5 29 2 11 31 3 23 23 4 17 17 5 28 16 6 13 21 7 12 26 8 4 28 9 21 14 10 16 25

4. qvemoT mocemulia 10 cdis piris SerCevis sxeulis wonisa (funtebSi) da TviTpativiscemis monacemebi

cdis piri

sxeulis wona

KTviTpativiscema

1 100 39 2 111 47 3 117 54 4 124 23 5 136 35 6 139 30 7 143 48 8 151 20 9 155 28 10 164 46

a. ujrian furcelze aageT wertilTa gabnevis diagrama.

monacemebs Soris kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar arsebobs?

b. monacemebs Soris kavSiri srulia Tu ara? g. cdis piris sxeulis wonis codna gaaumjobesebda Tu ara

am cdis piris TviTpativiscemis qulis winaswarmetyvelebis sizustes?

pirsonis korelaciis koeficienti wertilTa gabnevis diagramis Seswavla warmodgenas gviqmnis imis Sesaxeb, arsebobs Tu ara mimarTeba or cvlads Soris da rogoria am mimarTebis mimarTuleba. magram mxolod wertilTa gabnevis diagrama ar aris sakmarisi imis dasadgenad, arsebobs Tu ara kavSiri or cvlads Soris. diagramaze gamosaxuli mimarTeba raodenobrivad unda aRiweros. aRweriT statistikebs, romlebic or cvlads Soris kavSiris xarisxs gamoxatavs, korelaciis koeficientebi ewodeba. intervalebis an Sefardebis skalebisaTvis gavrcelebuli koeficientia pirsonis korelaciis koeficienti, romelic aRiniSneba r -iT. am statistikas mieniWa karl pirsonis (1857-1936), im britaneli statistikosis saxeli, romelmac am


8

koeficientis gamosaTvleli formula Seqmna. pirsonis korelaciis koeficienti ganisazRvreba, rogorc:

pirsonos korelaciis koeficienti = ( )( )

( )[ ] ( )[ ]∑∑∑

−−

−−=

22YY XX

YYXXr

sadac X aris cdis piris X cvladis Sesabamisi qula

X aris X cvladis Sesabamisi yvela qulis saSualo Y aris cdis piris Y cvladis Sesabamisi qula

Y aris Y cvladis Sesabamisi yvela qulis saSualo pirsonis korelaciis koeficienti (Pearson correlation coefficient): statistika, romelic aRiniSneba r -iT da asaxavs intervalebis an Sefardebis skalaze or cvlads Soris wrfivi kavSiris xarisxs. es formula SeiZleba Caiweros ufro martivi aRniSvnebis gamoyenebiT, rogorc:

( )( )YX

XY

SSSSCPr =

sadac XYCP aris ( )( )∑ −− YYXX , anu aris X –sa da Y –is jvaredina namravli

( X da Y indeqsebi niSnavs, rom jvaredina namravlebi gamoTvlilia X da Y cvladebisTvis)

XSS aris X monacemebis kvadratebis jami (anu ( )∑ −2

XX )

YSS aris Y monacemebis kvadratebis jami (anu ( )∑ −2

YY ) ganmartebiTi formula Cveulebriv r –is gamosaTvlelad ar gamoiyeneba. mis nacvlad gamoiyeneba an standartuli qulis formula, an gamTvliTi formula, romlebic am TavSi qvemoT aris warmodgenili. magram ganmartebiTi formula asaxavs, Tu ras zomavs korelaciis koeficienti. ganvixilavT am formulas da Semdeg warmovadgenT standartuli qulisa da gamoTvliT formulebs. korelaciis sazomis miReba: jvaredina namravlebi imisaTvis, rom davinaxoT, Tu rogor aris r imis sazomi, ramdenad Zlieria korelacia or cvlads Soris, davubrundeT cxrilis #13.1 monacemebs. am monacemebis wertilTa gabnevis diagrama (ix. suraTi 13.1) gamoxatavda, rom monacemebi dadebiT kavSirSia: martoobis dabali qulebi dakavSirebulia depresiis dabal qulebTan da martoobis maRali qulebi dakavSirebulia depresiis maRal qulebTan. me ganvixilav statistikas, romelic or cvlads Soris korelaciis siZlieres gamoxatavs sxvaobis qulis gamoyenebiT, romelic aCvenebs, Tu ramdenad gansxvavdeba qula Tavisi

saSualosagan, XX − an YY − . magaliTad, cxrilze #13.1 martoobis


9

qulebis saSualoa 17. me-2 cdis pirs aqvs martoobis 27 qula. Tu

gamovTvliT XX −2 , anu 27-17 = +10-s, davinaxavT, rom cdis pirma miiRo martoobis qulebis saSualoze 10-iT maRali qula. me-4 cdis pirma miiRo 7 qula martoobis skalaze. am cdis piris martoobis qula SeiZleba aseve

gamoisaxos, rogorc XX −4 ; am SemTxvevaSi 7 – 17 = - 10, anu 10 qula martoobis qulis saSualos qvemoT. qulis gardaqmna am qulis saSualosagan gansxvavebad imasac gamoxatavs, qula saSualoze metia Tu naklebi da imasac, Tu ramdenad metia an naklebia. Y qulebi SeiZleba gamoisaxos msgavsadve am qulebis saSualosTan,

Y -Tan, mimarTebiT. magaliTad, depresiis qulebis saSualoa 28. me-2 cdis pirma miiRo depresiis 37 qula. am qulis saSualosgan gansxvavebaSi

gadayvaniT viRebT: YY −2 = 37-28 = +9. meores mxriv, me-4 cdis pirma miiRo depresiis 23 qula. saSualosgan gansxvavebis saxiT gamosaxuli, es qula

udris: YY −4 = 23 – 28 = -5, anu depresiis saSualo qulaze 5 quliT dabalia. sxvaobis am monacemebidan Cven vxedavT, rom me-2 cdis pirs aqvs saSualoze maRali qulebi rogorc martoobis, aseve depresiis skalebze, maSin, roca 44-e cdis piris orive qula saSualoze dabalia. davuSvaT, rom me yvela cdis pirisaTvis viTvli martoobisa da depresiis sxvaobis qulebs, rogorc es cxrilis #13.2 b da d svetebSia naCvenebi. yuradReba miaqcieT imas, rom im 6 cdis piridan, romelTa martoobis qula saSualoze maRalia (sveti b, cdis pirebi 2, 3, 5 da 8), 4 cdis piris depresiis qulac saSualoze maRalia (sveti d). msgavsadve, 3 cdis pirs im 4-dan, romelTa martoobis qula saSualoze dabalia (sveti b, cdis pirebi 1, 4 da 10), depresiis qulac saSualoze dabali aqvs (sveti d). cxrili #13.2 X -sa da Y -is jvaredina namravlebis gamoTvla. a da g svetebSi mocemuli qulebi

cxrilis #13.1 martoobisa da depresiis qulebia.

(a) (b) (g) (d) (e)

martooba depresia

cdis piri X XX − Y YY − ( )( )YYXX −−

1 4 -13 16 -12 +156 2 27 +10 37 +9 +90 3 18 +1 33 +5 +5 4 7 -10 23 -5 +50 5 30 +13 34 +6 +78 6 12 -5 32 +4 -20 7 18 +1 24 -4 -4 8 23 +6 29 +1 +6 9 19 +2 26 -2 -4 10 12 -5 26 -2 +10

X 17 Y 28 jami +367


10

qulebis korelaciis siZlieris sazomi miiReba TiToeuli cdis piris

qulebis X cvladis saSualosgan sxvaobisa da Y cvladis saSualosgan

sxvaobis erTmaneTze gamravlebiT da Semdeg yvela cdis piris namravlis SekrebiT. es procesi naCvenebia cxrilis #13.2 e svetSi. magaliTad, me-4

cdis pirisaTvis, ( XX − )( YY − ) = (-10) (-5) = +50. am svetSi namravlebis jami aris +367.

( )( )∑ −− YYXX sidides ewodeba X -isa da Y -is jvaredina namravli,

rac aRiniSneba, rogorc XYCP da is warmoadgens r –is formulis

mricxvels. XYCP asaxavs monacemebis or simravles Soris kavSiris siZlieresa da mimarTulebas. Tu monacemebi dadebiT kavSirSia, rogorc 13.2 cxrilze, maSin jvaredina namravli dadebiT mniSvnelobas iRebs; rac ufro

Zlieria korelacia X –sa da Y –s Soris, miT ufro izrdeba XYCP –is

mniSvneloba. uaryofiTad dakavSirebuli monacemebisaTvis XYCP uaryofiTia da misi mniSvneloba miT ufro didia, rac ufro Zlieria kavSiri.

X -isa da Y -is jvaredina namravlebi (Cross products of X and Y ): ( )( )∑ −− YYXX mniSvneloba X da Y cvladebisTvis

cxrilze #13.3a naCvenebia XYCP –is mniSvneloba suraTis #13.3 Sesabamisi

X –isa da Y –is uaryofiTi kavSirisaTvis. am monacemebisaTvis XYCP aris -399. cxrili #13.3 a) X -isa da Y –is jvaredina namravlebi martoobisa da depresiis qulebisaTvis.

b) X -isa da Y –is jvaredina namravlebi im SemTxvevisaTvis, roca martoobisa da

depresiis qulebs Soris kavSiris ar aris.

Mmartooba depresia


1 5 -10 32 +7 -70 2 24 +9 23 -2 -18 3 20 +5 17 -8 -40 4 12 -3 32 +7 -21 5 14 -1 21 -4 +4 6 20 +5 24 -1 -5 7 28 +13 14 -11 -143 8 13 -2 25 0 0 9 6 -9 36 +11 -99 10 8 -7 26 +1 -7

X 15 Y 25 jami -399


11

rodesac X da Y qulebi erTmaneTTan kavSirSi ar aris, XYCP nuls

uaxlovdeba. XYCP im SemTxvevisaTvis, roca X da Y ar aris kavSirSi, naCvenebia cxrilze #13.3b. suraTze #13.5 am qulebis wertilTa gabnevis diagramaa naCvenebi. yuradReba miaqcieT imas, rom X da Y qulebi ar korelirebs: X -is maRali maCveneblebi SeTanxmebulad ar aris dakavSirebuli Y -is maRal an dabal qulebTan. msgavsadve, X -is dabali qulebi ar aris SeTanxmebulad dakavSirebuli Y -is maRal an dabal

qulebTan. cxrilze #13.3b gamoTvlili XYCP = -12 Seesabameba kavSiris ararsebobas. suraTi #13.5 10 cdis piris martoobisa da depresiis qulebs Soris kavSiri ar arsebobs. qulebi aRebulia 13.3b cxrilidan.

r : XYCP -is Sedareba monacemebis saerTo cvalebadobis sazomTan

XYCP gamoxatavs or cvlads Soris kavSiris mimarTulebas, magram XYCP –is

ricxviTi mniSvneloba damokidebulia X –isa da Y –is gazomvis erTeulze da Cvens xelT arsebuli qulebis raodenobaze. qulebis or gansxvavebul

simravles SeiZleba hqondes erTnairi siZlieris kavSiri, magram XYCP -is gansxvavebuli mniSvnelobebi, radgan am qulebiT gazomilia gansxvavebuli

martooba depresia


1 24 +7 14 -12 -84 2 16 -1 36 +10 -10 3 4 -13 24 -2 +26 4 12 -5 16 -10 +50

5 22 +5 28 +2 +10

6 30 +13 26 0 0 7 10 -7 34 +8 -56 8 28 +11 30 +4 +44 9 14 -3 28 +2 -6 10 10 -7 24 -2 +14

X 17 Y 26 jami _12


12

erTeulebi an qulebis erTi simravle ufro didi SerCevidanaa miRebuli,

vidre meore. imisaTvis, rom XYCP , rogorc X -isa da Y –is korelaciis sazomi gamosadegi iyos, is warmodgenili unda iyos iseTi formiT, romelic standartul mniSvnelobebs miiRebs gazomvis erTeulis da monacemebis raodenobis miuxedavad. jvaredina namravlebi standartul mniSvnelobebs maSin iRebs, rodesac maT qulebis TiToeuli simravlis saerTo cvalebadobis sazomebs vadarebT. cvalebadobis erT-erTi sazomi, romelsac Cveulebriv viyenebT, aris kvadratebis jami (SS). X cvladiT aRniSnuli monacemebisaTvis

XSS = ( )∑ −2

XX da Y cvladiT aRniSnuli monacemebisaTvis YSS = ( )∑ −2

YY . imisaTvis, rom SS gadaviyvanoT jvaredina namravlis erTeulebSi,

ToToeuli SS -dan viRebT kvadratul fesvs. Semdeg XSS -is kvadratuli

fesvis gamravlebiT YSS –is kvadratul fesvze viRebT monacemTa ori

simravlis saerTo cvalebadobis sazoms, ( )( )YX SSSS . pirsonis korelaciis

koeficienti, r , aris XYCP –is Sefardeba saerTo cvalebadobis am sazomTan. Sesabamisad, Cven vxedavT, rom Semdegi ori formula r -is misaRebad erTsa da imave ricxviT mniSvnelobebs iyenebs:

( )( )( )[ ] ( )[ ]∑∑∑

−−

−−=

22YYXX

YYXXr

da

( )( )YX

XY

SSSSCPr = .

yuradReba miaqcieT imas, rom r –is formulis mniSvneli yovelTvis

dadebiT mniSvnelobas iRebs, radgan sxvaobis kvadrats Seicavs. maSasadame, korelaciis koeficinetis dadebiT (+) Tu uaryofiT (_) niSans mricxvelis niSani gansazRvravs. cxrilSi #13.4 naCvenebia cxrilis #13.1 10 qulisaTvis ganmartebiTi formulis gamoyenebiT r –is gamoTvla. TiToeuli cdis pirisaTvis:

• sveti a aris martoobis qula

• sveti b aris cdis piris martoobis qulisa da martoobis qulebis

saSualos Soris sxvaoba (anu ( XX − )) • sveti g aris b svetis mniSvnelobis kvadrati

• sveti d aris depresiis qula

• sveti e aris cdis piris depresiis qulisa da depresiis qulebis

saSualos Soris sxvaoba (anu ( YY − )) • sveti v aris e svetis mniSvnelobis kvadrati

• sveti z warmoadgens b da e svetebis namravls (( XX − )( YY − )), anu

XYCP -s.

XSS miiReba g svetis monacemebis dajamebiT, YSS _ v svetis monacemebis

dajamebiT da XYCP _ z svetis monacemebis dajamebiT. Semdeg xdeba am jamebis formulaSi Casma r –is ricxviTi mniSvnelobis misaRebad, romelsac


13

realurir ewodeba (ix. cxrilis #13.4 me-2 safexuri). cxrilis #13.1

monacemebisaTvis, realurir = +0,78. sanam realurisr interpretacias ganvixilavde,

warmogidgenT or alternatiul formulas, romlebic xSirad gamoiyeneba korelaciis koeficientis gamosaTvlelad. cxrili #13.4 r –is gamoTvla cxrilis #13.1 martoobisa da depresiis qulebisaTvis ganmartebiTi formulis gamoyenebiT.

safexuri 1. viTvliT ( )∑ −=2

XXSS X -s, ( )∑ −=2

YYSSY -s da

( )( )∑ −−= YYXXCPXY -s.

(a) (b) (g) (d) (e) (v) (z)

martooba

depresia

cdis piri

X XX − ( )2XX − Y YY − ( )2YY − ( )( )YYXX −−

1 4 -13 169 16 -12 144 +156 2 27 +10 100 37 +9 81 +90 3 18 +1 1 33 +5 25 +5 4 7 -10 100 23 -5 25 +50 5 30 +13 169 34 +6 36 +78 6 12 -5 25 32 +4 16 -20 7 18 +1 1 24 -4 16 -4 8 23 +6 36 29 +1 1 +6 9 19 +2 4 26 -2 4 -4 10 12 -5 25 26 -2 4 +10

X 17

XSS 630 Y 28

YSS 352 XYCP +367,0

safexuri 2. viTvliT r -s:

( )( )( )( ) ,367

,352

,6302

2

∑∑∑

+==−−

==−

==−

XY

Y

X

CPYYXX

SSYY

SSXX

Dda

( )( )( )[ ] ( )[ ]∑∑∑

−−

−−=

22YYXX

YYXXr

an

( )( ).

YX

XY

SSSSCPr =

ricxviTi mniSvnelobebis CasmiT:


14

( )( ).78,0779,0

352630367

+=+=+

=realurir

pirsonis korelaciis koeficientis alternatiuli formulebi r -is standartuli qulebis formula r -is ganmartebiTi formula SeiZleba Caiweros mniSvnelis dayofiT mis Semadgenel kvadratebis or jamad:

( )( )( )[ ] ( )[ ]∑∑∑

−−

−−=

22YYXX

YYXXr

an

( )( )( )( )YX SSSS

YYXXr ∑ −−=

yuradReba miaqcieT, rom ( XX − ) / SSx Zalian hgavs standartul qulas, z = ( XX − )/S, romelic me-6 TavSia warmodgenili. imisaTvis, rom es

gamosaxuleba standartul monacemad gardavqmnaT da miviRoT XS , X

monacmebis SerCevis standartuli gadaxra, saWiroa XSS gavyoT N-ze, anu

XSS –is Semadgeneli monacemebis raodenobaze. igive msjeloba exeba ( YY − ) / YSS –sac. Sesabamisad, r -is standartuli qulis formulaa:

( )wyviliN

zzr YX∑= ,

sadac: Xz aris ( XX − )/ XS

Yz aris ( YY − ) / YS wyviliN aris monacemebis wyvilebis raodenoba

XS -s, X qulebis SerCevis standartul gadaxras, Semdegnairad vpoulobT:

( )N

XXS X

∑ −=

2

da YS -s, Y monacemebis SerCevis standartul gadaxras, Semdegnairad:


15

( )N

YYSY

∑ −=

2

cxrili #13.5 gviCvenebs, rogor xdeba am formulis gamoyeneba realurisr

gamosaTvlelad cxrilis #13.1 monacemebisaTvis. pirveli safexuria z qulebis gamosaTvlelad saWiro XS –is da YS –is miReba. am safexurze

mocemuli ( )2XX −∑ = 630 da ( )2YY −∑ = 352 aRebulia cxrilidan #13.4. am

gamoTvlebiT Cven vigebT, rom X qulebis standartul qulebad

gardasaqmnelad Xz = ( X - 17,0)/7,937. msgavsadve, Y monacemebisaTvis Yz = (Y - 28,0)/5,933 (ixileT cxrilis me-2 safexuri). mesame safexurze xdeba TiToeuli nedli qulis standartul qulad gardaqmna z-is formulebis gamoyenebiT. me-3 safexuris a svetSi warmodgenilia TiToeuli cdis piris X qula. b svetSi warmodgenilia

XX − sxvaoba TiToeuli cdis pirisaTvis da g svetSi - TiToeuli cdis piris z qula. d svetSi warmodgenilia TiToeuli cdis piris Y qula, e

svetSi warmodgenilia YY − sxvaoba da v svetSi _ z qula ToToeuli Y –

Tvis. z svetSi gamravlebulia ToToeuli cdis piris Xz da Yz qulebi da

am svetis monacemebis jami, +7,794, gvaZlevs ( )∑ YX zz sidides, rac saWiroa r –is formulis mricxvelisaTvis. cxrilis me-4 safexuri gviCvenebs, Tu rogor xdeba ( )∑ YX zz –isa da

wyvilisN mniSvnelobebis Casma formulaSi. miRebulia realurir = +0,78, anu igive

mniSvneloba, rac ganmartebiTi formulis gamoyenebiT cxrilSi #13.4. rodesac Tqveni monacemebi standartuli qulebiT aris warmodgenili, girCevT, standartuli qulebis meTodi gamoiyenoT. Tu monacemebi standartuli qulebiT ar aris warmodgenili, maSin girCevT, gamoiyenoT qvemoT mocemuli gamoTvliTi formula. cxrili #13.5 r –is gamoTvla cxrilis #13.1 martoobisa da depresiis qulebisaTvis standartuli qulis formulis mixedviT

safexuri I. XS -isa da YS –is miReba:

( )wyviliN

XXS X

∑ −=

2

, ( )wyviliN

YYSY

2−

= ∑

cxrilis #13.4 g svetidan

cxrilis #13.4 v svetidan

( ) 6302=−∑ XX

wyviliN =10

.937,710

0,630==XS

( ) 3522=−∑ YY

wyviliN =10

933,510

0,352==YS

safexuri II. Xz -is, Yz -sa da YX zz∑ -is miReba:

X qulebis standartul qulebSi gadasayvanad

Y qulebis standartul qulebSi gadasayvanad


16

0,17=X 937,7=XS

maSasadame,

937,70,17−

=Xz X

0,28=Y 933,5=YS

maSasadame,

.933,5

0,28−=

YzY

safexuri III. nedli qulebis z qulebSi gadayvana

(a) (b) (g) (d) (e) (v) (z) martooba depresia

cdis piri

X XX − Xz Y YY − Yz Xz Yz

1 4 -13,0 -1,638 16 -12,0 -2,023 +3,314 2 27 +10,0 +1,260 37 +9,0 +1,517 +1,911 3 18 +1,0 +0,126 33 +5,0 +0,843 +0,106 4 7 -10,0 -1,260 23 -5,0 -0,843 +1,062 5 30 +13,0 +1,638 34 +6,0 +1,011 +1,656 6 12 -5,0 -0,630 32 +4,0 +0,674 -0,425 7 18 +1,0 +0,126 24 -4,0 -0,674 -0,085 8 23 +6,0 +0,756 29 +1,0 +0,168 +0,128 9 19 +2,0 +0,252 26 -2,0 -0,337 -0,085 10 12 -5,0 -0,630 26 -2,0 -0,337 +0,212 X 17,0 Y 28,0 jami +7,794

safexuri IV. r -is gamoTvla:

( )∑ += 794,7YX zz (ixileT z sveti),

wyviliN =10,

( )wyviliN

zzr YX∑= .

ricxviTi mniSvnelobebis CasmiT,

.78,0779,0

10794,7

+=+=

+=realurir

cxrili #13.6 cxrilis #13.1 martoobisa da depresiis qulebisaTvis gamoTvliTi formulis mixedviT r –is gamoTvla

safexuri I. ∑ X -is, ∑Y -is, ∑ 2X -is, ∑ 2Y -sa da ∑ XY -is gamoTvla:

(a) (b) (g) (d) (e) cdis piri

X

Y

2X

2Y

XY 1 4 16 16 256 64 2 27 37 729 1369 999 3 18 33 324 1089 594 4 7 23 49 529 161 5 30 34 900 1156 1020 6 12 32 144 1024 384 7 18 24 324 576 432 8 23 29 529 841 667


17

9 19 26 361 676 494 10 12 26 144 676 312

jamebi 170,0 280,0 3520,0 8192,0 5127,0

safexuri II. r -is gamoTvla:

( ) ( )

105127

3520

900.28170

170

2

22

=

=

=

==

=

∑∑∑∑

wyviliNXY

X

X

X

( ) ( )8192

400.78280

280

2

22

=

==

=

∑∑∑

Y

Y

Y

( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑

∑∑∑−

−=

YYNXXN

YXXYNr

wyviliwyvili

wyvili

ricxviTi mniSvnelobebis CasmiT miviRebT:

( )( ) ( )( )( )( )[ ] ( )( )[ ]

( )( ).78,0779,0

14,47093670

400.78920.81900.28200.35600.47270.51

400.78819210900.28352010280170512710

+=+=+

=−−

−=

−−−

=realurir

r-is gamoTvliTi formula r-is mniSvnelobis gamoTvliTi formulaa:

( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑

∑∑∑−

−=

YYNXXN

YXXYNr

wyviliwyvili

wyvili

es formula iyenebs uSualod nedl qulebs da ar saWiroebs qulasa da qulebis saSulos Soris sxvaobis gamoTvlas. cxrili #13.6 gviCvenebs, Tu rogor xdeba am formulis gamoyeneba cxrilis #13.1 monacemebisaTvis. cxrilis #13.6 a da b svetebi warmoadgens Sesabamisad, martoobis ( X ) da

depresiis (Y ) qulebs. am svetebis monacemebis jamebi, ∑ X = 170 da ∑Y = 280,

gvaZlevs formulisaTvis saWiro ∑ X –sa ∑Y –s. g sveti gviCvenebs

ToToeul X 2 –s da am svetis monacemebis jami, 3520, gvaZlevs 2∑ X -s. d

sveti warmoadgens TiToeul Y 2 -s da am svetis monacemebis jami, 8192,

∑ 2Y -ia. e svetSi miRebulia X Y -is mniSvneloba TiToeuli cdis

pirisaTvis da am svetis monacemebis jami, 5127, ∑ XY -is mniSvnelobaa. cxrilis #13.6 me-2 safexuri gviCvenebs r –is formulaSi ricxviTi

mniSvnelobebis Casmas. realurir = +0,78. yuradReba miaqcieT imas, rom


18

miuxedavad imisa, Tu ra formula iqna gamoyenebuli (ganmartebiTi, z qulis,

Tu gamoTvliTi), cxrilis #13.1 monacemebisaTvis realurir = + 0,78.

aq saWiroa kidev erT mniSvnelovan sakiTxze gamaxvildes yuradReba. rogorc aRvniSneT, cvladebis X -iT an Y -iT aRniSvna absoluturad pirobiTia. Cvens magaliTSi martoobis qula aRiniSna X -iT da depresiis qula Y -iT. Tumca, aRniSvna rom Segvebrunebina da depresiis qula aRgveniSna X -iT da martoobis qula - Y -iT, TiToeuli formulis gamoyenebiT warmoebuli gamoTvlebis Sedegebi iseTive iqneboda, rogoric mocemulia cxrilebSi #13.4, 13.5 da 13.6. cxrilis #13.1 martoobisa da depresiis qulebs Soris korelacia aris 0,78 miuxedavad imisa, Tu gamoTvlis miznebisaTvis romeli cvladi aRiniSna X -iT da romeli - Y -iT. r -is formulebi - Sejameba

r -is 4 sxvadasxva formulis Sejameba mocemulia cxrilSi #13.7. yvela

formula realurisr erTsa da imave mniSvnelobas iZleva, roca maT qulebis

erTi simravlisTvis viyenebT. cxrili #13.7 r -is formulebis Sejameba ganaszRvrebiTi formula

( )( )( )[ ] ( )[ ]∑∑∑

−−

−−=

22YYXX

YYXXr

XYCP -iT, XSS -Ta da YSS -iT Cawerili ganmartebiTi formula

( )( )YX

XY

SSSSCPr =

standartuli qulis formula

( )wyviliN

zzr YX∑=

gamoTvliTi formula

( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑

∑∑∑−

−=

YYNXXN

YXXYNr

wyviliwyvili

wyvili



19

1. gansazRvreT: jvaredina namravli, XYCP , r , XSS , YSS 2. axseniT, Tu ratom aris jvaredina namravlebi monacemTa or

simravles Soris korelaciis sazomi. 3. mocemulia Semdegi monacemebi:

cdis piri

martooba

depresia

1 4 19 2 12 20 3 28 33 4 18 32 5 14 26 6 6 19 7 22 28 8 23 36 9 15 30 10 18 27

a. am qulebisaTvis ganmartebiTi formulis gamoyenebiT

gamoTvaleT realurir .

b. am kiTxvis qulebi gadmotanilia davalebidan SeamowmeT

Tqveni codna XIII-1, kiTxva #3, cxrili #1. realurir SeadareT im kiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT, raTa TvalsaCinod warmoidginoT

mimarTeba, romelsac realurir aRwers.

4. mocemulia Semdegi monacemebi:

cdis piri

martooba

depresia

1 5 29 2 11 31 3 23 23 4 17 17 5 28 16 6 13 21 7 12 26 8 4 28 9 21 14 10 16 25

a. am monacemebisaTvis standartuli qulis formulis

gamoyenebiT gamoTvaleT realurir .

am kiTxvis monacemebi gadmotanilia davalebidan SeamowmeT

Tqveni codna XIII_1, kiTxva #3, cxrili #2. realurir

SeadareT im kiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT, raTa TvalsaCinod

warmoidginoT mimarTeba, romelsac realurir aRwers.

b. mocemulia Semdegi monacemebi:


20

cdis piri

sxeulis wona

TviTpativiscema

1 100 39 2 111 47 3 117 54 4 124 23 5 136 35 6 139 30 7 143 48 8 151 20 9 155 28 10 164 46

a. am monacemebisaTvis ganmartebiTi formulis gamoyenebiT

gamoTvaleT realurir .

b. am kiTxvis monacemebi gadmotanilia davalebidan SeamowmeT

Tqveni codna XIII_1, kiTxva #4. realurir SeadareT im

SekiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT imisaTvis, raTa TvalsaConod

warmoidginoT mimarTeba, romelsac realurir aRwers.

r - is Tvisebebi korelaciis koeficienti gamoxatavs or cvlads Soris wrfivi kavSiris rogorc mimarTulebas, aseve xarisxs. kavSiris mimarTuleba gamoixateba korelaciis koeficientis dadebiTi an uaryofiTi niSniT. imis aRmniSvnel xarisxs, Tu ramdenad uaxlovdeba wertilTa gabnevis diagramis wertilebis ganlageba wrfes, gvaZlevs r -is absoluturi mniSvneloba, sidide, romelic SeiZleba iRebdes 0-dan 1-mde mniSvnelobebs. niSnisa (+ an _) da ricxviTi mniSvnelobis gaTvlasiwinebiT r iRebs mniSvnelobebs -1-dan 0-is gavliT +1-mde. kavSiris mimarTuleba zemoT ganvixileT wertilTa gabnevis diagramaze qulebs Soris dadebiTi da uaryofiTi kavSirebi. korelaciis koeficienti kavSiris mimarTulebas

plus an minus niSnis meSveobiT asaxavs. dadebiTi r (mag., realurir = +0,86),

asaxavs X –sa da Y –s Soris dadebiT kavSirs; X cvladis qulebis zrdasTan erTad izrdeba Y cvladis qulebic. qulebi 13.1 da 13.2 cxrilebze

asaxavs dadebiT kavSirebs; cxrilisTvis #13.1 realurir = +0,78 da

cxrilisTvis #13.2 realurir = +1,00.

uaryofiTi r (magaliTad, realurir = _0,86) aRniSnavs cvladebs Soris

Sebrunebul kavSirs: roca X cvladis qulebi izrdeba, Y cvladis qulebi klebulobs. uaryofiTi kavSiri naCvenebia cxrilebze #13.3 da 13.4.


21

cxrilisTvis #13.3 realurir udris _0,83-s da cxrilisTvis #13.4 udris _1,00-

s. kavSiris xarisxi r -is ricxviTi mniSvneloba gamoxatavs, imas, Tu ramdenad kargad aRwers wrfe X da Y cvladebs Soris mimarTebas. 1,00-is toli r niSnavs, rom kavSiri idealurad wrfivia; wertilTa gabnevis diagramaze yvela wertili wrfeze Zevs. suraTi #13.2 gviCvenebs sruli dadebiTi kavSiris amsaxvel

wertilTa gabnevis diagramas, realurir = +1,00 martoobisa da depresiis

qulebisaTvis. sruli uaryofiTi kavSiri, realurir = _1,00 warmodgenilia

suraTze #13.4. rodesac r udris plus an minus 1,00-s, X -is yoveli mniSvnelobisTvis arsebobs misi Sesabamisi Y –is mxolod erTi mniSvneloba. aseT kavSirs xedavT suraTebze #13.2 da 13.4. martoobis ToToeuli qulisaTvis arsebobs depresiis mxolod erTi qula. aseTi kavSiri gvaZlevs saSualebas, X qulebis mixedviT zustad viwinaswarmetyveloT Y . suraTebze #13.2 da 13.4 Tu iciT martoobis qula, zustad SegiZliaT iwinaswarmetyveloT am qulasTan dakavSirebuli depresiis qula. meore ukidures SemTxveaSi, Tu cvladebs Soris kavSiri ar arsebobs, maSin korelaciis koeficientis mniSvneloba nuls uaxlovdeba. rodesac r daaxloebiT 0-is tolia, erTi qulis mniSvneloba ar aris kanonzomierad dakvSirebuli meore qulis mniSvnelobasTan. aseTi magaliTi naCvenebia

suraTze #13.5, sadac realurir = _0,2.

korelaciis koeficientis 0-dan +1,00-mde mniSvnelobebi gamoxatavs, rom X –sa da Y –s Soris arsebobs kavSiri, magram ara sruli kavSiri. TvalsaCinoebisTvis aRvniSnavT, rom suraTze #13.1, romlisTvisac r = +0,78, X da Y cvladebi ar aris srul kavSirSi. martoobis qulebis zrdasTan erTad depresiis qulebic izrdeba, magram martoobis qulis codna depresiis qulis zusti winaswarmetyvelebis saSualebas ar gvaZlevs. magaliTad, orma cdis pirma miiRo 12 qula martoobis skalaze, magram depresiis skalaze erT cdis pirs hqonda 32 qula da meores – 26 qula. cxrilze #13.3 warmodgenilia msgavsi uaryofiTi kavSiri,

romlisTvisac realurir = _0,83. am SemTxvevaSi martoobis qulebis zrdisas

depresiis qulebi mcirdeba. magram kavSiri aqac ar aris sruli; martoobis qulis codna ar gvaZlevs masTan dakavSirebuli depresiis qulis zusti winaswarmetyvelebis saSualebas. magaliTisaTvis, yuradReba miaqcieT, rom orma cdis pirma miiRo 20 qula martoobis skalaze, magram maTgan erTma depresiis skalaze miiRo 17, meorem ki _ 24 qula. xSirad Znelia, TvalsaCinod warmovidginoT, Tu rogor saxes miiRebda r –is mniSvneloba, monacemebi wertilTa gabnevis diagramaze rom yofiliyo warmodgenili. es SesaZlebeli xdeba mxolod r –is mniSvnelobebis im qulebis wertilTa gabnevis diagramasTan SedarebiT, romlebisTvisac korelacia gamovTvaleT. suraTi #13.6 warmoadgens r –is gansxvavebuli mniSvnelobebis Sesabamis wertilTa gabnevis diagramebs. a-dan e-mde grafikebi asaxavs 1,00-dan 0,00-mde korelaciebs. yuradReba miaqcieT imas, rom r –is klebasTan erTad X –sa da Y –s Soris wrfivi


22

kavSiri naklebad Zlieri xdeba, sanam realurir = 0,00, roca kavSiri aRar

arsebobs. pirsonis r aris X da Y cvladebs Soris wrfivi kavSiris sazomi. grafiki #13.6v X da Y cvladebs Soris arawrfiv, U-s formis kavSirs warmoadgens. am kavSiris dros X –is mcire mniSvnelobebi dakavSirebulia Y –is did mniSvnelobebTan. garkveul wertilamde X -is matebasTan erTad Y mcirdeba; X –is Semdgomi mateba ki dakavSirebulia Y –is matebasTan. X –sa da Y –s Soris aseTi kavSiris wrfiT warmodgena SeuZlebelia da misi aRwerisaTvis r -is gamoyeneba ar SeiZleba. suraTi #13.6 wertilTa gabnevis diagramebi gamoxatavs X da Y cvladis monacemebs Soris sxvadasva korelaciebs 10 cdis piris qulebisaTvis. qulebis nakrebebs Soris

pirsonis korelaciis koeficientis mniSvneloba mocemulia realurir -iT.

(a) realurir =+1,0 (b) realurir =-1,0 (g) realurir =-0,49

(d) realurir =+0,25 (e) realurir =0,00 (v) arawrfivi

diapazonis SezRudva da r –is mniSvneloba r -is mniSvneloba mgrZnobiarea mniSvnelobaTa im diapazonisadmi, rac X –ma da Y –ma SeiZleba miiRos. erT-erTi an orive cvladis mniSvnelobaTa diapazonis SemcirebasTan erTad r –is mniSvnelobac mcirdeba. aseTi kavSiris sailustraciod ganvixiloT suraTis #13.1 monacemebi,

romelTaTvisac realurir = +0,78. davuSvaT, rom minda vikvlio kavSiri

martoobasa da depresias Soris mxolod im 6 cdis pirTan, romelTac martoobis yvelaze maRali qulebi aqvT. es aris martoobis qulebi 18-dan 30-mde (anu martoobis qulebi ## 2, 3, 5, 7, 8 da 9 cdis pirebisagan). depresiis qulebis diapazoni am cdis pirebisaTvis 24-dan 37-mdea. aTive cdis piris martoobisa da depresiis monacemebis diapazonTan SedarebiT, maRali qulebis mqone cdis pirebis martoobisa da depresiis monacemebis

diapazoni SeizRuda, anu damoklda. am eqvsi monacemisaTvis realurir aris

+0,69, rac ufro naklebia, vidre +0,78-is toli realurir , monacemebis sruli

simravlis SemTxvevaSi. am magaliTis mniSvneloba naTelia. SezRuduli diapazonis mqone monacemebisTvis gamoTvlilma korelaciis koeficientma SeiZleba gvafiqrebinos, rom or cvlads Soris kavSiri ar arsebobs, maSin, roca kavSiris aRmoCena SesaZlebeli iqneboda, monacemebis ufro didi diapazoni rom gamogveyenebina. amitom, korelaciis koeficientis gamoTvlamde kargad SeiswavleT mopovebuli monacemebi, raTa gansazRvroT, warmoadgenen Tu ara isini SezRudul diapazons. determinaciis koeficienti

determinaciis koeficienti (Coefficient of determination): 2r -is mniSvneloba, romelic X da Y cvladebis saerTo cvalebadobas asaxavs.


23

me-9 TavSi warmovadgine 2η (eta kvadrati), rogorc eqsperimentSi damoukidebeli cvladis damokidebul cvladze gavlenis siZlieris sazomi.

korelaciur kvlevebSi determinaciis koeficienti, 2r ( r -is mniSvnelobis kvadrati), warmoadgens X –sa da Y –s Soris kavSiris siZlieris analogiur sazoms.

2r -is interpretacia 2η –is interpretaciis msgavsia. kerZod, 2r gamoxatavs, qulebis erTi simravlis cvalebadobis ra proporciaa dakavSrebuli qulebis meore simravlis cvalebadobasTan. magaliTad,

cxrilis #13.1 monacemebisaTvis, r = +0,78; maSasadame, realuri2r -s

mniSvnelobaa (+0,78)², anu 0,61. es sidide niSnavs, rom qulebis am wyvilebs saerTo aqvT cvalebadobis 61 procenti. sxva sityvebiT, martoobis qulebis cvalebadobis 61 procenti depresiis qulebis cvalebadobasTan aris dakavSirebuli. korelacia da mizezobrioba bevri cvladi Zlier kavSirSia erTmaneTTan, Tumca X da Y cvladebs Soris korelaciis arseboba ar niSnavs, rom erTi iwvevs meores. Tu X da Y cvladebi korelaciaSia, maSin am cvladebs Soris mizez-Sedegobrivi kavSiris sul mcire sami varianti arsebobs:

• X iwvevs Y –s.

• Y iwvevs X –s

• arc X da arc Y mizezobriv gavlenas ar axdens erTmaneTze, orive gamowveulia mesame, Z cvladiT.

ganvixiloT magaliTi martoobisa ( X cvladi) da depresiis (Y cvladi) korelaciis Sesaxeb. Cvens magaliTSi miviReT +0,78 korelacia, rac niSnavs, rom SedarebiT martosuli adamianebi ufro depresiulebic arian. romeli cvladi axdens meoreze mizez-Sedegobriv gavlenas, Tu aseTi gavlena arsebobs? martooba iwvevs depresias? Tu depresiuloba iwvevs martoobas? Tu am SemTxvevaSi arc erTi aseTi gavlena ar arsebobs, da mesame cvladi, Z, magaliTad megobruli urTierTobebis CamoyalibebisaTvis saWiro socialuri unarebis nakleboba, iwvevs adamianis rogorc martosulobas, aseve depresiulobas? es sami mizez-Sedegobrivi mimarTeba naCvenebia cxrilze #13.8. or cvlads Soris korelacia miuTiTebs mxolod imaze, rom isini wrfivad arian dakavSirebuli; is ar gvawvdis informacias imis Sesaxeb, Tu ra iwvevs maT kavSirs. sxva magaliTSi sigaretis moweva ( X cvladi) da stresis done (Y cvladi) dadebiT kavSirSia; sigaretis mwevelebs Cveulebriv stresis ufro maRali done aqvT, vidre aramwevelebs (Parrott, 1999). es kavSiri korelaciuria da miRebulia adamianebis gamokiTxviT sigaretis mowevis Cvevisa da stresis donis Sesaxeb. am SemTxvevaSi saeWvoa ara TavisTavad kavSiri, aramed am kavSiris mizezi (Kassel, 2000). am kavSiris asaxsnelad gamoTqmuli erT-erTi varaudis Tanaxmad, mwevelebs stresis ufro maRali done aqvT sxvadasxva mizezis gamo (Z cvladi), rogoricaa aramwevelebTan SedarebiT ufro Zlieri samsaxureobrivi an ojaxuri stresebi da moweva am stresebis Sesustebis meTodia. meore SemoTavazebuli axsnis mixedviT, stresis maRal dones realurad moweva iwvevs da is nikotinze


24

damokidebulebis Sedegia. Tumca martiv korelaciur aRmoCenas ar SeuZlia upasuxos kiTxvas, Tu romeli axsnaa WeSmariti am oridan. sxva kvlevaSi miRebuli iyo uaryofiTi korelacia Ramis Zilis xangrZlivobasa ( X cvladi) da dRis ganmavlobaSi energiis dones (Y cvladi) Soris; is adamianebi, romelTac naklebad sZinavT, acxadeben, rom dRis ganmavlobaSi meti energia aqvT. SeiZleba Tu ara am mimarTebidan davaskvnaT, rom Tu dRis ganmavlobaSi energiis naklebobas ganicdi, RamiT naklebi unda iZino? pasuxia _ ara, radgan korelaciuri kavSiri ar gvaZlevs aseTi mizez-Sedegobrivi daskvnis gakeTebis saSualebas. albaT, miRebul mimarTebaze pasuxismgebelia mesame cvladi, rogoricaa fiziologiuri mdgomareoba, romelic gavlenas axdens dRis ganmavlobaSi energiis raodenobazec da Zilis saWiro xangrZlivobazec. am magaliTebidan gamomdinare, yovelTvis unda gvaxsovdes, rom korelaciuri kavSiri gamoxatavs mxolod or cvlads Soris kavSirs, magram arafers gveubneba am kavSiris mizezze. cxrili #13.8 ori cvladis, martoobisa ( X ) da depresiis (Y ), korelaciis Sesabamisi SesaZlo mimzez-Sedegobrivi kavSirebi.

a) miRebuli korelacia:

miRebulia X da Y cvladebs Soris dadebiTi korelacia

adamianebs, romelTac aqvT martoobis

maRali qula ( X ), aqvT depresiis maRali

qula (Y ).

adamianebs, romelTac aqvT martoobis

dabali qula ( X ), aqvT depresiis dabali

qula (Y ).

b) miRebuli korelaciis amxsneli SesaZlo mizez-Sedegobrivi kavSirebi:

pirveli SesaZlebloba X cvladi iwvevs Y cvlads

pirovnebis martosuloba ( X ) iwvevs mis depresiulobas (Y ).

meore SesaZlebloba Y cvladi iwvevs X cvlads

pirovnebis depresiuloba (Y ) iwvevs mis martosulobas ( X ). mesame SesaZlebloba

Z cvladi iwvevs X da Y cvladebs saTanado socialuri unarebis nakleboba ( Z cvladi) iwvevs pirovnebis martosulobasa ( X ) da depresiulobas (Y ).


25


1. ras gveubneba r –is plus an minus miSani X da Y cvladebis kavSiris Sesaxeb?

2. ras gveubneba r –is ricxviTi mniSvneloba X da Y cvladebis kavSiris Sesaxeb?

3. ra gavlena SeiZleba hqondes X an Y cvladis monacemebis diapazonis SezRudvas r –is mniSvnelobaze?

4. gamoTvaleT determinaciis koeficienti realurisr Semdegi

mniSvnelobebisaTvis:

a. +0,12 g. _0,74 e. _0,59 b. _0,36 d. +0,23 v. +0,19

5. Tqven akvirdebiT kavSirs X da Y cvladebs Soris.

gansazRvreT is sami SesaZlo mizez-Sedegobrivi kavSiri, riTac SeiZleba korelaciis axsna.

6. mravalma mkvlevarma daadgina qvemoT CamoTvlili korelaciebi. gansazRvreT mesame cvladi, romelic, SesaZloa, miRebul korelaciaze pasuxismgebeli iyos a. uaryofiTi korelacia Semosavalsa da sisxlis wnevas

Soris b. dadebiTi korelacia yavis smasa da sigaretis mowevas

Soris g. dadebiTi korelacia bavSvebis mier televiziiT Zaladobis

yurebasa da maT agresiul qcevas Soris.

pirsonis r koeficientis statistikuri mniSvnelovnobis Semowmeba

populaciis korelacia, ρ

pirsonis r koeficienti aRwers SerCevis X da Y qulebs Soris wrfiv kavSirs. rogorc qulebis nebismieri simravlis SemTxvevaSi, SerCeva miRebulia qulebis ufro didi populaciidan da SeiZleba gvsurdes, SevawmoT, arsebobs Tu ara X da Y monacemebs Soris korelacia populaciaSi. populaciaSi X da Y monacemebs Soris wrfiv kavSirs warmoadgens parametri ρ (berZnuli anbanis me-17 aso, ro ). martoobisa da depresiis qulebis korelaciis Sesaxeb magaliTSi miRebuli realurir = +0,78,

gvaZlevs ρ–s savaraudo mniSvnelobas. Tumca, SerCevis Secdomis gamo,

realurir ρ–s toli ar iqneba. radgan SerCevis safuZvelzea miRebuli,

sxvadasxva SerCevisaTvis gamoTvlili realurir , iseve rogorc nebismieri sxva

statistika, sxvadasxva mniSvnelobas mogvcems, miuxedavad imisa, rom es SerCevebi aRebulia EerTi da imave populaciidan. aqedan gamomdinare,

SesaZlebelia, SevamowmoT hopoTeza imis Sesaxeb, rom realurir axasiaTebs

garkveuli ρ–s mqone populaciidan aRebul SerCevas. Tumca, SesaZlebelia SevamowmoT hopoTeza ρ–s nebismier mniSvnelobasTan dakavSirebiT, yvelaze

ρ : ro, populaciis korelaciis koeficientis aRmniSvneli simbolo.


26

gavrcelebulia Semowmeba hopoTezisa imis Sesaxeb, rom im populaciaSi, saidanac aRebulia SerCeva, ρ nulis tolia. aranulovani korelaciis Semowmeba populaciaSi statistikuri Semowmeba imisa, SemTxveviTad gansxvavdeba Tu ara r populaciaSi nulovani korelaciisagan martivia. saWiroa hopoTezebis statistikuri Semowmebis nacnobi safexurebis gavla:

• vaxdenT nulovani hopoTezis, 0H -isa, da alternatiuli hipoTezis,

1H -is, formulirebas;

• vadgenT r -is SerCeviT ganawilebas im varaudiT, rom 0H WeSmaritia.

es ganawileba mocemulia cxrilSi #A.5; • virCevT mniSvnelovnobis dones;

• cxrilSi #A.5 mocemuli r –is SerCeviTi ganawilebis mixedviT vpoulobT r -is kritikul mniSvnelobas, romelic aRiniSneba,

rogorc kritikulir ;

• r -is SerCeviT ganawilebaSi ganvsazRvravT uaryofis ares.

• SerCevis monacemebis mixedviT viTvliT korelaciis koeficients,

romelic aRiniSneba realuriTr ;

• imis mixedviT, xvdeba Tu ara realurir uaryofis areSi, viRebT

gadawyvetilebas 0H –is miRebis an uaryofis Sesaxeb.

statistikuri hipoTeza nulovani hopoTeza. nulovani hopoTeza aseTia:

0H : ρ = 0 es hipoTeza ambobs, rom populaciaSi X da Y cvladebs Soris korelacia nulis tolia. alternatiuli hipoTeza. alternatiuli hipoTeza Semdegnairia:

1H : ρ ≠ 0 es hipoTeza ambobs, rom populaciaSi X da Y cvladebs Soris korelacia ar aris nulis toli. r -is SerCeviTi ganawileba da kritikuli mniSvnelobebi radgan r –s SeuZlia 0-dan 1-mde mniSvnelobebi miiRos, misi SerCeviTi

ganawileba 0H : ρ = 0–sTvis SeiZleba advilad ganisazRvros. Tumca, rogorc

sxva statistikuri kriteriumebis SemTxvevaSi, r -is SerCeviTi ganawileba

damokidebulia mis Tavisuflebis xarisxze. r –saTvis df udris wyviliN – 2,

sadac wyviliN aris im qulebis wyvilebis raodenoba, saidanac gamoTvlilia

r . r –is kritikuli mniSvnelobebi, anu kritikulir , r –is SerCeviTi


27

ganawilebebidan im SemTxvevebisTvis, roca df meryeobs 1-dan 100-mde, da roca α = 0,05 da 0,01-s, warmoadgenilia danarTis cxrilSi #A.5. uaryofis zonis dadgena

Tu 1H WeSmaritia, mosalodnelia, rom r –is iseT mniSvnelobebs miviRebT,

romlebic aRemateba nuls. Sesabamisad, realurisr uaryofis zona aris r –is is

mniSvneloba, romelic udris an metia kritikulzer . roca realurir absoluturi

mniSvnelobiT (anu –is + an _ niSans yuradRebas Tu ar mivaqcevT,) skritikuliir

tolia an masze metia, is xvdeba uaryofis areSi. Tu 0H WeSmaritia, aseTi

realuriisr albaToba alfas mniSvnelobis tolia an masze naklebia.

gadawyvetilebis miRebisaTvis saWiro wesebi statistikuri hipoTezis Sesaxeb gadawyvetilebis miRebis wesebi nacnobi nimuSis mixedviT xdeba:

• Tu realurir uaryofis areSia, maSin:

uaryaviT 0H : ρ = 0 miiReT 1H : ρ ≠ 0

• Tu realurir ar xvdeba uaryofis areSi, maSin

ar uaryoT 0H

ar miiRoT 1H

am process giCvenebT qvemoT mocemuli magaliTi. cxrili #13.9 ajamebs im gadawyvetilebebsa da daskvnebs, rasac r –is mixedviT viRebT. cxrili #13.9 Sejameba: statistikuri hipoTezebis r –is mixedviT Semowmebisas gadawyvetilebebis miReba da daskvnebis gakeTeba. gamoyenebulia mniSvnelovnobis 0,05 done.

Tu realurir xvdeba uaryofis areSi, roca

05,0=α , maSin:

Tu realurir ar xvdeba uaryofis areSi,

roca 05,0=α , maSin:

• realurisr albaToba 0,05-ze naklebia an

misi tolia, anu 05,0≤p

• realurisr albaToba 0,05-ze metia, anu

p > 0,05.

• uarvyofT 0H : ρ = 0-s

• uarvyofT 0H -s

• viRebT 1H : ρ ≠ 0-s. realurir gviCvenebs,

rom populaciis ρ nulisagan

gansxvavebulia.

• uarvyofT 1H -s. realurir gviCvenebs,

rom populaciis ρ nulisagan

gansxvavebulia.

• realurir statistikurad

mniSvnelovania 0,05 doneze.

• realurir ar aris statistikurad

mniSvnelovani 0,05 doneze.

• viRebT gadawyvetilebas, rom im populaciaSi, saidanac SerCeva iyo

aRebuli, X da Y cvladebi korelaciaSia.

• viRebT gadawyvetilebas, rom im populaciaSi, saidanac SerCeva iyo

aRebuli, X da Y cvladebi ar aris korelaciaSi.


28

daSvebebi r -is statistikuri mniSvnelovnobis SemowmebisaTvis r -is statistikuri mniSvnelovnobis Semowmebisas igulisxmeba, rom populaciaSi X da Y organzomilebian normalur ganawilebas Yqmnis. organzomilebiani normaluri ganawilebis maxasiaTeblebia:

• X qulebi, martoobis qulebi Cvens magaliTSi, normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva;

• Y qulebi, depresiis qulebi Cvens magaliTSi, normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva;

• ToToeuli X qulis Sesabamisi Y qulebis ganawileba populaciaSi normaluria. Cveni magaliTis mixedviT, es piroba niSnavs, rom martoobis TiToeuli qulisTvis populaciaSi arsebobs depresiis qulebis normaluri ganawileba;

• ToToeuli Y qulisaTvis X qulebis ganawileba populaciaSi normaluria. es piroba wina pirobis msgavsia, magram axla depresiis TiToeuli qulisTvis populaciaSi arsebobs martoobis qulebis normaluri ganawileba.

ori qcevis gazomvisas SeiZleba davuSvaT, rom isini populaciaSi oragnzomilebian normalur ganawilebas qmnian da Sesabamisad, SegviZlia SevamowmoT hipoTeza populaciis qulebis aranulovani korelaciis Sesaxeb. organzomilebiani normaluri ganawileba (Bivariate normal distribuition): X da Y cvladebis iseTi ganawileba, romlisTvisac (1) X qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (2) Y qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (3) TiToeuli X qulisaTvis Y qulebis ganawileba populaciaSi normaluria da (4) TiToeuli Y qulisaTvis X qulebis ganawileba populaciaSi normaluria.

magaliTi #13.1

martoobasa da depresias Soris realurisr statistikuri mniSvnelovnobis

Semowmeba

amocana cxrilis #13.1 martoobisa da depresiis qulebisTvis pirsonis korelaciis

koeficientia realurir = +0,78. gamoxatavs Tu ara SerCevisaTvis miRebuli es

mniSvneloba aranulovani korelaciis arsebobas depresiisa da martoobis qulebs Soris im populaciaSi, saidanac moxda cdis pirebis SerCeva? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna: amocana moiTxovs, rom SevamowmoT statistikuri hipoTeza imis Sesaxeb, rom martoobisa da depresiis qulebisTvis populaciis korelacia, ρ, nulis tolia.


29

saWiro statistika: realurir = +0,78

daSveba gamoyenebisaTvis: martoobisa da depresiis monacemebs organzomilebiani normaluri ganawileba aqvs im populaciaSi, saidanac aRebulia SerCeva.

statistikuri hipoTezebi: 0H : ρ = 0 1H : ρ ≠ 0 mniSvnelovnobis done: 05,0=α

Tavisuflebis xarisxi (df): df = wyviliN _ 2 = 10 _ 2 = 8.

r -is kritikuli mniSvneloba: cxrilis #A.5 mixedviT, kritikulir (8) = 0,632.

uaryofis zonebi: realurisr is mniSvnelobebi, romelTa absoluturi

mniSvneloba udris an metia 0,632-ze.

gadawyvetileba: realurir = +0,78-is absoluturi mniSvneloba metia, vidre

kritikulir = 0,632. aqedan gamomdinare, is xvdeba uaryofis areSi. maSasadame,

uarvyofT 0H : ρ = 0-s viRebT 1H : ρ ≠ 0-s daskvna: populaciaSi martoobisa da depresiis qulebs Soris

mniSvnelovani korelacia arsebobs. realurir = +0,78 mxolod SemTxveviT ar

gansxvavdeba populaciaSi WeSmariti nulovani korelaciisgan. populaciis

korelaciis arsebuli saukeTeso Sefaseba aris realurir = +0,78. martosuli

adamianebi depresiulebic arian. determinaciis koeficienti, realuri2r =

(+0,78)² = 0,61, gamoxatavs, rom martoobisa da depresiis qulebs cvalebadobis 61 procenti saerTo aqvT.

magaliTi #13.2

dakavSirebulia Tu ara iumoris grZnoba fizikur janmrTelobasTan? amocana erT-erTi Tviseba, romelsac sxvebis daxasiaTebisas viyenebT, aris iumoris grZnoba. qerolis da Smidtis (Caroll & Shmidt, 1992) hopoTezis mixedviT, iumoris kargi grnoba SeiZleba dakavSirebuli iyos karg fizikur janmrTelobasTan, radgan is adamianebi, romlebic SfoTvis gamomwvev situaciebs iumoriT umklavdebian, SesaZlebelia, ufro advilad axerxebdnen stresuli situaciebis kontrols. davuSvaT, rom am hopoTezas amowmebT da amisaTvis 16 cdis pirs iumoris grZobasa da fizikur


30

janmrTelobas uzomavT. iumoris grZnoba izomeba situaciaze iumoriT pasuxis kiTxvariT (Situation Humor Response Questionnaire, Martin & Lefcourt, 1984). am kiTxvarSi warmodgenilia 18 situaciis aRwera da cdis pirs moeTxoveba, TiToeuli situacia 1-dan 5-mde skalaze Seafasos imis mixedviT, Tu ramdenad saxalisod eCveneba is. Sesabamisad, am kiTxvarze miRebuli qulebi SeiZleba meryeobdes 18-dan 90-mde, sadac ufro maRali qulebi miuTiTebs adamianis mier situaciebSi meti iumoris gamoyenebaze. fizikuri janmrTelobis gasazomad cdis pirebs avsebineben 20-kiTxvian TviTSefasebis kiTxvars janmrTelobis problemebis Sesaxeb. am kiTxvaris qulebi SeiZleba meryeobdes nulidan (janmrTelobis problemebis ararseboba) 20-mde (janmrTelobis 20 problema). am or sazomze Tqven Semdeg monacemebs iRebT:

cdis piri iumoris kiTxvari:

X fizikuri

janmrTeloba: Y

1 36 9 2 68 3 3 61 12 4 47 12 5 35 15 6 48 17 7 42 10 8 30 17 9 56 11 10 39 17 11 51 14 12 54 8 13 60 3 14 29 13 15 65 7 16 47 8

aris Tu ara iumoris kiTxvaris qula dakavSirebuli janmrTelobis problemebis raodenobasTan? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna: amocana moiTxovs, SevamowmoT statistikuri hipoTeza imis Sesaxeb, rom populaciaSi iumoris kiTxvarisa da janmrTelobis problemebis monacemebs Soris korelacia, ρ nulis tolia. saWiro statistika: r. daSveba gamoyenebisaTvis: iumoris kiTxvarisa da janmrTelobis monacemebs organzomilebiani normaluri ganawileba aqvs im populaciaSi, saidanac aRebulia SerCeva.

statistikuri hipoTezebi: 0H : ρ = 0 1H : ρ ≠ 0 mniSvnelovnobis done: 05,0=α


31

Tavisuflebis xarisxi (df): df = wyviliN _ 2 = 16 _ 2 = 14.

r -is kritikuli mniSvneloba: cxrilis #A.5 mixedviT, kritikulir (14) = 0,497.

uaryofis zonebi: realurisr is mniSvnelobebi, romelTa absoluturi

mniSvneloba udris an metia 0,497–ze.

gamoTvla: realurisr gamosaTvlelad gamoviyenebT gamoTvliT formulas:

( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑

∑∑∑−

−=

YYNXXN

YXXYNr

wyviliwyvili

wyvili

am formulisaTvis saWiro ricxviTi mniSvnelobebia:

wyviliN = 16

∑ XY = 7908

∑ X = 768

∑Y = 176

∑ 2X = 39,092

∑ 2Y = 2242

( )2∑ X = 589.824

( )2∑Y = 30.976.

am formulaSi ricxviTi mniSvnelobebis CasmiT miviRebT:

( ) ( )( )( )[ ] ( )[ ]

65,0079,211.13

8640976.30224216824.589092.3916

176768790816−=

−=

−−−

=realurir

realuri2r = (_0,65) 2 =0,42

gadawyvetileba: realurir = _0,65-is absoluturi mniSvneloba metia, vidre

kritikulir = +0,497 da maSasadame, xvdeba uaryofis areSi. gadawyvetilebebi 0H –

sa da 1H -Tan dakavSirebiT aseTia:

uarvyofT 0H : ρ = 0-s viRebT 1H : ρ ≠ 0-s daskvna: populaciaSi, saidanac aRebulia SerCeva, arsebobs mniSvnelovani uaryofiTi korelacia iumoris koTxvaris monacemebsa da janmrTelobis problemebs Soris. im cdis pirebs, romelTac iumoris kiTxvarSi maRali qulebi miiRes, naklebi janmrTelobis problemebi hqondaT, vidre im cdis pirebs, romelTac ufro dabali qulebi miiRes iumoris skalaze.


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determinaciis koeficienti aRniSnavs, rom iumoris kiTxvarisa da janmrTelobis monacemebs saerTo aqvT cvalebadobis 42 procenti. korelaciuri kvlevis Sedegebis warmodgena

korelaciuri kvlevis angariSi unda Seicavdes realurisr mniSvnelobas,

Tavisuflebis xarisxs (df) visrealurisaTr , gamoyenebul mniSvnelovnobis dones

da informacias imis Sesaxeb, iyo Tu ara realurir statistikurad

mniSvnelovani. xSirad angariSSi mocemulia monacemebis saSualo da standartuli gadaxrac. ilustraciisaTvis gamoviyeneb martoobasa da depresias Soris korelaciis magaliTs. alfas 0,05 doneze, martoobisa (M = 17,0; SD = 8,4) da depresiis (M = 28,0; SD

= 6,3) monacemebs Soris arsebobs mniSvnelovani korelacia, r (8) = +0,78, p < 0,01, martoobisa da depresiis qulebi mniSvnelovan dadebiT kavSirSia.

___________________________________________________________________ spirmenis ranguli korelaciis koeficienti

spirmenis ranguli korelaciis koeficienti, r s (Spearman rank-order correlation coefficient): korelaciis koeficienti, romelic rigis skalaze gamoiyeneba. pirsonis r koeficienti Sesaferisia maSin, roca im cvladebis Sesabamisi qulebi, romelTa korelaciis dadgenasac vapirebT intervalebis an Sefardebis skalazea. Tumca, zogjer qulebi X da Y cvladebis rangebs warmoadgens. rodesac rigis mixedviT dalagebuli qulebi gvaqvs, Cveulebriv monacemTa or simravles Soris mimarTebis ricxviTi mniSvnelobis dasadgenad gamoiyeneba spirmenis ranguli korelaciis

koeficienti, rac aRiniSneba Sr –iT. spirmenis ranguli korlaciis

koeficienti gamoiTvleba formuliT:

( )( )

−−= ∑

16

1 2

2

wyviliwyvili NND

rs

sadac D aris cdis piris monacemebis rangebs Soris gansxvaveba wyviliN aris kvlevaSi rangebis wyvilebis raodenoba

rogorc pirsonis r koeficientis SemTxveaSi, Sr -is mniSvnelobebic

SeiZleba iyos dadebiTi an uaryofiTi da iRebdes mniSvnelobas -1,00-dan nulis gavliT +1,00-mde.

Sr -is gamoyenebis magaliTi

yvelas SegvimCnevia, rom erTi da imave saxeobis cxovelebi gansxvavebulad iqcevian. cxovelebis qcevis gansxvavebebis Sesaswavlad fiverma, mendlma da beiTsonma (Feaver, Mendl, & Bateson, 1986) Seqmnes iseTi qcevebis gazomvis


33

meTodi, rogoricaa TamaSi, devna, gaqceva, miwaze garTxma da ase Semdeg. davuSvaT, rom mkvlevars, romelic am meTods iyenebs, aqvs hopoTeza, rom katebis TamaSisa ( X ) da devnis (Y ) qcevebi korelaciaSia. garkveuli drois ganmavlobaSi akvirdebian 10 katas erTsa da imave garemoSi da maTi TamaSisa da devnis qcevebis rangebs aniWeben. is kata, romelic yvelaze mets TamaSobs, miiRebs rangs 1, xolo is, romelic yvelaze naklebs TamaSobs, _ rangs 10. igive wesis mixedviT, kata, romelic yvelaze xSirad avlens devnis qcevas, miiRebs rangs 1 da kata, romelic yvelze iSviaTad avlens devnis qcevas, rangs 10. 10 katis TamaSisa da devnis qcevebis qulebi warmodgenilia cxrilis #13.10 a da b svetebSi. 1-li da me-10 katebi devnis qcevis mixedviT erTnairad Seafases. maSasadame, orive katas mianiWes rangi 6,5, rac aris 6-isa da 7-is, im ori rangis saSualo, rac maT im SemTxvevaSi mieniWeboda, qulebi bmuli rom ar yofiliyo.

Sr -is gamoTvlisaTvis saWiro procedurebi naCvenebia cxrilis #13.10

momdevno nawilSi. pirveli safexuria sxvaobis, D–s, povna rangebis ToToeuli wyvilis erTi rangis meorisaTvis gamoklebiT. es safexuri naCvnebia g svetSi, romelSic TiToeuli katis TamaSis rangs ( X ) gamoklebulia devnis rangi (Y). Semdeg TiToeuli D ayvanilia kvadratSi,

rac naCvenebia d svetSi. am svetis monacemebis jami, 80,50, gvaZlevs ∑ 2D -s

mniSvnelobas. formulaSi vsvamT ∑ 2D -isa da wyvilisN mniSvnelobebs da

viTvliT Sr -s: realuriSr = +0,51.

spirmenis ranguli korelaciis koeficientis statistikuri mniSvnelovnoba

Sr -is statistikuri mniSvnelovnobis Semowmeba iseve xdeba, rogorc

pirsonis r –sa. nulovani hopoTezaa, rom populaciaSi ranguli korelacia,

ρs, nulis tolia, anu 0H : ρs = 0. alternatiuli hipoTezis mixedviT,

populaciaSi korelacia ar aris nuli toli, anu 1H : ρs ≠ 0. Sr –is

kritikuli mniSvnelobebi 0,05 da 0,01 doneebze warmodgenilia danarTis

cxrilSi #A-9. kritikuli mniSvnelobebi damokidebulia wyvilzeN da ara

Tavisuflebis xarisxze. Tu realuriSr metia an tolia kritikuli

mniSvnelobisa, maSin unda miviRoT 0H –is uaryofisa da 1H –is miRebis

gadawyvetileba.

magaliTisaTvis, Sr -is kritikuli mniSvneloba maSin, roca wyviliN = 10

da 05,0=α , aris 0,648. radgan realuriSr = +0,51 kritikul mniSvnelobaze

naklebia, 0,05 doneze nulovan hopoTezas ar uarvyofT. TamaSisa da devnis qcevebis rangebs Soris mimarTeba ar aris statistikurad mniSvnelovani; SerCevaSi, romelzec xdeboda dakvirveba, TamaSisa da devnis qcevebi erTmaneTTan dakavSirebuli ar iyo.

Sr -is Tvisebebi

spirmenis korelaciis koeficienti gviCvenebs ori cvladis rangebs Soris SeTanxmebas. Tu X da Y cvladebs erTnairi rangebi aqvs miniWebuli, maSin


34

Sr iqneba +1,00. Tu cvladebi sapirispirod aris Sefasebuli ise, rom X

cvladis rangi 1 Seesabameba Y cvladis yvelaze dabal rangs da X

cvladis yvelaze dabali rangi Seesabameba Y cvladis rangs 1, maSin Sr

iqneba _1,00. Tu rangebs Soris arsebobs garkveuli, magram ara sruli SeTanxmeba, maSin 0-sa da 1,00-s Soris mniSvnelobas miiRebs. kavSiris mimarTulebas gamoxatavs korelaciis plus an minus niSani. cxrili #13.10

spirmenis ranguli korelaciis koeficientis ( sr ) gamoTvla 10 katis TamaSisa da

devnis qcevebisaTvis. a da b svetebSi warmodgenilia TiToeuli katis TamaSisa da devnis qcevebis rangebi.

(a) (b) (g) (d) kata TamaSi: X devna: Y ( )YXD −= ( )22 YXD −=

1 2 3 4 5 6 7 8 9 10

2 6 8 4 7 5 10 3 1 9

6,5 8 4 1 5 9 10 2 3 6,5

-4,5 -2 4 3 2 -4 0 1 -2 2,5

20,25 4,00 16,00 9,00 4,00 16,00 0,00 1,00 4,00 6,25

jami 80,50

Sr -is formulaa

( )( )

−−= ∑

16

1 2

2

swyviliwyvili

realuri NND

r

Tu CavsvamT ∑ = 50,802D da wyviliN = 10 mniSvnelobebs,

( )( )( )( )

ebTdavamrgval demeasedebam Tu

realuri

1,5,0512,0448,01

1101050,8061 2 s

+=+=−=

−

−=r


1. gansazRvreT: organzomilebiani normaluri ganawileba, ρ. 2. CamoayalibeT statistikuri hopoTeza, romelic saWiroa

imisTvis, rom SevamowmoT, mniSvnelovnad gansxvavdeba Tu ara

realurir nulisagan.

3. wyvilisN mocemuli mniSvnelobebidan TiToeulisaTvis

gansazRvreT Tavisuflebis xarisxi (df) da Semdeg cxrilis #A.5 mixedviT ipoveT kritikulir , roca α = 0,05 da α = 0,01. wyviliN =


35

a. 8 g. 15 e. 20 b. 32 d. 43 v. 75

4. qvemoT mocemuli ToToeuli realurisar da iswyvilisaTvN CamoayalibeT

nulovani da alternatiuli hipoTezebi; Semdeg ipoveT kritikulir

mniSvnelovnobis 0,05 doneze. gansazRvreT, xvdeba Tu ara realurir

uaryofis areSi da Semdeg miuTiTeT, Tu ra gadawyvetilebas iRebT statistikur hipoTezebTan dakavSirebiT.

rrealuri

wyviliN

a. b. g. d. e. v.

+0,580 -0,346 -0,682 +0,204 +0,249 -0,731

12 12 20 24 80 9

5. kolejis profesorebi xSirad miiCneven, rom leqciebze daswreba da saboloo Sefaseba (niSani) korelaciaSia. davuSvaT, rom profesori aRricxavda 12 studentis daswrebas da miiRo gacdenebisa da gamocdis Sefasebebis Semdegi saSualoebi:

studenti

gacdenebi

Ggamocdis Sefasebebis saSualo

1 22 72 2 6 88 3 1 99 4 8 68 5 6 66 6 11 77 7 10 52 8 6 78 9 10 76 10 0 96 11 5 91 12 10 86

a. am monacemebisaTvis gamoTvliTi formulis meSveobiT ipoveT

realurisr mniSvneloba

b. SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara realurir

populaciaSi nulovani korelaciisagan. g. ra daskvnas akeTebT gacdenebisa da gamocdebis saSualo

maCveneblebis mimarTebis Sesaxeb?


36

6. Sr -isa da wyvilisN TiToeuli mniSnelobisaTvis gansazRvreT

statistikuri nulovani hipoTezebi; Semdeg ipoveT kritikulir

mniSvnelovnobis 0,05 doneze. miuTiTeT, xvdeba Tu ara miRebuli

Sr uaryofis areSi da Semdeg miuTiTeT, ra gadawyvetilebas

iRebT statistikur hipoTezebTan dakavSirebiT.

Sr wyviliN

a. b. g. d. e. v.

+0,401 -0,620 +0,541 -0,366 +0,544 -0,377

17 12 24 10 21 30

Sejameba

• korelaciuri kvlevis mizania or an met cvlads Soris kavSiris xarisxis dadgena.

• wertilTa gabnevis diagramaze, ori monacemidan erT-erTi ( X cvladi) warmodgenilia horizontalur RerZze (abscisaze) da meore (Y cvladi) gadazomilia vertikalur RerZze (ordinataze). cdis piris qulas oridan TiToeul cvladze wertilTa gabnevis diagramaze erTi wertili warmoadgens.

• dadebiTi anu pirdapiri kavSiris SemTxvevaSi X cvladis mniSvnelobis zrdasTan erTad izrdeba Y cvladis mniSvnelobac.

• uaryofiTi, anu Sebrunebuli kavSiris SemTxvevaSi X cvladis mniSvnelobis zrdasTan erTad Y cvladis mniSvneloba mcirdeba.

• pirsonis korelaciis koeficienti ganisazRvreba Semdegnairad:

( )( )( )[ ] ( )[ ]∑∑∑

−−

−−=

22YYXX

YYXXr

• ( )( )∑ −− YYXX aris X da Y cvladebis jvaredina namravli da

aRiniSneba XYCP –iT. Sesabamisad, korelaciis koeficienti SeiZleba aseve gamoisaxos, rogorc:

( )( )YX

XY

SSSSCPr =

• standartuli qulebis gamoyenebiT r –is mniSvneloba mocemulia formuliT:

( )wyviliN

zzr YXΣ=

• r -is mniSvneloba gamoTvliTi formulis gamoyenebiT Semdegia:


37

( )( ) ( )( )( )( ) ( )[ ]( )( )( )[ ]2222 ∑∑∑∑

∑∑∑−

−=

YYNXXN

YXXYNr

wyviliwyvili

wyvili

• kavSiris mimarTulebas gamoxatavs r –is dadebiTi an uaryofiTi niSani

• xarisxi imisa, Tu ramdenad qmnis wertilTa gabnevis diagramis wertilebi wrfes, warmodgenilia r –is absoluturi mniSvnelobiT, romelic SeiZleba icvlebodes 0,00-dan 1,00-mde.

• populaciaSi X da Y cvladebs Soris wrfivi kavSiri warmodgenilia ρ –Ti.

• r -is statistikuri mniSvnelovnobis SemowmebisaTvis gamoiyeneba cxrili #A.5.

• determinaciis koeficienti 2r gamoxatavs X cvladis cvalebadobis im proporcias, romelic dakavSirebulia Y cvladis cvalebadobasTan.

• Tu X da Y cvladebs Soris korelacia arsebobs, maSin cvladebs Soris mizez-Sedegobrivi mimarTebis, sul mcire, sami SesaZlebloba arsebobs: X aris Y –is mizezi Y aris X –is mizezi arc X da arc Y ar axdens erTmaneTze mizez-Sedegobriv

gavlenas da orives mizezia mesame, Z cvladi.

• spirmenis ranguli korelaciis koeficienti gamoiTvleba formuliT:

( )( )

−−= ∑

16

1 2

2

wyviliwyvili NND

rs

• Sr -is statistikuri mniSvnelovnobis Semowmeba xdeba cxrilis #A.6 gamoyenebiT.


• dadebiTi kavSiri

• determinaciis koeficienti ( 2r )

• korelacia

• korelaciis koeficienti

• korelaciuri kvleva

• organzomilebiani ganawileba

• organzomilebiani normaluri ganawileba

• pirsonis korelaciis koeficienti ( r )

• ρ

• Sρ

• spirmenis ranguli korelaciis

koeficienti ( Sr )

• XSS

• YSS

• uaryofiTi kavSiri • wertilTa gabnevis diagrama

• wrfivi kavSiri • X -sa da Y -is jvaredina namravlebi

( XYCP )


38

kiTxvebi

1. pirveli Tavis kiTxvaSi #3 mocemuli iyo kvleva mamakacebis

gabrazebis donesa da sisxlis wnevas Soris kavSiris Sesaxeb. davuSvaT, rom am sakiTxiT dainteresebuli eqsperimentatori 16 mamakacisagan moipovebs gabrazebis donis qulebs da sisxlis sistolur wnevas (merkuris milimetrebSi). gabrazebis donis qulebi miRebuli iyo Sefasebis skalis gamoyenebiT, romelzec SesaZlo qulebi meryeobda 10-dan 100-mde da maRali qulebi niSnavda brazis ufro maRal dones. miRebuli iyo Semdegi qulebi:

cdis piri Bbrazis done sisxlis wneva

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

64 30 37 39 17 29 36 34 44 32 37 47 50 41 46 61

170 132 129 144 145 122 124 145 137 115 127 148 148 144 133 158

a. aageT wertilTa gabnevis diagrama gabrazebis

qulisa (abscisaze) da sisxlis wnevis monacemebisaTvis (ordinataze).

b. werilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi Tu kavSiri ar vlindeba?

g. gamoTvliTi formulis gamoyenebiT gamoTvaleT

realurir da SeamowmeT, mniSvnelovnad gansxvavdeba Tu

ara is populaciaSi nulovani korelaciisagan 0,05 doneze.

d. SerCevis safuZvelze ra daskvnas akeTebT gabrazebis donesa da sisxlis sistolur wnevas Soris kavSiris Sesaxeb?

e. gamoTvaleT 2r . am ori sazomisaTvis gabrazebis donisa da sisxlis wnevis cvalebadobis ra proporciaa saerTo?

v. Tqveni kvlevis monacemebi warmoadgineT am Tavis Sesabamis ganyofilebaSi (korelaciuri kvlevis


39

Sedegebis warmodgena) mocemuli magaliTis mixedviT.

2. adamianebisadmi Cvens reaqcias xSirad maTi garegnoba gansazRvravs. rogorc Cans, mamakacebis SemTxvevaSi mainc, simaRle xSirad sxvaTa mxridan dadebiT reaqciebs iwvevs. rogorc Zveli gamonaTqvami ambobs, `maRali, Savgvremani da simpaTiuri~. aseTi tipis msjelobidan gamomdinare, vilimezs, aisenbergsa da kerols (Villimez, Eisenberg, & Carroll, 1986) ainteresebdaT daedginaT, arsebobda Tu ara raime kavSiri dawyebiTi skolis bavSvebis fizikur sididesa da akademiur moswrebas Soris. maT miiRes bavSvebis sididis ramdenime maCvenebeli da Seadares isini testis qulebsa da niSnebis furclis niSnebs. imisaTvis, rom gaekontrolebinaT simaRleSi asakTan dakavSirebuli gansxvavebebi, yvela gamokvleuli

bavSvis asaki erTi da igive iyo. maT miiRes –0,12-is toli realurir

simaRlesa da niSnebis furclis niSnebs Soris 53 mesameklasel

gogonasTan da +0,32-is toli realurir 42 mesameklasel biWTan.

a. daadgineT, mniSvnelovnad gansxvavdeba Tu ara TiToeuli korelaciis koeficienti populaciaSi nulovani korelaciisagan 0,05 doneze.

b. realurisr TiToeuli mniSvnelobisaTvis gamoTvaleT

determinaciis koeficienti. g. SerCevis safuZvelze ra daskvnas akeTebT dawyebiTi

skolis bavSvebis simaRlesa da niSnebis furclis niSnebs Soris mimarTebis Sesaxeb?

3. studentebs xSirad ainteresebT, iwinaswarmetyvelon Tavisi saboloo Sefaseba Sualeduri gamocdis Sefasebis mixedviT. aseTi winaswarmetyvelebis gasakeTeblad saWiroa Sualeduri da saboloo gamocdebis Sefasebebis korelacia. davuSvaT, rom elementaruli statistikis klasSi Sualedur da saboloo gamocdebze 10-ma studentma miiRo Semdegi Sefasebebi:

gamocda

studenti Sualeduri: X saboloo: Y

1 2 3 4 5 6 7 8 9 10

87 89 97 80 73 85 81 74 80 89

84 91 96 87 66 90 79 80 89 86

a. daxazeT wertilTa gabnevis diagrama Sualeduri

(gadazomeT abscisaze) da saboloo (gadazomeT ordinataze) gamocdebisaTvis.


40

b. wertilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar vlindeba?

g. gamoTvleT realurir gamoTvliTi formulis

gamoyenebiT da SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.

d. SerCevis safuZvelze ra daskvnas akeTebT Sualeduri gamocdis niSnebsa da saboloo gamocdis niSnebs Soris kavSiris Sesaxeb?

4. krancma (Krantz, 1987) daadgina, rom fizikuri momxibvleloba da popularoba dadebiT korelaciaSia sabavSvo baRis aRsazrdel gogonebSi, magram ara biWebSi. davuSvaT, rom Tqven CaatareT msgavsi kvleva, romlisTvisac damkvirvebeli sabavSvo baRis moswavleebs afasebda fizikuri momxibvlelobisa da popularobis mixedviT. 12 gogonasa da 14 biWis Sefasebis Sedegad miiReT Semdegi rangebi:

gogonebi

fizikuri mimzidveloba:

X

popularoba: Y

3 6 2 9 12 1 7 8 4 10 5 11

1 7 3 10 11 3 8 7 5 9 6 12

biWebi fizikuri

mimzidveloba:

X

popularoba: Y

5 11 3 9 4 8 12 1

9 1 10 4 6 12 5 13


41

14 10 6 13 2 7

8 14 2 11 7 3

a. gamoTvaleT realuriSr gogonebis monacemebisaTvis da

SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.

b. SerCevis safuZvelze ra daskvnas akeTebT gogonebSi fizikuri momxibvlelobisa da popularobis rangebs Soris kavSiris Sesaxeb?

g. gamoTvaleT realuriSr biWebis monacemebisaTvis da

SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.

d. SerCevis safuZvelze ra daskvnas akeTebT biWebSi fizikuri momxibvlelobisa da popularobis rangebs Soris kavSiris Sesaxeb?

5. uongis (Wong, 1993) kvleva afasebda kolejis 76 students veqsleris inteleqtis skaliT (Wechsler Adult Intelligence Scale - Revised) da iTvlida miRebuli IQ-s korelacias studentebis niSnebis saSualosTan. IQ-s qulebis diapazoni iyo 83-dan 133-mde.

IQ qulebsa da niSnebis saSualos Soris realurir iyo +0,18, rac

aramniSvnelovania 0,05 doneze. gansazRvreT erTi SesaZlo mizezi imisa, Tu ratom ar aris korelacia or cvlads Soris am kvlevaSi.

6. samsaxurSi da saxlSi kompiuterebis sul ufro da ufro xSirad gamoyenebasTan erTad izrdeba interesi imis mimarT, Tu ra gavlenas axdens kompiuteris ekranis xangrZlivad yureba aRqmis unarze. vaisi, qimeli da Staini (Weiss, Kimmel, & Stein, 1993) adgendnen korelacias siRrmis aRqmis maCvenebelsa da uSualod aRqmis testis Catarebamde kompiuterTan gatarebuli saaTebis raodenobas Soris. isini zomavdnen cdis piris mier kompiuterTan gatarebuli saaTebis raodenobas da im Secdomis sidides (santimetrebSi), rasac cdis piri uSvebda siRrmis aRqmis testSi ori vertikaluri xazis erTmaneTis paralelurad dalagebisas. Tu cdis piris siRrmis aRqma idealuri iyo, Secdomis qula nuli santimetri iqneboda. Secdomad CaiTvleboda SemTxveva, roca cdis piri ver SeZlebda ori xazis paralelurad ganlagebas; rac ufro didia Secdoma, miT ufro cudia siRrmis aRqma. davuSvaT, rom Tqven SeafaseT 13 cdis piri maT mier kompiuterTan gatarebuli saaTebisa da siRrmis aRqmis testis mixedviT, uSualod kompiuterTan muSaobis damTavrebis Semdeg da miiReT Semdegi monacemebi:

cdis piri kompiuterTan

muSaobis xangrZlivoba

siRrmis aRqmis Secdoma (sm.):

Y


42

(sT.): X

1 2 3 4 5 6 7 8 9 10 11 12 13

0,5 1,7 4,2 6,0 2,8 5,3 0,9 3,4 1,4 5,1 1,9 4,8 2,6

4,3 3,1 2,1 0,4 6,3 3,7 1,6 1,2 2,4 5,2 0,6 4,1 5,4

a. aageT wertilTa gabnevis diagrama kompiuterTan

muSaobis xangrZlivobisa (gadazomeT abscisaze) da siRrmis aRqmis Secdomis (gadazomeT ordinataze) monacemebisaTvis.

b. werilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar vlindeba?

g. gamoTvliTi formulis gamoyenebiT gamoTvaleT

realuriSr da SeamowmeT, mniSvnelovnad gansxvavdeba Tu

ara is populaciaSi nulovani korelaciisagan 0,05 doneze.

d. SerCevis safuZvelze ra daskvnas akeTebT kompiuterTan muSaobis saaTebsa da siRrmis aRqmis Secdomas Soris kavSiris Sesaxeb?

7. hubneri da makkalafi (Huebner & McCullough, 2000) ikvlevdnen mozardebis skoliT kmayofilebasTan dakavSirebul faqtorebs. erT-erTi nakvlevi cvladi iyo skolasTan dakavSirebuli yovedlRiuri uaryofiTi movlenebis raodenoba. yoveldRiuri uaryofiTi movlenebi moicavs iseT aspeqtebs rogoricaa Relva akademiuri moswrebis gamo, cudi maswavleblis yola, an aramosawon adamianebTan urTierToba. davuSvaT, rom am sakiTxiT dainteresebuli mkvlevari 15 mozards sTxovs, rom awarmoon skolasTan dakavSirebuli yoveldRiuri uaryofiTi movlenebis Cawera da skoliT kmayofileba Seafason skalaze, romelzec qulebi meryeobs 6-dan (Zalian dabali kmayofileba) 48-mde (Zalian maRali kmayofileba). mkvlevarma Semdegi monacemebi miiRo:

studenti uaryofiTi

movlenebi skoliT

kmayofileba 1 2 3 4 5

7 14 8 19 6

34 14 24 15 29


43

6 7 8 9 10 11 12 13 14 15

10 17 7 12 17 6 9 13 22 5

27 14 38 19 21 29 27 18 18 30

a. daxazeT wertilTa gabnevis diagrama uaryofiTi movlenebis raodenobisa (gadazomeT abscisaze) da skoliT kmayofilebisaTvis (gadazomeT ordinataze).

b. wertilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar vlindeba?

g. gamoTvaleT realurir gamoTvliTi formulis

gamoyenebiT da SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.

d. SerCevis safuZvelze ra daskvnas akeTebT uaryofiTi movlenebis raodenobasa da skoliT kmayofilebas Soris kavSiris Sesaxeb?

8. pirvel TavSi warmodgenili iyo voerisa da qeivelis (Woehr & Cavell, 1993) da qeivelisa da voeris (Cavell & Woehr, 1994) naSromi fsiqologiis Sesavlis kursis pirveli gamocdaze warmatebis korelatebis Sesaxeb. erT-erTi gazomili cvladi iyo im saaTebis raodenoba, romelTa ganmavlobaSic studenti kviris manZilze televizors uyurebda. davuSvaT, rom Tqvens 17 megobars sTxoveT, CaeweraT kviraSi ramdeni saaTis ganmavlobaSi uyurebdnen televizors, raTa am monacemebis maTi pirveli gamocdebis SedegebTan korelacia dagedginaT. monacemebis mixedviT miiReT Semdegi mniSvnelobebi:

televizoris yurebisaTvis XSS =652; pirveli gamocdis

niSnisaTvis YSS =2094 da XYCP = _859.

a. am monacemebis gamoyenebiT ipoveT realurir

televizoris yurebasa da gamocdis niSans Soris.

b. realurisr mniSvneloba statistikurad mniSvnelovania

0,05 doneze? g. davuSvaT, rom erTi megobari televozors uyurebs

kviraSi 4 saaTis ganmavlobaSi, xolo meore – 15 saaTis ganmavlobaSi. Tqvens mier gamoTvlili korelacia ra winaswarmetyvevlebis gakeTebisaken gibiZgebT?

9. maswavlebelma, romelic dainteresebuli iyo bavSvis klaselebis mier mowonebasa da bavSvis klasSi CarTulobas Soris kavSiriT, am ori ganzomilebis mixedviT mexuTeklaseli


44

bavSvebis ranJireba moaxdina. davuSvaT, rom 16 baSvisaTvis miRebulia Semdegi rangebi:

bavSvi mowonebadobis

done CarTulobis

done 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

6 9 1 2 16 3 8 4 10 5 15 12 7 13 14 11

8 12 2 4 14 1 11 3 5 6 13 9 7 10 16 15

a. am qulebisaTvis ipoveT Sr -is mniSvneloba.

b. SeamowmeT da gansazRvreT, mniSvnelovnad

gansxvavdeba Tu ara miRebuli Sr populaciaSi

nulovani korelaciisagan. g. ra daskvnas akeTebT mowonebadobasa da klasSi

CarTulobas Soris kaSiris Sesaxeb?

Tavi XIV. regresia da winaswarmetyveleba ________________________________________________________________________

1

Tavi XIV

regresia da winaswarmetyveleba __________________________________________________________

wrfivi kavSiri wrfivi kavSiris gansazRvreba wrfis daxriloba wrfis Y -TanakveTa wrfivi kavSiris magaliTebi da maTi gantolebebi

regresiis wrfis dadgena magaliTi ufro realuri magaliTi umcires kvadratTa meTodi

X -is mixedviT Y -is winaswarmetyvelebisaTvis umcires kvadratTa

regresiis wrfis daxriloba da Y -TanakveTa umcires kvadratTa regresiis wrfis morgeba monacemebze

winaswarmetyvelebis Secdoma naSTi: winaswarmetyvelebis Secdomis sazomi Sefasebis standartuli Secdoma

XYs ⋅ -is gamoTvliTi formula

XYs ⋅ -is gamoTvla r -is mixedviT informacia, romelsac gvaZlevs Sefasebis standartuli Secdoma

Y -is mixedviT X -is winaswarmetyveleba regresiis wrfis gamoyeneba mravlobiTi regresia


2

me-13 TavSi korelaciis ganxilvisas aRvniSneT, rom Tu ori cvladi srul kavSirSia, ise, rogorc monacemebi suraTebze #13.2 da 13.4, erTi cvladis mniSvnelobis codna meore cvladis mniSvnelobis zusti winaswarmetyvelebis saSualebas gvaZlevs. magram erTi cvladis qulebi meore cvladis qulebis winaswarmetyvelebisaTvis maSinac SegviZlia

gamoviyenoT, roca X da Y cvladebi srul kavSirSi ar arian, magram maT

Soris aris statistikurad mniSvnelovani kavSiri. bevri mizezi arsebobs imisa, Tu ratom SeiZleba gvindodes erTi cvladis mixedviT meores winaswarmetyveleba. magaliTad, davuSvaT, rom gindaT icodeT, ras SeiZleba niSnavdes pirvel gamocdaze miRebuli qula Tqveni saboloo niSnisaTvis. am SemTxvevaSi Tqven gindaT, pirveli gamocdis niSnis mixedviT iwinaswarmetyveloT saboloo Sefaseba. kolejis misaRebi komisiis wevrebs surT, standartizebuli miRwevis testebis mixedviT iwinaswarmetyvelon niSnebis saSualo. personalis menejerebs SeuZliaT TanamSromlebis samsaxurSi warmateba samsaxureobrivi testis qulebis mixedviT iwinaswarmetyvelon. Tqvens eqims ki SeuZlia Tqveni samedicino gamokvlevis Sedegebis mixedviT iwinaswarmetyvelos garkveuli daavadebis riski. sxva magaliTebi qvemoTaa moyvanili. es Tavi exeba erTi cvladis mixedviT meores winaswarmetyvelebas im SemTxvevaSi, roca es ori cvladi wrfiv kavSirSia. Cvens mier gamoyenebuli midgoma gulisxmobs erTi qulis mixedviT meores winaswarmetyvelebis mizniT ori cvladis wertilTa gabnevis diagramisTvis wrfis morgebas, romelsac regresiis wrfe ewodeba. or cvladi SeiZleba arawrfivadac iyos dakavSirebuli, magram Cveni msjeloba mxolod wrfiv kavSirs exeba. Semdeg gagacnobT martoobis kiTxvaris pasuxebis mixedviT depresiis qulebis winaswarmetyvelebis magaliTs.

wrfivi kavSiri wrfivi kavSiris gansazRvra

wrfivi kavSiri: kavSiri or cvlads Soris, romlis drosac X cvladis

yoveli erTi erTeuliT cvlilebisas Y cvladi mudmivi sididiT icvleba. or cvlads Soris wrfivi kavSiri aris cvladebs Soris kavSiri, romelic

wrfiT aRiwereba. suraTze #14.1 warmodgenilia ori cvladi, X da Y , romlebic srul wrfiv kavSirSia. es suraTi saSualebas gvaZlevs wrfivi kavSiri ufro kargad ganvsazRvroT. wrfivi kavSiris dros, yovelTvis, roca erTi cvladi erTi erTeuliT icvleba, meore cvladic ganicdis

mudmiv cvlilebas. suraTze #14.1 X -is yoveli erTi erTeuliT zrdisas Y

0,5 erTeuliT izrdeba. magaliTad, roca X izrdeba 2-dan 3-mde (erTi


3

erTeuliT X RerZze), Y izrdeba 1,0-dan 1,5-mde (0,5 erTeuli Y RerZze). msgavsadve, roca X izrdeba 4-dan 5-mde (erTi erTeuliT X RerZze),

Y izrdeba 2,0-dan 2,5-mde (0,5 erTeuli Y RerZze). Sesabamisad, am magaliTSi,

X -is erTi erTeuliT cvlilebas Sesabamis sazomze mosdevs Y sazomze 0,5 erTeuliT mudmivi cvlileba. suraTi #14.1

X da Y cvladebs Soris sruli dadebiTi wrfivi kavSiri +0,5 daxrilobiT

wrfivi kavSiris moTxovna, rom X cvladis erTi erTeuliT cvlilebas

Tan unda axldes Y cvladis mudmivi sididiT cvlileba, ZalaSia rogorc

dadebiTi, aseve uaryofiTi kavSiris SemTxvevaSi. suraTi #14.2 gviCvenebs X -

sa da Y -s Soris srul uaryofiT kavSirs. yovelTvis, roca X 1 erTeuliT

icvleba, Y mudmivi sididiT icvleba. magaliTad, X -is erTi erTeuliT, 2-

dan 3-mde cvlilebisas Y 9,4-dan 8,6 mde icvleba, anu Y cvladis _0,8-iT

cvlileba xdeba. msgavsadve, X -is cvlilebas 8-dan 9-mde Tan axlavs Y –is

4,6-dan 3,8-mde cvlileba, anu Y cvladi isev _0,8-iT icvleba. suraTis #14.2

wrfivi kavSirisaTvis, X -is yoveli 1 erTeuliT cvlilebas Tan axlavs Y–is Sesabamisi da mudmivi cvlileba _0,8 erTeuliT. wrfis gantoleba: abXY += wrfis gantolebas Semdegi saxe aqvs:

abXY += ,

sadac X aris X cvladis Sesabamisi qula

Y aris Y cvladis Sesabamisi qula b aris wrfis daxriloba a aris konstanta, romelsac Y–is TanakveTa ewodeba es gantoleba gamoiyeneba X cvladis mniSvnelobis mixedviT Y cvladis mniSvnelobis winaswarmetyvelebisaTvis. wrfis daxriloba wrfis daxriloba: wrfis daxriloba mocemulia tolobiT

cvlileba ismniSvnelob is

cvlileba ismniSvnelobis

−−

=XY

b


4

suraTebze #14.1 da 14.2, X -is yovel 1 erTeuliT cvlilebas Tan axlavs

Y cvladis Sesabamisi da mudmivi cvlileba. X cvladis cvlilebasa da mis

Sesabamisad Y cvladis cvlilebis kavSirs warmoadgens im wrfis

daxriloba, romelic am ori cvladis mimarTebas asaxavs. kerZod, X da Y

cvladebis damakavSirebeli wrfis daxriloba aRiniSneba b–Ti da ganisazRvreba, rogorc

daxriloba = cvlileba ismniSvnelob is


−−

=XY

b

suraTi #14.2

X da Y cvladebs Soris sruli uaryofiTi wrfivi kavSiri _0,8 daxrilobiTa da +11

Y -TanakveTiT. es kavSiri gamoisaxeba, rogorc:

12

12

XXYYb

−−

=

sadac 1X da 2X aris X -is ori mniSvneloba

1Y da 2Y aris 1X -is da 2X –is Sesabamisi Y-is mniSvnelobebi suraTze #14.1 warmodgenili ricxviTi mniSvnelobebi saSualebas

gvaZlevs, 1X = 2 da 2X = 3 mniSvnelobebis gamoyenebiT X -isa da Y-is mniSvnelobebze gamavali wrfis daxriloba ganvsazRvroT. X –is am

mniSvnelobebis Sesabamisi Y –is mniSvnelobebia 1Y = 1 da 2Y = 1,5. maSasadame,

.5,015,0

2315,1

12

12 +=++

=−−

=−−

=XXYYb

am wrfis daxriloba aris +0,5; X -is yoveli erTi erTeuliT gazrdisas Y izrdeba 0,5 erTeuliT. wrfis daxriloba mudmivia mis mTel sigrZeze.

maSasadame, miuxedavad imisa, Tu X -is ra mniSvnelobebs gamoviyenebT, am

magaliTSi b mainc +0,5 iqneba. magaliTad, davuSvaT, rom gamoviyeneT

mniSvnelobebi 1X = 3 da 2X = 5. X -is am mniSvnelobebisTvis, Y -is Sesabamisi

mniSvnelobebia 1Y = 1,5 da 2Y = 2,5. aqedan gamomdinare,

5,021

355,15,2

+=++

=−−

=b .


5

roca viciT suraTze #14.1 gamosaxuli wrfis daxriloba, SegviZlia

davweroT am wrfis X da Y mniSvnelobebis damakavSirebeli gantoleba. es

gantolebaa:

XY 5,0+=

es gantoleba gvaZlevs X -is TiToeuli mniSvnelobis Sesabamisi Y-is mniSvnelobas da SegviZlia XY 5,0+= X -is sxvadasxva mniSvnelobebisTvis

Semdegnairad amovxsnaT:

Tu X = ____, maSin Y = ____. 0 1 2 3 4 5 6 7 8

0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

SeadareT Y –is es mniSvnelobebi grafikze #14.1 datanil Y-is mniSvnelobebs

X –is TiToeuli mniSvnelobis SemTxvevaSi. mxedvelobaSi miiReT, rom

XY 5,0+= aris gantoleba, romlic aRwers suraTze #14.1 warmodgenil wrfiv kavSirs. maSasadame, rom gvcodnoda mxolod es gantoleba da cdis

piris X qula, zustad SevZlebdiT Y qulis winaswarmetyvelebas.

bXY = gantoleba warmoadgens wrfis zogadi gantolebis nawils da

aRwers X -sa da Y-s Soris nebismier kavSirs im pirobiT, rom roca X = 0,

Y = 0. magram Tu Y ar udris nuls, roca X = 0? am problemas qvemoT davubrundebiT. wrfis Y -TanakveTa suraTi #14.3

X da Y cvladebs Soris wrfivi kavSiri +0,5 daxrilobiT da +3 Y -TanakveTiT

daakvirdiT suraTze #14.3 gamosaxul kavSirs X da Y cvladebs Soris. am

wrfis daxrilobac +0,5-ia, X -is yoveli erTi erTeuliT cvlilebisas Y

+0,5 erTeuliT icvleba. magram ar SeiZleba, es wrfe XY 5,0+= gantolebiT

aRiweros, radgan, am gantolebis mixedviT, Y unda iyos nulis toli, roca


6

X = 0. magram, suraTze #14.3 vxedavT, rom Y = 3, roca X = 0. modiT,

XY 5,0+= gantolebaSi CavsvaT X –is sxva mniSvneloba, magaliTad, X = 10.

amis Sedegad aRmovaCenT, rom Y unda iyos 5-is toli, roca X = 10. Tumca,

suraTze #14.3 vxedavT, rom Y = 8, roca X = 10. XY 5,0+= gantolebidan

miRebuli Y -isa da grafikze #14.3 datanili Y -is SedarebiT davadgenT, rom XY 5,0+= gantoleba gvaZlevs Y -is iseT mniSvnelobas, romelic 3 erTeuliT

naklebia grafikze datanil Y -Tan SedarebiT. maSin, davuSvaT, rom viyenebT XY 5,0+= gantolebas da misgan miRebuli Y -is TiToeul mniSvnelobas

vumatebT 3-s. maSin gantoleba aseTia:

35,0 ++= XY

es gantoleba ukve aRwers suraTze #14.3 X -sa da Y -s Soris kavSirs. 35,0 ++= XY gantolebis amoxsniT X -is sxvadasxva mniSvnelobebisaTvis

viRebT Semdegs:

Tu X = ____, maSin Y = ____. 0 2 4 10 13 19 24 26 30

3,0 4,0 5,0 8,0 9,5 12,5 15,0 16,0 18,0

Y -is es mniSvnelobebi zustad Seesabameba suraTze #14.3 aRniSnul Y-is mniSvnelobebs. maSasadame, 35,0 ++= XY aris gantoleba, romelic aRwers suraTze #14.3 gamosaxul wrfes.

es magaliTi gvaZlevs X -sa da Y -s Soris wrfivi kavSiris

gantolebis kerZo SemTxvevas. wrfis zogadi gantolebis gamosaxvisaTvis, am 35,0 ++= XY gantolebaSi +0,5 daxrilobas CavanacvlebT b asoTi da

konstanta 3-s _ a asoTi. Sedegad miviReT abXY += , wrfis zogadi gantoleba b daxrilobiTa da a-s toli Y –TanakveTiT. Y-TanakveTa aris Y-is mniSvneloba, roca X = 0. roca X = 0, wrfe kveTs Y RerZs da am wertilSi Y-is mniSvneloba udris Y-TanakveTas. suraTze #14.3 Y-TanakveTa aris +3, maSin roca suraTze #14.1 Y-TanakveTa aris 0.

Y TanakveTa: abXY += gantolebaSi Y-is mniSvneloba, roca X = 0.


7

wrfivi kavSirebis magaliTebi da maTi gantolebebi wrfivi kavSirebis da maTi aRmweri gantolebebis gasacnobad suraTi #14.4

warmoadgens X –sa da Y –s Soris rogorc dadebiTi, aseve uaryofiTi

wrfivi kavSirebis magaliTebs. TiToeuli naCvenebi kavSirisaTvis warmodgenilia wrfis gantoleba daxrilobiTa da Y TanakveTiT. magaliTad,

(b) grafiki X -sa da Y -s Soris uaryofiTi kavSiris saCveneblad

warmoadgens suraTze #14.2 gamoyenebul wrfiv kavSirs. am kavSiris Sesabamisi gantolebaa 118,0 +−= XY . aq Cven vxedavT, rom daxriloba aris

_0,8, minus niSani ki miuTiTebs, rom kavSiri uaryofiTia. erTi erTeuliT

X -is yoveli matebisas Y 0,8 erTeuliT mcirdeba. roca X nulis tolia,

wrfe Y RerZs kveTs +11 mniSvnelobaze; maSasadame, am gantolebisaTvis Y-TanakveTa aris +11. SeiswavleT TiToeuli magaliTi da TiToeuli warmodgenili kavSirisaTvis gansazRvreT b da a. suraTi #14.4

X da Y cvladebs Soris sxvadasxvagvari wrfivi kavSirebi: a) Y =+1,2 X +2;

b) Y = - 0,8 X +11; g) Y = -0,1 X + 30; d) Y = +0,2 X + 25; e) Y = +1,5 X _ 2; v) Y = -1,4 X +100

SeamowmeT Tqveni codna XIV-1

1. gansazRvreT: a, b, wrfivi kavSiri, wrfis daxriloba, Y-TanakveTa. 2. dawereT wrfis gantoleba. 3. dawereT wrfis daxrilobis gantoleba. 4. qvemoT mocemuli TiToeuli grafikisaTvis ipoveT daxriloba da Y-

TanakveTa. Semeg dawereT TiToeuli wrfis gantoleba. Aa. b.

g. d.

regresiis wrfis dadgena magaliTi


8

wina magaliTebi exeboda cvladebs, romlebic aRniSnuli iyo, rogorc

ubralod X da Y da abXY += gantoleba gamoiyeneboda X -is mixedviT

Y -is winaswarmetyvelebisaTvis. imisaTvis, rom gagacnoT, Tu rogor vadgenT gantolebas, romelic monacemTa or simravles Soris wrfiv kavSirs aRwers, davubrundebi me-13 TavSi warmodgenil amocanas martoobisa da depresiis qulebis kavSiris Sesaxeb. Cveni mizania, SevadginoT iseTi gantoleba, romelic saSualebas mogvcems, martoobis qulebis mixedviT depresiis qulebi viwinaswarmetyveloT.

davuSvaT, rom cxrilis #14.1 monacemebi miRebuli iyo 10

adamianisagan. es qulebi srul dadebiT kavSirSia, realurir = +1,00.

Sesabamisad, moveliT, rom gabnevis diagrama gamoavlens, rom qulebi wrfezea ganlagebuli, rogorc amas suraTi #14.5 gviCvenebs. martoobisa da depresiis qulebs Soris wrfivi kavSiris gantolebas viRebT wrfis daxrilobisa da Y-TanakveTis dadgeniT. orive es mniSvneloba vipoveT grafikis mixedviT: b = +1 da a = +10. aqedan gamomdinare, gantoleba,

romelic cxrilis #14.1 depresiis qulebs (Y ) martoobis qulebTan ( X )

akavSirebs, aris:

101 += XY anu

depresiis qula = (+1)(martoobis qula) + 10.

cxrili #14.1 martoobisa da depresiis hopoTeturi qulebi cdis 10 monawilisaTvis

monawile martoobis qula: X depresiis qula: Y

1 2 3 4 5 6 7 8 9 10

6 15 28 2 24 8 25 30 14 18

16 25 38 12 34 18 35 40 24 28

suraTi #14.5 cxrilSi #14.1 mocemuli martoobisa da depresiis qulebis wertilTa gabnevis diagrama


9

radgan qulebs Soris korelacia aris +1,00, Tu viciT ToToeuli cdis piris mxolod martoobis qula, am gantolebis gamoyenebiT SegviZlia zustad viwinaswarmetyveloT misi depresiis qula. cxrili #14.2 martoobis qulebis mixedviT nawinawarmetyvelebi depresiis qulebi Semdegi gantolebis gamoyenebiT:

depresiis qula = (+1,00) (martoobis qula) +10 a svetSi mocemulia martoobis realuri qula, b svetSi _ gantolebis mixedviT nawinaswarmetyvelebi depresiis qula da g svetSi _ depresiis realuri qula cxrilidan #14.1.

monawile

(a) martoobis qula: X

(b) nawinaswarmetyvelebi depresiis qula: Y′

Y (g) depresiis

realuri qula: Y 1 2 3 4 5 6 7 8 9 10

6 15 28 2 24 8 25 30 14 18

16 25 38 12 34 18 35 40 24 28

16 25 38 12 34 18 35 40 24 28

nawinaswarmetyvelebi depresiis qulebi mocemulia cxrilis #14.2 b svetSi. magaliTad, Ppirvel monawiles aqvs depresiis 6 qula. am gantolebis gamoyenebiT,

nawinaswarmetyvelebi depresiis qula = (+1) (martoobis qula) +10

da gamovTvliT, rom

nawinaswarmetyvelebi depresiis qula = (+1)(6) + 10 = 6 + 10 = 16.

depresiis qula, romelic martoobis 6-is toli mniSvnelobis mixedviT viwinaswarmetyveleT, 16-ia. martoobis sxva monacemebis mixedviT nawinaswarmetyvelebi depresiis qulebi warmodgenilia cxrilis #14.2 (b) svetSi. es nawinaswarmetyvelebi qulebi aRniSnulia Y′–iT, (ikiTxeba, rogorc `Y primi~ an `Y nawinaswarmetyvelebi~), imis gamosaxatad, rom isini nawinaswarmetyvelebia martoobis qulebis mixedviT Semdegi gantolebis meSveobiT:


10

101 +=′ XY . TiToeuli monawilis mier miRebuli realuri depresiis qula (rac aRiniSneba Y–iT), warmodgenilia (g) svetSi. yuradReba miaqcieT imas, rom nawinaswarmetyvelebi da realuri depresiis qulebi identuria. radgan

realurir = +1,00, Y ′ da Y erTi da igivea da martoobis qulebis mixedviT

depresiis qulebis winaswarmetyveleba SecdomasTan ar aris dakavSirebuli. ufro realuri magaliTi

cxrili #14.3 10 monawilis martoobisa da depresiis hipoTeturi qulebi. qulebis TiToeuli nakrebisaTvis mocemulia saSualo da standartuli gadaxra.

monawile

Mmartoobis qula:

X

Kdepresiis qula:

Y

1 4 16 2 27 37 3 18 33 4 7 23 5 30 34 6 12 32 7 18 24 8 23 29 9 19 26 10 12 26

X 17,0 Y 28,0

Xs 8,37 Ys 6,25

gamoyenebiTi TvalsazrisiT, cxrilis #14.1 monacemebi (qulebi) uCveuloa. iSviaTad an arasdros aris qulebi iseT srul kavSirSi, rogorc cxrilis #14.1 monacemebia. ufro realistur SemTxvevebSi ori cvladis qulebi gamoavlens srulze nakleb korelacias, rogorc es cxrilis #14.3 monacemebis SemTxvevaSia. es monacemebi gamoviyeneT me-13 TavSi pirsonis korelaciis koeficientis warmosadgenad (ix. cxrili #13.1).

realurir am monacemebisTvis aris +0,78, rac statistikurad mniSvnelovania

0,05 doneze.

suraTi #14.6 cxrilSi #14.3 mocemuli martoobisa da depresiis qulebis wertilTa gabnevis

diagrama. wertilta gabnevis diagramaze A da B wrfeebi pirobiTad aris gavlebuli


11

cxrilis #14.3 monacemebi warmodgenilia suraTze #14.6 gamosaxul

wertilTa gabnevis diagramaze. rogorc realurir = +0,78 gviCvenebs, es

monacemebi dadebiT kavSirSia; depresiis maRali qulebi dakavSirebulia martoobis maRal qulebTan da depresiis dabali qulebi dakavSirebulia martoobis dabal qulebTan. miuxedavad imisa, rom kavSiri martoobisa da depresiis qulebs Soris suraTze #15.6 ar aris sruli, is wrfivia. magram romeli iqneboda am monacemebis aRsaweri saukeTeso wrfe? cxadia, SeiZleba, rom gabnevis diagramaze ramdenime wrfe gaivlos. magaliTad, Cven am grafikze ori, A da B wrfe gavavleT. wrfeebi pirobiTad aris

gavlebuli garda im pirobisa, rom TiToeuli wrfe gadis X = 17 da Y = 28 wertilze. A wrfe ukeTesad Seesabameba da aRwers monacemebs Tu B wrfe? am kiTxvaze pasuxis gasacemad, raTa gadavwyvitoT ramdenad kargad Seesabameba TiToeuli wrfe korelaciaSi myofi monacemebis nakrebs, saWiroa garkveuli meTodi. umcires kvadratTa meTodi suraTze #14.6 TiToeuli, A da B wrfe, X –is TiToeuli mniSvnelobisTvis Y –is mniSvnelobis (anu Y ′ -is) winaswarmetyvelebis saSualebas gvaZlevs. magaliTad, martoobis 7 qulas A wrfis mixedviT Seesabameba 24,0-is toli Y ′ da 27-is tol martoobis qulas Seesaameba 32,0-is toli Y ′ . meores mxriv, B wrfe prognizirebs 20-is tol Y ′–s 36,0-is toli martoobis 7 qulisaTvis da 36-is tol Y ′ -s martoobis 27 qulisaTvis. es nawinaswarmetyvelebi Y qulebi warmodgenilia cxrilis #14.4 g svetSi A wrfisaTvis da v svetSi B wrfisaTvis. cxrili #14.4

A an B wrfeebis gamoyenebiT suraTis #14.6 X qulebis mixedviT Y qulebis

winaswarmetyvelebis Secdoma. Secdoma warmodgenilia realur Y qulasa da X -is

mixedviT nawinawarmetyveleb Y ′qulas Soris sxvaobiT, anu YY ′− -iT. a da b svetebSi warmodgenilia realuri X da Y qulebi. g svetSi warmodgenilia A

wrfis mixedviT nawinaswarmetyvelebi Y , svetebSi d da e - A wrfis mixedviT nawinaswarmetyvelebi Y qulis Secdoma da Secdomis kvadrati, v svetSi – B wrfis

mixedviT nawinaswarmetyvelebi Y , z da T svetebSi – B wrfis mixedviT nawinaswarmetyvelebi qulebis Secdoma da Secdomis kvadrati.

(a) (b) (g) (d) (e) (v) (z) (T)

A wrfis mixedviT

nawinaswarmetyvelebi

B wrfis mixedviT

nawinaswarmetyvelebi

realuri qulebi

qulebi Secdoma qulebi Secdoma


12

monawile X Y Y ′ YY ′− ( )2YY ′−

Y ′ YY ′− ( )2YY ′−

1 4 16 22,8 _6,8 46,24 17,6 _1,6 2,56 2 27 37 32,0 +5,0 25,00 36,0 +1,0 1,00 3 18 33 28,4 +4,6 21,16 28,8 +4,2 17,64 4 7 23 24,0 _1,0 1,00 20,0 +3,0 9,00 5 30 34 33,2 +0,8 0,64 38,4 _4,4 19,36 6 12 32 26,0 +6,0 36,00 24,0 +8,0 64,00 7 18 24 28,4 _4,4 19,36 28,8 _4,8 23,04 8 23 29 30,4 _1,4 1,96 32,8 _3,8 14,44 9 19 26 28,8 _2,8 7,84 29,6 _3,6 12,96 10 12 26 26,0 0,0 0,00 24,0 +2,0 4,00

jami 0,0 159,20 0,0 168,00 TiToeuli Y ′qula cdomilia realurad miRebul Y qulasTan SedarebiT da

YY ′− sxvaoba aris X qulidan Y qulis winaswarmtyvelebisas daSvebuli Secdomis sazomi. cxrili #14.4 gviCvenebs suraTis #14.6 A (d sveti) da B (z sveti) wrfeebis gamoyenebiT X qulebis mixedviT TiToeuli Y qulis winaswarmetyvelebasTan dakavSirebul Secdomebs. TiToeuli am wrfis mixedviT winaswarmetyveleba SecdomasTan aris dakavSirebuli, magram romelime wrfes mivyavarT Tu ara ufro mcire winaswarmetyvelebis SecdomasTan, vidre meores? SegviZlia, vcadoT, am kiTxvas vupasuxoT cxrilis #14.4 e da T svetebSi mocemuli Secdomebis SekrebiT. magram Tu svetebs SevajamebT, rogorc es cxrilSi #14.4,

aRmovaCenT, rom Secdomebis (cdomilebebis) jami ( )∑ ′−YY TiToeuli

wrfisaTvis nulis tolia. amitom Secdomebis jams ver gamoviyenebT, rogorc kriteriums wertilTa gabnevis diagramisa da wrfis Sesabamisobis dasadgenad; is yovelTvis nulis toli iqneba. igive problema gvqonda me-5 TavSi, rodesac vcdilobdiT saSualos garSemo qulebis cvalebadobis

sazomis dadgenas. im SemTxvevaSi davadgineT, rom ( )∑ − XX yovelTvis

nulis toli iyo. problemis gadasaWrelad XX − sxvaobebis dajamebamde

moxda TiToeulis kvadratSi ayvana. Semdeg ( )∑ −2

XX gamoviyeneT qulebSi

cdomilebis cvalebadobis sazomis misaRebad. amJamindeli problemac SeiZleba msgavsadve gadaiWras TiTeuli

YY ′− sxvaobis kavadratSi ayvaniT da YY ′− sxvaobebis kvadratebis SejamebiT. Semdeg ( )∑ ′− 2YY –s gamoviyenebT, rom ganvsazRvroT, romeli

wrfe Seesabameba saukeTesod wertilTa gabnevis diagramas. cxrili #14.4 e

da z svetebSi gviCvenebs TiToeuli, A da B , wrfisaTvis ( )∑ ′− 2YY –is

miRebas. ( )∑ ′− 2YY A wrfisaTvis aris 159,20 (ix. sveti e), maSin, roca B

wrfisaTvis es aris 168,0 (ix. sveti z). ( )∑ ′− 2YY -ze, rogorc


13

winaswarmetyvelebis Secdomis sazomze dayrdnobiT, A wrfe nakleb cdomilebas iwvevs, vidre B wrfe. umcires kavadratTa meTodiT miRebuli regresiis wrfe: wrfe,

romelic minimumamde amcirebs ( )∑ ′− 2YY -is mniSvnelobas.

wrfes, romelic minimumamde amcirebs ( )∑ ′− 2YY –is mniSvnelobas,

umcires kavadratTa meTodiT miRebuli regresiis wrfe ewodeba, radgan is minimalur mniSvnelobas aniWebs TiToeuli prognizirebuli qulis realuri qulisagan sxvaobis kvadratebis jamis mniSvnelobas. anu, is minimumamde

amcirebs ( )∑ ′− 2YY –s. suraTze #14.7 naCvenebia umcires kvadratTa regresiis

wrfe, romelic Seesabameba cxrilis #14.3 monacemebis gabnevis diagramas. am wrfis gantolebaa 1,18583,0 ++=′ XY . am wrfis maxasiaTebelia is, rom,

( )∑ ′− 2YY minimaluria; cxrilis #14.3 monacemebis Sesabamis sxva arc erT

wrfes ar eqneba ( )∑ ′− 2YY –is ufro mcire mniSvneloba. axla ki avxsni, Tu

rogor unda vipovoT umcires kvadratTa regresiis wrfis daxriloba da Y –TanakveTa. umcires kvadratTa regresiis wrfis daxriloba da Y –TanakveTa Y -is X –is mixedviT winaswarmetyvelebisaTvis

imisaTvis, rom X cvladis mixedviT Y cvladis winaswarmetyvelebisaTvis

vipovoT umcires kvadratTa regresiis wrfe, unda ganvsazRvroT im wrfis

daxriloba (b ) da Y –TanakveTa ( a ), romelic ( )∑ ′− 2YY –is mniSvnelobas

minimumamde amcirebs. suraTi #14.7 umcires kvadratTa meTodiT miRebuli regresiis wrfe 14.3 cxrilSi mocemuli martoobisa da depresiis qulebisaTvis

umcires kvadratTa meTodiT miRebuli wrfis daxriloba arsebobs umcires kvadratTa meTodiT miRebuli wrfis daxrilobis povnis ramdenime alternatiuli formula; TiToeuli formula gvaZlevs b –s erTsa da imave mniSvnelobas. gadaxraTa da jvaredina namravlebis formula. ori cvladis umcires kvadratTa regresiis wrfis daxriloba am qulebis maTi saSualodan gadaxrebis gamoyenebiT mocemulia formuliT:


14

daxriloba = ( )( )

( )∑∑

−

−−= 2

XX

YYXXb

gaixseneT, rom am formulis mricxveli ( )( )∑ −− YYXX , aris X –sa da Y –is

jvaredina namravli da aRiniSneba XYCP –iT. am formulis mniSvneli,

( )∑ −2

XX ,

aris X –is kvadratebis jami, anu XSS . Sesabamisad, es gantoleba SemiZlia

aseve Cavwero b –s jvaredina namravlebisa da kvadratebis jamis saxiT, rogorc:

daxriloba = X

XY

SSCPb =

gamoTvliTi formula. formula, romelic gamoTvlis ufro advil xerxs gvTavazobs, Semdegia:

daxriloba = ( )( ) ( )( )( )( ) ( )22 ∑∑

∑∑∑−

−=

XXN

YXXYNb

wyvili

wyvili

sadac wyviliN aris qulebis wyvilebis raodenoba.

korelaciuri formula. b –s mniSvneloba SeiZleba agreTve miviRoT r -is meSveobiT Semdegi gantolebis gamoyenebiT:

daxriloba =

=

X

Y

ssrb

sadac r aris realurir monacemTa or wyebas Soris

Ys aris populaciisaTvis gamoTvlili Y qulebis standartuli gadaxra

Xs aris populaciisaTvis gamoTvlili X qulebis standartuli gadaxra. am gantolebis gamoyenebas giCvenebT magaliTi #14.1. romeli formula unda gamoiyenoT b –s gamosaTvlelad? cxrilSi #14.5 warmodgenilia daxrilobis gamosaTvleli TiToeuli zemoT mocemuli


15

formula. monacemTa simravleze gamoyenebisas TiToeuli formula gvaZlevs b –s erTsa da imave mniSvnelobas. romeli formula unda amovirCioT monacemTa simravlis mixedviT b –s gamosTvlelad? zogadad, gamoTvliTi formulisaTvis saWiro gamoTvlebi ufro martivi Sesasrulebelia da naklebad mosalodnelia, rom Secdomamde migviyvans, vidre is gamoTvlebi, romlebic saWiroa gadaxraTa da jvaredina namravlebis formulebisaTvis. aqedan gamomdinare, gamoTvliTi formula umjobesia gamoiyenoT monacemTa mixedviT b –s gamosaTvlelad da misi gamoyeneba qvemoT iqneba naCvenebi.

Tumca, Tu Tqven ukve ipoveT realurisr mniSvneloba, yvelaze martivi iqneba

korelaciuri formula, radgan am SemTxvevaSi realurisr mniSvnelobis garda

Tqven mxolod Xs da Ys unda ipovoT. cxrili #14.5 umcires kvadratTa meTodiT miRebuli wrfis daxrilobis formulebi

gadaxraTa formula

( )( )( )∑

∑−

−−= 2

XX

YYXXb

jvaredina namravlebisa da kvadratebis jamis formula

X

XY

SSCPb =

gamoTvliTi formula

( )( ) ( )( )( )( ) ( )22 ∑∑

∑∑∑−

−=

XXN

YXXYNb

wyvili

wyvili

wyviliN aris qulebis wyvilebis raodenoba.

korelaciis formula

=

X

Y

ssrb

b -s povna gamoTvliTi formulis meSveobiT cxrili #14.6 gviCvenebs, Tu rogor xdeba gamoTvliTi formulis gamoyeneba imisaTvis, rom vipovoT suraTis #14.7 umcires kvadratTa regresiis wrfis daxriloba, romelic cxrilis #14.3 depresiisa da martoobis qulebs Seesabameba. cxrilis #14.7 a da b svetebi warmoadgens X da Y qulebs. am

svetebis jamebi gvaZlevs saWiro ∑ X –sa da ∑Y –s: ∑ X =170 da ∑Y = 280. g


16

svetSi miRebulia TiToeuli 2X da am svetis monacemebis jami, 3520,

gvaZlevs ∑ 2X -s. d sveti gvaZlevs TiToeuli XY -is mniSvnelobas (anu

TiToeuli X qulis mis Sesabamis Y qulaze namravls) da am svetis

monacemebis jami, 5127, gvaZlevs ∑ XY -s.

cxrilis #14.6 meore safexuri gviCvenebs am gantolebaSi ∑ X –is,

∑Y -is, ∑ XY -isa da ∑ 2X –is mniSvnelobebis Casmas. cxrilis #14.3

qulebisaTvis miRebuli b –s mniSvneloba aris +0,583. plus niSani miuTiTebs, rom daxriloba dadebiTia. Y -TanakveTa da umcires kvadratTa regresiis wrfe umcires kvadratTa regresiis wrfis Y -TanakveTa gamoiTvleba b –s mniSvnelobis mixedviT Semdegi formuliT:

XbYa −=

sadac Y aris Y qulebis saSualo

X aris X qulebis saSualo b aris umcires kvadratTa regresiis wrfis daxriloba, miRebuli

wina qveTavSi mocemuli erT-erTi formulis meSveobiT. vaCvenoT, Tu rogor xdeba am formulis gamoyeneba 14.3 cxrilis

qulebisaTvis, sadac Y = 28, X = 17 da b = +0,583. Sesabamisad,

( )( ) 1,18089,18911,92817583,028 ==−=+−=a meaTedebamde damrgvalebis Semdeg. am qulebisaTvis umcires kvadratTa meTodiT dadgenili regresiis wrfe Y RerZs kveTs Y =18,1 mniSvnelobaze. cxrili #14.6

b -s gamoangariSeba gamoTvliTi formulis gamoyenebiT. martoobis qulebi ( X ) a svetSi da depresiis qulebi (Y ) b svetSi aRebulia 14.3 cxrilidan.

safexuri I. ∑ X -is, ∑Y -is, ∑ 2X -isa da

∑ XY -is gamoTvla:

(a) (b) (g) (d) cdis piri

X

Y

2X

XY 1 4 16 16 64 2 27 37 729 999 3 18 33 324 594


17

4 7 23 49 161 5 30 34 900 1020 6 12 32 144 384 7 18 24 324 432 8 23 29 529 667 9 19 26 361 494 10 12 26 144 312

jamebi 170 280 3520 5127 safexuri II. b -s gamoTvla:

( )

,10,5127

,3520

,280

,900.28170

,170

2

22

=

=

=

=

==

=

∑∑∑∑∑

wyviliNXY

X

Y

X

X

da

( )( ) ( )( )( )( ) ( )22 ∑∑

∑∑∑−

−=

XXN

YXXYNb

wyvili

wyvili.

Tu CavsvamT ricxviT mniSvnelobebs,

( )( ) ( )( )( )( )

.583,063003670

900,28352010280170512710

+=+

=

−−

=b

umcires kvadratTa meTodiT dadgenili regresiis wrfis gantoleba roca a -sa da b -s mniSvlelobebs CavsvamT wrfis gantolebaSi

abXY +=′ ,

miviRebT, rom 1,18583,0 ++=′ XY .

maSasadame, umcires kvadratTa regresiis wrfes cxrilis #14.3 qulebisaTvis aRwers gantoleba 1,18583,0 ++=′ XY . imisaTvis, rom miviRoT Y –is

nawinaswarmetyvelebi mniSvnelobebi, am gantolebaSi unda CavsvaT X –is mniSvneloba da gantoleba amovxsnaT Y ′–is mimarT. Y –is nawinaswarmetyvelebi mniSvnelobebi cxrilis #14.3 qulebisaTvis Semdegia:


18

cdis piri

Tu X = ____, maSin Y ′= ____

1 4 20,4 2 27 33,8 3 18 28,6 4 7 22,2 5 30 35,6 6 12 25,1 7 18 28,6 8 23 31,5 9 19 29,2 10 12 25,1

Y ′ -is es mniSvnelobebi Zevs suraTze #14.7 gamosaxul wrfeze. umcires kvadratTa regresiis wrfis morgeba monacemebisaTvis axla giCvenebT, Tu rogor xdeba 1,18583,0 ++=′ XY -iT warmodgenili umcires kvadratTa regresiis wrfis morgeba suraTze #14.7 datanili qulebisaTvis. Tu cnobilia nebismieri ori wertili wrfeze, maSin Cven sakmarisi informacia gvaqvs grafikze wrfis asagebad. regresiis wrfisaTvis, wrfis erT-erTi wertilia Y -TanakveTa, _ Y ′ -is mniSvneloba, roca X =0. Cveni magaliTisaTvis Y –TanakveTa aris 18,1. maSasadame, wrfis erTi wertilis koordinatebia X =0, Y = 18,1. meore wertilis gageba SegviZlia gantolebaSi X –is sxva mniSvnelobis CasmiT da Y ′–is mimarT amoxsniT. miuxedavad imisa, rom SegviZlia CavsvaT X –is nebismieri mniSvneloba, viyenebT X –is

mniSvnelobas, roca, magaliTad, X =17,0. gantolebaSi X =17,0-is CasmiT viRebT:

( ) 0,281,1817583,0 =++=′Y

meaTedebamde damrgvalebiT.

Sesabamisad, roca X =17, 0,28=′Y . Y –is nawinaswarmetyvelebi mniSvneloba,

0,28=′Y , udris Y –is mniSvnelobas. umcires kvadratTa meTodiT miRebuli

nebismieri wrfisaTvis, roca X –is mniSvneloba X qulebis saSualos tolia, Y ′ toli iqneba realuri Y qulebis saSualosi. maSasadame, meore

wertili regresiis wrfeze yovelTvis iqneba X ,Y koordinatebis mqone

wertili. suraTis #14.7 qulebisaTvis, X =17, Y =28 da suraTze #14.7 regresiis wrfe am koordinatebze gadis.


19


1. gansazRvreT: umcires kvadratTa regresiis wrfe, Y ′ . 2. CawereT wrfis meSveobiT X –is mixedviT Y –is mniSvnelobis

gansasazRvri zogadi gantoleba. 3. testi #2-is niSnebis (Y ) testi #1-is niSnebze ( X ) regresiis wrfis

daxriloba da Y-TanakveTa aris b = +0,08 da a =9. a. dawereT testi #2-is niSnebis testi #1-is niSnebze regresiis

gantoleba b. ipoveT testi #2-is nawinaswarmetyvelebi niSnebis mniSvnelobebi

(anu Y ′ ) testi #1-is Semdegi qulebisaTvis (anu X -sTvis): 60, 65, 80, 85, 90.

g. ujrian furcelze aageT regresiis wrfe. 76=X . 4. davaleba #2-is qulebis (Y ) davaleba #1-is qulebze ( X ) regresiis

wrfis daxriloba da Y-TanakveTa aris b = _1,4 da a =23. a. dawereT davaleba #2-is qulebis davaleba #1-is qulebze

regresiis gantoleba b. ipoveT davaleba #2-is nawinaswarmetyvelebi qulebis

mniSvnelobebi (anu Y ′ ) davaleba #1-is Semdegi qulebisaTvis: (anu X -Tvis): 5, 8, 11, 14, 17.

g. ujrian furcelze aageT regresiis wrfe. .11=X winaswarmetyvelebis Secdoma

naSTi: winaswarmetyvelebis Secdomis sazomi naSTi: YY ′− -is mniSvneloba. rogorc suraTidan #14.7 davinaxeT, Y cvladis winaswarmetyvelebisaTvis X cvladis gamoyeneba prognozirebis garkveul SecdomasTan aris dakavSirebuli, Tu, ra Tqma unda, X -sa da Y –s Soris sruli korelacia ar aris. sxvaoba qulasa da mis nawinaswarmetyveleb mniSvnelobas Soris, YY ′− , aris am Secdomis sazomi da mas naSTi ewodeba. naSTi gamoiyeneba, rogorc safuZveli winaswarmetyvelebis sizustis sazomisa, romelsac Sefasebis standartuli Secdoma ewodeba. Sefasebis standartuli Secdomis gansaxilvelad davubrundeT cxrils #14.7. cxrilis #14.7 a da b svetebSi warmodgenilia cxrilis #14.3 martoobisa da depresiis qulebi. regresiis gantolebidan 1,18583,0 ++=′ XY

miRebuli depresiis nawinaswarmetyvelebi qulebi (anu Y ′ ) naCvenebia g svetSi. magaliTad, Y ′ me-6 cdis pirisTvis, romlis martoobis qulaa 12, udris

( )( ) 10,251,1812583,06 =++=′Y


20

imisaTvis, rom minimumamde Segvemcirebina damrgvalebis Secdoma Semdegi gamoTvlebisas, g svetis nawinaswarmetyvelebi mniSvnelobebi measedebamde davamrgvaleT. kargi iqneba, Tu gamoTvliT danarCen Y ′ mniSvnelobebs g svetSi, raTa darwmundeT, rom SegiZliaT am qulebisTvis regresiis gantolebaze muSaoba. TiToeuli cdis pirisaTvis naSTi, YY ′− , naCvenebia d svetSi. am naSTebis jami nulis tolia (damrgvalebis Secdomis farglebSi), amitom am jams winaswarmetyvelebis Secdomis sazomad ver gamoviyenebT. maSasadame, winaswarmetyvelebis Secdomis sazomis misaRebad naSTebi kvadratSi agvyavs, rogorc es e svetSia naCvenebi. e svetSi YY ′− mniSvnelobebis kvadratebis

SejamebiT viRebT kvadratebis jams, rasac naSTiSS , naSTebis kvadratebis

jami, ewodeba. am magaliTisaTvis naSTiSS udris ( )∑ ′− 2YY -s, anu 138,1597-s.

naSTiSS : ( )∑ ′− 2YY -is mniSvneloba.

cxrili #14.7 regresiis 1,18583,0 ++=′ XY gantolebis gamoyenebiT cxrilis #14.3 martoobis

qulebis ( X , a sveti) mixedviT depresiis nawinaswarmetyvelebi qulebis (Y ′ , g sveti) naSTis gamoTvla. naSTi, anu winaswarmetyvelebis Secdoma, mocemulia d svetSi da naSTis kvadratebi warmodgenilia e svetSi.

Sefasebis standartuli Secdoma

(a) (b) (g) (d) (e)

cdis piri X Y Y ′ YY ′− ( )2YY ′−

1 4 16 20,43 -4,43 19,6249 2 27 37 33,84 3,16 9,9856 3 18 33 28,59 4,41 19,4481 4 7 23 22,18 0,82 0,6724 5 30 34 35,59 -1,59 2,5281 6 12 32 25,10 6,90 47,6100 7 18 24 28,59 -4,59 21,0681 8 23 29 31,51 -2,51 6,3001 9 19 26 29,18 -3,18 10,1124 10 12 26 25,10 0,90 0,8100

jami: 138,1597


21

Sefasebis standartuli Secdoma ( XYs ⋅ ): ( )

2

2

−

′−∑wyviliN

YYgamosaxulebis

mniSvneloba me-5 TavSi qulebis simravlisaTvis cdomilebis cvalebadobis sazomi, standartuli gadaxra, s , miviReT qulebis am nakrebis SS -is gayofiT mis Tavisuflebis xarisxze da Semdeg miRebuli mniSvnelobidan kvadratuli fesvis amoRebiT. Y –is X –is mixedviT winaswarmetyvelebis Secdomis cvalebadobis msgavs sazoms, Sefasebis standartuli Secdomas, viRebT

naSTsSS mis Tavisuflebis xarisze ganayofidan kvadratuli fesvis amoRebiT.

naSTisTvisSS Tavisuflebis xarisxi (df ) udris wyviliN _ 2-s, sadac wyviliN aris

qulaTa wyvilebis raodenoba. aqedan gamomdinare, gazomvis standartuli

Secdoma, romelic aRiniSneba rogorc XYs ⋅ da gamoxatavs, rom

Y nawinaswarmetyvelebia X –is mixedviT, ganisazRvreba, rogorc:

gazomvis standartuli Secdoma = 2−

=⋅wyvili

naSTi

NSS

s XY

es igivea, rac

gazomvis standartuli Secdoma = ( )

2

2

−

′−= ∑

⋅wyviliN

YYs XY .

am magaliTSi, naSTiSS =138,1597 da wyviliN = 10; Sesabamisad, martoobis qulebis

mixedviT, depresiis qulebis winaswarmetyvelebis Sefasebis standartuli Secdoma aris

demeasadebam ulidamrgvaleb ,16,4210

1597,138=

−=⋅XYs

XYs ⋅ -is gansazRvreba

( )2

2

−

′−= ∑

⋅wyviliN

YYs XY

gviCvenebs, rom is warmoadgens nawinaswarmetyvelebi qulebis realur (miRebul) qulebTan Sedarebis Secdomis cvalebadobis sazoms. Tumca, radgan am formulisaTvis saWiroa YY ′− sxvaobis gamoTvla, is Zalian bevr dros moiTxovs da gamoTvlis SecdomasTan aris dakavSirebuli. amitom Seiqmna ori alternatiuli formula: erTia gamoTvliTi formula, romelic SeiZleba gamoviyenoT YY ′− sxvaobis gamoTvlis gareSe da meorea formula r –is gamoyenebiT.


22

XYs ⋅ -is gamoTvliTi formula

gamoTvliTi formula XYs ⋅ -is uSualod qulebis mixedviT gamoTvlisaTvis, naSTis povnis gareSe Semdegia:

( ) ( ) ( ) ( )( )[ ]( )

−

−−−

−=

∑ ∑∑∑∑∑ ∑⋅ 22

222

21

XXN

YXXYNYYN

NNs XY

wyvili

wyvili

wyvili

wyviliwyvili

sadac wyviliN aris qulaTa wyvilebis raodenoba.

es gantoleba ar aris iseTi rTuli, rogoric erTi SexedviT SeiZleba Candes. is gamoTvlebi, rac am formulis gamoyenebiT martoobisa da

depresiis qulebisTvis XYs ⋅ -is misaRebad aris saWiro, naCvenebia cxrilSi #14.8. am cxrilis pirveli safexuri moicavs am formulis amosaxsnelad saWiro ricxviT mniSvnelobebs. ToToeuli cdis pirisaTvis:

• a svetSi mocemulia martoobis qula, X

• b svetSi mocemulia depresiis qula, Y

• g svetSi mocemulia martoobis qulis kvadrati, ( 2X )

• d svetSi mocemulia depresiis qulis kvadrati ( 2Y )

• e svetSi mocemulia martoobisa da depresiis qulis namravli ( XY ) cxrili #14.8

XYs ⋅ -is gamoangariSeba gamoTvliTi formulis gamoyenebiT. martoobis qulebi ( X , a

sveti) da depresiis qulebi (Y , b sveti) aRebulia cxrilidan #14.3.

safexuri I. ∑ X -is, ∑Y -is, ∑ 2X -is, ∑ 2Y -sa da ∑ XY -is gamoTvla:

(a) (b) (g) (d) (e) cdis piri

X

Y

2X

2Y

XY 1 4 16 16 256 64 2 27 37 729 1369 999 3 18 33 324 1089 594 4 7 23 49 529 161 5 30 34 900 1156 1020 6 12 32 144 1024 384 7 18 24 324 576 432 8 23 29 529 841 667 9 19 26 361 676 494 10 12 26 144 676 312

jamebi 170 280 3520 8192 5127


23

safexuri IIi. XYs ⋅ -is gamoTvla:

( )

,10,5127

,3520

,280

,900.28170

,170

2

22

=

=

=

=

==

=

∑∑∑∑∑

wyviliNXY

X

Y

X

X

( )8192

400.78280

280

2

22

=

==

=

∑∑∑

Y

Y

Y

da

( ) ( ) ( ) ( )( )[ ]( )

( ) ( ) ( ) ( )( )[ ]( )

( )

Sedegad isdamrgvaleb demeasedebam

wyvili

wyvili

wyvili

wyviliwyvili

4,16

079,1382801

630036703520

801

900.28352010280170512710400.78819210

8101

21

2

2

22

222

=

=

−

=

−−

−−

=

−

−−−

−=

∑ ∑∑∑∑∑ ∑⋅

XXN

YXXYNYYN

NNs XY

∑ X aris a svetis jami, ∑Y aris b svetis jami, 2∑ X aris g svetis

jami, ∑ 2Y aris d svetis jami da 2∑ X aris e svetis jami. me-2 safexuri

gviCvenebs am formulaSi mniSvnelobebis Casmas. XYs ⋅ –is miRebuli mniSvneloba aris 4,16, igive, rac ganmartebiTi formulis gamoyenebiT miviReT.

XYs ⋅ -is gamoTvla r –is mixedviT gazomvis standartuli Secdoma SeiZleba agreTve gamoiTvalos X -sa da Y -s

Soris realurisar da Y cvladis standartuli gadaxris gamoyenebiT Semdegi

formuliT:

( )2realuri

wyvili

wyvili rNN

ss YXY −

−

−=⋅ 1

21

cxrilis #14.3 monacemebisaTvis,

25,6=Ys ,

78,0+=realurir ,


24

10=wyviliN .

maSasadame,

( ) ( )[ ] ( )( ) ,15,46638,025,678,0121011025,6 2 ==+−

−−

=⋅XYs

damrgvalebis Secdomis sazRvrebis gaTvaliswinebiT es igivea, rac ganmartebiTi an gamoTvliTi formulis meSveobiT miRebuli mniSvneloba.

rodesac wyviliN -s mniSvneloba didia, wyviliN _ 1-is wyviliN _ 2-Tan

Sefardeba uaxlovdeba 1,00-s da formula xSirad Caiwereba, rogorc

21 realurirss YXY −=⋅

XYs ⋅ -is gamosaTvleli formulebi Sejamebulia cxrilSi #14.9. cxrili #14.9

Sefasebis standartuli Secdomis ( XYs ⋅ ) formulebi

naSTebis formula

2−=⋅

wyvili

naSTi

NSS

s XY

an

( )2

2

−

′−= ∑

⋅wyviliN

YYs XY

gamoTvliTi formula

( ) ( ) ( ) ( )( )[ ]( )

−

−−−

−=

∑ ∑∑∑∑∑ ∑⋅ 22

222

21

XXN

YXXYNYYN

NNs XY

wyvili

wyvili

wyvili

wyviliwyvili

formula realurisr gamoyenebiT

( )2realuri

wyvili

wyvili rNN

ss YXY −

−

−=⋅ 1

21

informacia, romelsac gvaZlevs Sefasebis standartuli Secdoma


25

imisaTvis, rom gavigoT Tu ra informacias gvaZlevs Sefasebis standartuli Secdoma, cota xniT mivubrundebiT standartuli gadaxris, s -is, ganxilvas. qulebis simravlis standartuli gadaxra gvaZlevs am qulebis Secdomis (cdomilebis) cvalebadobas. Tu qulebis Sesaxeb araferi viciT garda maTi saSualosi, saukeTeso winaswarmetyveleba am qulebisaTvis aris maTi saSualo. ra Tqma unda, savaraudoa, rom es winaswarmetyveleba mcdari iqneba da saSualod, es Secdoma (cdomileba) standartuli gadaxris mniSvnelobis toli iqneba. magaliTad, davuSvaT, Tqven giTxres, rom cxrilis #14.3 monacemebis depresiis saSualo qula aris 28. Tu gTxoven, rom iwinaswarmetyveloT calkeuli qula, maSin, roca araferi iciT imis

garda, rom 0,28=Y , Tqveni saukeTeso winaswarmetyveleba iqneba 28,0. magram cxrilis #14.3 nebismieri qulisaTvis es winaswarmetyveleba mcdaria; arc erT cdis pirs miuRia depresiis 28 qula. qulaTa am simravlisaTvis aseTi winaswarmetyvelebis saSualo Secdoma aris standartuli gadaxris

mniSvneloba, 25,6=Ys . axla davuSvaT, TqvenTvis cnobili gaxda, rom martoobisa da depresiis qulebs Soris arsebobs wrfivi kavSiri, gamoxatuli, rogorc

1,18583,0 ++=′ XY . garda amisa, davuSvaT, TqvenTvis aseve cnobili gaxda, rom pirvel cdis pirs aqvs depresiis 4 qula. am qulis mixedviT da regresiis gantolebis gamoyenebiT Tqven iwinaswarmetyvelebT, rom pirovnebis Y qula iqneba 20,43. miuxedavad imisa, rom es winaswarmetyvelebac mcdaria (cdis piris realuri qula iyo 16), nawinaswarmetyvelebi 20,43 ufro axlosaa realur qulasTan,

vidre Y –is mixedviT nawinaswarmetyvelebi 28,0. Sefasebis standartuli Secdoma aris nawinaswarmetyvelebi qulebis Secdomis standartuli

gadaxra. maSasadame, XYs ⋅ gamoxatavs regresiis wrfis mixedviT prognozirebisas daSvebul saSualo Secdomas. yuradReba miaqceT imis, rom

Cvens magaliTSi XYs ⋅ = 4,16 naklebia 25,6=Ys -ze. aqedan gamomdinare, Tu Y –s

viwinaswarmetyvelebT X –is mixedviT, winaswarmetyvelebis Secdoma,

Cveulebriv, naklebi iqneba imasTan SedarebiT, ubralod Y rom gamogveyenebina yvela Y mniSvnelobis winaswarmetyvelebisaTvis. cxrilSi

#14.10 warmodgenilia Ys -sa da XYs ⋅ –s Soris kavSiri, rogorc X da

Y cvladebs Soris korelaciis funqcia. cxrili Sedgenilia realurisr

mniSvnelobebis CasmiT Semdeg formulaSi:

2 XY1 realurirss YXY −=⋅

yuradReba miaqcieT imas, rom roca realurir nulis tolia, maSin XYs ⋅ = Ys da

cdis piris martoobis qulis codna misi depresiis qulis prognozirebis sizustes ar zrdis. r –is mcire mniSvnelobebisTvis (magaliTad, r -is mniSvnelobaa 0,1, 0,2 an 0,3), X cvladis codna Y cvladis qulis winaswarmetyvelebis sizustis gazridisaTvis bevrs arafers akeTebs. meores mxriv, r -is didi mniSvnelobebi (magaliTad, r =0,7; 0,8 an 0,9) mniSvnelovnad


26

aumjobesebs winaswarmetyvelebis sizustes. magaliTad, Tu X –sa da Y –s

Soris korelacia aris 0,8, maSin XYs ⋅ 0,6-jer naklebia Ys –ze. roca r udris 1,00-s, maSin XYs ⋅ = 0; Y qulis winaswarmetyveleba X qulis mixedviT zustia da winaswarmetyvelebisas Secdomas ar vuSvebT. cxrili #14.10

XYs ⋅ -sa da Ys –s Soris kavSiri, rogorc r –is funqcia.

Tu realurir = _____ , maSin XYs ⋅ =_____ Ys . 0 1,000 0,1 0,995 0,2 0,980 0,3 0,954 0,4 0,917 0,5 0,866 0,6 0,800 0,7 0,714 0,8 0,600 0,9 0,436 1,0 0,000

X -is winaswarmetyveleba Y -is mixedviT

aqamde moyvanil magaliTebSi martoobis qulebi ( X ) gamoiyeneboda depresiis qulebis (Y ) winaswarmetyvelebisaTvis. magram SegveZlo, Segvebrunebina procesi da depresiis qulebi gamogveyenebina martoobis winaswarmetyvelebisaTvis. es mogvcemda imisgan gansxvavebuol regresiis gantolebas, rac depresiis martoobis mixedviT winaswarmetyvelebisaTvis gvqonda. cxrilSi #14.11 warmodgenilia martoobisa da depresiis qulebi cxrilis #14.3 Sebrunebuli X da Y aRniSvnebiT; depresiis qulebi axla aRniSnulia X -iT (a sveti) da martoobis qulebi Y -iT (b sveti). g da d

svetebSi mocemulia 2∑ X -sa da ∑ XY -is mniSvnelobebi, rac saWiroa

gamoTvliTi formulis gamoyenebiT wrfis daxrilobis gamoTvlisaTvis, rogorc amas cxrilis me-2 safexuri gviCvenebs. me-3 safexurze miRebulia Y -TanakveTa, a . depresiis mixedviT martoobis winaswarmetyvelebis regresiis wrfe warmodgenilia me-4 safexurze. es regresiis wrfe gamoiyeneboda e svetSi naCvenebi Y ′ mniSvnelobebis gamosaTvlelad. naSTebi warmodgenilia v svetSi da naSTebis kvadratSi ayvanili mniSvnelobebi – z svetSi. me-5

safexurze naCvenebia naSTebis mixedviT XYs ⋅ -is gamoTvla.

yuradReba miaqcieT imas, rom regresiis wrfis daxriloba da Y -TanakveTa damokidebulia imaze, Tu romeli cvladia aRniSnuli X -iT da


27

romeli - Y -iT. Tu martooba aris aRniSnuli X -iT, maSin regresiis wrfea 1,18583,0 ++=′ XY . magram, Tu depresiaa aRniSnuli X -iT, maSin regresiis

wrfea 2,12043,1 −+=′ XY . Tumca orive wrfe Y -iT aRniSnuli cvladis erTnairad zusti winaswarmetyvelebis saSualebas gvaZlevs. es Tviseba

SeiZleba davinaxoT TiToeuli wrfis XYs ⋅ -is Ys -Tan SedarebiT. Tu martooba

aris X cvladi, maSin XYs ⋅ aris 4,16 da Ys aris 6,25. am SemTxvevaSi

XYs ⋅ / Ys udris 4,16/6,25, rac udris 0,67-s. meores mxriv, Tu depresiaa X

cvladi, XYs ⋅ udris 5,56 da Ys aris 8,37. aq XYs ⋅ / Ys udris 5,56/8,37, rac udris 0,66-s. damrgvalebis Secdomis farglebSi, TiToeuli wrfe qulebis winaswarmetyvelebisas erTnair SecdomasTan aris dakavSirebuli. SemTxvevaTa umetesobaSi mkvlevars garkveuli mizezi aqvs imisaTvis, rom romelime cvladi aRniSnos X -iT da is Y -is winaswarmetyvelebisaTvis gamoiyenos. maSasadame, monacemebis umetesobisaTvis, xdeba mxolod erTi regresiis wrfis dadgena. cxrili #14.11 umcires kvadratTa regresiis wrfis, nawinaswarmetyvelebi qulebis da naSTebis gamoTvla martoobis qulebis depresiis qulebis mixedviT winaswarmetyvelebisaTvis.

safexuri I. ∑ X -is, ∑Y -is, ∑ 2X -isa da ∑ XY -is gamoTvla:

cdis piri

(a) (b) (g) (d) (e) (v) (z) depresia

( X )

martooba

(Y ) 2X XY Y ′ YY ′− ( )2YY ′−

1 16 4 256 64 4,488 -0,488 0,238 2 37 27 1369 999 26,391 0,609 0,371 3 33 18 1089 594 22,219 -4,219 17,800 4 23 7 529 161 11,789 -4,789 22,935 5 34 30 1156 1020 23,262 6,738 45,401 6 32 12 1024 384 21,176 -9,176 84,199 7 24 18 576 432 12,832 5,168 26,708 8 29 23 841 667 18,047 4,953 24,532 9 26 19 676 494 14,918 4,082 16,663 10 26 12 676 312 14,918 -2,918 8,515

jamebi 280 170 8192 5127 247,362


28

safexuri II. viTvliT b -s gamoTvliTi formulis gamoyenebiT:

( )

.105127

8192

170

400.78280

280

2

22

=

=

=

=

==

=

∑∑∑∑∑

wyviliNXY

X

Y

X

X

( )( ) ( )( )( )( ) ( )22 ∑∑

∑∑∑−

−=

XXN

YXXYNb

wyvili

wyvili

ricxviTi mniSvnelobebis CasmiT:

( )( ) ( )( )( )( ) 043,1

35203670

78400819210170280512710

+=+=−

−=b

safexuri 3. a -s gamoTvla:

.2,12)28(043,117

17

,28

−=−=−=

=

=

XbYa

Y

X

safexuri 4. regresiis wrfea:

2,12043,1 −+=′ XY

safexuri 5. naSTebis mixedviT XYs ⋅ -is gamoangariSeba:

( )

( ) ( )56,58362,247

2

82362,247

2

2

=÷=

−÷′−=

=−

=′−

∑

∑

⋅ wyvili

wyvili

NYYs

NYY

XY

regresiis wrfis gamoyeneba wrfivi regresiis saTanadod gamoyenebisaTvis saWiroa ori piroba. pirveli,

am qulebis realurir statistikurad mniSvnelovani unda iyos. Tu qulebs Soris

korelacia ar aris, maSin Y qulis saukeTeso winaswarmetyveleba iqneba Y qulebis saSualos gamoyeneba. Tu X da Y cvladebs Soris korelacia ar aris da Tqven gaZleven X qulas da gTxoven Y qulis winaswarmetyvelebas,

Y -is waukeTeso winaswarmetyveleba iqneba ubralod Y . X -is codna araviTar informacias ar gaZlevT Y -is mniSvnelobis Sesaxeb. meore piroba aris is, rom kavSiri X da Y cvladebs Soris wrfivi unda iyos. umcires kvadratTa meTodiT miRebuli wiri aris wrfe. Tu X da Y cvladebi ar aris wrfivad dakavSirebuli, maSin maTTvis umcires kvadratTa regresiis wrfis misadageba marTebuli ar aris. magaliTad, Tu X da Y cvladebs Soris kavSiri aris U –s an Sebrunebuli U –s (anu ) formis, maSin am qulebisaTvis umcires kvadratTa regresiis wrfis ageba ar migviyvanda zust winaswarmetyvelebamde, radgan es qulebi wrfivad dakavSirebuli ar aris. aseve frTxilad unda viyoT da ar gavavrceloT winaswarmetyveleba im SerCevis qulebis miRma, romlidanac winaswarmetyvelebis gantoleba iqna miRebuli. Cveni SerCevis sazRvrebs miRma mniSvnelobebisaTvis es kavSiri SeiZleba arawrfivi iyos.


29


1. ganmarteT: umcires kvadratTa meTodiT miRebuli regresiis wrfe,

naSTi, naSTiSS , Sefasebis standartuli Secdoma, XYs ⋅ , YY ′− . 2. ratom ar gamoiyeneba ( )∑ ′−YY , rogorc X –is mixedviT Y –is

mniSvnelobis winaswarmetyvelebis Secdomis sazomi?

3. ratom aris SesaZlebeli gamoviyenoT ( )∑ ′− 2YY , rogorc X –is

mixedviT Y –is mniSvnelobis winaswarmetyvelebis Secdomis sazomi? 4. dawereT Sefasebis standartuli Secdomis ganmartebiTi formula. 5. kvlevaSi, romlis mizani iyo gamocdis niSnebis winaswarmetyveleba

mecadineobisaTvis daTmobili saaTebis mixedviT, qulaTa 16

wyvilisaTvis miiRes Semdegi mniSvnelobebi: 72=∑ X , ∑ = 4082X ,

( ) 51842=∑ X , 1264=∑Y , ∑ = 636,1022Y , ( ) 696.597.12

=∑Y da ∑ XY =6140.

ras udris XYs ⋅ -is mniSvneloba am kvlevis monacemebisaTvis?

6. fiziologma 45 adamianze 3 Tvis dakvirvebis Sedegad daadgina realurir =

+0,41-is toli korelacia fizikuri varjiSis odenobasa da wonis

klebas Soris. Ys -is mniSvneloba iyo 5,2 funti. am klvevis

monacemebisaTvis ras udris XYs ⋅ ? 7. maswavlebelma, romelsac surs, rom Sualeduri gamocdis Sedegebis

mixedviT iwinaswarmetyvelos saboloo gamocdis niSnani, 27

studentisagan Sedgenil klasSi daadgina, rom naSTiSS =462,25. ras udris

am klevis monacemebisaTvis XYs ⋅ ?

magaliTi #14.1

regresiis wrfis dadgena korelaciis koeficientis mixedviT amocana kvebiTi darRvevebis - anoreqsiisa da bulimiis - kvlevis erTi mimarTuleba Seiswavlis pirovnebis sxeulis xatsa da mis sxva maxasiaTeblebs (sqesobrivi roli, aRqmuli momxibvlvloba, an TviTSefaseba) Soris mimarTebas. magaliTad, minci da betci (Mintz & Betz, 1986) kolejis studendebSi TviTSefasebis skalebis gamoyenebiT zomavdnen sakuTari sxeuliT kmayofilebasa da depresiisadmi midrekilebas. sxeuliT kmayofilebis qulebi meryeobda 1-dan (rac gamoxatavda sakuTari sxeulis mimarT Zlier dadebiT damokidebulebas) 7-mde (rac gamoxatavda sakuTari sxeulis mimarT Zlier uaryofiT damokidebulebas). depresiisadmi midrekilebis qulebic aseve meryeobda 1-dan (rac gamoxatavda depresiisadmi Zlier mdgradobas) 7-mde (gamoxatavda depresiisadmi midrekilebas). 23


30

saSualoze odnav naklebi wonis mdedrobiTi sqesis studentis jgufSi mincma da betcma aRmoaCines statistikurad mniSvnelovani korelacia sxeuliT kmayofilebis Sefasebasa ( X cvladi) da depresiisadmi

midrekilebis maCvenebels (Y cvladi) Soris, realurir = +0,44. davuSvaT, rom

gavimeoreT es kvleva saSualoze odnav naklebi wonis mqone 15 qalze da miviReT sxeuliT kayofilebasa da depresiisadmi midrekilebas Soris +0,58-

is toli korelacia realurir . roca Tavisuflebis xarisxi (df ) aris 13,

kritikulir = 0,514, maSasadame, realurir statistikurad mniSvnelovania. sxeuliT

kmayofilebis qulebis saSualo iyo X = 3,7 da standartuli gadaxra iyo 1,8.

depresiisadmi midrekilebisaTvis Y = 3,8 da Ys =1,9. ipoveT: (a) depresiisadmi midrekilebis sxeuliT kmayofilebaze umcires kvadratTa regresiis wrfe, (b) depresiisadmi midrekilebis nawinaswarmetyvelebi mniSvneloba, rogorc sxeuliT kmayofilebis qulebis funqcia da (g) Sefasebis standartuli Secdoma. amoxsna: amocana moiTxovs regresiis wrfis daxrilobisa da Y-TanakveTis mniSvnelobebis povnas. radgan realurir mocemulia, daxrilobas vpoulobT

korelaciis formulis gamoyenebiT, romelic warmodgenilia cxrilze #14.5. daxrilobis dadgena: daxrilobas, b -s, vpoulobT r –is mixedviT Semdegnairad:

=

X

Y

ssrb

realurisr , Xs -isa da Ys -is mniSvnelobebis CasmiT viRebT:

61,08,19,158,0 +=

+=b .

Y-TanakveTis povna: Y-TanakveTas vpoulobT Semdegnairad

( )( ) 54,17,361,08,3 =+−=−= XbYa

regresiis gantolebis miReba: depresiisadmi midrekilebis (Y ) sxeuliT kmayofilebaze ( X ) regresiis gantoleba aseTia:

54,161,0 ++=′ XY .

depresiisadmi midrekilebis nawinaswarmetyvelebi qulebis (Y ′ ) miReba: sxeuliT kmayofilebis qulebis ( X ) mixedviT depresiisadmi midrekilebis nawinaswarmetyvelebi qulebi (Y ′ ) miiReba regresiis gantolebis X –is


31

mimarT amoxsniT. magaliTad, sxeuliT kmayofilebis 1-is toli SefasebisaTvis regresiis ganotoleba Semdeg saxes iRebs:

15,254,1)1(61,0 =++=′Y .

1-is toli sxeulis kmayofilebis Sefasebis mixedviT nawinaswarmetyvelebi depresiisadmi midrekilebis qula aris 2,15, anu, Tu meaTedebamde davamrgvalebT, 2,2. regresiis wrfis ageba: regresiis wrfe, romlis gantolebaa

54,161,0 ++=′ XY , gavlebulia suraTze #14.8. sxeuliT kmayofilebis mniSvneloba, _ is cvladi, romlis mixedviTac xdeba winaswarmetyveleba ( X ), gadazomilia abscisaze da nawinaswarmetyvelebi cvladis, depresesiisadmi midrekilebis (Y ′ ), mniSvneloba gadazomilia ordinataze. regresiis wrfis gavleba xdeba

grafikze Y-TanakveTisa da X ,Y koordinatebis (Sesabamisad 3,7 da 3,8) aRniSvniT da am ori wertilis SemaerTebeli wrfis gavlebiT. depresiisadmi midrekilebis nawinaswarmetyvelebi qulebi X –is ToToeuli mniSvnelobisaTvis wrfeze wertilebiT aris aRniSnuli.

Sefasebis standartuli Secdomis povna: XYs ⋅ gamoiTvleba Semdegnairad:

( )2121

realuri

wyvili

wyvili rNN

ss YXY −

−

−=⋅

Tu CavsvamT: Ys =1,9, 15=wyviliN , realurir = +0,58-s, miviRebT

( ) ( )( ) 6,158,012151159,1 2 =+−

−−

=⋅XYs

meaTedebamde davamrgvalebis Sedegad. suraTi #14.8 umcires kvadratTa meTodiT miRebuli regresiis wrfe sxeulis xatisa da depresiisadmi midrekilebis qulebisaTvis. depresiisadmi midrekilebis nawinaswarmetyvelebi mniSvnelobebi regresiis wrfeze wertilebiT aris aRniSnuli.

es XYs ⋅ naklebia, vidre Ys =1,9. pirovnebis sxeuliT kmayofilebis qulis codna aumjobesebs depresiisadmi midrekilebis qulis winaswarmetyvelebis

sizustes Y =3,8–s, yvela cdis pirisaTvis nawinaswarmetyvelebi mniSvnelobis gamoyenebasTan SedarebiT.


32

magaliTi #14.2

regresiis wrfisa da Sefasebis standartuli Scdomis dadgena gamoTvliTi formulis mixedviT amocana amocanaSi #14.1 moyvanili iyo mincisa da betcis (Minz & Betz, 1986) kvleva sxeuliT kmayofilebasa da depresiisadmi midrekilebis qulebis mimarTebis Sesaxeb. cdis pirebisagan maT aseve miiRes socialuri TviTpativiscemis monacemebi. TviTpativiscemis skalebi iZleoda 20-dan (rac aRiniSnavda dabal TviTpativiscemas) 100-mde (aRniSnavda maRal TviTpativiscemas) qulebs. sxeuliT kmayofilebis qulebi icvleboda 1-dan (rac aRniSnavda Zlier dadebiT damokidebulebas sakuTari sxeulis mimarT) 7-mde (rac aRniSnavs Zlier uaryofiT damokidebulebas sakuTari sxeulis mimarT). saSualoze odnav naklebi wonis mqone qalebis kvlevisas mincma da betcma miiRes sxeuliT kmayofilebis qulebsa ( X ) da socialur TviTpativiscemas

(Y ) Soris statistikurad mniSvnelovani korelacia realurir = _ 0,60. davuSvaT,

rom am kvlevis 15 studentze gameorebisas miviReT sxeuliT kmayofilebisa da socialuri TviTSefasebis Semdegi qulebi:

studenti

sxeuliT kmayofileba: X

TviTpativiscema:

Y 1 2 76 2 4 48 3 6 27 4 1 70 5 3 41 6 5 38 7 6 37 8 3 61 9 7 33 10 1 91 11 4 36 12 2 55 13 3 49 14 5 51 15 4 52

gvinda, rom vipovoT: (a) socialuri TviTpativiscemis qulebis sxeuliT kmayofilebis qulebze umcires kvadratTa regresiis wrfe, (b) TviTpativiscemis nawinaswarmetyvelebi qulebi, rogorc sxeuliT

kmayofilebis funqcia, (g) XYs ⋅ –is mniSvneloba YY ′− naSTebis mixedviT da (d)

XYs ⋅ –is mniSvneloba gamoTvliTi formulis gamoyenebiT.


33

amoxsna: cxrili #14.12 gvaZlevs moTxovnili informaciis mosaZieblad saWiro mniSvnelobebs. am cxrilSi mocemulia Semdegi mniSvnelobebi:

sveti Aa sxeuliT kmayofilebis qulebi ( X ) da ∑ X

Bb TviTkmayofilebis qulebi (Y ) da ∑Y

Gg sxeuliT kmayofilebis qulebis kvadrati ( 2X ) da

∑ 2X

Dd TviTpativiscemis qulebis kvadrati ( 2Y ) da ∑ 2Y

Ee sxeuliT kmayofilebisa da TviTpativiscemis

qulebis jvaredina namravlebi da ∑ XY

v dadgenili regresiis wrfis mixedviT nawinaswarmetyvelebi Y ′ -is mniSvnelobebi

z naSTi TviTpativiscemis TiToeuli qulisaTvis cxrili #14.12 umcires kvadratTa regresiis wrfis, nawinaswarmetyvelebi qulebis da naSTebis dadgena TviTpativiscemisa da sxeuliT kmayofilebis qulebisaTvis

(a) (b) (g) (d) (e) (v) (z) studenti X Y 2X 2Y X Y Y ′ YY ′− 1 2 76 4 5776 152 64,859 +11,141 2 4 48 16 2304 192 48,837 -0,837 3 6 27 36 729 162 32,815 -5,815 4 1 70 1 4900 70 72,870 -2,870 5 3 41 9 1681 123 56,848 -15,848 6 5 38 25 1444 190 40,826 -2,826 7 6 37 36 1369 222 32,815 +4,185 8 3 61 9 3721 183 56,848 +4,152 9 7 33 49 1089 231 24,804 +8,196 10 1 91 1 8281 91 72,870 +18,130 11 4 36 16 1296 144 48,837 -12,837 12 2 55 4 3025 110 64,859 -9,859 13 3 49 9 2401 147 56,848 -7,848 14 5 51 25 2601 255 40,826 +10,174 15 4 52 16 2704 208 48,837 +3,163 jamebi 56 765 256 43,321 2480 saSualo 3,73 51,00 s 1,83 17,54


34

regresiis wrfis povna b -s gamoangariSeba gamoTvliTi formulis saSualebiT:

( )

,2480

,765

,256

,313656

,56

2

22

=

=

=

==

=

∑∑∑∑∑

XY

Y

X

X

X

15=wyviliN ,

da

( )( ) ( )( )( )( ) ( )22 ∑∑

∑∑∑−

−=

XXN

YXXYNb

wyvili

wyvili


( )( ) ( )( )( )( ) ( ) .011,8

7045640

31362561576556248015

−=−

=−−

=b

gamovTvaloT a : X = 3,73 da Y = 51,00. am mniSvnelobebis CasmiT

XbYa −= –Si viRebT

( )( ) .881,8073,3011,800,51 =−−=a L regresiis wrfe: regresiis wrfe warmodgenilia gantolebiT:

881,80011,8 +−=′ XY .

TviTpativiscemis nawinaswarmetyvelebi qulebi: regresiis gantolebidan nawinaswarmetyvelebi TviTpativiscemis qulebi mocemulia cxrilis v svetSi. imisaTvis, rom minimumamde SevamciroT damrgvalebis Secdoma naSTebis

mixedviT XYs ⋅ -is gamoTvlisas, Y ′ -is mniSvnelobebs vamrgvalebT meaTasedebamde.

XYs ⋅ naSTebis mixedviT: naSTebis mniSvnelobebi mocemulia cxrilis z

svetSi. XYs ⋅ -is mniSvneloba am naSTebidan Semdegnairad miiReba:

( )2

2

−

′−= ∑

⋅wyviliN

YYs XY

( )∑ ′− 2YY -is misaRebad z svetSi mocemuli TiToeuli naSTi agvyavs

kvadratSi da Semdeg vkrebT kvadratSi ayvanil 15 mniSvnelobas.

( )∑ =′− 7380,12932YY . wyviliN _ 2 = 15 _ 2 = 13.


.98,913

7380,1293==⋅XYs

XYs ⋅ gamoTvliTi formulis mixedviT: gamoTvliTi formulaa:


35

( ) ( ) ( ) ( )( )[ ]( )

−

−−−

−=

∑ ∑∑∑∑∑ ∑⋅ 22

222

21

XXN

YXXYNYYN

NNs XY

wyvili

wyvili

wyvili

wyviliwyvili


( )( ) ( ) ( ) ( )( ) ( )( )[ ]( )( ) ( )

( )

.98,9909,405.19195

1

7045640590.64

1951

562561576556248015765321.4315

13151

2

2

22

=

=

−−

=

−−

−−

=⋅XYs

am formuliT miRebuli XYs ⋅ -is mniSvneloba igivea, rac ganmartebiTi formulisa da naSTebis gamoyenebiT miRebuli mniSvneloba. daskvna: regresiis gantoleba

881,80011,8 +−=′ XY saSualebas gvaZlevs, winaswarmetyvelebis mniSvnelovnad Semcirebuli SecdomiT viwinaswarmetyveloT TviTpativiscemis qulebi sxeuliT

kmayofilebis qulebis mixedviT. XYs ⋅ = 9,98 sidide, regresiis gantolebis mixedviT gamoTvlili TviTpativiscemis qulis winaswarmetyvelebasTan

dakavSirebuli saSualo Secdoma ufro mcirea, vidre 54,17=Ys - Secdomis

is saSualo sidide, rac Y -is (anu 51,0), rogorc yvela cdis pirisaTvis TviTSefasebis nawinaswarmetyvelebi qulis gamoyenebasTan aris dakavSirebuli.

magaliTi #14.3

samsaxureobrivi warmatebis winaswarmetyveleba regresiis gantolebis mixedviT amocana Sromis fsiqologiis ZiriTadi aspeqtia biznes da samTavrobo organizaciebSi samsaxuris dawyebis msurvelebis (aplikantebis) samsaxureobrivi warmatebis winaswarmetyveleba. am winaswarmetyvelebebis gasakeTeblad xSirad regresiuli analizi gamoiyeneba. aplikantebs utardebaT samuSaosTan dakavSirebuli testi da am testis qulebi gamoiyeneba samuSaoze potenciuri warmatebis winaswarmetyvelebisaTvis. ganvixiloT martivi magaliTi. davuSvaT, rom kompania samuSaos msurvels


36

(aplikants) utarebs tests `samuSaos nimuSi~. testi `samuSaos nimuSi~ aris kompaniaSi Sesabamisi samsaxuris miniaturuli versia. testis `samuSaos nimuSi~ Sedegebi ( X cvladi) gamoiyeneba samsaxurSi realuri warmatebis (Y ′ ) prognozirebisaTvis regresiis gantolebaze dayrdnobiT:

557,0 ++=′ XY

davuSvaT, rom 12 aplikants Cautarda testi `samuSaos nimuSi~ da miRebulia Semdegi qulebi:

aplikanti testis `samuSaos nimuSi~ qula: X 1 12 2 37 3 27 4 34 5 17 6 39 7 24 8 40 9 47 10 15 11 31 12 21

kompania ar daiqiravebs iseT adamians, romlis samuSao kvalifikacia 75 procentze naklebia. romel aplikantebs SeuZliaT samsaxurSi moxvedra maTi `samuSaos nimuSze~ qulebis mixedviT? amoxsna: amoxsnisaTvis saWiroa TiToeuli aplikantis `samuSaos nimuSis~ testis qulis mixedviT regresiis gantolebis gamoyenebiT miviRoT samuSao kvalifikaciis nawinaswarmetyvelebi qula. qvemoT naCvenebia TiToeuli aplikantis samuSao kvalifikaciis nawinaswarmetyvelebi qula:

aplikanti NsamuSaoze efeqturobis

nawinaswarmetyvelebi qula: Y ′ 1 64,1 2 83,1 3 75,5 4 80,8 5 67,9 6 84,6 7 73.2 8 85,4 9 90,7


37

10 66,4 11 78,6 12 71,0

# 1, 5, 7, 10 da 12 aplikantebis nawinaswarmetyvebi qulebi saWiro 75 qulaze dabalia; Sesabamisad, es aplikantebi samuSaoze misaRebad Sesaferisebi ar arian. yvela sxva aplikantis qula 75 an ufro maRalia, maSasadame, es aplikantebi samuSaoze misaRebad Sesaferisebi arian.

557,0 ++=′ XY regresiis wrfe naCvenebia suraTze #14.9. suraTi #14.9 regresiis wrfe 557,0 ++=′ XY , romelic gamoiyeneba samuSaoSi efeqturobis

dasadgenad samsaxuris nimuSis testis miedviT. samuSaos nimuSisaTvis nawinaswarmetyvelebi samuSaoSi efeqturobis qulebis regresiis wrfeze wertilebiT aris aRniSnuli. gaiTvaliswineT, rom samuSao kvalifikaciis nawinaswarmetyvelebi 75 qula Seesabameba `samuSaos nimuSis~ testis 26,3 qulas. nebismieri, 26,3-ze dabali samuSaos nimuSis testis qula niSnavs samuSao kvalifikaciis efeqturobis nawinaswarmetyveleb 75 qulaze naklebs. aqedan gamomdinare, im aplikantebs, romelTac eqnebaT 26,3 qulaze naklebi samuSaos nimuSis testSi, samsaxurSi ar miiReben (ar daiqiraveben). mravlobiTi regresia mravlobiTi regresia: Y qulis mniSvnelobebis winaswarmetyveleba X cvladis ramdenime mniSvnelobis gamoyenebiT regresiuli analizi farTod gamoiyeneba bihevioristul mecnierebebsa da mTel rig praqtikul SemTxvevebSi. maSin, roca Cvens yvela amocanaSi warmodgenili iyo X da Y cvladebi, praqtikaSi X cvladis mixedviT Y cvladis winaswarmetyvelebisaTvis SesaZlebelia cnobili iyos mxolod X cvladi da regresiis gantoleba, sadac regresiis gantoleba miRebulia im SemTxvevebidan, roca cnobili iyo X –ca da Y –c. magaliTad, saTbobi sawvaviT movaWres wina kviris saSualo temperaturis ( X cvladi) mixedviT SeuZlia iwinaswarmetyvelos, ramdeni sawvavis SekveTa dasWirdeba mas Semdegi kvirisaTvis (Y cvladi). anda gamyidvels SeuZlia, iwinaswarmetyvelos vaWrobis moculoba (Y cvladi) rekalmaze daxarjuli Tanxis odenobis ( X cvladi) mixedviT. Cvens TiToeul magaliTSi Y cvladis winaswarmetyvelebisTvis gamoiyeneboda mxolod erTi X cvladi. Tumca praqtikaSi regresiis gantolebebi ufro rTulia imasTan SedarebiT, rac warmogidgineT da Y -is winaswarmetyvelebisaTvis iyenebs ramdenime X cvlads. am midgomas mravlobiTi regresia ewodeba, radgan gamoiyeneba


38

mravali prediqtori. magaliTad, damqiravebels SeiZleba ainteresebdes potenciuri TanamSromlis samuSaoze warmateba, rogorc miRwevis testis qulis, standartizebuli intervius qulisa da samuSaos nimuSis testis qulis funqcia. am midgomis gamomxatveli zogadi regresiis gantoleba aseTia:

aXbXbXbY +++=′ 332211

sadac Y ′aris samsaxurSi warmatebis nawinaswarmetyvelebi qula, 1X , 2X da

3X aris prediqtori cvladebi (am SemTxvevaSi 1X aris miRwevis testis

qula, 2X aris standartizebuli intervius qula da 3X aris samuSaos

nimuSis testis qula); 1b , 2b da 3b aris Sesabamisad 1X -is, 2X -sa 3X -is

regresiis koeficientebi, xolo a aris Y -TanakveTa. regresiis koeficientebi warmoadgens TiToeuli X cvladis wonas Y

cvladis winaswarmetyvelebisas. magaliTad, Tu 1b = +0,18, 2b = +0,10, 3b = +0,34

da a = 30,3, maSin gantoleba aseT saxes iRebs:

3,3034,010,018,0 321 ++++=′ XXXY

Tu aplikantis miRwevis testze aqvs 75 qula, interviuze 20 qula daagrova da samuSaos nimuSis testze - 90 qula, maSin

( )( ) ( )( ) ( )( ).4,763,306,300,25,13

3,309034,02010,07518,0=+++=

+++=′Y

mravlobiTi regresiis gantolebebi farTod gamoiyeneba qceviT mecnierebebSi winaswarmetyvelebisaTvis. magaliTad, fridmeni da kolegebi (Friedman et al., 1993) sicocxlis xangrZlovobis winaswarmetyvelebisaTvis iyenebdnen bavSvis pirovnebis Svid ganzomilebas: komunikabelurobas, fizikur energias, mxiarulobas, keTilsindisierebas, motivacias, guneba-ganwyobilebis mdgradobas da inteleqts. sxva SemTxvevaSi mozardebis mier sigaretis mowevis winaswarmetyvelebisaTvis uiliamsma da kovingtonma (Williams & Covington, 1997) mravlobiTi regresiis gantolebisaTvis gamoiyenes Semdegi cvladebi: rasa, asaki, TanatolebTan urTierToba, ojaxuri siaxlove, mweveli megobrebis raodenoba, mweveli ojaxis wevrebis raodenoba da mozardis mier janmrTelobisaTvis riskis aRqma. Tumca am rTuli gantolebebisaTvis regresiis koeficientebis gamoTvla iseT codnas moiTxovs, rac am Sesavali Tavis farglebs scildeba.


39


1. pirovnuli Tvisebebi da garkveuli davalebebis warmatebiT Sesruleba

xSirad dakavSirebulia. magaliTad, gormlim da gormlim (Gormly & Gormly, 1986) ikvlies korelacia socialuri introversiis Sefasebebsa da kubikebis awyobis tests Soris, romelic sivrciT unarebs zomavda. maT miiRes statistikurad mniSvnelovani dadebiTi korelacia: socialurad ufro introvertma adamianebma ufro male aawyves kubikebiT saWiro konstruqcia. davuSvaT, rom Tqven gaimeoreT es kvleva da cdis pirebs afasebdiT socialuri introversiis 9-qulian skalaze, sadac 1 niSnavs Zlier socialurad intoverts da 9 aRniSnavs sruliad araintroverts. aRricxavdiT dros (wamebSi), rac TiToeul cdis pirs 9 feradi kubikis gamoyenebiT garkveuli konstruqciis asawyobad dasWirda. 12 cdis pirisagan miRebuli qulebi Semdegnairia:

cdis piri

introversia: ( X )

Kkubikebis awyobis dro (Y )

1 6 217 2 3 230 3 9 315 4 1 224 5 4 271 6 2 198 7 3 256 8 5 263 9 8 321 10 7 289 11 8 265 12 6 291

kubikebis awyobis saSualo dro iyo 261,67 wami da Ys = 38, 82 wami. am

qulebis realurir statistikurad mniSvnelovania 0,05 doneze. b –s gamoTvliTi formulis gamoyenebiT gaavleT kubikebis awyobis itroversiis qulebze regresiis wrfe.

a. rogoria regresiis wrfis daxriloba? b. ras udris regresiis wrfis Y –TanakveTa? g. CawereT regresiis wrfis gantoleba. d. aageT wertilTa gabnevis diagrama introversiisa da kubikebis

awyobis qulebisaTvis. diagramaze gaavleT regresiis wrfe. e. ras udris introversiis TiToeuli qulisaTvis kubikebis agebis

nawinaswarmetyvelebi qula (Y ′ )? v. TiToeuli cdis pirisaTvis mocemuli Y ′–is gamoyenebiT ipoveT

naSTi.


40

z. ipoveT naSTiSS .

T. naSTiisSS gamoyenebiT ipoveT XYs ⋅ .

i. SeadareT am am qulebis XYs ⋅ da Ys = 38,82. introversiis qulebis mixedviT kubikebis awyobis qulebis winaswarmetyvelebisaTvis regresiis gantolebis gamoyeneba amcirebs Tu ara winaswarmetyvelebis Secdomas imasTan SedarebiT, roca kubikebis awyobis yvela qulis nawinaswarmetyveleb mniSvnelobad kubikebis awyobis saSualo qula (anu 261,7 wami) gamoiyeneboda?

2. frinvelebze damkvirveblebs SemCneuli aqvT, rom frinvelTa bevri saxeoba garemos daTvalierebas sakmao dros uTmobs. elgari, makkei da vuni (Elgar, McKey, & Woon, 1986) akvirdebodnen beRurebs, raTa daedginaT kavSiri garemos daTvalierebis sixSiresa da kvebis siswrafes Soris. maT am or cvlads Soris statistikurad mniSvnelovani kavSiri aRmoaCines. davuSvaT, rom Tqven gaimeoreT es kvleva 15 beRuraze dakvirvebiT. Tqven iwerdiT TiToeuli beRuris mier wuTSi ganxorcielebuli garemos daTvalierebis raodenobas ( X cvladi) da TiToeuli frinvelis mier wuTSi ganxorcielebuli sakvebis akenkvis

raodenobas (Y cvladi). daTvalierebis saSualo raodenoba ( X ) iyo

26,3 daTvaliereba wuTSi, Xs = 8,7. akenkvebis saSualo raodenoba iyo

13,3 wuTSi da Ys =4,5. am or cvlads Soris realurir aris _0,84. b –s gamosTvleli korelaciuri formulis gamoyenebiT aageT akenkvis sixSiris (Y ) daTvalierebebis sixSireze ( X ) regresiis wrfe.

a. rogoria regresiis wrfis daxriloba? b. ras udris regresiis wrfis Y -TanakveTa? g. CawereT regresiis wrfis gantoleba. d. rogoria akenkvis nawinaswarmetyvelebi sixSire im

SemTxvevisaTvis, roca garemos daTvalierebis raodenoba wuTSi 41-iaA? roca wuTSi 18-ia?

Sejameba

• wrfivi regresiis SemTxvevaSi X cvladis yoveli erTi erTeuliT cvlilebisas Y cvladi ganicdis mudmiv cvlilebas

• wrfis zogadi gantoleba gadmocemulia formuliT: abXY +=

• daxriloba b , ganisazRvreba, rogorc:



−−

=XY

b

• Y -TanakveTa, a , aris Y –is mniSvneloba, roca X =0.


41

• wrfis zofadi gantoleba SeiZleba gamoviyenoT X is cnobili qulebis mixedviT Y qulebis winaswarmetyvelebisaTvis: abXY +=′ . Y ′aris Y -is nawinaswarmetyvelebi mniSvneloba.

• naSTi, YY ′− , aris sxvaoba Y realur qulasa da X –is mixedviT nawinaswarmetyveleb qulas Y ′Soris.

• umcires kvadratTa regresiis wrfe minimumamde amcirebs naSTebis kvadratebis jamis mniSvnelobas.

• Sefasebis standartuli Secdoma Semdegnairad gamoiTvleba:

2−=⋅

wyvili

naSTi

NSS

s XY

an

( )2

2

−

′−= ∑

⋅wyviliN

YYs XY .

• Sefasebis standartuli Secdoma gamoxatavs X –is mixedviT Y qulebis winaswarmetyvelebis Secdomis sidides.

• mravlobiTi regresia gulisxmobs Y cvladis winaswarmetyvelebisaTvis ramdenime X cvladis gamoyenebas.


• mravlobiTi regresia

• Sefasebis standartuli

Secdoma ( XYs ⋅ )

• wrfis daxriloba (b )

• naSTi ( YY ′− ) • wrfivi kavSiri • naSTiSS

• umcires kvadratTa regresiis wrfe

• wrfis gantoleba

• Y -TanakveTa ( a ) • Y ′

kiTxvebi

1. jeimim gaigo, Tu ra Sualeduri Sefaseba aqvs statistikis kursSi.

misma maswavlebelma aRniSna, rom wina semestrSi jgufis 10 studentis mier miRebuli Sualeduri da saboloo gamocdebis niSnebi Semdegia:

studenti gamocda

Sualeduri: X saboloo: Y

1 87 84 2 89 91 3 97 96 4 80 87


42

5 73 66 6 85 90 7 81 79 8 74 80 9 80 89 10 89 86

a. gamoTvliTi formulis gamoyenebiT ipoveT im regresiis wrfis

daxriloba, romlis mixedviTac xdeba Sualeduri Sefasebis mixedviT saboloo gamocdis Sefasebis winaswarmetyveleba.

b. am qulebisaTvis realurir aris +0,781, Xs = 7,37 da Ys = 8,34.

korelaciuri formulis gamoyenebiT gamoTvaleT b . g. am qulebisaTvis daadgineT umcires kvadratTa regresiis wrfis

gantoleba X -is mixedviT Y qulis winaswarmetyvelebisaTvis.

d. jeimim Sualedur gamocdaze miiRo 77 qula. g punqtSi dadgenili regresiis wrfis mixedviT, rogoria misi nawinaswarmetyvelebi saboloo gamocdis qula?

e. rogoria XYs ⋅ -is mniSvneloba regresiis am wrfisaTvis? 2. mkvlevari akvirdeboda kviris ganmavlobaSi studentebis mier video-

TamaSebiT garTobis xangrZlovobas ( X cvladi) da iTvlida

korelacias maTi semestruli niSnebis saSualo ariTmetikulTan (Y cvladi). videoTamaSebiT garTobis 0-dan 15 saaTamde

mniSvnelobebisaTvis realurir iyo _0,632 ( p < 0,05). Xs = 4,10 saaTi da

Ys =0,75 saaTi. regresiis wrfis Y –TanakveTa aris 2,23. a. ipoveT videoTamaSebiT garTobis xangrZlivobis mixedviT

niSnebis saSualo ariTmetikulis gamosaTvleli umcires kvadratTa regresiis wrfe.

b. braiani videoTamaSebiT kviraSi daaxloebiT 3 saaTis ganmavlobaSi erToba. ra aris misi niSnebis saSualo ariTmetikulis nawinaswarmetyvelebi mniSvneloba?

g. Cvens SerCevaSi videoTamaSebiT garTobis saSualo xangrZlivoba iyo 5,1 saaTi kviraSi. ras udris niSnebis saSualo ariTmetikuli am SerCevisaTvis?

3. kvlevis Sedegad dadginda wrfivi kavSiri orsuli qalis mier sigaretis mowevasa da axalSobilis mcirewonianobas Soris (`Birth Weight~, 1986). dedis mier orsulobis dros dReSi erTi Reri sigaretis mowevasTan erTad bavSvis dabadebis wona mcirdeba 4,3 unciiT. maSasadame, Tu deda orsulobis dros eweoda saSualod 2 Rer sigarets dReSi, bavSvis dabadebis wona Semcirdeboda 8,6 unciiT. CaTvaleT, rom Tu orsuli deda sigarets ar eweva, axalSobilis saSualo wona aris 128 uncia.

a. dawereT wrfivi kavSiris gantoleba am monacemebisTvis (miniSneba: mocemulia daxrilobaca da Y -TanakveTac)


43

b. daxazeT regresiis wrfe ujrebian furcelze. abscisaze

gadazomeT moweuli sigaretis Rerebis raodenoba ( X ).

g. rogoria im axalSobilis nawinaswarmetyvelebi dabadebis wona, romlis dedac orsulobis dros eweoda Semdegi raodenobis sigarets dReSi:

0 1 5 8 10 13 20

4. me-13 TavSi, kiTxvaSi gameorebisTvis #1, 16 mamakacis gabrazebis

qulebisa da sisxlis sitoluri wnevis Semdegi mniSvnelobebi warmogidgineT:

cdis piri Bbrazis done sisxlis wneva

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

64 30 37 39 17 29 36 34 44 32 37 47 50 41 46 61

170 132 129 144 145 122 124 145 137 115 127 148 148 144 133 158

a. am monacemebis mixedviT daadgineT sisxlis wnevis (Y ) gabrazebis

doneze ( X ) regresiis gantoleba. gamoiyeneT gamoTvliTi

formula. b. ras udris TiToeuli adamianisaTvis nawinaswarmetyvelebi

sisxlis wneva, rogorc gabrazebis donis funqcia? g. daxazeT wertilTa gabnevis diagrama gabrazebis donisa

(abscisaze) da sisxlis wnevisaTvis (ordinataze). amave diagramaze gaavleT regresiis wrfec.

d. am monacemebisaTvis Ys -is ra proporcias warmoadgens XYs ⋅ ? 5. SeamowmeT Tqveni codna XIV-4-is savarjiSoSi #2 warmodgenili iyo

kvleva wuTSi garemos daTvalierebis raodenobasa ( X ) da sakvebis

akenkvis raodenobas (Y ) Soris mimarTebis Sesaxeb 15 beRurisaTvis. davuSvaT, rom msgavsi dakvirveba awarmoeT 13 WinWraqaze da miiReT Semdegi monacemebi:


44

frinveli

garemos

daTvaliereba: X

sakvebis akenkva: Y

1 29 12 2 34 8 3 24 14 4 14 20 5 37 11 6 22 13 7 11 23 8 33 9 9 24 11 10 30 14 11 36 9 12 20 23 13 32 7

a. rogoria am regresiis wrfis daxriloba? b. rogoria am regresiis wrfis Y –TanakveTa? g. CawereT regresiis wrfis gantoleba. d. aageT wertilTa gabnevis diagrama garemos daTvalierebisa da

sakvebis akenkvis qulebisaTvis. am diagramazeve gaavleT regresiis wrfe.

e. am monacemebisaTvis gamoTvliTi formulis gamoyenebiT

gamoTvaleT XYs ⋅ .

v. SeadareT am monacemebis mixedviT gamoTvlili XYs ⋅ da Ys = 5,41. garemos daTvalierebis qulebis mixedviT akenkvis qulebis winaswarmetyvelebisaTvis regresiis gantolebis gamoyeneba amcirebs Tu ara winaswarmetyvelebis Secdomas imasTan SedarebiT, roca garemos daTvalierebis yvela qulis nawinaswarmetyveleb mniSvnelobad am qulebis saSualo (anu 13,4) gamoiyeneboda?

6. kompania samsaxuris msurvelTa samuSaoSi warmatebis winaswarmetyvelebisaTvis iyenebs standartizebul interviuSi miRebul qulas. gamoyenebuli regresiis gantoleba aris 182,1 ++=′ XY ,

sadac Y ′ aris nawinaswarmetyvelebi warmateba samsaxurSi da X aris

aplikantis mier standartizebul interviuSi miRebuli qula. kompania samsaxurSi miiRebs mxolod im aplikantebs, visi samsaxurSi warmatebis nawinaswarmetyvelebi qula 60 an metia. qvemoT moyvanilia rva aplikantis standartizebuli intervius qulebi. romeli aplikanti daakmayofilebs miRebis pirobebs?

aplikanti standartizebuli intervius qula: X


45

daiana 31 rodrigo 36 kolini 30 maria 41 rubeni 39 luka 29 holi 27 blaiTi 44

7. me-13 Tavis me-8 kiTxva warmoadgenda kvlevas kviraSi televizoris yurebis xangrZlivobasa da pirveli gamocdis niSans Soris

korelaciis Sesaxeb 17 studentisaTvis. am qulebisaTvis realurir iyo

_0,735, televizoris yurebis saSualo xangrZlivoba ( X ) iyo 10,0 da Xs

iyo 6,38-s. gamocdaze miRebuli saSualo niSani (Y ) iyo 76,0 da Ys iyo

11,44. XSS iyo 652,0 da XYCP iyo _859,0. a. ipoveT regresiis wrfis gantoleba televizoris yurebis

xangrZlivobis mixedviT gamocdis niSnebis winaswarmetyvelebisaTvis.

b. ipoveT nawinaswarmetyvelebi gamocdis niSnebi im studentebisaTvis, romlebic televizors kviraSi 4 da 15 saaTis ganmavlobaSi uyureben.

g. am monacemebisaTvis ras udris XYs ⋅ ? Ys -is ra nawils warmoadgens

XYs ⋅ ?

Tavi XV. araparametruli statistikuri kriteriumebi _______________________________________________________________________________

1

Tavi XV araparametruli statistikuri kriteriumebi __________________________________________________________ sixSireTa analizi: xi-kvardat kriteriumi

damoukideblobis xi-kvadrat kriteriumi Tanxmobis χ² kriteriumi xi-kvardat kriteriumis Sedegebis warmodgena man-uitnis U kriteriumi

statistikuri hipoTezebis Semowmeba man-uitnis U kriteriumis gamoyenebiT man-uitnis U kriteriumis daSvebebi U-s gamosaTvleli formulebi

man-uitnis U kriteriumis Sedegebis warmodgena uilkoqsonis niSniani rangebis kriteriumi

statistikuri hipoTezebis Semowmeba uilkoqsonis niSniani rangebis kriteriumis gamoyenebiT

uilkoqsonis niSniani rangebis kriteriumis Sedegebis warmodgena araparametruli kriteriumebis gamoyeneba


2

araparametruli statistikuri kriteriumi (Nonparametric statistical test): statistikur kriteriums ewodeba araparametruli, Tu hipoTezaSi ar aris gamoTqmuli varaudi populaciis parametris Sesaxeb. me-8-dan me-12 Tavamde ganixileboda statistikuri hipoTezis Semowmeba, romelic yuradRebas amaxvilebs t kriteriumsa da dispersiul analizze. es orive kritieriumi parametruli kriteriumia, radgan populaciis parametrebis (μ) Sesaxeb daskvnebis gasakeTeblad SerCevis statistikebs

(magaliTad, X -s) iyenebs. araparametruli statistikuri kriteriumebi amowmeben hipoTezebs, romlebic ar exeba populaciis parametrebs. maT ufro naklebi daSvebebi aqvT populaciis formis Sesaxeb, vidre parametrul kriteriumebs. am mizezis gamo araparametrul kriteriumebs zogjer ganawilebisgan Tavisufal kriteriumebs uwodeben. gaixseneT, rom parametruli kriteriumebis daSvebebia:

• populaciebSi, romlebidanac aRebulia SerCeva, qulebi normalurad aris ganawilebuli;

• sxvadasxva populaciaSi qulebis dispersiebi erTmaneTis tolia. praqtikaSi miRebuli monacemebi, SesaZloa, ar Seesabamebodes am daSvebebs. miuxedavad amisa, radgan parametruli kriteriumebi sandod iTvleba, mkvlevrebs SeuZliaT gamoiyenon t kriteriumi an dispersiuli analizi iseTi monacemebisTvisac ki, romlebic daSvebebs srulyofilad ar akmayofilebs. Tumca, zogierT SemTxvevaSi am daSvebebTan Sesabamisobis xarisxi sakmarisi ar aris imisaTvis, rom SeiZlebodes parametruli kriteriumis gamoyeneba. magaliTad, adamianis reaqciis drois gazomvisas reaqciis droebis umravlesoba Zalian xanmoklea, (magaliTad, daaxloebiT 500 milisekundi an naklebi). reaqciis dro 0 wamze naklebi ver iqneba; magram zeda zRvari SeiZleba nebismieri iyos. Zalian cota adamians SeiZleba hqondes sxva adamianebTan SedarebiT reaqciis Zalian didi dro. maSasadame, reaqciis drois monacemebis sixSireebis ganawileba, savaraudod, dadebiTad gadaxrili iqneba; reaqciis droebis umravlesoba Zalian mcirea da cota Tu iqneba bevrad xangrZlivi. cxadia, rom aseTi ganawileba ar aris miaxloebuli normalur ganawilebasTan. parametruli kriteriumebi aseve akeTebs daSvebas, rom azri aqvs

miRebuli qulebis saSualos ( X ) gamoTvlas. es moTxovna gulisxmobs, rom monacemebi intervalebis an Sefardebis skalaze unda iyos. magram xSirad qcevis gazomva am doneebze ver xdeba. eqsperimentatori, SesaZlebelia, ubralod iTvlides im adamianebis raodenobas, romlebic TiToeul qcevas avlenen da amdagvarad iRebdes qcevis gamovlenis sixSires, romelic saxeldebis skalazea. an miRebuli qulebi SeiZleba warmoadgendes rigis skalas, magaliTad, garkveuli pirovnuli niSniT individebis rigis


3

mixedviT ganlagebisas. radgan parametruli kriteriumebisTvis saWiroa X -is gamoTvla, isini Sesaferisi ar aris im SemTxvevebisTvis, roca qulebi saxeldebis an rigis skalazea. amgvarad, araparametruli kriteriumebi gamoiyeneba maSin, roca:

• parametruli kriteriumebis daSvebebi ar aris dakmayofilebuli an

• qulebi saxeldebis an rigis skalazea warmodgenili

cxrili #15.1 warmoadgens parametrul kriteriumebs da Sesabamis araparametrul kriteriumebs erTfaqtoriani sqemisaTvis. am TavSi ganvixilavT xi-kvadrats, man-uitnis U da uilkoqsonis niSniani rangebis kriteriumebs. korelaciis araparametruli kriteriumi, spirmenis ranguli korelaciis koeficienti warmodgenilia me-13 TavSi. cxrili #15.1 parametruli da araparametruli kriteriumebi erT-faqtoriani sqemisaTvis, rogorc sqemis tipisa da gazomvis skalis funqcia

sqemis tipi cdis pirebs Soris cdis pirTa Sida monacemebis

gazomvis skala

ori done

sami an meti done

ori done sami an meti done

parametruli kriteriumebi

Sefardebis an intervalebis

lobisdamoukidebt

an cdis pirebs Soris ANOVA

cdis pirebs Soris ANOVA

lidamokidebut an cdis pirTa Sida ANOVA

cdis pirTa Sida ANOVA


rigis

man-uitnis U

kruskal-uolisis ANOVA rangebis mixedviT

uilkoqsonis niSniani rangebi

fridmenis ANOVA rangebis mixedviT

saxeldebis

xi-kvadrati

xi-kvadrati

____________________________________________________________________

sixSireTa analizi: xi-kvadrat kriteriumi

xi-kvadrat kriteriumi (Chi-square test): statistikuri kriteriumi, romelic gamoiyeneba saxeldebis skalis monacemebis gasaanalizeblad, roca mocemulia sxvadasxva kategoriis Sesabamisi sixSireebi.


4

bevri kvlevis Sedegad viRebT iseT monacemebs, romlebic mxolod imas gviCvenebs, qceva ganxorcielda Tu ara. magaliTad, bunebriv pirobebSi cxovelebze dakvirvebisas mecnierma SeiZleba Caiweros, axorcielebs Tu ara cxoveli garkveul qcevas. aseTi monacemebi saxeldebis skalazea; isini ubralod aRniSnaven qcevis urTierTgamomricxavi kategoriebis xdomilebas. SesaZlebelia, aRiricxos sxvadasxva kategoriis qcevis gamovlenis sixSire, magram TiToeuli qula gamoxatavs mxolod imas, qceva gamovlinda Tu ara; is ar Seicavs pasuxis Sesaxeb ricxvobriv informacias. araparametruli kriteriumi, xi-kvadrat kriteriumi, am monacemebisaTvis Sesaferisi sazomia, rodesac cdis pirebs Soris sqemaa gamoyenebuli, monacemebi saxeldebis skalazea da mxolod qcevis gamovlenis sixSires aRniSnavs. ganvixilavT xi-kvadratis or formas: damoukideblobis xi-kvadrat kriteriums da Tanxmobis xi-kvadrat kriteriums. damoukideblobis xi-kvadrat kriteriumi

damoukideblobis xi-kvadrat kriteriumi (Chi-square test of independence): xi-kvadrati, romelic gamoiyeneba maSin, roca monacemebis kategorizacia ori damoukidebeli niSnis mixedviT xdeba. damoukideblobis xi-kvadrat kriteriumi (romelsac aseve ewodeba ormimarTulebiani xi-kvadrati), gamoiyeneba, roca qulis kategorizacia xdeba ori gansxvavebuli ganzomilebis mixedviT. am kriteriumis gamoyenebas qvemoT moyvanili magaliTi giCvenebT. samyaros Sesaxeb codnisa da mis mimarT damokidebulebebis mniSvnelovan wyaros televiziaSi nanaxi warmoadgens. bihevioristi mecnierebi SeSfoTebulebi arian imiT, rom televiziam, misi informaciuli SesaZleblobebis gamo, SeiZleba warmoadginos sazogadoebis arazusti saxe. ganvixiloT iseTi martivi ram, rogoricaaa satelevizio gmiris Tmis feri. populaciaSi garkveuli Tmis feris mqone mamakacebisa da qalebis Tanafardoba daaxloebiT Tanabaria (magaliTad, WaRara Tmis mqone mamakacebisa da qalebis Tanafardoba daaxloebiT Tanabaria). Tumca, deivisma (Davis, 1990) aRmoaCina, rom satelevizio gmirebis SemTxvevaSi xuTjer meti iyo albaToba imisa, rom qalebs qera Tma eqnebodaT da eqvsjer meti iyo albaToba imisa, rom mamakacebs WaRara Tma eqnebodaT. msgavsadve, elasmarma, hasegavam da breinma (Elasmar, Hasegawa, & Brain, 1999) daadgines, rom siuJetis mixedviT, ufro meti qeraTmiani qali personaJi aRwevda mizans, vidre muqTmiani. davuSvaT, rom mkvlevars ainteresebs daadginos, arsebobs Tu ara kavSiri satelevizio gadacemebis personaJi qalebis Tmis fersa da karieras Soris. misi hipoTezis Tanaxmad, satelevizio gadacemebis qeraTmian personaJ qalebs ufro xSirad eqnebodaT profesiuli donis samuSao, vidre muqTmian personaJebs. am hipoTezis Sesamowmeblad man SemTxveiTad SearCia satelevizio gadacemebis garkveuli raodenoba, romlebSic monawileobdnen zrdasruli personaJi qalebi da Semdeg es personaJebi daajgufa Tmis feris mixedviT (qera an muqTmiani) da karieris mixedviT (profesiuli an


5

araprofesiuli). TiToeuli persoinaJi SerCevaSi mxolod erTxel aRiricxeboda. am kvlevis hipoTeturi Sedegebi naCvenebia cxrilSi #15.2. am cxrilSi satelevizio personaJis kategorizacia ori ganzimilebis mixedviT xdeba: (1) Tmis feri: qera an muqi da (2) samuSaos done: profesiuli an araprofesiuli. am cxrilSi O–Ti aRniSnuli mniSvnelobebi (observed frequency – realuri sixSire) warmoadgens TiToeul kategoriaSi miRebuli Sedegebis sixSires. am kvlevaSi daakvirdnen sul 180 qals da TiTeuli maTganisagan am cxrilisaTvis mxolod erTi pasuxi miiRes: personaJi an qera iyo, an muqTmiani da misi samuSao an profesiuli iyo, an – araprofesiuli.

niSanTa SeuRlebis cxrilebi cxrili #15.2 satelevizio gadacemebis qali personaJebis Tmis ferisa da karieris mixedviT

kategorizaciis hipoTeturi monacemebi. O -s saxiT mocemuli mniSvnelobebi

warmoadgens realur sixSireebs, xolo E -s saxiT warmodgenilia marginaluri

sixSireebis mixedviT miRebuli mosalodneli sixSireebi.

Tmis feri

Qqera muqi M(marginaluri sixSire)

karieris done

profesiuli O=36 E =28

O=48 E =56

84

araprofesiuli

O=24 E =32

O=72 E =64

96

(marginaluri sixSire)

60

120

(sul) 180

niSanTa SeuRlebis cxrili (Contingency table): striqonebiani da svetebiani cxrili, romelSic warmodgenilia yoveli kategoriis sixSireebi mocemuli striqonisa da svetis cvladebis gadakveTis Sedegad. cxrils #15.2 ewodeba striqonebiani ( r ) da svetebiani ( c ) niSanTa SeuRlebis cxrili an sixSireebis ormimarTulebiani cxrili. svetebi (aRiniSneba c -Ti) warmoadgens erT kategorias, romelSic SesaZloa cdis pirebi moxvdnen. striqonebi (aRiniSneba r -iT) warmoadgens pasuxis meore katagorias. maSasadame, cxrili #15.2 aris 2 striqoni ( r = samsaxuris donis Sesabamisi ori kategoria: profesiuli da araprofesiuli) 2 svetze ( c=


6

Tmis feris Sesabamisi ori kategoria: qera da muqi) niSanTa SeuRlebis cxrili. cxrils ewodeba niSanTa SeuRlebis cxrili, radgan Cven vcdilobT davadginoT, cdis piris samsaxuris done, profesiuli an araprofesiuli, SesabamisobaSia an SeuRlebulia Tu ara misi Tmis ferTan. marginaluri (Sejamebuli) sixSireebi, romlebic naCvenebia cxrilSi, aris striqonebis jamebi (anu striqonis marginaluri sixSireebi) da svetebis jamebi (anu svetebis marginaluri sixSireebi) miRebuli sixSireebisaTvis.

marginaluri (Sejamebuli) sixSireebi (Marginal Frequencies): niSanTa SeuRlebis cxrilSi sixSireebis jami striqonis an svetis mixedviT. mosalodneli siSireebi

mosalodneli siSireebi (Expected Frequencies): niSanTa SeuRlebis cxrilidan miRebuli sixSireebi im daSvebis safuZvelze, rom cxrilSi warmodgenili ori ganzomileba damoukidebelia. cxrilSi #15.2 warmodgenilia 60 qeraTmiani da 120 muqTmiani personaJi. sul 180 personaJidan 84-s, 46,7%-s, hqonda profesiuli donis samsaxuri da 96-s, anu 53,3%-s, hqonda araprofesiuli samsaxuri. Tu Tmis feri ar aris dakavSirebuli samsaxuris donesTan, maSin mosalodnelia, rom 60 qeraTmiani personaJis daaxloebiT 46,7%-s (anu 84/180) anu 28 personaJs, unda hqondes profesiuli donis samsaxuri. ZalaSia igive molodini, Tu Tmis fersa da samsaxuris dones Soris kavSiri ar arsebobs. am SemTxvevaSi moveliT, rom profesiuli donis samsaxuri eqneba 120 muqTmiani personaJis daaxloebiT 46,7 procents, anu 56 qals, da araprofesiuli donis samsaxuri eqneba 120 muqTmiani personaJebis 53,3 procents, anu 64 qals. am mniSvnelobebs mosalodneli sixSireebi ewodeba, _ pasuxis mosalodneli sixSireebi im SemTxvevaSi, Tu Tmis feri ar aris dakavSirebuli samsaxuris donesTan. mosalodneli sixSireebi gamoxatulia E aRniSvniT (Expected frequency, mosalodneli sixSiris aRsaniSnavad) cxrilis TiToeul ujraSi. amgvarad, mosalodneli sixSireebi miiReba im varaudis safuZvelze, rom ar arsebobs kavSiri im or ganzomilebas Soris, romlis mixedviTac xdeba cdis monawilis kategorizacia. am magaliTSi mosalodneli sixSireebi uSvebs, rom Tmis feri da samsaxuris done erTmaneTisgan damoukidebelia. xi-kvadrat kriteriumis misaRebad realuri da mosalodneli sixSireebi gamoiyeneba. Tu SerCevis or ganzomilebas Soris kavSiri ar aris, miRebuli sixSireebi da mosalodneli sixSireebi daaxloebiT toli unda iyos. Tu svetisa da striqonis cvladebi ar aris dakavSirebuli erTmaneTTan, miRebulsa da mosalodnel sixSireebs Soris didi gansxvavebebi mxolod iSviaTad da SemTxveviT miiReba. magram Tu SerCevaSi svetisa da striqonis cvladebi dakavSirebulia, maSin miRebuli sixSireebi ufro metad unda gansxvavdebodes mosalodneli sixSireebisagan. magaliTad, Tu Tmis feri dakavSirebulia karieris donesTan, rogorc hipoTeza


7

varaudobda, maSin mosalodnelia, rom qeraTmian personaJebs ufro meti profesiuli donis samsaxuri eqnebaT, vidre SemTxveviT mosalodneli 28 da ufro naklebi araprofesiuli samsaxuri, vidre SemTxveviT mosalodneli 32. msgavsadve, Tu hipoTezaSi gamoTqmuli varaudi Tmis ferisa da samsaxuris donis kavSiris Sesaxeb gamarTlda, aseve mosalodnelia, rom muqTmian personaJebs SemTxveiT mosalodnel 56-ze naklebi profesiuli donis da 64-ze meti araprofesiuli donis samsaxuri eqnebaT.

2χ statistika xi-kvadrat kriteriumi (aRiniSneba 2χ -iT) zomavs miRebuli sixSireebis mosalodneli sixSireebisagan gansxvavebas:

( )∑∑= =

−=

r

r

c

c rcErcErcO

1 1

2

2χ

sadac χ aris `xi~, berZnuli anbanis ocdameerTe aso. am formulaSi

rcO aris rc ujris realuri sixSire, sadac r striqonis nomeria da c - svetis nomeri.

rcE aris rcujris mosalodneli sixSire

r aris striqonis cvladis kategoriebis raodenoba (Cvens magaliTSi samsaxuris done, r = 2)

c aris svetis cvladis doneebis raodenoba (Cvens magaliTSi Tmis feri, c = 2).

ormagi jamis niSnebi da zRvrebi (∑=

r

r 1∑=

c

c 1

) miuTiTebs, rom niSanTa

SeuRlebis cxrilis yvela svetisa da striqonis ( ) rcrcrc EEO 2−

mniSvnelobebi jamdeba.

statistikuri hipoTezebis Semowmeba χ²-iT Cvens magaliTSi 2χ -is gamoyeneba imis dasadgenad, dakavSirebulia Tu ara Tmis feri samsaxuris donesTan, moiTxovs stastikuri hipoTezis Semowmebas Semdegi nacnobi safexurebis mixedviT:

• nulovani, 0H , hipoTezisa da alternatiuli, 1H , hipoTezis

Camoyalibeba. nulovani hipoTeza saSualebas gvaZlevs, avagoT 2χ statistikis SerCeviTi ganawileba

• mniSvnelovnobis donis SerCeva

• cxrilSi #A.7 mocemuli 2χ -is SerCeviTi ganawilebidan 2χ -is kritikuli mniSvnelobis moZebna, rac aRiniSneba, rogorc

kritikuli2χ .

• kritikulis2χ gamoyenebiT 2χ is SerCeviTi ganawilebaSi uaryofis

zonis gansazRvra


8

• realuris2χ gamoTvla niSnanTa SeuRlebis cxrilis sixSireebis

mixedviT

• gadawyvetilebis miReba 0H -is uaryofis an miRebis Sesaxeb imis

mixedviT, xvdeba Tu ara realuri2χ uaryofis areSi.


nulovani hipoTeza. nulovani hipoTeza, romlis farglebSic

yalibdeba 2χ -is SerCeviTi ganawileba da romlis farglebSic dgindeba TiToeuli ujrisaTvis mosalodneli sixSireebi, aris:

0H : populaciaSi striqonis cvladi da svetis cvladi

damoukidebelia.

nimuSad moyvanili kvlevis SemTxvevaSi 0H SeiZleba konkretulad

Camoyalibdes, rogorc:

0H : satelevizio personaJis samsaxuris done ar aris damokidebuli

misi Tmis ferze. alternatiuli hipoTeza. alternatiuli hipoTezaa:

1H : striqonisa da svetis cvladebi populaciaSi dakavSirebulia.

nimuSad moyvanili kvlevisaTvis 1H SeiZleba konkretulad Camoyalibdes, rogorc

1H : satelevizio personaJis samsaxuris done dakavSirebulia misi Tmis ferTan. mniSvnelovnobis donis SerCeva mniSvnelovnobis donis SerCevis procesi identuria imisa, rac wina TavebSia ganxiluli. miRebuli praqtikis Sesabamisad, virCevT α = 0, 05-s. χ²kritikulis povna da uaryofis zonis dadgena

2χ -is Tavisuflebis xarisxi. 2χ -is, da Sesabamisad kritikulis2χ , konkretuli

SerCeviTi ganawileba damokidebulia statistikuri sididis Tavisuflebis

xarisxze. Tavisuflebis xarisxi (df ) 2χ -Tvis aris

Tavisuflebis xarisxi = ( )( )11 −− cr

sadac r SeuRlebis cxrilSi striqonebis raodenobaa da c - svetebis raodenoba. magaliTisaTvis, r = 2 da c= 2, maSasadame Tavisuflebis xarisxi

df = (2 - 1)(2 - 1), rac 1-is tolia. kritikulis2χ mniSvnelobebi α = 0, 05 da 0,01

doneebze sxvadasxva Tavisuflebis xarisxis mqone 2χ -is SerCeviTi

ganawilebisTvis mocemulia damatebis cxrilSi #A-5. cxrilis #A.7 mixedviT, roca Tavisuflebis xarisxi 1-ia da α = 0, 05, 2χ -is kritikuli mniSvneloba aris 3,84.


9

uaryofis zonis dadgena. Tu nulovani hipoTeza WeSmaritia da pasuxis

siswore studentis tipze ar aris damokidebuli, maSin realuri2χ SedarebiT

mcire unda iyos; mosalodneli da miRebuli sixSireebi niSanTa SeuRlebis cxrilis TiToeuli ujrisaTvis bevrad msgavsi unda iyos. meores mxriv, Tu

1H WeSmaritia da striqonis cvladi dakavSirebulia svetis cvladTan,

maSin realuris2χ mniSvneloba gaizrdeba da realuri da mosalodneli

sixSireebi erTmaneTisagan gansxvavebuli iqneba. Sesabamisad, 2χ -is SedarebiT didi mniSvnelobebi iSviaTi unda iyos, Tu nulovani hipoTeza WeSmaritia, magram xSiri unda iyos, Tu alternatiuli hipoTezaa WeSmariti.

maSasadame, realuris2χ mniSvneloba, romelic kritikulze

2χ metia an tolia, statistikurad mniSvnelovania α-s SerCeul mniSvnelobaze. Sesabamisad, Cvens

magaliTSi Tu realuri2χ 3,84-ze metia an tolia, is xvdeba uaryofis areSi da

moxdeba 0H –is uaryofa da 1H –is miReba.

realuris2χ gamoTvla

realuris2χ gamoTvlisaTvis saWiroa pasuxebis realuri da mosalodneli

sixSireebis codna. mosalodneli sixSireebi niSanTa SeuRlebis cxrilis TiToeuli ujrisaTvis marginaluri sixSireebisagan Semdegi formuliT miiReba:

( )( )raodenoba saerTo pasuxebis

ujrisaTvis sixSire imarginalur svetisujrisaTvis sixSire imarginalur striqonis

sixSire

imosalodnel

ujris

=

magaliTad, mosalodneli sixSire ujrisaTvis `qera Tma/profesiuli donis samsaxuri~ (striqoni 1, sveti 1) gamoiTvleba Semdegnairad:

( )( )jami

sixSire imarginalur svetis pirvelisixSire imarginalur striqonis pirveli=11E

anu

( )( ) 281805040

1806084

11 ===E

es aris mosalodneli sixSire , E , cxrilis #15.2 `qera Tma/profesiuli donis samsaxuri~ ujrisaTvis.


10

mas Semdeg, rac gavigebT mosalodnel sixSireebs TiToeuli

ujrisaTvis, realuri2χ gamoiTvleba Semdegnairad:

( )∑∑= =

−=

r

r

c

c rcErcErcO

1 1

2

2χ

realuri2χ -is gamosaTvlelad,

• TiToeuli ujrisaTvis miRebul sixSires (O ) gamoakeliT mosalodneli sixSire ( E )

• TiToeuli O - E sxvaoba aiyvaneT kvadratSi • O - E sxvaobis kvadrati gayaviT am ujris mosalodnel sixSireze • realuris

2χ misaRebad SeajameT niSanTa SeuRlebis cxrilSi TiToeuli

ujrisaTvis miRebuli (O - E )² ∕ E mniSvnelobebi

realuris2χ ricxviTi mniSvneloba cxrilSi #15.2 mocemuli sixSireebisaTvis

Semdegnairad gamoiTvleba:

( ) ( ) ( ) ( )

( ) ( )

Sedegad. isdamrgvaleb demeasedebam

realuri

,43.6,000,1143,1000,2286,2

,648

568

328

288

,64

647256

564832

322428

2836

2222

22222

=+++=

+−

+−

+=

−+

−+

−+

−=χ

gadawyvetilebis miReba statistikuri hipoTezis Sesaxeb

realuri2χ , romlis mniSvnelobaa 6,43, metia kritikulze

2χ 3,84-is toli mniSvnelobiT da Sesabamisad, uaryofis areSi xvdeba. aqedan gamomdinare,

xdeba 0H -is uaryofa da 1H -is miReba. striqonisa da svetis cvladebs

Soris arsebobs statistikurad mniSvnelovani kavSiri. kriteriumis mixedviT gakeTebuli daskvnebi

0H -is uaryofa miuTiTebs, rom Tmis feri da samsaxuris done

urTierTdakavSirebulia. kavSiris buneba Cans cxrilidan #15.2: profesiuli donis samsaxuris SemTxveviTze meti sixSire dakavSirebulia qera TmasTan da araprofesiuli samsaxuris SemTxveviTze meti sixSire dakavSirebulia muq TmasTan. daSvebebi xi-kvadrat kriteriumis gamoyenebisaTvis damoukideblobis xi-kvadrat kriteriumis gamoyeneba SesaZlebelia nebismieri zomis niSanTa SeuRlebis cxrilis SemTxvevaSi (magaliTad, 2×3, 3×4, an 4×5). Tumca misi gamoyenebisaTvis arsebobs garkveuli daSvebebi:


11

• TiToeuli cdis piri gvaZlevs mxolod erT pasuxs niSanTa SeuRlebis cxrilisaTvis

• miRebuli pasuxebis raodenoba imdenad didi unda iyos, rom arcerTi mosalodneli sixSire ar iyos 10-ze naklebi 2×2 Sesabamisobis cxrilSi an 5-ze naklebi im cxrilSi, romelic 2×2-ze didia. Tu es piroba dakmayofilebuli ar aris, an meti pasuxi unda Segrovdes an gamoyenebul unda iqnas raime alternatiuli kriteriumi, magaliTad fiSeris zusti kriteriumi.

xi-kvadratisa da sixSiris monacemebis sxva kriteriumebis gamoyenebis Sesaxeb dawvrilebiTi informaciis moZieba SeiZleba statistikis ufro rTul saxelmZRvaneloebSi.

magaliTi #15.1

xi-kvadrat kriteriumis gamoyeneba 3×2 niSanTa SeuRlebis cxrilisaTvis amocana ojaxSi pirveli Sviloba bavSvs xSirad garkveul upiratesobebs aniWebs. sxvadasxva kvelvebma aCvena, rom pirvel Svilebs, rogorc wesi, ukeTesi Sedegebi aqvT akademiur testebSi, vidre rigiT momdevno Svilebs, aisenmanma (Eisneman, 1987) ki aRmoaCina kavSiri kreatulobasa da dabadebis rigs Soris kolejis student mamakacebSi. mis SerCevaSi pirveli Svili mamkacebis qulebi xSirad kreatulobis maCveneblebis medianaze zeviT iyo da arapirveli Svili mamakacebis qulebi - medianaze qveviT. davuSvaT, rom Tqven ganavrcobT am kvlevas da ikvlevT zrdasrul mamakacebs: 100 pirvel Svils da 100 momdevno Svils, da sTxovT, moifoqron Cveuli sagnebis kreatuli gamoyenebis gzebi. maTi pasuxebis kategorizacias axdenT imis mixedviT, maTi qulebi kreatulobis qulebis ganawilebis zeda, Sua Tu qveda mesamedSia. dabadebis rigisa da kreatulobis testis mixedviT kategorizaciis Sedegad miRebulia Semdegi realuri sixSireebi:

dabadebis rigi pirveli Svili momdevno Svili

kreatulobis testis qula

zeda mesamedi Sua mesamedi qveda mesamedi

47 29 29 35 24 36

aris Tu ara kavSiri dabadebis rigsa da kreatulobas Soris mamakacebSi? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna am kvlevaSi cdis pirebis dajgufebis erTi kategoria aris dabadebis rigi ori mniSvnelobiT: pirveli Svili da momdevni Svili. meore kategoria


12

dafuZnebulia imaze, kreatulobis testis qula kolejis populaciis qulebis zeda, Sua Tu qveda mesamedSia. TiToeuli cdis piri gvaZlevs sixSirisaTvis mxolod erT qulas, anu ricxvs, maSasadame, am monacemebisaTvis xi-kvadratis gamoyeneba mizanSewonilia.

saWiro kriteriumi: 2χ daSvebebi χ²–is gamoyenebisaTvis:

1. TiToeuli cdis pirisagan viRebT mxolod erT pasuxs niSanTa SeuRlebis cxrilisaTvis.

2. arc erTi mosalodneli sixSire ar unda iyos 5-ze naklebi. es minimaluri mosalodneli sixSire aviReT imitom, rom niSanTa SeuRlebis 3×2 cxrili ufro didia, vidre 2×2.

statistikuri hipoTeza:

0H : SerCevaSi dabadebis rigi da kreatuloba urTierTdamoukidebelia;

1H : SerCevaSi dabadebis rigsa da kreatulobas Soris aris kavSiri. mniSvnelovnobis done: α = 0,05 Tavisuflebis xarisxi, df : df = ( r -1)( c -1) = (3-1)(2-1) = 2 kritikuli mniSvneloba: cxrilis #A.7 mixedviT, χ²kritikuli(2) = 5,99146, anu Tu measedebamde davamrgvalebT, 5,99.

uaryofis are: realuris2χ 5,99-is toli an meti mniSvneloba.

gamoTvla: realuris2χ formulisaTvis saWiroa mosalodneli sixSireebis

povna; amisaTvis ki saWiroa marginaluri sixSireebis gamoTvla. miRebuli (realuri) sixSireebi (aRniSnuli O-Ti), mosalodneli sixSireebi (aRniSnuli E-Ti) da marginaluri sixSireebi mocemulia Semdeg cxrilSi: Ddabadebis rigi pirveli

Svili momdevno Svili

(striqonis marginaluri sixSire)

kreatulobis testis qula

zeda mesamedi O=47

E =38

O=29

E =38

76

Sua mesamedi O=29

E =32 O=35

E =32

64

qveda mesamedi O=24

E =30 O=36

E =30

60


13

(svetis marginaluri sixSire)

100 100 (jami) 200

svetis marginaluri sixSireebi miRebulia TiToeul svetSi mocemuli realuri sixSireebis SejamebiT. striqonis marginaluri sixSireebic miRebulia msgavsadve TiToeuli striqonis realuri sixSireebis SejamebiT. marginaluri sixSireebis codnis SemTxvevaSi mosalodneli sixSireebis gamoTvla Semdegnairad xdeba:

( )( )raodenoba saerTo pasuxebis

ujrisaTvis sixSire imarginalur svetisujrisaTvis sixSire imarginalur striqonis

sixSire

imosalodnel

ujris

=

mosalodneli sixSireebi cxrilis TiToeuli ujrisaTvis aseTia:L

ujra: pirveli Svili - zeda ⅓: ( )( ) 38200

7600200

1007611 ===E

ujra: momdevno Svili - zeda ⅓: ( )( ) 38200

7600200

1007612 ===E

ujra: pirveli Svili - Sua ⅓: ( )( ) 32200

6400200

1006421 ===E

ujra: momdevno Svili - Sua ⅓: ( )( ) 32200

6400200

1006422 ===E

ujra: pirveli Svili - qveda ⅓: ( )( ) 30200

6000200

1006031 ===E

ujra: momdevno Svili - qveda ⅓: ( )( ) 30200

6000200

1006032 ===E

realuris2χ mniSvneloba gamoiTvleba formuliT:

( )∑∑

==

−=

2

1

23

1

2

c rc

rcrc

r EEO

realuriχ

ricxviTi mniSvnelobebis Casmis Semdeg miviRebT:

( ) ( ) ( ) ( ) ( ) ( )

22,7,20,120,128,028,013,213,2

,30

303630

302432

323532

322938

382938

3847 2222222

=+++++=

−+

−+

−+

−+

−+

−=realuriχ


14

gadawyvetileba: realuri2χ = 7,22 aRemateba kritikuli

2χ = 5,99-s; maSasadame, is

xvdeba uaryofis areSi. uarvyofT 0H -s da viRebT 1H -s. daskvna: Cvens SerCevaSi dabadebis rigi da kreatuloba erTmaneTTan dakavSirebulia. realuri da mosalodneli siSireebis cxrilze dakvirvebis Sedegad vxedavT, rom pirveli Svili mamakacebis qulebi ufro xSirad da momdevno Svili mamkacebis qulebi ufro iSviaTad xvdeba kreatulobis qulebis zeda mesamedSi, vidre SemTxveviT iqneboda mosalodneli. aseve vamCnevT, rom pirveli Svili mamakacebis qulebi ufro iSviaTad xvdeba qveda mesamedSi momdevno Svili mamakacebis qulebTan SedarebiT, vidre amas SemTxveviTobis hipoTeza winaswarmetyvelebs. Tanxmobis χ²²kriteriumi

Tanxmobis χ² kriteriumi (Chi-square test of goodness of fit): xi-kvadrati, roca monacemebis kategorizacia xdeba mxolod erTi ganzomilebis mixedviT. damoukideblobis xi-kvadrat kriteriumis SemTxvevaSi qulebis dajgufeba (kategorizacia) xdeboda ori damoukidebeli ganzomilebis mixedviT: satelevizio personaJi qera iyo Tu muqTmiani da am personaJis kariera profesiuli iyo Tu ara. Tanxmobis χ² kriteriumis SemTxvevaSi qulebis kategorizacia xdeba cvladis mxolod erTi ganzomilebis mixedviT. ganvixiloT magaliTi. bevri adamiani miiCnevs, rom paTologiuri qceva savse mTvaris dros ufroa mosalodneli. am mosazrebis empiriulad Semowmebis mizniT, davuSvaT, rom moaxdineT erTi wlis ganmavlobaSi fsiqiatriul ganyofilebaSi Semosuli axali klientebis miRebis kategorizacia mTvaris fazebis mixedviT da miiReT Semdegi ganawileba:

mTvaris faza savse mTvare axali mTvare pirveli meoTxedi mesame meoTxedi

62 50 60 56 gansxvavebulia Tu ara miRebuli klientebis raodenoba mTvaris fazis

mixedviT? am kiTxvaze pasuxis gasacemad SeiZleba gamoviyenoT Tanxmobis 2χ kriteriumi.

mocemuli sixSireebi axdenen qcevis kategorizacias mxolod mTvaris fazis mixedviT. xi-kvadrat kriteriumis gamosayeneblad saWiroa, davadginoT mTvaris TiToeuli fazis dros klientebis miRebis mosalodneli sixSireebi. radgan striqonisa da svetebis marginaluri sixSireebi ar gvaqvs, raime sxva safuZveli unda gamaCndes mosalodneli sixSireebis winaswarmetyvelebisaTvis. Tu fsiqiatriul ganyofilebaSi miReba ar aris kavSirSi mTvaris fazasTan, maSin axali pacientebis miRebis sixSire TiToeul fazaSi erTnairi unda iyos da raime gamovlenili gansxvaveba


15

mxolod SerCevis Secdomas unda asaxavdes. es msjeloba SeiZleba gamoixatos nulovani hipoTeziT:

0H : SerCevaSi fsiqiatriul ganyofilebaSi axali klientebis miReba

da mTvaris faza erTmaneTisgan damoukidebelia.

maSin alternatiuli hipoTeza iqneba:

1H : SerCevaSi fsiqiatriul ganyofilebaSi axali klientis miReba da mTvaris faza urTierTdamokidebulia.

Tu es alternatiuli hipoTeza WeSmaritia, maSin axali miRebebis raodenoba mTvaris sxvadasxva fazebis SemTxvevaSi erTnairi ar unda iyos. magram Tu

WeSmaritia 0H , maSin TiToeuli fazis dros pacientebis meoTxedis miReba

unda xdebodes. maSasadame, miRebebis mosalodneli sixSireebi SeiZleba miviRoT, Tu miRebebis saerTo raodenobas gavyofT 4-ze, anu mTvaris fazebis raodenobaze. Tu aseve vimsjelebT Cvens magaliTzec, radgan 228 axali pacientia miRebuli, mosalodneli sixSire TiToeuli fazisTvis aris

4228 , anu 57. mTvaris TiToeuli fazisaTvis realuri da mosalodneli

sixSireebi aseTia:

mTvaris faza savse mTvare Aaxali mTvare Ppirveli meoTxedi mesame meoTxedi

O=62 O=50 O=60 O=56 E=57 E=57 E=57 E=57

am monacemebis mixedviT xi-kvadrati miiReba Semdegnairad:

( )∑=

−=

c

c c

cc

EEO

1

22χ

Tavisuflebis xarisxi am isrealurisTv

2χ aris c – 1. aqedan gamomdinare,

( ) ( ) ( ) ( )

47,15784

,571

579

5749

5725

,571

573

577

575

,57

575657

576057

575057

5762

2222

22222

==

+++=

−++

−+=

−+

−+

−+

−=realuriχ


16

χ²²-is kritikuli mniSvneloba 3-is toli Tavisuflebis xarisxisaTvis da

roca α = 0,05, mocemulia A.7 cxrilSi, kritikuli2χ (3) = 7,81. realuri

2χ = 1,47 da

naklebia kritikuli2χ -ze, amitom ar xvdeba uaryofis zonaSi. anu ver vaxerxebT

nulovani hipoTezis uaryofas. ar arsebobs damamtkicebeli sabuTi imisa, rom gamokvleul SerCevaSi fsiqiatriul ganyofilebaSi axali miRebebis raodenoba mTvaris fazebzea damokidebulia. daSvebebi Tanxmobis xi-kvadrat kriteriumisaTvis iseTivea, rogoric damoukideblobis xi-kvadrat kriteriumisaTvis: TiToeulma cdis pirma unda mogvces erTi pasuxi da mosalodneli sixSireebi ar unda iyos 10-ze naklebi, - Tu gvaqvs mxolod ori kategoria da ar unda iyos 5-ze naklebi, Tu gvaqvs sami an meti kategoria.

xi-kvardat kriteriumis Sedegebis warmodgena amerikis fsiqologiuri asociaciis saxlmZRvanelo publikaciebis Sesaxeb (The Publication Manual of the American Psychological Association, 1994) moiTxovs, rom xi-kvadrat kriteriumis Sesaxeb warmodgenili Sedegebi Seicavdes Tavisuflebis xarisxsa (df ) da SerCevis zomas, rasac χ²realuri eyrdnoba. sailustraciod gamoviyeneb satelevizio personaJebis Sesaxeb amocanisaTvis gamoTvlili xi-kvadratis magaliTs. 1-li cxrili warmoadgens Tmis ferisa da karieris donis Sesabamis

sixSireebs. alfas 0,05 mniSvnelovnobis donis SemTxvevaSi xi-kvadrati am

sixSireebisaTvis statistikurad mniSvnelovani aRmoCnda. 2χ (1, N =180)²=6,43,

p < 0,05. ufro meti iyo albaToba imisa, rom satelevizio gadacemis

qeraTmian personaJebs eqnebodaT profesiuli donis kariera, vidre muqTmianebs.

cxrili 1 satelevizio gadacemebis qali personaJebis kategorizaciis hipoTeturi monacemebi.

Tmis feri

qera muqi

kariera profesiuli 36 48 araprofesiuli

24

72

Sedegebis warmodgenaSi:

cxrili 1 gvaZlevs miRebul sixSireebs α = 0,05 aRniSnavs SemowmebisaTvis arCeul


17

mniSvnelobis dones χ² (1, N = 180)² aRniSnavs gazomvis statistikas, xi-kvadrats.

frCxilebSi mocemuli cifri 1 miuTiTebs Tavisuflebis xarisxs df= 1 da N= 180 gviCvenebs SerCevis zomas.

= 16,57 gviCvenebs realuri2χ -s mniSvnelobas (da ara

kritikuli2χ -s mniSvnelobas A.7 cxrilidan)

p < 0,01. gamoxatavs, rom

a. Tu 0H WeSmaritia, realuri2χ -s albaToba

naklebia an tolia 0,01-sa. es mniSvneloba

aris realuri2χ -s an ufro ukiduresi

mniSvnelobis albaToba, Tu 0H WeSmaritia;

es albaToba SeiZleba α-s arCeuli mniSvnelobisgan gansxvavebuli iyos.

b. uaryofil iqna 0H :,,satelevizio gadacemis

personaJis samsaxuris done ar aris damokidebuli personaJis Tmis ferze”

dadasturda 1H : ,,satelevizio personaJis samsaxuris done dakavSirebulia personaJis Tmis ferTan”.

SeamowmeT Tqveni codna 15.1

1. ganmarteT c, xi-kvadrat kriteriumi, mosalodneli sixSireebi,

marginaluri sixSireebi, araparametrulikriteriumi, parametruli kriteriumi, r , niSanTa SeuRlebis cxrili, ormimarTulebiani sixSireebis cxrili.

2. ra SemTxvevebSi ixmareba araparametruli kriteriumi eqsperimentis monacemebis analizisaTvis?

3. gansazRvreT saTanado statistikuri kriteriumi CamoTvlilTagan TiToeuli erT-faqtoriani sqemisTvis, damoukidebeli cvladis ori donisaTvis da miTiTebuli gazomvis skalisaTvis: a. cdis pirebs SigniT, Sefardebis skala b. cdis pirebs Soris, saxeldebis skala g. cdis pirebs Soris, Sefardebis skala d. cdis pirebs Soris, rigis skala e. cdis pirebs SigniT, rigis skala

4. ra saxis monacemebis analizi aris xi-kvadrat kriteriumisTvis

Sesaferisi? 5. ra aris warmodgenili niSanTa SeuRlebis cxrilis svetebSi? 6. ra aris warmodgenili niSanTa SeuRlebis cxrilis striqonebSi?


18

7. qvemoT mocemulia realuri2χ -s mniSvnelobebi Tavisuflebis

sxvadasxva xarisxisaTvis (romelic miTiTebulia frCxilebSi).

realuri

2χ -s da Tavisuflebis xarisxis (df) ToToeuli

mniSvnelobisaTvis ipoveT kritikuli2χ -s Sesabamisi mniSvneloba 0,05

doneze. Semdeg gansazRvreT, xvdeba Tu ara realuri2χ uaryofis areSi

da ra gadawyvetilebas iRebT 0H -sa da 1H -Tan dakavSirebiT.

8. gamoTvaleT mosalodneli sixSireebi qvemoT mocemuli TiToeuli

niSanTa SeuRlebis cxrilisaTvis, Semdeg ipoveT realuri2χ da

mouTiTeT, aris Tu ara statistikurad mniSvnelovani 0,05 doneze. ToToeul ujraSi mocemuli mniSvnelobebi warmoadgens miRebul sixSireebs.

a.

sveti

1 2

striqoni

1 12 32

2 16 38

b.

sveti

1 2

striqoni

1 27 22

2 17 46

g.

sveti

realuri2χ (2) = 7,26 realuri

2χ (4) = 8,33

realuri2χ (7) = 14,19 realuri

2χ (20) = 26,94


19

1 2

striqoni

1 33 66

2 27 14

9. nimuSis saxiT mocemuli amocanis magaliTze xi-kvadratis

gamoyenebiT amoxseniT Semdegi amocana. gamoiyeneT 0,05 mniSvnelovnobis done. yofila Tu ara iseTi SemTxveva, roca SuqniSanis wiTel Suqze ar gaCerebulxarT? SesaZlebelia - ara, magram albaT sxvebis amgvari qcevis momswre mainc yofilxarT. ramdenad xSirad xdeba, rom mZRolebi ar Cerdebian wiTel Suqze da aris Tu ara aseTi qcevis sixSire damokidebuli dRis droze da kviris dReze? davuSvaT, rom mkvlevari ikvlevda am sakiTxs da akvirdeboda im avtomobilebis raodenobas, romlebic gzajvaredinze `sdeq~ niSanze ar gaCerdnen. is akvirdeboda transports dilis 10-dan 11 saaTamde da SuadRis 2-dan 3 saaTamde erTi kviris samSabaTs, oTxSabaTs da xuTSabaTs. mkvlevari iniSnavda manqanis nomrebs da TiToeul manqanas aRricxavda mocemuli kviris ganmalobaSi mocemul gzajvaredinze mxolod pirvelad gavlisas. davuSvaT, rom mkvlevarma im manqanebis sixSireebi mogvawoda, romlebic ar Cerdebodnen `sdeq~ niSanze mocemuli kviris dReebis or monakveTSi.

Kkviris dRe

samSabaTi

oTxSabaTi MxuTSabaTi

dro

dilis 10:00-dan 11:00-mde

38 46 58

SuadRis 2:00-dan 3:00-mde

46

54

63

aris Tu ara kavSiri kviris dRes an dRis monakveTsa da mZRolebis mier `sdeq~ niSnis ignorirebas Soris?

10. nimuSis saxiT mocemuli amocanis formatis mixedviT xi-kvadratis gamoyenebiT amoxseniT Semdegi amocana. gamoiyeneT 0,05 mniSvnelovnobis done. fsiqologebi varaudoben, rom qalebSi kvebiTi darRvevebis ganviTarebis erT-erTi mizezi isaa, rom arsebobs bevri iseTi signali, romelic xazs usvams sigamxdres, rogorc qalisaTvis sasurvel Tvisebas. smiTma, ualdorfma da trembaTma (Smith, Waldorf, & Trembath, 1990) ikvlies aseTi signalis erT-erTi forma, piradi gancxadeba (romelsac partnioris mosaZebnad


20

iyeneben), raTa SeemowmebinaT, am gancxadebebSi naxsenebi iyo Tu ara sigamxdre, rogorc sasurveli partnioris mniSvnelovani Tviseba. isini varaudobdnen, rom mamakacebis mier qalebis mosaZebnad ganTavsebul gancxadebebSi ufro meti albaTobiT iqneboda aRniSnuli wonis moTxovna, vidre qalebis mier mamkacebis mosaZebnad ganTavsebul gancxadebebSi. am hipoTezis Sesamowmeblad maT gamoikvlies piradi gancadebebis garkveuli raodenoba da moaxdines maTi kategorizacia gancxadebis avtoris sqesisa da gancxadebaSi sasurveli wonis aRniSvna-araRniSvnis mixedviT. maTi kvlevis Sedegebs gamoxatavs qvemoT mocemuli hipoTeturi sixSireebi. arsebobs Tu ara am monacemebs Soris is kavSiri, rasac hopoTeza varaudobs?

aRniSnuli iyo Tu ara

sasurveli wona diax

ara

mamakacis mier ganTavsebuli gancxadeba

89 186

qalis mier ganTavsebuli gancxadeba

6

219

man-uitnis U kriteriumi

man-uitnis U kriteriumi (Mann-Whitney U test): araparametruli kriteriumi cdis pirebs Soris sqemisaTvis, roca gamoyenebulia damoukidebeli cvladis ori done da monacemebi sul mcire, rigis skalazea.

man-uitnis U kriteriumi aris araparametruli kriteriumi cdis pirebs Soris sqemisaTvis damoukidebeli cvladis ori donis gamoyenebiT, roca qulebi, sul mcire, rigis skalazea. man-uitnis U kriteriums ganvixilavT orazrovani figurebis aRqmis kvlevis magaliTze (Reisberg, 1983, Strueber and Stadler, 1999). suraTi 15.1 Sroderis kibe, orazrovani figura orazrovani figura aris ori an meti SesaZlo perceptuli organizaciis mqone stimuli. stimulis magaliTad 15.1 suraTze gamosaxulia Sroderis kibe. am figuras gaaCnia ori SesaZlo perceptuli organizacia: pirvel SemTxvevaSi Cans Cveulebrivi kibe, meore SemTxvevaSi ki kibe amobrunebuli gveCveneba. safexurebi amobrundeba, Tu Tqven maT garkveuli


21

xnis ganmavlobaSi uyurebT. Tu erTi perceptuli organizaciis meoreTi Sesacvlelad yuradReba da mexsiereba aris saWiro, maSin yuradRebisa da mexsierebis momcveli damatebiTi amocana gazrdis Sebrunebis aRqmisaTvis saWiro dros. am hipoTezis Sesamowmeblad Seiqmna cdis pirebis ori ekvivalenturi jgufi. Srederis kibe orive jgufs aCvenes da cdis pirebma aRniSnes, Tu rodis dainaxes stimulis orive organizacia. damoukidebeli cvladi iyo xelis SemSleli amocanis tipi. sakontrolo jgufis cdis pirebi ubralod uyurebdnen naxats da acxadebdnen gamosaxulebis Sebrunebis Sesaxeb. eqsperimentuli jgufis cdis pirebi stimulis danaxvis paralelurad iTvlidnen samniSna ricxvidan klebis mimarTulebiT da 3-is bijiT, anu asrulebdnen amocanas, romelic moiTxovs yuradRebisa da mexsierebis CarTvas. Tu am figuris Sebrunebuli variantis danaxva moiTxovs yuradRebasac da mexsierebasac, maSin klebis mimarTulebiT Tvlam xeli unda SeuSalos figuris aRqmas da misi SebrunebiT danaxvis dro unda gazardos. eqsperimentatorebi iniSnavdnen, Tu ramdeni dro dasWirdaT cdis pirebs figuris Sebrunebulad dasanaxad. Tu cdis piri ver dainaxavda am figuris Sebrunebas meore wuTis dasasrulisaTvis, eqsperimenti Cerdeboda da cdis pirs mianiWebdnen ,,usasrulo” (∞) latentobas. 15.3 cxrili warmoadgens am eqsperimentis hipoTetur monacemebs im SemTxvevisaTvis, roca TiToeul jgufSi cxra cdis piria. qulebi naCvenebia TiToeuli jgufisaTvis mzardi latentobis mixedviT, Tumca isini am TanmimdevrobiT ar Cawerila, radgan sinamdvileSi cdis pirebisgan iyo miRebuli. yuradReba miaqcieT, rom eqsperimentul jgufSi erT-erTma cdis pirma 2 wuTis ganmavlobaSi ver dainaxa figuris Sebruneba. am cdis pirs mianiWes usasrulo latentoba ricxviTi mniSvnelobis miniWebis gareSe. radgan t kriteriumica da dispersiuli analizic moiTxovs TiToeuli cdis piris ricxviT mniSvnelobas, maT ver gamoviyenebT am monacemebis analizisaTvis. magram man-uitnis U kriteriumi mxolod imas moiTxovs, rom qulebi rigis mixedviT iyos dalagebuli, usasrulo latentobis rigis mixedviT dalageba ki SesaZlebelia. maSasadame, am monacemebis analizi xdeba man-uitnis U kriteriumis meSveobiT. statistikuri hipoTezebis Semowmeba man-uitnis U kriteriumiT Tavdapirvelad warmovadgenT U kriteriumis cnebis mimoxilvas da Semdeg misi gamoTvlis sqemas. orive midgoma moiTxovs monacemebis rigis moxedviT dalagebas gazomvis statistikis misaRebad da am statistikis SerCeviTi ganawilebis misaRebad. es dalageba SesaZlebelia maSinac, roca monacemebi intervalebis an Sefardebis skalazea; qulebis rigis mixedviT dalagebiT xdeba maTi rigis skalaze gadayvana. cxrili 15.3 perceptuli Sebrunebis hipoTeturi dro (wamebSi) Srederis kibisaTvis. sakontrolo jgufis cdis pirebi ubralod uyurebdnen am figuras. eqsperimentuli jgufis cdis pirebi figuris yurebisas iTvlidnen klebis mimarTulebiT samis toli bijiT.


22

jgufi

sakontrolo

( 1A ) Eeqsperimentuli

( 2A ) 2 4 5 10 6 11 8 12 9 14 13 17 15 85 21 98 42 a∞

msjelobis gaadvilebis mizniT, TiToeuli pirobisaTvis qulebi warmodgenilia latentobis zrdadi TanmimdevrobiT. aes qula warmoadgens Sebrunebis `usasrulo~

latentobas; am cdis pirma ver dainaxa figuris Sebruneba 2-wuTiani Sualedis ganmavlobaSi

qulebis rigis mixedviT dalageba (ranJireba) qulebis rigis mixedviT dalagebis pirveli safexuria orive jgufis qulebis gaerTianeba (N raodenobis qulebis jamis misaRebad) da Semdeg am qulebis zrdadi mniSvnelobis mixedviT, umciresidan udidesisaken dalageba. es procedura 15.3 cxrilis monacemebisaTvis naCvenebia 15.4 cxrilis pirvel safexurSi. rigis mixedviT dalagebisas SenarCunebulia qulebis jgufisadmi mikuTvnebuloba (ix. me-2 safexuri 15.4 cxrilSi). Semdeg xdeba am qulebisaTvis 1-dan N-mde rangis miniWeba, umciresi maCveneblisaTvis 1-is da udidesisaTvis N rangis miniWebiT. rangebis miniWebis procesi naCvenebia 15.4 cxrilis me-3 safexurSi. xSirad SesaZlebelia bmuli qulebis miReba, magaliTad, roca ori adamianis latentobaa 12 wami. 15.5 cxrilSi aRwerilia procedurebi bmuli qulebis SemTxvevisaTvis, romelic gamoyenebulia 15.2 amocanaSi. U statistika U statistika aris is raodenoba, ramdenjerac erT jgufis qulis rangi win uswrebs meore jgufis qulis rangs. es gamoTvlebi naCvenebia 15.4 cxrilis me-4 da me-5 safexurebiT. magaliTad, 2 qula (pirvel safexurSi naCvenebi) A1 jgufidan (naCvenebia me-2 safexurSi) rangis mixedviT win uswrebs yvela A2 maCvenebels (gansazRvrulia me-3 safexurSi). maSasadame, es qula rangis mixedviT win uswrebs cxrave A2 qulas. A1 jgufis qulebi: 5, 6, 8 da 9 rangis mixedviT win uswrebs A2 jgufis rva qulas. 15.4 cxrilis me-4 safexuri gviCvenebs, Tu rangis mixedviT ramdenjer uswrebs win A1 qula A2 qulas.


23

daakvirdiT ricxviT maCveneblebs am safexurisTvis da gaiazreT, Tu rogor iqna isini miRebuli. saerTo jamSi 56 SemTxveva iyo imisa, rom A1 qula A2

qulas rangis mixedviT win uswrebda, rac aris me-4 safexurze naCvenebi ricxvebis jami (anu 9 + 8 + 8 + 8 + 8 + 5 + 4 + 3 + 3 = 56). Sesabamisad, UA1realuri (A1 jgufisaTvis miRebuli U) aris 56. gvaqvs U -s mniSvneloba A2 jgufisTvisac. es ricxvi warmoadgens im raodenobas, Tu ramdenia iseTi SemTxveva, roca A2 qulis rangi win uswrebs A1 jgufis qulis rangs. 15.4 cxrilis me-5 safexuri gviCvenebs, Tu ramdenjer uswrebs win rangis mixedviT A1 qula A2 qulas. am safexurze ricxvebis jami gvaZlevs UA2realuri= 25 (anu 8 + 4 + 4 + 4+ + 3 + 2 = 25). cxrili 15.4 15.3 cxrilis qulebis rigis mixedviT dalageba 1. umciresidan udidesisaken dalagebuli latentobis qulebi 2 4 5 6 8 9 10 11 12 13 14 15 17 21 41 85 98 ∞ 2. jgufis

wevrobaa 1 2 1 1 1 1 2 2 2 1 2 1 2 1 1

2 2 2

3. rangi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 4. ramdenjer

uswrebs win 1A

qula 2A qulas 9 8 8 8 8 5 4 3 3

5. ramdenjer uswrebs win

2A qula

1A qulas 8 4 4 4 3 2

a1 aRniSnavs, rom qula 1A , sakontrolo jgufidanaa; 2 aRniSnavs, rom qula 2A ,

eqsperimentuli jgufidanaa.

U statistikis Tvisebebi Tu eqsperimentul Carevas Sedegi ar mohyveba, maSin mosalodnelia, rom qulebis gaerTianebisa da rigis mixedviT dalagebis Semdeg, rogorc es 15.4 cxrilzea mocemuli, TiToeuli jgufis qulebi rangebis mixedviT daaxloebiT erTnairad iqneba ganawilebuli. ar aris mosalodneli, rom erTi jgufis qulebi warmodgenili iqneba mxolod qveda mcire rangebiT da meore jgufis qulebi mxolod zeda rangebiT. aqedan gamomdinare, UA1 realuri

daaxloebiT toli iqneba UA2 realuri-sa. magaliTad, eqsperimentul Carevas Sedegi rom ar hqonoda Cvens magaliTSi, UA1 realuri-ca da UA2realuri-c daaxloebiT 40-is toli iqneboda. Carevas gansakuTrebiT Zlieri efeqti rom hqonoda da yvela A1 qulas yvela A2 qulaze dabali rangi hqonoda, UA1realuri

iqneboda 81 da UA2 realuri iqneboda 0. UA1 realuri-sa da UA2realuri-s mniSvnelobebi


24

idealur Sebrunebul kavSirSia. roca UA1 realuri izrdeba, UA2realuri Tanabari sididiT mcirdeba. maSasadame, U-s mxolod erTi, umciresi mniSvneloba gamoiyeneba statistikuri gazomvisas. Semdegi mimarTebis saSualebiT advilad SegviZlia imis gansazRvra, moxda Tu ara am umciresi mniSvnelobis gamoTvla:

realurirealuri 1212 AAAA UnnU −= sadac nA1 aris monacemebis raodeonoba A1 jgufSi da nA2 aris monacemebis raodenoba A2 jgufSi. Cvens magaliTSi UA2realuri gamoiTvleba UA1 realuri-s mixedviT Semdegnairad:

UA2 realuri = (9)(9) – 56 = 81 – 56 = 25, da 25 aris 15.3 cxrilis monacemebisTvis gamoTvlili U -s ori mniSvnelobidan umciresi. maSasadame, UA2 realuri = 25. U statistika gamoiyeneba hopoTezebis statistikuri Semowmebisas hopoTezebis statistikuri Semowmebis Tanmimdevrobis mixedviT. statistikuri hipoTeza

nulovani hipoTeza. nulovani hipoTeza, romlis farglebSic U -s SerCeviTi ganawileba miiReba, aris Semdegi:

0H : A1 monacemebis populaciaSi ganawileba da A2 monacemebis

populaciaSi ganawileba erTnairia.

Tu 0H WeSmaritia, maSin UA1realuri-sa da UA2realuri-s mniSvnelobebi daaxloebiT

toli unda iyos. alternatiuli hopoTeza. alternatiuli hipoTeza U kriteriumisaTvis aris:

1H : A1 monacemebis populaciaSi ganawileba da A2 monacemebis populaciaSi ganawileba erTnairi ar aris.

Tu 1H WeSmaritia, maSin UA1realuri da UA2realuri erTmaneTisgan gansxvavebuli unda iyos. U -s SerCeviTi ganawileba da kritikuli mniSvnelobebi U -s SerCeviTi ganawileba gansxvavebulia nA1–sa da nA2–sTvis. U statistikis gamoyenebis gasaadvileblad Seiqmna U –s SerCeviTi ganawilebisa da kritikuli mniSvnelobebis cxrilebi sandoobis 0,01 da 0,05 doneebze nA1–sa da nA2–is sxvadasxva zomisaTvis. radgan U –s maqsimaluri mniSvneloba ganisazRvreba nA1–sa da nA2–is meSveobiT, U -s SerCeviTi ganawilebis


25

cxrilebSi warmodegenilia U –s is umciresi mniSvnelobebi, romelTac kvlevebis Sedegad viRebT. 1-dan 20-mde cdis pirisagan Sedgenili jgufebisaTvis U –s mniSvnelobebi mniSvnelvnebis α= 0,01 da α= 0,05 doneebze warmodgenilia danarT cxrilSi A.8. am cxrilSi mocemuli mniSvnelobebi warmoadgens umciresi U realuris Sesabamis U –s kritikul mniSvnelobebs, rac aRniSnulia U kritikuliT. α – s arCeuli mniSvnelobisaTvis statistikurad mniSvnelovania umciresi U realuris is mniSvnelobebi, romlebic U kritikul –ze naklebia an tolia. yuradReba miaqcieT imas, rom t , F da

2χ ²statistikebisagan gansxvavebiT, U realuris uaryofis zona warmodgenilia

im mniSvnelobebiT, romlebic naklebia an tolia U kritikulis. Cveni magaliTis monacemebis SemTxvevaSi nA1= 9 da nA2 = 9, Sesabamisad U kritikuli = 17 mniSvnelvnebis 0,05 doneze. gadawyvetilebis miReba statistikuri hipoTezis Sesaxeb statistikuri hipoTezis Sesaxeb gadawyvetilebis miReba xdeba Urealuris ori mniSvnelobidan umciresis gamoyenebiT, Semdegnairad:

umciresi Urealuris mniSvneloba


Urealuri-ze naklebi an Urealuris toli

uarvyofT H0-s viRebT H1- s

Urealuri-ze meti ar uarvyofT H0-s ar viRebT H1- s

Cveni magaliTis SemTxvevaSi, 15.3 cxrilSi warmodgenili Sebrunebuli gamosaxulebis danaxvis latentobis maCveneblebisaTvis, UA1realuri = 56 da UA2realuri = 25. UA2realuri or miRebul mniSvnelobas Soris umciresia da swored es mniSvneloba gamoiyeneba hopoTezis SemowmebisaTvis. Urealuri = 25 metia, vidre Ukritikuli = 17. aqedan gamomdinare, Urealuris mniSvneloba ar

xvdeba uaryofis zonaSi; Cven ver uarvyofT 0H –s d ar viRebT 1H –s.

kriteriumis mixedviT gakeTebuli daskvnebi

radgan ver vaxerxebT 0H –is uaryofas, mivdivarT im daskvnamde, rom Cveni

magaliTis SemTxvevaSi ar arsebobs imis damadasturebeli monacemebi, rom sakontrolo jgufis qulebis populaciaSi ganawileba gansxvavdeba eqsperimentuli jgufis qulebis populaciaSi ganawilebisagan. sakontrolo pirobasTan SedarebiT klebis mimarTulebiT Tvlam ar Secvala Sebrunebuli gamosaxulebis danaxvisaTvis saWiro dro (latentoba).


26

man-uitnis U kriteriumis daSvebebi

U kriteriumis daSvebebia: • TiToeuli cdis piri iZleva mxolod erT qulas

monacemebisaTvis

• monacemebi, sul mcire, rigis skalazea da warmoadgens uwyvet cvlads. bevr SemTxvevaSi, rogorc es Cvens magaliTSi iyo, monacemebi iqneba an intervalebis an Sefardebis skalaze da U

kriteriumis gamoyenebisaTvis moxdeba am monacemebis rangebSi gadayvana.

U-s gamosaTvleli formulebi 15.4 cxrilSi UA1–sa da UA2 - is misaRebad gamoyenebuli midgoma rTulia, magram TvalsaCinod asaxavs, Tu ras warmoadgens UA1 da UA2. gamoTvlis miznebisTvis Seqmnilia gamartivebuli formulebi. UA1 realuri

UA1 realuri-is mniSvneloba gamoiTvleba formuliT:

( ) ∑−++= 1

1121 A1 2

1A

AAAA RnnnnU realuri

sadac nA1 aris A1 jgufSi monacemebis raodenoba

nA 2 aris A2 jgufSi monacemebis raodenoba

∑ 1AR aris A1 jgufis monacemebisaTvis miniWebuli rangebis jami

∑ 1AR miiReba 15.4 cxrilis me-3 safexurze A1 monacemebisaTvis miniWebuli

rangebis SekrebiT. magaliTad, A1 jgufSi monacemebis rangebia 1, 3, 4, 5, 6, 10,

12, 14 da 15. am rangebis jamia 70; Sesabamisad ∑ 1AR = 70. UA1 –is

gantolebaSi ricxvebis CasmiT viRebT:

( )( ) ( ) 56702

19999 A1 =−+

+=realuriU

UA2 realuri

UA2 realuri-is mniSvneloba gamoiTvleba formuliT:

( ) ∑−++= 2

2221 A2 2

1A

AAAA RnnnnU realuri


27

sadac ∑ 2AR aris A2 jgufis monacemebisaTvis miniWebuli rangebis jami. A2 jgufSi monacemebis rangebia 2, 7, 8, 9, 11, 13, 16, 17 da 18 (ix. 13.4 cxrilis me-3

safexuri). am rangebis jamia 101; anu ∑ 2AR = 101. formulaSi Casmis Sedegad,

( )( ) ( ) 251012

19999 A2 =−+

+=realuriU

13.5 cxrili warmoadgens U kriteriumis Sejamebas am gamoTvliTi formulebis gamoyenebiT. es cxrili agreTve Seicavs procedurebs, romlebic unda Sesruldes bmuli rangebis SemTxvevaSi. 13.2 magaliTSi naCvenebia am cxrilSi mocemuli safexurebis gamoyeneba. cxrili 15.5

man-uitnis U kriteriumis gamoTvliTi formulis gamoyeneba

safexuri 1. orive jgufis qulebi daalageT rigis mixedviT umciresi

mniSvnelobidan (rangi = 1) udides mniSvnelobamde (rangi = N ).

SeinarCuneT TiToeuli qulis jgufis wevroba ( 1A da 2A ). bmul qulebs mianiWeT im rangebis saSualo, rasac mianiWebdiT im SemTxvevaSi, bmuli rom ar yofiliyo.

safexuri 2. ipoveT ∑ 1AR da ∑ 2AR .

safexuri 3. gamoTvaleT UA1 realuri

( ) ∑−++= 1

1121 A1 2

1A

AAAA RnnnnU realuri

safexuri 4. gamoTvaleT UA2 realuri

( ) ∑−++= 2

2221 A2 2

1A

AAAA RnnnnU realuri

an

realurirealuri 1212 AAAA UnnU −=

safexuri 5. amoirCieT U realuri-s mniSvnelobebidan umciresi. SeadareT es U realuri Ukritikuli-s.

magaliTi 15.2

man-uitnis U kriteriumis gamoyeneba amocana bevr kolejsa da universitetSi janmrTelobis problemebs Soris yvelaze mniSvnelovania alkoholis borotad gamoyeneba. davuSvaT, rom fsiqologma


28

studentebisaTvis SeimuSava alkoholis Sesaxeb ganaTlebis programa. saswavlo wlis dasawyisSi fsiqologma SemTxveviTad SearCia 8 pirvelkurseli mamakaci da maT gaacno ganaTlebis programa. am programis gacnobidan 6 kviris Semdeg man studentebs Seavsebina kiTxvari alkoholis moxmarebis Sesaxeb. amavdroulad man SemTxveviTad SearCia 10 sxva pirvelkurseli mamakaci, romlebic ar gascnobian alkoholis ganaTlebis programas da maTac Seavsebina kiTxvari. yoveldRiurad moxmarebuli alkoholis raodenoba sufTa alkoholis unciebSi Semdegnairi aRmoCnda:

ganaTlebis programis piroba

gaiara programa (A1)

ar gauvlia programa (A2)

0,31 0,53 0,58 0,14 0,16 0,52 0,53 0,02

0,41 0,63 1,14 0,21 0,89 0,55 0,89 0,91 0,08 0,59

radgan es maCveneblebi asaxavs qcevis gazomvas da ara TviTon qcevas, fsiqologma CaTvala, rom isini intervalebis skalis moxovnebs ar akmayofilebda. Sesabamisad, fsiqologi am monacemebs mxolod rigis skalaze ganixilavda. moaxdina Tu ara alkoholis Sesaxeb ganaTlebis programam gavlena im adamianebis alkoholis moxmarebaze, vinc am programas gaecno? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna :kvlevis sqema aris cdis pirebs Soris sqema damoukidebeli cvladis, ganaTlebis programis pirobis, ori doniT. miCneulia, rom damokidebuli cvladi, moxmarebuli alkoholis raodenoba, rigis skalazea. Sesabamisad, man-uitnis U kriteriumi am monacemebis Sesabamisi statistikuri kriteriumia. saWiro kriteriumi: umciresi U realuri.

daSvebebi man-uitnis U kriteriumis gamoyenebisTvis: 1. TiToeuli cdis piris Semowmeba xdeba mxolod erTi

eqsperimentuli Carevis pirobaSi 2. damokidebuli cvladi, sul mcire, rigis skalazea da uwyvetia.

statistikuri hipoTeza


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0H : populaciaSi im adamianebis qulebis ganawileba, romelTac

gaiares saganmanaTleblo programa (A1), identuria populaciaSi im adamianebis qulebis ganawilebisa, romelTac ar gauvliaT saganmanaTleblo programa (A1).

1H : populaciaSi im adamianebis qulebis ganawileba, romelTac

gaiares saganmanaTleblo programa (A1), ar aris identuri populaciaSi im adamianebis qulebis ganawilebisa, romelTac ar gauvliaT saganmanaTleblo programa (A2).

mniSvnelovnobis done: α= 0,05 U-s kritikuli mniSvneloba: roca nA1 = 8, nA 2 = 10 da α= 0,05, A.8 cxrilis mixedviT Ukritikuli = 17. uaryofis are: umciresi U realuri-s mniSvnelobebi, romlebic udris an naklebia 17-ze. gamoTvla: U realuri-s gamosaTvlelad unda mivyveT 15.5 cxrilSi mocemul safexurebs

safexuri 1: orive jgufis gaerTianebuli monacemebi daalageT rigis mixedviT umciresidan udidesisaken (zrdadobis mixedviT).

rigis mixedviT dalagebuli monacemebi: 0,02 0,08 0,14 0,16 0,21 0,31 0,41 0,52 0,53 0,53 0,55 0.58 0,59 0,63 0,89 0,89 0,91 1,14 jgufTan mikuTvnebuloba: 1 2 1 1 2 1 2 1 1 1 2 1 2 2 2 2 2 2

rangi: 1 2 3 4 5 6 7 8 9,5 9,5 11 12 13 14 15,5 15,5 17 18

safexuri 2: gamoTvaleT TiToeuli jgufis rangebis jami

∑ 1AR = 1+3+4+6+8+9,5+9,5+12 = 53 ∑ 2AR = 2+5+7+11+13+14+15,5+15,5+17+18= 118

safexuri 3: gamoTvaleT U A1realuri

( )( ) ( )

( ).63533680

532/7280

532

188108 A1

=−+=−+=

−+

+=realuriU

safexuri 4: gamoTvaleT U A2realuri


30

( )( ) ( )

( ).171185580

1182/11080

1182

11010108 A2

=−+=−+=

−+

+=realuriU

safexuri 5: airCieT umciresi U realuri

U A1realuri = 17 aris umciresi U A realuri

gadawyvetilebis miReba: miRebuli U realuri = 17 udris Ukritikuli = 17-s.

maSasadame, is xvdeba uaryofis areSi, uarvyofT 0H -s da viRebT 1H -s.

daskvna: 0H -is uaryofa niSnavs, rom saganmanaTleblo programis jgufis

monacemebis ganawileba gansxvavebulia arasaganmanaTleblo jgufis monacemebis ganawilebisgan. alkoholis Sesaxeb ganaTlebis programam Seamcira moxmarebuli alkoholis raodenoba, rac izomeboda studentebis mier mowodebuli maCveneblebiT maT mier dReSi moxmarebuli alkoholis raodenobis Sesaxeb. man-uitnis U kriteriumis Sedegebis warmodgena

U kriteriumis Sedegebis warmodgena Sedegebis statistikuri warmodgenis zogadi principebis Sesabamisad xdeba. sailustraciod gamoviyenebT amocanas imis Sesaxeb, Tu rogoria klebis mimarTulebiT Tvlis gavlena Sroderis kibis Sebrunebebis danaxvaze. sakontrolo (n = 9) da eqsperimenul (n = 9) jgufebSi figuris SebrunebisaTvis saWiro drois ganawilebebis Sesadareblad gamoiynes man-uitnis U krteriumi. alfas 0,05

mniSvnelobisas ganawilebebi mniSvnelovnad ar gansxvavdeboda erTmaneTisgan, U = 25, p > 0,05

Sedegebis aseTi warmodegena gvawvdis Cveul informacias gamoyenebuli kriteriumis Sesaxeb. U = 25 umciresi U realuri–ia da p > 0,05 gamoxatavs, rom U aramniSvnelovania 0,05 doneze. mowodebuli informacia jgufis zomis Sesaxeb (nA1 = nA 2 = 9) saWiroa imisaTvis, rom U kritikuli movZebnoT A.8 cxrilSi.


1. gaaerTianeT A1 da A2 jgufebis qvemoT mocemuli monacemebi da daalageT rigis mixedviT 1-dan N-mde. SeinarCuneT TiToeuli qulis jgufTan mikuTvnebuloba. Semdeg gamoTvaleT: UA1realuri,

ramdenjer uswrebs win A1 monacemis rangi A2 monacemis rangs,


31

UA2realuri, ramdenjer uswrebs win A2 monacemis rangi A1 monacemis rangs.

jgufi

1A 2A

15 23 10 19

7 26 14 16

a. U realuri–s mniSvnelobebidan romelia umciresi? b. aris Tu ara U realuri statistikurad mniSvnelovani 0,05

doneze? 2. pirveli kiTxvis qulebisaTvis ipoveT UA1realuri da UA2realuri U–s

gamosaTvleli formulis gamoyenebiT. a. formulebiT miRebuli UA1realuri -sa da UA2realuri-s mniSvnelobebi eTanxmeba Tu ara pirvel kiTxvaSi miRebul mniSvnelobebs?

3. CamoayalibeT statistikuri hipoTeza man-uitnis U kriteriumisaTvis.

4. Tu 0H WeSmaritia, rogori unda iyos UA1realuri–isa da UA2realuri–is

mimarTeba?

5. Tu 1H WeSmaritia, rogori unda iyos UA1realuri–isa da UA2realuri –is mimarTeba?

6. Semdeg cxrilSi mocemulia Urealuri-is umciresi mniSvnelobebi sxvadasxva zomis jgufebisaTvis. U realuri–s da jgufis zomis TiToeuli nakrebisaTvis ipoveT Ukritikuli–s mniSvneloba 0,01 doneze. Semdeg miuTiTeT, xvdeba Tu ara U realuri uaryofis

zonaSi da ra gadawyvetilebs miiRebT 0H –sa da 1H -Tan

dakavSirebiT.

Urealuri 1An 2An

a. b. g. d. e. v.

3 12 28 21 4 126

6 10 16 7 13 18

4 10 9 19 19 19

7. amocanis nimuSis formatis mixedviT man-uitnis kriteriumis gamoyenebiT amoxseniT Semdegi amocana. gamoiyeneT mniSvnelovnobis 0,05 done. musikas sul ufro da ufro xSirad iyeneben stomatologiuri pacientebis relaqsaciisaTvis.


32

davuSvaT, rom somatologma Seamowma pacientebis relaqsaciisaTvis musikis efeqturoba, risTvisac pacientebi SemTxveviT miakuTvna ori jgufidan erT-erTs. cameti pacienti musikis pirobaSi stomatologiuri procedurebis dros usmenda musikas, 12 pacienti musikis arqonis pirobaSi igive stomatologiuri procedurebis dros ar usmenda musikas. stomatologTan vizitis naxevari drois gasvlis Semdeg TiToeul pacients sTxovdnen, ganesazRvraT sakuTari SfoTva 1-dan (sruliad ar vSfoTav) 100-mde (Zalian vSfoTav) skalaze. SfoTvis miRebuli monacemebi Semdegia:

Emusikis piroba

moismina musika

( 1A ) ar mousmenia

musika

( 2A ) 52 75 66 33 17 78 94 89 27 49 53 78

44 21 66 54 37 84 30 29 54 49 33 10 78

gansxvavdeba Tu ara jgufebi SfoTvis maCveneblebis ganawilebis mixedviT?

uilkoqsonis niSniani rangebis kriteriumi

uilkoqsonis niSniani rangebis kriteriumi (Wilcoxon signed-ranks test): araparametruli kriteriumi cdis pirebs Soris sqemisaTvis damoukidebeli cvladis ori doniT.

uilkoqsonis niSniani rangebis kriteriumi araparametruli kriteriumia cdis pirebs SigniT sqemisaTvis damoukidebeli cvladis ori doniT.

maSasadame, is aris lidamokidebut kriteriumis araparametruli analogi.

uilkoqsonis kriteriumis daSvebis Tanaxmad, qcevis ganzomileba uwyvetia da monacemebi rigis mixedvis SeiZleba dalagdes. uilkoqsonis niSniani rangebis kriteriumis sailustraciod warmovadgenT eqsperiments xmaSi emociis gamoxatvis Sesaxeb (Cosmides, 1983).


33

davuSvaT, rom fsiqologis hipoTezaa, rom xmis sixSireebi icvleba sxvadasxva fsiqologiuri mdgomareobis dros. am hopoTezis Sesamowmeblad is iyenebs erTfaqtorian, cdis pirebs SigniT sqemas damoukidebeli cvladis ori doniT. damoukidebeli cvladia scenaris tipi, romelic cdis pirma unda waikiTxos: mxiaruli situaciis amsaxveli scenari (A1) an sevdiani situaciis amsaxveli scenari (A2). scenaris mizania cdis pirSi mxiaruli an sevdiani emociis gamowveva. ToToeul pirobaSi cdis piris amocanaa, Cumad waikiTxos scenari da eqsperimentatoris miniSnebaze xmamaRla warmoTqvas: ,,me jer kidev vkiTxulob~. TiToeuli cdis pirisaTvis damokidebuli cvladia gamoTqmis ,,mer jer kidev vkiTxulob~ xmovani sixSire da izomeba wamSi rxevebis raodenobiT (rxeva wamSi anu herci). maSasadame, monacems warmoadgens diapazoni cdis piris mier am sityvebis warmoTqmis umaRles xmovan sixSiresa da yvelaze dabal xmovan sixSires Soris. davuSvaT, 15.6 cxrilSi warmodgenilia scenaris orive pirobaSi monawile 10 cdis piris sixSireebi. aq naCvenebia am monacemebis saSualoebi, standartuli gadaxrebi da dispersiebi. yuradReba miaqcieT imas, rom gansxvavdeba ara mxolod sixSireebis diapazonebis saSualoebi, aramed didi

gansxvavebaa ori pirobis cvalebadobas Soris. gaixseneT, rom lidamokidebut moiTxovs, rom ori populaciis dispersia Tanabari iyos, am magaliTSi ki ori dispersia erTmaneTisgan Zlier gansxvavdeba (mxiaruli scenarisaTvis 704,37 da sevdiani scenarisaTvis 10281,96). maSasadame, es monacemebi arRvevs

lidamokidebut is erT-erT daSvebas. uilkoqsonis niSniani rangebis kriteriums ar

aqvs moTxovna dispersiebis Sesaxeb da aqedan gamomdinare, is am monacemebisaTvis Sesaferisi kriteriumia. cxrili 15.6 xmovani sixSiris hipoiTeturi qulebi (hercebSi), rogorc wakiTxuli scenaris tipis funqcia

scenaris tipi cdis piri mxiaruli ( 1A ) sevdiani ( 2A )

1 2 3 4 5 6 7 8 9 10

305 275 360 299 317 326 284 347 331 314

214 385 519 307 483 454 501 463 382 297

AX

As

315,80 26,54

400,50 101,40


34

2s 704,37 10.281,96

statistikuri hipoTezebis Semowmeba uilkoqsonis niSniani rangebis kriteriumiT T statistika uilkoqsonis kriteriumi rigis mixedviT dalagebul monacemebs iyenebs statistikis misaRebad, romelsac ewodeba T da romelic ori eqsperimentuli pirobidan aRebuli rangebis umciresi jamia. statistikuri hipoTeza nulovani hipoTeza. nulovani hipoTeza, romelic T–s gamoyenebiT mowmdeba, aris:

0H : erTmaneTTan dakavSirebuli A1 da A2 qulebis populaciaSi

ganawilebebi identuria.

alternatiuli hopoTeza. alternatiuli hipoTezaa:

1H : erTmaneTTan dakavSirebuli A1 da A2 qulebis populaciaSi ganawilebebi ar aris identuri.

SerCeviTi ganawileba da T-s kritikuli mniSvnelobebi

kritikuliT -s mniSvnelobebi, roca α= 0,05 da α= 0,01 da SerCevis zoma 6-dan 50-

mdea, warmodgenilia danarT A.9 cxrilSi. am cxrilSi Segvaqvs wyviliN -s

mniSvnelobebi, rac aris qulebis wyvilebis raodenoba, anu cdis pirebis

raodenoba. realuriT –s is mniSvnelobebi, romlebic metia an udris kritikuliU –s,

xvdeba uaryofis areSi da statistikurad mniSvnelovania α–s arCeul doneze.

magaliTisaTvis, roca wyviliN = 10 da α= 0,05, kritikuliT = 8. maSasadame, uaryofis

areSi xvdeba kritikuliT -s is mniSvnelobebi, romlebic metia an tolia 8-is.

T -s gamoTvla T -s mniSvnelobis gamoTvla moicavs qvemoT CamoTvlil safexurebs. TiToeuli safexuris Sedegi xmovani sixSiris diapazonis Sesaxeb eqsperimentis moncemebisaTvis naCvenebia 15.7 cxrilSi.

safexuri 1

TiToeuli cdis pirisaTvis ipoveT

d = 21 AA XX − , anu monacemebs Soris gansxvaveba, rogorc es 15.7 cxrilSia mocemuli. sxvaobas


35

miuwereT + an – niSani. safexuri 2

ipoveT TiToeuli sxvaobis absoluturi

mniSvneloba ( d , mniSvneloba niSnis

gaTvaliswinebis gareSe), rogorc es naCvenebia b svetSi

safexuri 3

zrdadobis mixedviT (umciresi mniSvnelobis rangia 1 da udidesis _ N) daalageT absoluturi mniSvnelobebi, rogorc es g svetSia naCvenebi. 0-is toli sxvaobebis

ranJireba ar xdeba da kritikuliT –s

gamosaTvlelad gamoyenebuli N - is mniSvneloba mcirdeba 0-s toli sxvaobebis raodenobiT. d –s bmul qulebs eniWeba im rangebis saSualo, rasac mivaniWebdiT maT, bma rom ar momxdariyo.

safexuri 4

gGsvetis ToToeul rangs miuwereT a svetis sxvaobis niSani. niSniani rangebi naCvenebia d svetSi.

safexuri 5

SeajameT msgavsi niSnis mqone rangebi d svetidan. magaliTad, plus niSnis mqone rangebis jami d svetSi aris 6 (anu, (+4) + (+2)) da minus niSnis mqone rangebis jami aris – 49 (anu, (-5) + (-8) + (-1) + (-9) + (-7) + (-10) + (-6) + (-3)).

safexuri 6 ipoveT rangebis TiToeuli jamis absoluturi mniSvneloba. Semdeg amorCieT rangebis ori jamidan umciresi. am SemTxvevaSi es aris 6, dadebiTi rangebis jami. swored es jami aris

realuriT ; maSasadame, realuriT = 6.

cxrili 15.7 15.6 cxrilis qulebisTvis uilkoqsonis T -s gamoTvla

scenaris tipi (a) (b) (g) (d) cdis piri

mxiaruli

( 1A )

sevdiani

( 2A ) d d ranJirebuli

d

niSniani rangi

1 2 3 4 5 6 7 8 9

305 275 360 299 317 326 384 347 331

214 285 519 307 483 454 501 463 382

+91 _110 _159 _8 _166 _128 _217 _116 _51

91 110 159 8 166 128 217 116 51

4 5 8 1 9 7 10 6 3

+4 _5 _8 _1 _9 _7 _10 _6 _3


36

10 314 297 +17 17 2 +2 statistikuri hipoTezis Sesaxeb gadawyvetilebis miReba T realuri= 6 ufro mcirea, vidre Tkritikuli= 8; aqedan gamomdinare, T realuri

xvdeba uaryofis areSi. uarvyofT 0H -s da viRebT 1H -s.

daskvna kriteriumis mixedviT

0H -is uaryofas da 1H –is miRebas mivyavarT daskvnamde, rom monacemebis ori

ganawileba erTmaneTisagan gansxvavdeba; scenaris tipma ganapiroba sxvaobebi xmis sixSiris dipazonSi. xmis sixSiris diapazoni ufro didia sevdiani scenaris SemTxvevaSi, vidre mxiaruli scenaris SemTxvevaSi.

magaliTi 15.3

uilkoqsonis niSniani rangebis kriteriumis gamoyeneba amocana hipoTezis Tanaxmad, musikis tempi gavlenas axdens qcevis siswrafeze; kerZod, swrafi tempis musika zrdis qcevis ganxorcielebis SesaZlo siswrafes (McElrea & Standing, 1992; Roballey et al., 1985). warmoidgineT, rom am hopoTezis Sesamowmeblad mkvlevarma Caatara Semdegi kvleva: ori dRis ganmavlobaSi akvirdebodnen kafeteriis erTi da igive 11 klientis sadilobas musikis fonis ori gansxvavebuli pirobis SemTxvevaSi: pirvel dRes neli musikis pirobaSi (A1) da meore dRes swrafi musikis pirobaSi (A2). Caiweres TiToeuli cdis piris mier musikis TiToeul pirobaSi wuTSi miRebuli lukmebis raodenoba. davuSvaT, rom TiToeuli cdis piris mier musikis tempis TiToeuli pirobis SemTxvevaSi wuTSi miRebuli lukmebis raodenobebi Semdegia:

musikis tempi cdis piri neli ( 1A ) swrafi ( 2A )

1 2 3 4 5 6 7 8 9 10 11

3 4 2 3 3 4 1 5 3 5 2

4 6 3 3 8 2 4 4 6 9 8


37

eqsperimentatori miiCnevda, rom es monacemebi rigis skalazea. axdens Tu ara gavlenas musikis tempi miRebuli lukmebis raodenobaze? gamoiyeneT mniSvnelovnobis 0,05 done. amoxsna: kvlevis sqema warmoadgens erTfaqtorian, cdis pirTa Sida sqemas damoukidebeli cvladis, musikis tempis, ori doniT. damokidebuli cvladi izomeba, sul mcire, rigis skalaze. maSasadame, uilkoqsonis niSniani rangebis kriteriumi am monacemebisaTvis Sesabamisi kriteriumia. saWiro kriteriumi: T, rangebis jamebs Soris umciresi jami. daSvebebi kriteriumis gamoyenebisaTvis:

1. TiToeul cdis pirs vafasebT damoukidebeli cvladis orive doneze.

2. damokidebuli cvladi izomeba, sul mcire, rigis skalaze statistikuri hipoTeza:

0H : erTmaneTTan dakavSirebuli A1 da A2 monacemebis populaciaSi

ganawilebebi identuria.

1H : erTmaneTTan dakavSirebuli A1 da A2 monacemebis populaciaSi ganawilebebi ar aris identuri.

mniSvnelovnobis done: α = 0,05 T-s kritikuli mniSvneloba: im SemTxvevaSi, roca wyviliN = 11 da α = 0,05,

kritikuliT = 11. T-s gamoTvlisas cdis piri #4 gviCvenebs nulovan cvlilebas

musikis or pirobas Soris. es cdis piri gamoeTiSeba analizs. Sesabamisad,

wyviliN xdeba 10 da N = 10-is SemTxvevisaTvis, A.9 cxrilis mixedviT,

kritikuliT gaxdeba 8.

uaryofis are: realuriT -s is mniSvnelobebi, romlebic naklebia an tolia 8-is.

gamoTvla:

safexuri 1

TiToeuli cdis pirisaTvis viTvliT d = XA 1 – XA (15.8 cxrilis a sveti)

safexuri 2

vpoulobT d–s absolutur mniSvnelobebs (15.8 cxrilis b sveti)

safexuri 3

zrdadobis mixedviT, 1-dan 10-mde valagebT d–s absolutur mniSvnelobebs; d = 0 ar


38

monawiloeobs ranJirebaSi (g sveti). yuradReba miaqcieT bmuli d monacemebis ramdenime SemTxvevis ranJirebas. bmul qulebs aniWeben maT mier dakavebuli rangebis saSualos. magaliTad,

magaliTad, b svetSi aris sami sxvaoba d =1. es

mniSvnelobebi ikavebs rangebs 1, 2 da 3. am

rangebis saSualo aris 2; maSasadame, d =1-is

TiToeul mniSvnelobas eniWeba rangi 2. safexuri 4

d-s rangs (d sveti) vuwerT TiToeuli d –s niSans (a svetidan).

safexuri 5

vajamebT msgavsi niSnis mqone rangebs. plus niSnis mqone rangebis jamis absoluturi mniSvneloba aris 6,5 (anu, (+4,5) + (+2) = 6,5). minus niSnis mqone rangebis jamis absoluturi mniSvneloba aris 48,5 (anu, (-2) + (-4,5) + (-2) + (-9) + (-6,5) + (-6,5) + (-8) + (-10) = 48,5).

safexuri 6 plus niSnis mqone rangebis jamis absoluturi mniSvneloba rangebis jamebs Soris umciresia,

amitom realuriT = 6,5.

cxrili 15.8 uilkoqsonis T -s gamoTvla lukmebis raodenobis qulebis, rogorc musikis tempis funqciisaTvis

musikis tempi (a) (b) (g) (d) cdis piri

swrafi

( 1A )

neli

( 2A ) d d ranJirebuli

d

niSniani rangi

1 2 3 4 5 6 7 8 9 10 11

3 4 2 3 3 4 1 5 3 5 2

4 6 3 3 8 2 4 4 6 9 8

_1 _2 _1 0 _5 +2 _3 +1 _3 _4 _6

1 2 1 0 5 2 3 1 3 4 6

2 4,5 2 _ 9 4,5 6,5 2 6,5 8 10

_2 _4,5 _2 _ _9 +4,5 _6,5 +2 _6,5 _8 _10

gadawyvetileba: realuriT = 6,5 ufro naklebia, vidre kritikuliT = 8 da is xvdeba

uaryofis areSi, uarvyofT 0H -s da viRebT 1H –s.


39

daskvna: musikis ori tempis pirobebSi wuTSi miRebuli lukmebis raodenobis ganawilebebi mniSvnelovnad gansxvavdeba; nel tempTan SedarebiT swrafi tempis dros met lukmas iReben. uilkoqsonis niSniani rangebis kriteriumis Sedegebis warmodgena uilkoqsonis niSniani rangebis kriteriumis Sedegebis warmodgenas giCvenebT im magaliTis gamoyenebiT, romelSic xdeboda scenaris tipiT manipulireba da cdis pirebis xmis sixSiris diapazonis gazomva.

mxiaruli da sevdiani scenarebis pirobebSi xmis sixSireebis diapazonebis Sesadareblad gamoiyeneboda uilkoqsonis niSniani rangebis kriteriumi. alfas 0,05 doneze scenaris pirobebi mniSvnelovnad gansxvavdeboda erTmaneTisagan, T = 6 (N = 10), p < 0,05. mxiaruli scenaris pirobasTan SedarebiT sevdiani scenaris pirobaSi sixSireTa diapazoni ufro didi iyo.

ase warmodgenili Sedegebi gvaZlevs realuriT –s mniSvnelobis

interpretaciisaTvis saWiro informacias. radgan wyviliN -s mniSvneloba

viciT, SegviZlia saWiroebis SemTxvevaSi movZebnoT kritikuliT .

araparametruli kriteriumebis gamoyeneba rogorc vnaxeT, araparametruli kriteriumebi ar ar akeTeben daSvebebs SerCevis qulebis populaciaSi ganawilebis Sesaxeb. ufro metic, isini Cveulebriv moiTxoven mxolod imas, rom miRebuli monacemebis rigis mixedviT dalageba SegveZlos. parametruli kriteriumebi, rogoricaa t an dispersiuli analizi, ufro mkacr daSvebebs akeTeben eqsperimentis monacemebis bunebis Sesaxeb, _ iseT daSvebebs, romelTa dakmayofileba kvlevaSi miRebuli monacemebisaTvis zogjer saeWvoa. miuxedavad amisa, araparametruli statistikuri kriteriumebi ufro naklebad gamoiyeneba qceviTi mecnierebebis kvlevebSi, vidre parametruli kriteriumebi. ratom?

am kiTxvaze ramdenime pasuxi arsebobs. zogadad, Tu gasaanalizebeli monacemebi parametruli kriteriumebis daSvebebs akmayofilebs, maSin parametruli kriteriumi ufro aRmoaCens damoukidebeli cvladis efeqts, vidre misi Sesabamisi araparametruli kriteriumi. anu araparametruli kriteriumebi parametrul kriteriumebTan SedarebiT naklebad mZlavria da amitom, roca monacemebi akmayofilebs parametruli kriteriumis daSvebebs, ufro mosalodnelia, rom isini meore gvaris Secdomamde migviyvans. radgan iTvleba, rom parametruli kriteriumebis umravlesoba daSvebebTan Sesabamisobis TvalsazrisiT ufro mdgradia, bevri mkvlevari monacemebs parametruli kriteriumebis gamoyenebiT aanalizebs.


40

araparametruli kriteriumebi ufro sxva saxis informacias gvaZlevs,

vidre parametruli kriteriumebi. magaliTad, lobisdamoukidebt –as mivyavarT

daskvnamde populaciebis saSualoebis toloba-aratolobis Sesaxeb, xolo U kriteriumiT gakeTebuli daskvnebi exeba populaciis ganawilebis toloba-aratolobas. garda amisa, ufro rTuli sqemebis SemTxvevaSi, rogoricaa me-11 TavSi ganxiluli faqtoruli sqema, ar arsebobs iseTi araparametruli kriteriumebi, romlebic imdenive informacias gvaZlevs, ramdensac dispersiis parametruli faqtoruli analizi. rogorc aRvniSneT, parametruli kriteriumis gamoyenebis mkacrad dadgenili wesebi ar arsebobs. Sroderis kibis magaliTi, romelic moyvanili iyo U kriteriumis gasacnobad, warmoadgens araCveulebrivad naTel magaliTs im SemTxvevebisa, roca saWiroa araparametruli kriteriumis gamoyeneba. am eqsperimentSi miRebuli erT-erTi monacemi iyo `usasruloba~. roca aseTi monacemi gvaqvs, monacemebis saSualos an standartuli gadaxras ver gamovTvliT. amitom, cxadia, rom am monacemebisaTvis parametreuli kriteriumis gamoyeneba ar SeiZleba. sxva magaliTebi xSirad aseTi erTmniSvnelovani ar aris. magaliTad, ramdenad unda gansxvavdebodes populaciis qulebis ganawileba normaluri ganawilebisagan, rom parametruli kriteriumis gamoyeneba Seusabamod CaiTvalos? rodis iTvleba ori an meti populaciis dispersiebi tolad? qceviTi mecnierebi ver Tanxmdebian am kiTxvebis pasuxebze. Sesabamisad, isini SeiZleba imazec ver SeTanxmdnen, garkveuli monacemebis analizisaTvis parametruli kriteriumi unda gamoviyenoT, Tu araparametruli.


1. rogori tipis kvlevis sqemisaTvis aris Sesaferisi uilkoqsonis

kriteriumi? 2. romel gazomvis statistikas iyenebs uilkoqsonis niSniani

rangebis kriteriumi? 3. CamoayalibeT statistikuri hipoTeza uilkoqsonis niSniani

rangebis kriteriumisaTvis. 4. qvemoT mocemulia niSniani rangebis jamebis umciresi

absoluturi mniSvnelobebi sxvadasxva zomis jgufebze Catarebuli eqsperimentebidan. TiToeuli umciresi rangisa da

wyviliN –s mniSvnelobisaTvis gansazRvreT realuriT is mniSvneloba.

Semdeg ipoveT kritikuliT mniSvnelovnobis 0,05 doneze da miuTiTeT,

xvdeba Tu ara uaryofis areSi. aseve aRniSneT, ra

gadawyvetilebas iRebT 0H –sa da 1H -Tan dakavSirebiT.

Uumciresi rangi

wyviliN


41

a. b. g. d. e.

2 9 17 40 98

8 10 13 19 25

5. amocanis nimuSis formatis mixedviT uilkoqsonis kriteriumis

gamoyenebiT amoxseniT Semdegi amocana. gamoiyeneT mniSvnelovnobis 0,05 done.

sportis fsiqologebi dainteresebulebi arian sportuli Sejibrebebis dros SfoTvasa da sportul Sedegebs Soris kavSiriT. magaliTad, meinardma da houim (Maynard & Howe, 1987) gazomes ragbis moTamaSeebis SfoTva finalis pirveli da meore TamaSebis win. davuSvaT, rom Tqven CaatareT msgavsi kvleva da miiReT SfoTvis qvemoT moyvanili monacemebi. SesaZlo monacemebis diapazoni meryeobs 12-dan (Zalian dabali SfoTva) 84-mde (Zalian maRali SfoTva).

TamaSi moTamaSe 1 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

68 56 78 49 66 43 77 68 28 69 75 80 75 65 71

72 65 77 54 64 53 70 78 29 75 78 72 75 76 70

mosalodnelia, rom SfoTvis qulebis ganawileba populaciaSi uaryofiTad gadaxrili iqneba (moTamaSeebis umravlesobas eqneba Zalian maRali SfoTva da mxolod ramdenime maTgans eqneba Zalian dabali SfoTva). amitom Tqven apirebT araparametruli statistikuri kriteriumis gamoyenebas. gansxvavdeba Tu ara pirveli da meore TamaSis win miRebuli SfoTvis maCveneblebi?


42

Sejameba

• xi-kvadrat kriteriumi gamoiyeneba im monacemebisaTvis, romlebic warmoadgenen qcevis (pasuxis) gamovlenis sixSires.

• damoukideblobis xi-kvadrat kriteriumi gamoiyeneba ori damoukidebeli ganzomilebis kategorias Soris mimarTebis dasadgenad.

• damoukideblobis xi-kvadrat kriteriumi gamoiTvleba formuliT:

( )∑∑= =

−=

r

r

c

c rcErcErcO

1 1

2

2realuriχ

• Tanxmobis xi-kvadrat kriteriumi gamoiyeneba imisaTvis, rom dadgindes, Tu ramdenad kargad Seesabameba erT kategoriaSi moTavsebuli monacemebi am kategoriisaTvis Teoriulad miRebul mosalodnel sixSireebs.

• Tanxmobis xi-kvadrat kriteriumi gamoiTvleba formuliT:

( )∑=

−=

c

c c

cc

EEO

1

22realuriχ

• man-uitnis U kriteriumi araparametruli kriteriumia cdis pirebs Soris sqemisaTvis damoukidebeli cvladis ori donis gamoyenebiT, roca qulebi, sul mcire, rigis skalazea.

• U statistika is raodenobaa, ramdenjerac erTi jgufis qulis rangi win uswrebs meore jgufis qulis rangs.

• U A1realuri -s mniSvneloba gamoiTvleba formuliT:

( ) ∑−++= 1

1121 A1 2

1A

AAAA RnnnnU realuri

xolo U A2realuri-s _ formuliT:

( ) ∑−++= 2

2221 A2 2

1A

AAAA RnnnnU realuri

realuriU -s umciresi mniSvnelobebi gamoiyeneba, rogorc gazomvis

statistika.

• uilkoqsonis niSniani rangebis kriteriumi aris araparametruli kriteriumi cdis pirTa Sida sqemisaTvis, damoukidebeli cvladis ori doniT.

• uilkoqsonis niSniani rangebis kriteriumisTvis gamoiyeneba gazomvis statistika, T , ori eqsperimentuli pirobis monacemebis rangebis jamebs Soris umciresi jami.

• roca gasaanalizebeli monacemebi parametruli kriteriumis daSvebebs akmayofilebs, parametruli kriteriumi ufro mdgradia, vidre misi Sesabamisi araparametruli kriteriumi.


43


kiTxvebi

1. bevri adamiani fsiqoTerapiis kurss winaswar dagegmil vadamde

wyvets. aseTi adamianebi fsiqoTerapiiT mosargebleebis umravlesobas warmoadgenen (Stahler & Eisenman, 1987). davuSvaT, rom am problemis ukeT Sesaswavlad Tqven moipovebT monacemebs fsiqikuri janmrTelobis centrebidan da maT, vinc Terapias vadamde wyvets da maT, vinc ar wyvets, ajgufebT sqesisa da fsiqologiuri problemis simZimis mixedviT. a. Tqven miiReT im mamakacebisa da qalebis Semdegi sixSireebi,

romlebic vadamde wyveten an ar wyveten Terapiis kurss: sqesi

mamrobiTi mdedrobiTi ar Sewyvita Sewyvita

99 75

130 78

arsebobs Tu ara mimarTeba sqessa da Terapiis Sewyvetis sixSires Soris mniSvnelovnobis 0,05 doneze?

b. Semdeg moxda cdis pirebis dajgufeba maTi fsiqologiuri problemis simZimis mixedviT: msubuqi, zomieri, mZime. miRebulia Semdegi sixSireebi:

Ffsiqologiuri problemis

• araparametruli statistikuri kriteriumi

• damoukideblobis xi-kvadrat kriteriumi

• ganawilebisagan Tavisufali kriteriumebi

• Tanxmobis xi-kvadrat kriteriumi

• man-uitnis U kriteriumi

• marginaluri sixSireebi

• mosalodneli siSireebi

• niSanTa SeuRlebis cxrili

• parametruli kriteriumi

• niSanTa SeuRlebis cxrili

• T • kritikuliT

• realuriT

• uilkoqsonis niSniani rangebis kriteriumi

• U • kritikuliU

• realuriA1U

• realuriA2U

• xi-kvardat kriteriumi

• 2χ

• kritikuli2χ

• realuri2χ


44

simZime msubuqi zomieri mZime

ar Sewyvita Sewyvita

51 64

66 54

112 35

arsebobs Tu ara mimarTeba TerapiisTvis vadamde Sewyvetis maCvenebelsa da fsiqologiuri problemis simZimes Soris mniSvnelovnobis 0,05 doneze? Tu aseTi mimarTeba arsebobs, rogoria is?

2. 12.2 magaliTSi ganxiluli iyo kvleva imis Sesaxeb, aris Tu ara mTqnareba mowyenilobis niSani. erTfaqtoriani cdis pirTa Sida sqemis farglebSi, cdis pirebs aCvenebdnen rokmusikis 30-wuTian videorgols (damainteresebeli stimuli) da feradzolian sacdel satelevizio gamosaxulebas xmis gareSe (aradamainteresebeli stimuli) 30 wuTis ganmavlobaSi. akvirdebodnen da aRricxavdnen, Tu ramdenjer daamTqnarebda cdis piri TiToeuli sesiis ganmavlobaSi. davuSvaT, rom es eqsperimenti 11 cdis pirze gavimeoreT da ToToeuli sesiisaTvis miviReT damTqnarebis Semdegi raodenoba:

3. videorgolis tipi ( A ) cdis piri saintereso uintereso

1 2 3 4 5 6 7 8 9 10 11

5 2 4 3 0 4 7 6 3 1 8

7 1 7 8 2 5 6 23 3 4 9

a. uilkoqsonis niSniani rangebis kriteriumis gamoyenebiT mniSvnelovnobis 0,05 doneze daadgineT, axdens Tu ara davalebis tipi (esqperimentuli piroba) gavlenas damTqnarebis raodenobaze.

b. ratom SeiZleba iyos uilkoqsonis niSniani rangebis kriteriumi ufro Sesaferisi am monacemebisaTvis, vidre

lidamokidebut ? (miniSneba: daadgineT monacemebis ganawileba.)

g. ra daskvnebs akeTebT uilkoqsonis kriteriumis mixedviT? 3. me-9 Tavis #1 kiTxva exeboda klvevas imis Sesaxeb, Tu ra

gavlenas axdenda xmauri sisxlis wnevaze. Seqmnes cdis pirebis ori jgufi. siwynaris jgufis cdis pirebi 30 wuTis ganmavlobaSi isvenebdnen mosaxerxebel savarZelSi da 30 wuTis


45

gasvlis Semdeg maT uzomavdnen sisxlis sistolur wnevas. xmauris jgufis cdis pirebi aseve 30 wuTis ganmavlobaSi isxdnen savarZelSi, Tumca am 30 wuTis ganmavlobaSi isini usmendnen transportis xmaurs, romelic Cawerili iyo did qalaqSi, pikis saaTze. 30 wuTis ganmavlobaSi am xmauris mosmenis Semdeg maTac uzomavdnen sisxlis sistolur wnevas. davuSvaT, rom miRebulia sisxlis wnevis Semdegi maCveneblebi (merkuris milimetrebSi):

jgufi

siwynare xmauri 106 117 124 129 115 147 121 115 128 136 127

141 136 124 139 121 119 147 128 115 134 140

gamoiyeneT U kriteriumi da mniSvnelovnobis 0,05 doneze gamoTvaleT, gansxvavdeba Tu ara ganawilebebi. axseniT Tqveni daskvnebi.

4. imis dasadgenad, arsebobda Tu ara sqesebs Soris gansxvavebebi SfoTvaSi, rogorc pirovnul TvisebaSi, mkvlevarma 9 qalsa da 13 mamakacs Seavsebina SfoTvis kiTxvari. kiTxvarSi miRebuli qulebi SesaZloa meryeobdes 20-dan (dabali SfoTva) 80-mde (maRali SfoTva). davuSvaT, rom miRebulia Semdegi qulebi:

sqesi

mdedrobiT mamrobiTi 36 25 47 39 55 41 29 59 38

54 70 34 53 41 56 32 44 40 61 50 30


46

37 gamoiyeneT man-uitnis kriteriumi da mniSvnelovnobis 0,05 doneze daadgineT, gansxvavdeba Tu ara qalebis SfoTvis qulebis ganawileba mamakacebis SfoTvis qulebis ganawilebisagan. axseniT Tqveni daskvnebi.

5. me-9 TavSi mocemuli iyo amocana, romelic exeboda wonis matebas sigaretis mowevis Sewyvetis Semdeg. im kvlevaSi gazomes 8 adamianis wina maSin, roca isini mowevis Sewyvetis programaSi gawevriandnen da erTi wlis Semdeg. miiRes Semdegi wonebi:

wonis gazomvis dro cdis piri programis

dawyebisas Mmowevis

Sewyvetidan erTi wlis Semdeg

1 2 3 4 5 6 7 8

122,4 117,1 127,8 123,2 112,5 118,7 121,4 131,7

127,9 119,4 127,1 126,5 114,6 125,3 119,8 134,2

gaaanalizeT es monacemebi uilkoqsonis niSniani rangebis kriteriumis meSveobiT, raTa daadginoT, gansxvavdeba Tu ara ganawilebebi 0,05 doneze. aRwereT Tqveni daskvnebi.

danarTebi

A statistikuri cxrilebi B maTematikis mimoxilva C terminTa leqsikoni D statistikuri simboloebi E amocanebis pasuxebi F xSirad gamoyenebuli formulebi

danarTi A statistikuri cxrilebi cxrili # A.1 standartuli normaluri ganawilebis qveS mdebare

farTobis wili cxrili # A.2 t ganawilebis kritikuli mniSvnelobebi, rodesac

α = .05 da α = .01. cxrili # A.3a da b F ganawilebis kritikuli mniSvnelobebi,

rodesac (a) α = .05 da (b) α = .01. cxrili # A.4a da b studentis rangobrivi statistikis - q-s

mniSvnelobebi, rodesac (a) α = .05 da (b) α = .01.

cxrili # A.5 r-s kritikuli mniSvnelobebi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi).

cxrili # A.6 rS-s kritikuli mniSvnelobebi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi).

cxrili # A.7 xi kvadrat ganawilebis kritikuli mniSvnelobebi,

rodesac α = .05 da α = .01. cxrili # A.8a da b U-s kritikuli mniSvnelobebi man-uitnis U

kriteriumSi, rodesac (a) α = .05 da (b) α = .01. cxrili # A.9 T-s kritikuli mniSvnelobebi uilkoqsonis

kriteriumSi, rodesac α = .05 da α = .01.

cxrili # A.1 standartuli normaluri ganawilebis qveS mdebare farTobis wili cxrili # A.1 (gagrZeleba) wyaro: amoRebulia cxrilidan II.1, normaluri albaTobis funqcia da masTan dakavSirebuli funqciebi, CRC Handbook of Tables for Probability and Statistics (2nd ed.). Copyright 1968, CRC Press, Inc., Boca Raton, Florida. gamoiyeneba nebarTviT. cxrili # A.2 t ganawilebis kritikuli mniSvnelobebi, rodesac α = .05 da α = .01. trealuri-s (df) mniSvneloba, romelic tolia an metia cxrilSi moyvanili mniSvnelobisa, statistikurad mniSvnelovania SerCeul doneze. wyaro: amoRebulia cxrilidan 12, t-ganawilebis procentuli wertilebi, Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.3a F ganawilebis kritikuli mniSvnelobebi, rodesac α = .05. Frealuris mniSvneloba (dfmricxveli, dfmniSvneli), romelic tolia an metia cxrilSi moyvanili mniSvnelobisa, statistikurad mniSvnelovania .05 mniSvnelovnobis doneze. wyaro: amoRebulia cxrilidan 18, F ganawilebis procentuli wertilebi, (dispersiis koeficienti), zeda 5% wertilebi. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.3b F ganawilebis kritikuli mniSvnelobebi, rodesac α = .01. Frealuris mniSvneloba (dfmricxveli, dfmniSvneli), romelic tolia an metia cxrilSi moyvanili mniSvnelobisa, statistikurad mniSvnelovania .01 mniSvnelovnobis doneze. wyaro: amoRebulia cxrilidan # 18, F ganawilebis procentuli wertilebi, (dispersiis koeficienti), zeda 1% wertilebi. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.4a studentis rangobrivi statistikis - q-s mniSvnelobebi, rodesac α = .05. wyaro: amoRebulia cxrilidan # 29, studentis rangobrivi statistikis procentuli wertilebi, q = (xn - x1)/sv. zeda 5% wertilebi. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. Tu danarTi A.4 gamoiyeneba faqtorul sqemaSi ubralo efeqtebis kriteriumis CD-saTvis q-s mosaZebnad, saWiro svetis sapovnelad gamoiyeneT Semdegi gardaqmna.

sqemis tipi cxrilidan # A.4 gamoiyeneT q, romelic moTavsebulia damoukidebeli cvladis doneebis svetSi

2x2 2x3 3x2 3x3 3x4 4x3 4x4

3 5 5 7 8 8 10

cxrili # A.4b studentis rangobrivi statistikis - q-s mniSvnelobebi, rodesac α = .01. wyaro: amoRebulia cxrilidan # 29, studentis rangobrivi statistikis procentuli wertilebi, q = (xn - x1)/sv. zeda 1% wertilebi. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. Tu danarTi A.4 gamoiyeneba faqtorul sqemaSi ubralo efeqtebis kriteriumis CD-saTvis q-s mosaZebnad, saWiro svetis sapovnelad gamoiyeneT Semdegi gardaqmna.

sqemis tipi cxrilidan # A.4 gamoiyeneT q, romelic moTavsebulia damoukidebeli cvladis doneebis svetSi

2x2 2x3 3x2 3x3 3x4 4x3 4x4

3 5 5 7 8 8 10

cxrili # A.5 r-s kritikuli mniSvnelobebi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi). rrealuris mniSvneloba, romelic tolia an metia cxrilSi

moyvanili mniSvnelobisa, statistikurad mniSvnelovania α-s SerCeul doneze. adf udris Nwyvilebi - 2, sadac Nwyvilebi aris dawyvilebuli dakvirvebebis raodenoba. wyaro: amoRebulia cxrilidan # 13, korelaciis koeficientis, r, ganawilebis

procentuli wertilebi, rodesac ρ=0. Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.6

rS-s kritikuli mniSvnelobebi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi). rsrealuris mniSvneloba, romelic cxrilSi moyvanili

mniSvnelobis tolia an metia, statistikurad mniSvnelovania α-s SerCeul doneze. wyaro: E.G. Olds, ranguli gasxvavebebis kvadratebis jamebis ganawileba individebis mcire raodenobebisTvis. Annals of Mathematical Statistics, Volume 9, 1938, 133-148 da E.G. Olds, ranguli gasxvavebebis kvadratebis jamebis 5%-iani mniSvnelovnobis doneebi da koreqcia. Annals of Mathematical Statistics, Volume 20, 1949, 117-118. nebarTviT amoRebulia cxrilidan # 12.12,

spirmenis ranguli korelaciis koeficientis kritikuli mniSvnelobebi, rodesac ρ=0, CRC Handbook of Tables for Probability and Statistics (2nd ed.). Copyright 1968, CRC Press, Inc., Boca Raton, Florida. cxrili # A.7 xi kvadrat ganawilebis kritikuli mniSvnelobebi, rodesac α = .05 da α = .01. χ2realuris (df) mniSvneloba, romelic tolia an metia cxrilSI

moyvanili mniSvnelobisa, statistikurad mniSvnelovania α-s SerCeul doneze. wyaro: amoRebulia cxrilidan # 8, xi-kvadratis ganawilebis procentuli wertilebi, Biometrika Tables for Statisticians, Volume 1, Copyright 1976, Cambridge University Press.. gamoiyeneba Biometrika Trustees-is nebarTviT. cxrili # A.8a U-s kritikuli mniSvnelobebi man-uitnis U kriteriumSi, rodesac (a) α = .05 (ormxrivi kriteriumi). Tu jgufis zomebi toli ar aris, n1–iT ufro mcire zomis jgufi aRiniSneba. Urealuris mniSvneloba, romelic cxrilSi moyvanili mniSvnelobis tolia an naklebia, statistikurad mniSvnelovania .05 mniSvnelovnobis doneze. wyaro: D. Auble, U-s kritikuli mniSvnelobebi uilkoqsonis (man-uitnis) statistikaSi ori SerCevisaTvis, Institute of Educational Research, Indiana University, Bloomington, Indiana, Volume 1, Number 2, 1953, romelic nebarTviT amoRebulia cxrilidan # 10.4, U-s kritikuli mniSvnelobebi uilkoqsonis (man-uitnis) statistika ori SerCevisaTvis, CRC Handbook of Tables for Probability and Statistics (2nd ed.). Copyright 1968, CRC Press, Inc., Boca Raton, Florida. cxrili # A.8b U-s kritikuli mniSvnelobebi man-uitnis U kriteriumSi, rodesac (a) α = .01 (ormxrivi kriteriumi). Tu jgufis zomebi toli ar aris, n1–iT ufro mcire zomis jgufi aRiniSneba. Urealuris mniSvneloba, romelic tolia an naklebia cxrilSi moyvanili mniSvnelobisa, statistikurad mniSvnelovania .01 mniSvnelovnobis doneze. wyaro: D. Auble, U-s kritikuli mniSvnelobebi uilkoqsonis (man-uitnis) statistikaSi ori SerCevisaTvis, Institute of Educational Research, Indiana University, Bloomington, Indiana, Volume 1, Number 2, 1953, romelic nebarTviT amoRebulia cxrilidan # 10.4, U-s kritikuli mniSvnelobebi uilkoqsonis (man-uitnis) statistika ori SerCevisaTvis, CRC Handbook of Tables for Probability and Statistics (2nd ed.). Copyright 1968, CRC Press, Inc., Boca Raton, Florida. cxrili # A.9

T-s kritikuli mniSvnelobebi uilkoqsonis kriteriumSi, rodesac α = .05 da α = .01 (ormxrivi kriteriumi). Trealuris mniSvneloba, romelic tolia an naklebia cxrilSi moyvanili mniSvnelobisa, statistikurad

mniSvnelovania α-s SerCeul doneze. tireebi aRniSnavs, rom am doneze jgufis am zomisaTvis ver viRebT statistikur mniSvnelovnobas. wyaro: Robert L. McCornack, Extended Tables of the Wilcoxon Matched Pair Sighed Rank Statistic. Journal of the American Statistical Association, 1965, 60, 864-871. amoRebulia nebarTviT The Journal of the American Statitical Assocation, Copyright 1965 by the American Statistical Association. yvela ufleba daculia.

danarTi B maTematikis mimoxilva aq mokled aris mimoxiluli is maTematikuri simboloebi da operaciebi, romelic gvWirdeba am saxelmZRvaneloSi warmodgenili statistikis gasagebad. statistikuri simboloebi simbolo, romelsac xandaxan niSansac uwodeben, aris aso an aRniSvna, romelsac iyeneben raimes aRsaniSnad. Semdeg cxrilSi warmodgenilia statistikaSi gamoyenebuli zogadi simboloebi. simbolo da magaliTi

aRwera

X an Xi X gamoiyeneba cdis pirisagan miRebuli qulis an monacemis aRsaniSnad. indeqsi i, rodesac mas iyeneben, aRniSnavs konkretuli cdis piris qulas. magaliTad, X1 aRniSnavs 1-li cdis piris qulas.

= tolobis niSani.

X1 = 3 1-li cdis piris qula udris 3-s.

X5 = 27 me-5 cdis piris qula udris 27-s.

a = 2 a-s mniSvneloba 2-is tolia.

≠ utolobis niSani.

6 ≠ 3 6 ar udris 3-s.

+ plusi - mimatebis niSani. es niSani miuTiTebs, rom + niSniT dakavSirebuli ricxvebi unda daematos erTmaneTs. mimatebis Sedegs jami ewodeba. iseTi debuleba, rogoricaa "5 + 1-is jami" niSnavs, rom es ricxvebi unda mivumatoT erTmaneTs.

5 + 1 = 6 5-isa da 1-is jami udris 6-s. 5 + 1 + 6 = 12 5 plus 1 plius 6 udris 12-s.

a + b a plus b: b-s mniSvneloba unda mivumatoT a-s mniSvnelobas.

_ minusi - gamoklebis niSani. es niSani miuTiTebs, rom ricxvi, romelic mosdevs _-is niSans, unda gamoakldes ricxvs, romelic dgas _ niSnis win. gamoklebis Sedegs sxvaobas uwodeben. iseTi debuleba, rogoricaa "5-isa da 1-is sxvaoba" niSnavs, rom erTi ricxvi meores unda gamoakldes.

5 _ 1 = 4 5-s gamovakloT 1 udris 4-s.

a _ b a minus b: b-s mniSvneloba unda gamovakloT a-s mniSvnelobas.

()() an x gamravlebis niSani. frCxilebSi moTavsebuli an x-iT gamoyofili mniSvnelobebi unda gavamravloT erTmaneTze. gamravlebis Sedegs namravls uwodeben.

(2)(3) = 6 2-isa da 3-is namravli aris 6. orjer sami aris eqvsi.

2 x 3 = 6 2-isa da 3-is namravli aris 6. orjer sami aris eqvsi.

(a)( b) an a x b a-jer b: a-s mniSvneloba unda gamravldes b-s

mniSvnelobaze. arsebobs am wesis gamonaklisi: faqtorul sqemaSi A

da B faqtorebis urTierTqmedeba aRiniSneba AxB. am SemTxvevaSi, aRniSvna unda wavikiTxoT ase: "A-sa da B-s urTierTqmedeba". es ar niSnavs, rom A unda gamravldes B-ze.

÷ an / an __ gayofis niSani. ricxvi, romelic win uswrebs ÷ an / an ricxvi, romelic __ xazis zeviT weria (mricxveli), iyofa ricxvze, romelic mosdevs ÷ an / an ricxvze, romelic __ xazis qveviT weria (mniSvneli). gayofis Sedegs ganayofs uwodeben.

6 ÷ 2 = 3 eqvsi, gayofili orze, aris sami.

6/2 = 3 eqvsi, gayofili orze, aris sami.

26 = 3

eqvsi, gayofili orze, aris sami.

a ÷ b, an a/b, an ba

a, gayofili b-ze

< naklebobis simbolo. ricxvi, romelic weria <-is win, naklebia, vidre ricxvi, romelic weria <-is Semdeg.

2 < 3 ori naklebia samze

a < b a naklebia b-ze X3 < 10 mesame cdis piris qula naklebia aTze.

> metobis simbolo. ricxvi, romelic weria >-is win, metia, vidre ricxvi, romelic weria >-is Semdeg.

3 > 2 sami metia orze. a > b a metia b-ze. X3 > 10 mesame cdis piris qula metia aTze.

≤ "naklebia an tolia" simbolo. ricxvi, romelic weria ≤ -is win, naklebia an tolia, vidre ricxvi, romelic weria ≤ -is Semdeg.

2 ≤ 3 ori naklebia an tolia samis. 4 ≤ 4 oTxi naklebia an tolia oTxis. a ≤ b a naklebia an tolia b-si. X3 ≤ 10 mesame cdis piris qula naklebia an tolia 10-is.

≥ "metia an tolia" simbolo. ricxvi, romelic weria ≥-is win, metia an tolia, vidre ricxvi, romelic weria

≥-is Semdeg. 3 ≥ 2 sami metia an tolia oris.

4 ≥ 4 oTxi metia an tolia oTxis.

a ≥ b a metia an tolia b-s. X3 ≥ 10 mesame cdis piris qula metia an tolia 10-is.

5 < X1 < 10 rogorc xedavT, metobis an naklebobis siimboloebi SesaZlebelia erT aRniSvnaSi moTavsdes. es aRniSvna ase ikiTxeba: "pirveli cdis piris qula 5-ze metia da 10-ze naklebi".

5 ≤ X1 ≤ 10 "naklebia an tolia" da "metia an tolia" simboloebi aseve SeiZleva moTavsdes erT aRniSvnaSi. es aRniSvna ase ikiTxeba: "pirveli cdis piris qula metia an

tolia 5-ze da naklebia an tolia 10-is".

absoluturi mniSvnelobis simbolo. es simbolo niSnavs, rom Cven ignorirebas vukeTebT + an _ niSnebs, romlebic mibmulia ricxvebze.

_ 6 = 6 minus eqvsis absoluturi mniSvneloba aris eqvsi.

+ 6 = 6 plius eqvsis absoluturi mniSvneloba aris eqvsi.

( )2 frCxilebSi moTavsebuli ricxvi ayvanilia kvadratSi, anu gamravlebulia Tavis Tavze. zogjer frCxilebi ar gamoiyeneba da kvadratis indikatori, 2, ubralod mosdevs kvadratSi asayvan ricxvs.

(5)2 = 25 xuTis kvadrati, anu 5 x 5, aris 25. 52 = 25 xuTis kvadrati udris 25-s.

kvadratuli fesvis simbolo. niSnavs, vipovoT

ricxvi, romelic, Tu mas Tavis Tavze gavamravlebT

(anu kvadratSi aviyvanT) udris niSnis qveS

moTavsebul ricxvs. 25 = 5 kvadratuli fesvi 25-dan udris 5-s, radganac 5

gamravlebuli Tavis Tavze (anu 5 x 5) 25-is tolia. 36 = 6 kvadratuli fesvi 36-dan udris 6-s, radganac Tavis

Tavze gamravlebuli 6 (anu 6 x 6) 36-is tolia.

∑ dajamebis simbolo. ricxvebi, romlebic mosdeven ∑ simbolos, unda mivumatoT erTmaneTs.

∑ (3+2) = 5 3-isa da 2-is jami udris 5-s.

∑ (3+2+4) = 9 3 plius 2 plius 4 udris 9-s.

∑ X an ∑ Xi X-is jami. daajameT yvela qula, romelic warmodgenilia X-iT. magaliTad, Tu sami cdis piris

qulebia: X1 = 3, X2 = 6 da X13 = 4, maSin ∑ X = 3 + 6 + 4 = 13. xandaxan dajamebis simbolos Tavze zRvrebia

miTiTebuli, magaliTad, ∑=

3

1iiX . es zRvrebi miuTiTeben,

rom unda davajamoT pirvelidan mesame cdis piramde

qulebi. Tu zRvrebi ar gamoiyeneba, maSin ∑ X niSnavs, rom unda davajamoT X-iT aRniSnuli yvela qula.

maTematikuri operaciebi uaryofiTi ricxvebi uaryofiTi ricxvi SeiZleba miviRoT gamoklebis Sedegad, magaliTad 6 _ 8 = _2. 8, romelic 6-ze 2-iT metia, gvaZlevs _2 sxvaobas, rodesac mas eqvss vaklebT. uaryofiTi ricxvebi xSirad gvxvdeba statistikaSi da gamoiyeneba ZiriTad maTematikur operaciebSi. Tu ricxvs ar aqvs win minusis niSani, igulisxmeba, rom is dadebiTia. uaryofiTi ricxvebis mimateba 6 + (_4) = 2 uaryofiTi ricxvis dadebiTisaTvis mimateba igivea, rac

dadebiTi ricxvidan uaryofiTis gamokleba. ase rom, 6 + (_4) = 6 _ 4 = 2.

(_6) + (_4) = _10 am SemTxvevaSi, _4 akldeba _6-s, anu (_6) + (_4) = _6 _4 =

_10. uaryofiTi ricxvebis gamokleba 6 _ (_4) = 10 uaryofiTi ricxvis gamokleba igivea, rac uaryofiTi

ricxvis absoluturi mniSvnelobis mimateba. ase rom, 6 _ (_4) = 6 + 4 = 10.

uaryofiTi ricxvebis gamravleba (_6)(_4) = 24 Tu orive gasamravlebeli ricxvi uaryofiTia, maSin

namravli dadebiTia. (6)(_4) = _24 Tu ori ricxvidan mxolod erTia uaryofiTi, maSin (_6)(4) = _24 namravli uaryofiTia. uaryofiTi ricxvebis gayofa

_10 ÷ 5 = _2 Tu mricxveli an mniSvneli uaryofiTia, maSin ganayofi

10 ÷ _5 = _2 uaryofiTia.

_10 ÷ _5 = _2 Tu mricxvelic da mniSvnelic uaryofiTia, maSin ganayofi dadebiTia.

wiladebi wiladebTan muSaobis yvelaze martivi midgomaa mricxvelis mniSvnelze

gayofis gziT maTi aTwiladad gardaqmna, magaliTad, 41 = 0.25,

42 = 0.50,

da 43 = 0.75. Tu aTwiladi Semdgom gamoTvlebSi unda gamoviyenoT, maSin

is unda davamrgvaloT sul mcire meaTasedebamde (mag., 114 = 0.3636). ase

rom, 52 +

81 +

31 = 0.4000 + 0.1250 + 0.3333 = 0.8583.

wiladebi da procentebi wiladebi wiladi aris mTelis nawili. magaliTad, Tu 100 adamianma Seavso kiTxvari

da maTgan 45 mamakacia, maSin mamakaci respondentebis wiladi aris 10045

anu .45. qali respondentebis wiladi ki aris 10055

= .55.

procenti procenti miiReba, rodesac wiladi 100-ze mravldeba. amdenad, mamakaci respondentebis wiladi .45 udris 45 x 100 anu mamakaci respondentebis 45 procents. xolo qali respondentebis procenti ki aris 55 x 100 = 55 procenti.

maTematikuri operaciebis Tanmimdevroba momdevno magaliTebi gviCvenebs, Tu ra TanmimdevrobiT sruldeba maTematikuri operaciebi Cveulebriv statistikur aRniSvnebSi:

∑ − )( XX 1. saSualo akldeba yovel qulas qulaTa erTobliobaSi.

2. sxvaobebi jamdeba. 2

)(∑ − XX 1. saSualo akldeba yovel qulas qulaTa erTobliobaSi.

2. yoveli sxvaoba aiyvaneba kvedratSi. 3. kvadratSi ayvanili sxvaobebi jamdeba.

∑ 2X 1. yoveli X mniSvneloba aiyvaneba kvadratSi. 2. kvadratSi ayvanili X mniSvnelobebi jamdeba.

(∑ 2X ) 1. X mniSvnelobebi jamdeba.

2. X mniSvnelobebis jami (anu ∑ X ) aiyvaneba

kvadratSi.

∑ XY 1. qulaTa erTobliobaSi Sesatyvisi X da Y mniSvnelobebi mravldeba.

2. gadamravlebuli X Y mniSvnelobebi jamdeba qulaTa erTobliobisaTvis.

(∑ X )(∑Y ) 1. X mniSvnelobebi jamdeba, raTa miviRoT ∑ X .

2. Y mniSvnelobebi jamdeba, raTa miviRoT ∑Y .

3. X mniSvnelobebis jami (anu ∑ X ) mravldeba Y

mniSvnelobebis jamze (anu ∑Y ).

ტერმინები 1

თავი I • aRweriTi statistika (Descriptive statistics): _ statistikuri procedurebi, romlebic

gamoiyeneba SerCevidan miRebuli monacemebis Sesajameblad da aRsawerad.

• damokidebuli cvladi (Dependent variable): _ cvladi eqsperimentSi, romelic damokidebulia damoukidebel cvladze. SemTxvevaTa umravlesobaSi damokidebuli cvladi qcevis garkveuli sazomia.

• damoukidebeli cvladi (Independent variable): _ cvladi, romliTac manipulireben eqsperimentSi, damokidebul cvladze misi gavlenis gansazRvris mizniT.

• ekvivalenturi jgufebi (Equivalent groups): _ cdis pirTa jgufebi, romlebic, savaraudod, ar gansxvavdebian erTmaneTisgan raime mudmivi maxasiaTebliT eqsperimentis damoukidebeli cvladis miwodebamde.

• kvlevis hipoTeza (Research hypothesis): _ debuleba or an met cvlads Soris mosalodneli an nawinaswarmetyvelebi kavSiris Sesaxeb. eqsperimentSi kvlevis hipoTeza aris nawinaswarmetyvelebi kavSiri damoukidebel da damokidebul cvladebs Soris.

• korelacia (Covary): _ ori cvladi korelaciaSi imyofeba erTmaneTTan, rodesac erTi cvladis cvlileba ukavSirdeba meore cvladis SeTanxmebul sistematur cvlilebas.

• korelaciis koeficienti (Correlation coefficient): _ statistika, romelic ricxobrivad gviCvenebs maCvenebelTa ori jgufis dakavSirebulobis xarisxs da am kavSiris mimarTulebas.

• monacemebi (Data _ mravlobiTi, datum _ mxolobiTi): _ is qulebi an qcevis Tu maxasiaTeblebis gazomvis Sedegebi, romelic miiReba adamianebze an cxovelebze dakvirvebis Sedegad.

• nedli monacemebi (Raw data): _ yvela cdis pirisgan miRebuli qula, romelTac jer ar gauvliaT statistikuri analizi.

• parametri (Parameter): _ ricxvi, romelic aRwers populaciis maxasiaTebels.

• populacia (Population): _ aris saerTo maxasiaTeblis mqone adamianTa, cxovelTa an movlenaTa sruli erToblioba.

• regresiuli analizi (Regression analysis): _ statistikuri meTodebis gamoyeneba qulaTa erTi erTobliobis mixedviT qulaTa meore erTobliobis Sesaxeb winaswarmetyvelebisTvis.

• statistika (mecniereba) (Statistics): _ is meTodebi Tu procedurebi, romlebic gamoiyeneba monacemebis sistematizaciisTvis, gasaanalizeblad da maTgan daskvnebis gamosatanad.

• statistika (statistikuri sidide) (Statistic): _ monacemebze dayrdnobiT gamoTvlili calkeuli ricxvi, romelic SeiZleba gamoyenebul iqnes SerCevis monacemTa erTobliobis aRsawerad, an am monacemTa ufro sruli gaanalizebisTvis.

• statistikuri daskvna (Statistical inference): _ populaciis maxasiaTeblis Sefaseba SerCevidan miRebul statistikebze dayrdnobiT.

• statistikuri hipoTezis Semowmeba (Statistical hypothesis testing) _ procedurebi, romlebic jgufebs Soris SemTxveviTi gansxvavebis zomis winaswarmetyvelebisTvis gamoiyeneba.

• qula (Score): _ cdis piris mier davalebis Sesrulebis gazomviT miRebuli ricxviTi mniSvneloba.


• SemTxveviTi gansxvaveba (Chance difference): _ gansxvaveba ekvivalentur jgufebs Soris, romelic miiReba SemTxveviT an gauTvaliswinebeli mizeziT.

• SemTxveveiTi SerCeva (Random sample): _ SerCeva, romelSic individebi SerCeulni arian ise, rom populaciis TiToeuls wevrs SerCevaSi moxvedris toli Sansi aqvs da erTi wevris arCeva ar aris damokidebuli populaciis sxva wevrebis arCevaze.

• SerCeva (Sample): _ populaciidan mTeli rigi wesebis dacviT (raTa SerCeva populaciis reprezentaciuli iyos) arCeuli qvesimravle anu qvejgufi.

• cdis piri (Subject): _ pirovneba, romelic monawileobs eqsperimentSi.

• cdis pirTa Soris sqema (Between-subject design): _ kvlevis sqema, romelSic monawileobs ori an meti jgufi

• cvladi (variable): _ garemos nebismieri mdgomareoba, nebismieri stimuli, pirovnuli maxasiaTebeli an qceva, romelmac SeiZleba miiRos sxvadasxva mniSvneloba sxvadasxva dros an sxvadasxva adamianTan.

თავი II • bunebrivi dakvirveba (Naturalistic observation): _ kvleva, romelic gulisxmobs bunebriv

pirobebSi ganxorcielebul qcevebze dakvirvebas.

• gazomva (Measurement): _ garkveuli wesebis mixedviT cvladebisTvis ricxviTi mniSvnelobebis miniWeba.

• gamokiTxva (Survey research): _ kvleva, romelSic monacemebi adamianebis zepiri an werilobiTi interviuebiT miiReba.

• diskretuli (wyvetili) cvladi (Discrete variable): _ cvladi, romelmac SeiZleba miiRos sasruli an Tvladi mniSvneloba.

• empiriuli monacemebi (Empirical data): _ dakvirvebasa da sensorul gamocdilebaze dafuZnebuli gazomvis Sedegad miRebuli qulebi.

• eqsperimenti (Experiment): _ zedmiwevniT gakontrolebuli situacia, sadac mecnieri axdens erTi an meti damoukidebeli cvladis manipulirebas da akvirdeba, Tu ra gavlenas axdens es damokidebul cvladze.

• Tvisebrivi monacemebi (Qualitative data): _ saxeldebis skalaze miRebuli monacemebi, romlebic miuTiTeben, rom cvladebi Tvisebrivad gansxvavdebian erTmaneTisgan.

• intervalebis skala (Interval measurement): _ cvladis mniSvnelobebi dalagebulia skalaze da miniWebul ricxvebs Soris gansxvaveba asaxavs cvladis mniSnelobebs Soris erTi da igive sididis sxvaobas. intervalTa skalaze nuli pirobiTad sawyisi wertilia.

• kvlevis meTodi (Research method): _ midgoma monacemebis mogrovebisadmi.

• operaciuli gansazRvreba (Operational definition): _ dakvirvebisas, damoukidebeli cvladiT manipulirebisas an damokidebuli cvladis gazomvisas gamoyenebuli operaciebis detaluri aRwera.

• rigis skala (Ordinal measurement): _ skalaze cvladis sidide dalagebulia mniSvnelobebis mixedviT.

• raodenobrivi monacemebi (Quantitative data): _ rigis, intervalebisa an Sefardebis skalaze miRebuli monacemebi, romlebic miuTiTeben, Tu ra sididis aris cvladi.

• ricxvis realuri zeda sazRvari(Upper real limit of a number) _ ricxvisa da misi momdevno ricxvis Suawertili.


• ricxvis realuri sazRvrebi (Real limits of a number): _ ricxvisa da misi wina ricxvis Suawertili an ricxvisa da misi momdevno ricxvis Suawertili im skalaze, romelsac viyenebT mocemuli cvladis gasazomad.

• ricxvis realuri qveda sazRvari (Lower real limit of a number) _ ricxvisa da misi wina ricxvis Suawertili.

• saarqivo Canawerebi (Archival recodrs): _ kvleva, romelSic gamoiyeneba arsebuli Canawerebi.

• saxeldebis skala (Nominal measurement): _ gazomili cvladis sxvadasxva kategoriebSi gaerTianeba.

• uwyveti cvladi (Continuous variable): _ cvladi, romelmac SeiZleba miiRos usasrulo mniSvneloba cvladis nebismier or dones Soris.

• Sefardebis skala (Ratio measurement): _ cvladis sidide dalagebulia skalaze da miniWebul ricxvebs Soris gansxvaveba asaxavs cvladis mniSnelobebs Soris igive sididis gansxvavebas. Sefardebis skalaze arsebobs WeSmaritad nulovani wertili, romelic Seesabameba gasazomi maxasiaTeblis srul ararsebobas.

თავი III • asimetriuli ganawileba (Skewed distribution): _ sixSireTa ganawileba, sadac qulebi

Tavmoyrilia ganawilebis erT boloSi; ganawilebis meore boloSi (anu kudSi) ki ganlagebulia iSviaTad dafiqsirebuli qulebi.

• daujgufebel sixSireTa ganawileba (Ungroup frequency distribution): _ sixSireTa ganawileba, romelic agebulia miRebul minimalur da maqsimalur qulaTa Soris yvela SesaZlo mniSvnelobis dalagebiT da Semdeg qulis gaswvriv Tvlis Sedegis aRniSvnis CxiriT (/) imdenjer moTavsebiT, ramdenjerac iqna miRebuli es qula.

• dagrovili intervaluri sixSire (Cumulative frecuency of a class interval): _ intervalSi qulebis Sexvedris raodenobas damatebuli yvela ufro dabal intervalSi qulebis sixSireTa jami.

• dagrovili intervaluri sixSiris ganawileba procentebSi ( Cumulative grouped percentage frecuency of a class interval): _ intervalSi qulebis sixSires procentebSi damatebuli yvela ufro dabal intervalSi qulaTa sixSireTa jami procentebSi.

• dagrovili intervaluri fardobiTi sixSire (Cumulative grouped relative frecuency of a class interval): _ intervalSi qulebis fardobiT sixSires damatebuli yvela ufro dabal intervalSi qulaTa fardobiT sixSireTa jami.

• dajgufebul sixSireTa ganawileba (Grouped frequency distribution): _ ganawileba, sadac qulebi dajgufebulia intervalebSi da TiToeuli maTganSi moxvedrili qulebis sixSireebi gamoisaxeba cxrilis saxiT.

• dajgufebul sixSireTa ganawileba procentebSi (Grouped percentage frecuancy distribution): _ dajgufebul sixSireTa ganawileba, sadac intervalSi qulaTa sixSire Sefardebulia ganawilebaSi qulaTa saerTo raodenobasTan.

• dajgufebul fardobiT sixSireTa ganawileba (Grouped relative frequancy distribution): _ dajgufebul

• sixSireTa ganawileba miiReba qulaTa fardobiTi sixSiris 100-ze gamravlebiT.

• intervali (Class interval): _ intervalis sidide, romelic gamoiyeneba dajgufebul sixSireTa ganawilebaSi nedli qulebis dasajgufeblad. intervalis sidide aRiniSneba i-iT.


• intervalis dadgenili sazRvari (Stated limits of a class interval): _ maqsimaluri da minimaluri qulebi, romelic SeiZleba moxvdes am intervalSi.

• intervalis realuri sazRvari (Real limits of a class interval): _ intervalis dadgenil sazRvrebsa da am intervalis wina an momdevno intervalebis dadgenil sazRvrebs Soris Suawertili.

• intervalis Suawertili (Midpoint of a class interval): _ intervalis realur sazRvrebs Soris Suawertili.

• moda (Mode): _ qulaTa ganawilebaSi yvelaze xSirad dafiqsirebuli qula.

• nedli monacemebi an nedli qulebi (raw data or raw scores) _ kvlevis monawileebisgan miRebuli qulebi, romelTac jer ar gauvliaT statistikuri analizi.

• procentili (Percentile): _ qula, romelzec qulaTa garkveuli procenti ganawilebaSi tolia an naklebia am qulaze.

• svetebiani diagrama (Bar graph) _ raodenobrivi monacemebis sixSireTa ganawilebis grafikulad gamoxatvis saSualeba. mocemul kategoriaSi qulaTa gamoCenis sixSires svetis simaRle Seesabameba.

• sixSireTa ganawileba (Frequency distribution): _ cxrili, romelSic mocemulia TiToeuli qula da is sixSire, ra sixSiriTac gvxvdeba es qula.

• sixSireTa simetriuli ganawileba (Symmetrical frecuency distribution): _ sixSireTa ganawileba, romlis Suawertilze orad gadakecvis SemTxvevaSi miiReba erTnairi formis ori naxevari.

• sixSireTa poligoni (Frequency polygon): _ grafiki, romlis abscisebia intervalTa Suawertilebi, ordinatebi ki _ am intervalTa Sesabamisi sixSireebi.

• sixSiris procenti (%ƒ) (Relative frequency): _ qulis fardobiTi sixSire gamravlebuli 100-ze.

• fardobiTi sixSire (rƒ) (Relative frequency): _ qulis sixSire gayofili miRebul qulaTa saerTo raodenobaze.

• foTlebiani Reroebis diagrama (Stem-and-leaf display): _ monacemTa gamosaxvis forma, sadac qulis pirveli (an pirveli da meore) cifri aris Rero, meore (an bolo) ki _ foToli.

• qulis dagrovili sixSire (Cumulative frecuency of a score): _ qulis Sexvedris sixSires damatebuli yvela ufro naklebi qulis sixSireTa jami.

• qulis procentuli rangi (Percentile rank of a score): _ im qulaTa procenti ganawilebaSi, romlebic tolia an naklebia am qulaze.

• histograma (Histogram): _ svetebiani grafikis nairsaxeoba, sadac intervalebSi qulaTa Sexvedris sixSire gamosaxulia svetis simaRliT, TiToeuli intervalis sigrZe ki _ abscisaze svetis siganiT.

• ƒ: _ sixSireTa ganawilebaSi TiToeuli qulis Sexvedris martivi sixSire.

• i _ intervalis zoma dajgufebul sixSireTa ganawilebaSi.

• N: _ jgufSi miRebuli qulebis saerTo raodenoba

• X _ simbolo, romelic calkeuli, individualuri qulis aRsaniSnad gamoiyeneba.


თავი IV • bimodaluri (Bimodal): _ ganawileba, romelsac aqvs ori moda.

• kvadratebis jami (SS) (Sum of squares): _ ricxvi, romelic miiReba ganawilebaSi Semavali TiTouli qulidan ganawilebis saSulos gamoklebiT, TiToeuli sxvaobis kvadratSi ayvaniTa da miRebuli sidideebis SejamebiT.

• mediana (Median, Mdn): _ qula ganawilebaSi, romlis zemoT da qvemoT ganlagebulia qulaTa toli raodenoba, mediana aris 50-e procentili ganawilebaSi.

• moda (Mode): _ qula, romelic yvelaze xSirad gvxvdeba ganawilebaSi.

• multimodaluri (Multimodal): _ ganawileba, romelsac aqvs orze meti moda.

• populaciis saSualo (Population mean – µ) _ populaciaSi yvela SemTxvevis jamis fardoba qulebis saerTo raodenobasTan.

• unimodaluri (Unimodal): _ ganawileba, romelsac aqvs erTi moda.

• SerCevis saSualo ( X ) (Sample mean): _ miRebul qulaTa jami gayofili qulaTa mTlian raodenobaze.

• centraluri tendenciis sazomebi (Measures of central tendency): _ ricxvebi, romlebic warmoadgenen SerCevidan miRebul saSualo an tipur qulas.

• cvalebadobis sazomi (measures of central tendency) _ ricxvi, romelic miuTiTebs, Tu ramdenad gansxvavdebian qulebi erTmaneTisgan da ramdenad gansxvavdeba maTgan scentraluri tendenciis sazomi.

• M _ simbolo, romelic amerikis fsiqologiuri asociaciis publikaciebis saxelmZRvaneloSi (1994) miTiTebuli normebiT Sesrulebul naSromSi gamoiyeneba SerCevis saSualos aRsaniSnad.

თავი V • diapazoni (Range): _ sxvaoba ganawilebis maqsimalur da minimalur qulebs Soris.

• kvartilTSorisi gabnevis diapazoni (Interquartile range, IQR): _ ganawilebaSi Semaval qulaTa SuaTana 50%.

• kvartilTSorisi gabnevis diapazonis naxevari (Semi-interquartile range, SIQR) _ kvartilTSorisi gablevis diapazonis naxevari.

• populaciis dispersia (Population variance, σ2) _ populaciis yvela qulis gazomviT miRebuli dispersia.

• populaciis dispersiis Sefaseba (Estimated population variance, s2): _ cvalebadobis sazomi, romelic miiReba ganawilebaSi Semavali ToToeuli qulisa da am ganawilebis saSualos sxvaobebis kvadratSi ayvaniT, miRebuli sidideebis dajamebiTa da miRebuli jamis gayofiT ganawilebaSi Semavali qulebis raodenobaze erTiT nakleb ricxvze.

• populaciis standartuli gadaxra (σ) _ kvadratuli fesvi populaciis dispersiidan.

• populaciis standartuli gadaxris Sefaseba (Estimated populaation standard deviation) _ kvadratuli fesvi populaciis dispersiis Sefasebidan.

• SerCevis dispersia (Sample variance, S2): _ cvalebadobis sazomi, romelic miiReba ganawilebaSi Semavali ToToeuli qulisa da am ganawilebis saSualos sxvaobebis


kvadratSi ayvaniT, miRebuli sidideebis dajamebiTa da miRebuli jamis gayofiT ganawilebaSi Semavali qulebis raodenobaze.

• SerCevis standartuli gadaxra (Sample standard deviation, S): _ kvadratuli fesvi SerCevis dispersiidan.

• cvalebadoba (Variability): _ ramdenad gansxvavdebian qulebi erTmaneTisgan da ganawilebis centraluri tendenciis sazomisgan.

• X75 _ ganawilebis 75-e procentilis Sesabamisi qula.

• X25 _ ganawilebis 25-e procentilis Sesabamisi qula.

თავი VI • albaTuri Teoriuli ganawileba (Theoretical probability distribution) _ Teoriuli

albaTobis modelis gamoyenebiT miRebuli albaTuri ganawileba.

• albaTuri klasikuri ganawileba (Empirical probability distribution) _ xdomilobis realurad ganxorcielebis sixSiris daTvliT miRebuli albaTuri ganawileba.

• diskretuli Sedegebi (discrete outcomes) _ mniSvnelobebaTa sasruli raodenobis mqone Sedegebi ganawilebaSi.

• normaluri ganawileba (Normal distribution): _ Teoriuli maTematikuri ganawileba, romelic aRwers populaciis qulebis fardobiT sixSireebs.

• standartuli normaluri gadaxra (Standard normal deviate): _ normaluri ganawileba, romlis μ = 0, σ = 1.

• standartuli normaluri ganawileba (Standard normal distribution): _ normaluri ganawileba, romlis μ = 0, σ = 1.

• standartuli qula (Standard score): _ qula, romelic miiReba formuliT ( ) SXXz /−= .

• urTierTgamomricxavi Sedegebi (Mutually exclusive outcomes) _ Sedegebi, romelic ver ganxorcieldeba erTdroulad.

• Sedegi (outcome) _ albaTobis ganawilebaSi TiToeuli SesaZlo varianti.

• xdomiloba (event) _ albaTobis ganawilebaSi Sedegebis konkretuli simravlis ganxorcieleba.

• xdomilobis ganxorcielebis albaToba (probability of occurrence of en event – p) _

raodenoba saereTo

Sedegebis SesaZlo

raodeoba Sedegebis

iSemadgenel xdomilobis

xdomiloba =)(p

თავი VII • martivi SemTxveviTi SerCeva (Simple random sampling): _ populaciidan wevrebis SerCevis

iseTi procesi, rodesac populaciis TiToeul wevrs SerCevaSi moxvedris toli Sansebi aqvT da erTi wevris arCeva populaciis nebismieri sxva wevris arCevisgan damoukideblad xdeba.

• ndobis intervali (Confidence interval): _ qulis mniSvnelobaTa diapazoni, romelic, savaraudod, moicavs μ-is mniSvnelobas ndobis garkveul doneze.


• ndobis sazRvrebi (Confidence limits): _ ndobis intervalis SemomsazRvreli qveda da zeda qulebi.

• saSualos standartuli Secdoma, Xσ (Standard error of the mean): _ saSualos SerCeviTi

ganawilebis standartuli gadaxra, romelic miiReba σ-as SerCevis moculobis kvadratul fesvTan SefardebiT.

• saSualos standartuli Secdomis Sefaseba, Xs (Estimated standard error of the mean) _ s-is gamoyenebiT σ-as, saSualos standatruli Secdomis Sefaseba.

• saSualos SerCeviTi ganawileba (Sampling distribution of the mean): _ X -is mniSvnelobebis ganawileba, rodesac populaciidan aRebulia N moculobis mqone yvela SesaZlo SerCeva.

• statistikuri daskvna (Statistical inference) _ SerCevisgan miRebuli statistikis mixedviT populaciis Sefaseba.

• stratificirebuli SemTxveviTi SerCeva (Stratified random sampling): _ SerCevis meTodi, rodesac populaciis wevrebi erTiandeba homogenur qvejgufebSi, romelTac stratebi ewodeba. populaciis wevrebi SemTxveviT airCevian stratidan im proporciiT, romliTac es strata populaciaSi gvxvdeba.

• SerCeviTi ganawileba (Sampling distribution): _ populaciidan aRebuli N moculobis mqone yvela SesaZlo SerCevidan miRebuli statistikis mniSvnelobebis albaTuri Teoriuli ganawileba.

• SerCevis Secdoma (Sampling error): _ sidide, romliTac konkretuli SerCevis saSualo gansxvavdeba populaciis saSualosgan.

• Caunacvlebeli Sefaseba (Unbiased estimation): _ statistika, romlis mniSvnelobaTa saSualo, gamoTvlili usasrulod bevri SerCevidan, Sesafasebeli parametris mniSvnelobis tolia.

• centraluri zRvris Teorema (Central limit theorem): _ maTematikuri Teorema, romlis Tanaxmad, SerCevis moculobis zrdasTan erTad saSualos SerCeviTi ganawileba normalur ganawilebas uaxlovdeba.

• Zalmosili Sefaseba (Consistent estimation): _ statistika, romlisTvisac albaToba imisa, rom es statistika ufro axlos iqneba Sesafasebel parametrTan, izrdeba SerCevis moculobis zrdasTan erTad.

• wertilovani Sefaseba (Point estimation): _ parametris mniSvnelobis Sefaseba statistikis mniSvnelobiT (anu erTi ricxviT).

• Xµ _ saSualos Teoriuli SerCeviTi ganawilebis saSualo.

• XX _ saSualos empiriuli SerCeviTi ganawilebis saSualo.


VIII Tavi. statistikuri hipoTezebis Semowmebis Sesavali: z kriteriumi da erTi SerCevis t kriteriumi

parametruli kriteriumi (parametric test) - statistikur kriteriums ewodeba parametruli, Tu hipoTezaSi aris gamoTqmuli varaudi populaciis parametris Sesaxeb.

araparametruli kriteriumi (nonparametric test) - statistikur kriteriums

ewodeba araparametruli, Tu hipoTezaSi ar aris gamoTqmuli varaudi populaciis parametris Sesaxeb.

statistikuri hipoTeza (statistical hypothesis) _ debuleba populaciis

parametris Sesaxeb(parametruli kriteriumis Sesaxeb). nulovani hipoTeza (null hypothesis) - aris im pirobis Camoyalibeba, romelic

mecniers populaciis Sesaxeb apriorulad WeSmaritad miaCnia. es aris hipoTeza, romelic mowmdeba statistikuri kriteriumiT.

statistikuri alternatiuli hipoTeza (statistical alternative hypothesis) _ aris

debuleba imis Sesaxeb, Tu ra unda iyos WeSmariti, Tu nulovani hipoTeza mcdaria.

kriteriumis statistika (test statistic) aris ricxvi, romelic nulovani

hipoTezis Semowmebis saSualebas gvaZlevs da gamoiTvleba SerCevis monacemebze dayrdnobiT.

alfa(α): mniSvnelovnobis donis sidide (albaToba). uaryofis are (rejection region) - kriteriumis statistikis (SerCeviTi

ganawilebis) is mniSvnelobebi, romelTa albaToba tolia an naklebia α-ze nulovani hipoTezis WeSmaritebis pirobebSi. Tuki kriteriumis statistika moxvdeba uaryofis areSi, maSin H0 uaryofili iqneba.

ormxrivi kriteriumi (two-tailed test) - statistikuri kriteriumi, roca

uaryofis aris asagebad gamoyenebulia kriteriumis statistikis SerCevis ganawilebis orive kudi.

calmxrivi kriteriumi (one-tailed test)- statistikuri kriteriumi, roca

uaryofis aris asagebad gamoyenebulia kriteriumis statistikis SerCevis ganawilebis mxolod erTi kudi.

kritikuli mniSvnelobebi (critical value) - konkretuli ricxviTi

mniSvnelobebi, romlebic adgenen uaryofis aris sazRvrebs.

mniSvnelovnobis done (significance level) esaa albaTobis is mniSvneloba, romelic iZleva nulovani hipoTezis uaryofis pirobas statistikur kriteriumSi.


statistikurad mniSvnelovani gansxvaveba (statistically significant difference) -

kriteriumis statistikis realuri mniSvneloba xvdeba uaryofis areSi da H0 uaryofilia.

aramniSvnelovani gansxvaveba: (nonsignificant difference) - kriteriumis

statistikis realuri mniSvneloba ar xvdeba uaryofis areSi da nulovani hipoTeza ar aris uaryofili.

erTi SerCevis t kriteriumi (one-sample t test): t kriteriumi, romelic

gamoiyeneba SerCevis saSualosa da populaciis hipoTetur saSualos Soris gansxvavebis statistikuri mniSvnelovnobis Sesamowmeblad, roca

σ X Sefasebulia s X -is saSualebiT. Tavisuflebis xarisxi (degrees of freedom): qulaTa raodenoba, romelic

Tavisufalia cvalebadobis dros statistikis gamoTvlisas. pirveli gvaris Secdoma (Type I error) - Secdoma statistikuri

gadawyvetilebis miRebis dros, romelic gamoixateba nulovani hipoTezis uaryofaSi, maSin roca is realurad WeSmaritia.

meore gvaris Secdoma (Type II error) - Secdoma statistikuri

gadawyvetilebis miRebis dros, romelic gamoixateba imiT, rom ar xdeba nulovani hipoTezis uaryofa, maSin roca is realurad mcdaria da alternatiuli H1 hipoTeza ki WeSmariti.

beta (β): meore gvaris Secdomis albaToba. simZlavre (Power) – H0 uaryofis albaToba, roca H0 ar aris samarTliani

da H1 ki aris WeSmariti. statistikuri kriteriumis simZlavre mocemulia

1-β-s saSualebiT.

IXFTavi. eqsperimentis safuZvlebi da saSualoTa Soris gansxvavebis Semowmeba


cdis pirTa Soris sqema (Between–subject design) – eqsperimenti, romelSic

ori an meti jgufi.

SemTxveviTi SerCeva (Random sampling) - SerCevis meTodi, romelSic individebi iseTnairad arian arCeulni, rom populaciis TiToeul wevrs aqvs am SerCevaSi moxvedris erTnairi Sansi da erTi wevris SerCeva ar aris damokidebuli populaciis nebismieri sxva wevris SerCevaze.

xelmisawvdomi SerCeva (Convenience sampling): cdis pirebis iseTi SerCeva, romelic mkvlevrisaTvis ufro xelmisawvdomi an xelsayrelia.

ekvivalenturi jgufebi (Equivalent groups): cdis pirTa jgufebi, romlebic, savaraudod, ar gansxvavdebian erTmaneTisgan raime mudmivi an sistematuri maxasiaTebliT eqsperimentis damoukidebeli cvladis miwodebamde.

SemTxveviTi ganawileba (Random assignment): cdis pirTa jgufebSi

ganawilebis meTodi, romlis mixedviT, eqsperimentisaTvis SerCeul nebismier individs aqvs nebismier jgufSi ganawilebis toli albaToba da erTi cdis piris ganawileba jgufSi ar moqmedebs nebismieri sxva individis ganawilebaze imave jgufSi.

damoukidebeli cvladi ( Independent variable) _ cvladi, romliTac manipulireben eqsperimentSi, damokidebul cvladze misi gavlenis gansazRvris mizniT.

damoukidebeli cvladis done (Level of an independent variable) _

damoukidebeli cvladis erTi mniSvneloba. imisaTvis, rom cvladi gaxdes, damoukidebeli cvlads unda hqondes sul cota ori gansxvavebuli done.

faqtori (Factor) _ damoukidebeli cvladis alternatiuli saxelwodeba.

cdis pirTa Soris erTfaqtoriani sqema (One-factor between subject design): kvlevis sqema, romelSic erTi damoukidebeli cvladis manipulacia xdeba da iqmneba ori an meti jgufi.

damatebiTi cvladebi (Extraneous variables) - damoukidebeli cvladisagan gansxvavebuli nebismieri cvladi, romelsac SeuZlia imoqmedos damokidebul cvladze eqsperimentSi.

Sereuli eqsperimenti (Confounded experiment) aris eqsperimenti, romelSic damatebiTi cvladi sistematurad icvleba damoukidebel cvladTan erTad.

placebos kontroli (Placebo control) - simulirebuli eqsperimentuli

(Carevis) piroba.

damokidebuli cvladi (Dependent variable): aris cvladi eqsperimentSi, romelic damokidebulia damoukidebel cvladze.

saSualoTa Soris sxvaobis SerCeviTi ganawileba (Sampling distribution of the difference between means): SerCevis saSualoebs Soris sxvaobis ganawileba,


roca populaciidan amoRebulia n moculobis SerCevis yvela SesaZlo wyvili.

saSualoTa Soris sxvaobis standartuli Secdoma (Standard eror of the difference berween means): 21 AA XX − mniSvnelobaTa SerCeviTi Teoriuli ganawilebis standartuli gadaxra.

saSualoTa Soris sxvaobis standartuli Secdomis Sefaseba (Estimated standard error of the differnce between means): saSualoTa Soris sxvaobis

standartuli Secdomis Sefaseba miiReba s2-is gamoyenebiT σ2-is SefasebisaTvis.

mdgradoba (Robustness): es termini gamoiyeneba imis aRsaniSnavad, rom statistikuri kriteriumis daSvebebis darRvevas aqvs mcire gavlena pirveli gvaris Secdomaze.

efeqtis siZliere (Strength of effect): damoukidebeli cvladis siZliere

gazomili statistikis gavlenis siZlieriT. cdis pirTa Sida sqema (Within-subject design): kvlevis sqema, romelSic cdis

pirTa erTi jgufia da izomeba damoukidebeli cvladis TiTeuli donis mixedviT. cdis pirTa Sida sqemaSi TiToeul cdis pirs miewodeba eqsperimentis orive piroba.

X Tavi. erTfaqtoriani dispersiuli analizi cdis pirTa Soris

sqemisaTvis

erTfaqtoriani mravaldoniani sqema (One-factor multilevel design): eqsperimenti erTi damoukidebeli cvladiT da am damoukidebeli cvladis sami an meti doniT.

dispersiuli analizi (Analysis of variance ANOVA): statistikuri kriteriumi,

romelic gamoiyeneba mravaldoniani sqemebis analizisaTvis. sruli saSualo (Grand mean): yvela qulis saSualo eqsperimentSi.

kvadratebis saSualo (mean square): dispersiis saxelwodeba, romelic

gamoiyeneba dispersiul analizSi.


jgufSida Secdomis dispersia (Within-groups error variance): monacemTa dispersia jgufSi, romelic jgufis saSualos garSemoa gamoTvlili.

jgufTa Soris dispersia (Between-groups variance): dispersia, romelic

gamoiTvleba sruli saSualos garSemo jgufebis saSualoebis cvalebadobis gamoyenebiT.

a: A faqtoris doneebis raodenoba

nA: qulaTa raodenoba erTfaqtoriani sqemis erT doneze

mravaljeradi Sedarebis kriteriumebi (Multiple comparison tests):

statistikuri kriteriumebi, romlebic gamoiyeneba wyvilTaSoris SedarebebSi imis dasadgenad, Tu romeli saSuloebi gansxvavdeba mniSvnelovnad erTmaneTisagan erTfaqtorian mravaldonian sqemaSi.

wyvilTaSoris Sedarebebi (Pairwise comparisons): statistikuri Sedarebebi, romlebic erTmaneTs adareben or saSualos.

Secdomis xarisxi eqsperimentSi (Error rate in an experiment) aris sul cota

erTi pirveli gvaris daSvebis albaToba mainc statistikur SedarebebSi, romelic Catarebulia eqsperimentSi.

saboloo Sedarebebi (Post hoc comparisons): statistikuri kriteriumebi,

romlebic asruleben yvela SesaZlo wyvilTa Sedarebebs, mas Semdeg, rac statistikurad mniSvnelovani Frealuri dadasturdeba sruli dispersiuli analizisaTvis.

kritikuli gansxvaveba (CD Critical difference): aris or saSualos Soris

raodenobrivi gansxvaveba, romelic statistikurad mniSvnelovania.


terminebi (Tavebi 11, 12)

A faqtoris mTavari efeqtis saSualo (Main effect of factor A): gansxvavebaXA1-sa

daXA2-s Soris, gamoxatuliXA1-XA2. artefaqturi mTavari efeqti (Artifactual main effect): mTavari efeqti, romelic ar iZleva mniSvnelobis mqone informacias damoukidebeli cvladis efeqtis Sesaxeb faqtorul sqemaSi.

B faqtoris mTavari efeqtis saSualo (Main effect of factor B): gansxvavebaXB1-sa

daXB2-s Soris, gamoxatuliXB1-XB2. gavarjiSebis efeqti (Practice effect): Carevis mravaljeradi efeqti, romelic xdeba imis gamo, rom cdis pirebi SeiZleba gavarjiSdnen an daiRalnen eqsperimentis davalebis Sesrulebisas. damoukidebeli cvladis martivi efeqti (Simple effect of an independent variable): faqtorul sqemaSi erTi damoukidebeli cvladis efeqti meore damoukidebeli cvladis mxolod erTi donis dros. mTavari efeqtis saSualo (Main effect mean): faqtorul analizSi damoukidebeli cvladis erTi donisaTvis yvela cdis piris saSualo, meore damoukidebeli cvladis doneebis mixedviT klasifikaciis ignorirebiT. pirobebisa da cdis pirTa urTierTqmedeba (Treatment-by-subject interaction): situacia cdis pirTa Sida erTfaqtorian sqemaSi, rodesac damoukidebeli cvladis efeqti damokidebulia cdis pirze. ujredi an Carevis piroba (Cell or treatment condition): faqtorul eqsperimentSi TiToeuli damoukidebeli cvladis erTi donisagan warmoqmnili kombinacia. ujredis saSualo (Cell mean): ujredSi nAB qulebis saSualo faqtorul sqemaSi. urTierTqmedeba (Interaction): situacia faqtorul sqemaSi, rodesac erTi damoukidebeli cvladis efeqti damokidebulia masTan kombinaciaSi myofi meore damoukidebeli cvladis mniSvnelobaze. faqtoruli analizi (Factorial design): kvlevis iseTi sqema, romelSic erTdroulad icvleba ori an meti damoukidebeli cvladi. cdis pirTa SigniT sqema (Within-subjects design): kvlevis sqema, rodesac cdis pirTa erTi jgufi ganicdis damoukidebeli cvladis zemoqmedebas, rac izomeba am cvladis yoveli donis Sesabamisad. Carevis gadatanis efeqti (Treatment carry-over effect): Carevis mravaljeradi efeqti, romelic xdeba maSin, rodesac damoukidebeli cvladis erTi donis erTi efeqti gadaitaneba, gavlenas axdens Sesrulebaze damoukidebeli cvladis meore doneze.


Carevis mravaljeradi efeqti (Multiple treatment effects): cdis pirTa Soris sqemaSi Sesrulebis cvlilebebi, romlebic xdeba imis gamo, rom mowmdeba damoukidebeli cvladis yovel doneze.


terminebi Tavebi 13-15

Tavi 13

korelaciuri kvlevebi (Correlational Studies) 367: kvlevebi, romlebSic izomeba ori an meti cvladi maTi korelaciis mimarTulebisa da xarisxis dasadgenad. organzomilebiani ganawileba (Bivariate Distribution) 367: ganawileba, romelSic TiToeuli cdis pirisagan miiReba ori monacemi (qula).

wertilTa gabnevis diagrama (Scatterplot) 368: organzomilebiani ganawilebis grafiki, romelSic X cvladi gadazomilia hirizontalur RerZze da Y cvladi gadazomilia vertikalur RerZze. korelacia (Covary) 369: ori cvladi korelirebs, rodesac erTi cvladis cvlileba dakavSirebulia meore cvladis SeTanxmebul cvlilebasTan dadebiTi kavSiri (Positive relationship) 369: or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad izrdeba meore cvladis mniSvnelobac. wrfivi kavSiri (Linear relationship) 369: or cvlads Soris kavSiri, romelic SeiZleba aRiweros wrfiT. uaryofiTi kavSiri (Negative relationship) 370: or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad meore cvladis mniSvneloba mcirdeba. pirsonis korelaciis koeficienti (Pearson correlation coefficient) 374: statistika, romelic aRiniSneba r -iT da asaxavs intervalebis an Sefardebis skalaze or cvlads Soris wrfivi kavSiris siZlieres. X -isa da Y -is jvaredina namravlebi (Cross products of X and Y ) 375: ( )( )∑ −− YYXX

mniSvneloba X da Y cvladebisTvis determinaciis koeficienti (Coefficient of determination) 388: 2r -is mniSvneloba, romelic X da Y cvladebis saerTo cvalebadobas asaxavs. ρ (390): ro, populaciis korelaciis koeficientis aRmniSvneli simbolo. organzomilebiani normaluri ganawileba (Bivariate normal distribuition) 392: X da Y cvladebis iseTi ganawileba, romlisTvisac (1) X qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (2) Y qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (3) TiToeuli X qulisaTvis Y qulebis ganawileba populaciaSi normaluria da (4) TiToeuli Y qulisaTvis X qulebis ganawileba populaciaSi normaluria.


spirmenis ranguli korelaciis koeficienti ( sr ) (Spearman rank-order correlation coefficient) 396: korelaciis koeficienti, romelic rigis skalaze gamoiyeneba.

Tavi 14

wrfivi kavSiri (Linear Relation) 408: kavSiri or cvlads Soris, romlis drosac

X cvladis yoveli erTi erTeuliT cvlilebisas, Y cvladi mudmivi sididiT

icvleba. wrfis gantoleba (General Equation of a straight line) 410: abXY += wrfis daxriloba (Slope of a straight line) 410: wrfis daxriloba mocemulia tolobiT



−−

=XY

b

Y TanakveTa (Y- intercept) 412: abXY += gantolebaSi Y-is mniSvneloba, roca X = 0.

Y ′ (416): wrfis gantolebis meSveobiT, X –is mixedviT gamoTvlili Y –is mniSvneloba. umcires kavadratTa meTodiT miRebuli regresiis wrfe (Least-squares regression line) 420: wrfe, romelic minimumamde amcirebs ( )∑ ′− 2YY -is mniSvnelobas.

naSTi (Residual) 425: YY ′− -is mniSvneloba.

naSTiSS (SSResidual) 426: ( )∑ ′− 2YY -is mniSvneloba.

Sefasebis standartuli Secdoma ( XYs ⋅ ) (Standard error of estimate) 426:

( )2

2

−

′−∑wyviliN

YYgamosaxulebis mniSvneloba

mravlobiTi regresia (Multiple regression) 440: Y qulis mniSvnelobebis winaswarmetyveleba X cvladis ramdenime mniSvnelobis gamoyenebiT.

Tavi 15

araparametruli statistikuri kriteriumi (Nonparametric Statistical Tests) 449: statistikur kriteriums ewodeba araparametruli, Tu hipoTezaSi ar aris gamoTqmuli varaudi populaciis parametris Sesaxeb. xi-kvadrat kriteriumi (Chi-square test) 450: statistikuri kriteriumi, romelic gamoiyeneba saxeldebis skalis monacemebis gasaanalizeblad, roca mocemulia sxvadasxva kategoriis Sesabamisi sixSireebi.


damoukideblobis xi-kvadrat kriteriumi (Chi-square test of independence) 450: xi-kvadrati, roca monacemebis kategorizacia ori damoukidebeli niSnis mixedviT xdeba. niSanTa SeuRlebis cxrili (Contingency Table) 451: striqonebiani da svetebiani cxrili, romelSic warmodgenilia yoveli kategoriis sixSireebi mocemuli striqonisa da svetis cvladebis gadakveTis Sedegad. marginaluri (Sejamebuli) sixSireebi (Marginal Frequencies) 452: niSanTa SeuRlebis cxrilSi sixSireebis jami striqonis an svetis mixedviT. mosalodneli siSireebi (Expected Frequencies) 452: niSanTa SeuRlebis cxrilidan miRebuli sixSireebi im daSvebis safuZvelze, rom cxrilSi warmodgenili ori ganzomileba damoukidebelia.

Tanxmobis χ² kriteriumi (Chi-square test of goodness of fit) 458: xi-kvadrati, roca monacemebis kategorizacia xdeba mxolod erTi ganzomilebis mixedviT man-uitnis U kriteriumi (Mann-Whitney U test) 462: araparametruli kriteriumi cdis pirebs Soris sqemisaTvis, roca gamoyenebulia damoukidebeli cvladis ori done da monacemebi sul mcire, rigis skalazea. uilkoqsonis niSniani rangebis kriteriumi (Wilcoxon signed-ranks test) 473: araparametruli kriteriumi cdis pirebs Soris sqemisaTvis damoukidebeli cvladis ori doniT.

danarTi D statistikuri simboloebi A A faqtori, damoukidebeli cvladi eqsperimentSi. A1, A2, A3 A faqtoris doneebi erTfaqtorian an orfaqtorian sqemaSi. α alfa, pirveli gvaris Secdomis albaToba. A x B A da B faqtorebis urTierTqmedebis aRniSvna faqtorul

sqemaSi. a (1) A faqtoris doneebis raodenoba an (2) wrfis gantolebis Y-

TanakveTa regresiul analizSi. B B faqtori, meore damoukidebeli cvladi faqtorul sqemaSi. B1, B2, B3 B faqtoris doneebi faqtorul sqemaSi. b (1) B faqtoris doneebis raodenoba an (2) wrfis daxriloba

regresiul analizSi.

β beta, meore gvaris Secdomis albaToba.

χ2 xi-kvadrat statistika.

χ2kritikuli χ2-is kritikuli mniSvneloba.

χ2realuri monacemebidan miRebuli χ2-is mniSvneloba. CD kritikuli sxvaoba mravaljeradi Sedarebis kriteriumSi. CPXY X-isa da Y-is jvaredini namravli korelaciur kvlevaSi. cum f an cf qulis dagrovili sixSire. cfL qulebis dagrovili sixSire intervalis qveda realur

sazRvramde sixSireTa dajgufebul ganawilebaSi. cum rf an crf qulis dagrovili fardobiTi sixSire. cum %f an c%f procentebSi gamoxatuli qulis dagrovili sixSiris

ganawileba. D (1) ranJirebuli qulebis wyvilis sxvaoba individisaTvis;

gamoiyeneba spirmenis ranguli korelaciis koeficientSi an (2) cdis pirTa Sida sqemaSi sxvaoba cdis pirisagan miRebul or qulas Soris.

df Tavisuflebis xarisxebi. dispersiul analizSi df-s Cveulebriv indeqsebi moyveba, rogoricaa dfA, dfB, dfAxB, dfSecdoma an dfsruli, raTa miuTiTos, cvalebadobis romel wyaros Seesabameba Tavisuflebis es xarisxi.

η2 eta kvadrati. Erc qulis mosalodneli sixSire xi-kvadrat kriteriumis niSanTa

SeuRlebis cxrilis r striqonsa da c svetSi. f qulis sixSire. fi qulebis sixSire intervalSi sixSireTa dajgufebul

ganawilebaSi. F F kriteriumi dispersiul analizSi. Fkritikuli F-is kritikuli mniSvneloba. Frealuri monacemebidan miRebuli F-is mniSvneloba. H0 nulovani statistikuri hipoTeza. H1 alternatiuli statistikuri hipoTeza.

IQR kvartilTSorisi gabnevis diapazoni i intervalis klasis zoma an sigane sixSireTa dajgufebul

ganawilebaSi. M SerCevis saSualo. simbolo gamoiyeneba publikaciebSi,

romlebic misdeven amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils.

Mdn mediana. MS kvadratis saSualo, dispersiis Sefaseba dispersiul

analizSi. gadaxrebis kvadratebis saSualoebs, rogorc wesi, mohyveba indeqsebi, rogoricaa MSA, MSB, MSAxB, an MSSecdoma, raTa aRiniSnos, cvalebadobis romel wyaros asaxaven isini.

MSE Secdomis kvadratis saSualo. es simbolo gamoiyeneba dispersiul analizSi MSSecdomis aRsaniSnavad iseT publikaciebSi, romlebic misdeven amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils.

µ miu, populaciis saSualo. Xµ saSualos Teoriuli SerCeviTi ganawilebis saSualo.

N SerCevaSi qulebis sruli raodenoba, an eqsperimentSi qulebis sruli raodenoba.

N wyvilebi qulaTa wyvilebis raodenoba.

n qulebis raodenoba SerCevis qvejgufSi.

n A qulebis raodenoba erTi donisaTvis erTfaqtorian sqemaSi.

n AB orfaqtoriani sqemis ujredSi qulebis raodenoba.

rcO xi-kvadrat kriteriumis niSanTa SeuRlebis cxrilSi qulis

realuri sixSire r striqonsa da c svetSi. %f qulis sixSire procentebSi. P proporciiT gamoxatuli procentilis wertili. gamoiyeneba

qulis misaRebad procentilis konkretul wertilze.

xP X-is qulis procentiluri rangi.

p albaToba. q tukis HSD kriteriumSi gamoyenebuli studentis rangobrivi

statistika. r pirsonis korelaciis koeficienti SerCevisaTvis.

kritikulir pirsonis korelaciis koeficientis kritikuli mniSvneloba.

realurir X da Y cvladebs Soris pirsonis korelaciis koeficientis

miRebuli mniSvneloba. rf qulis fardobiTi sixSire. ρ ro, populaciis korelaciis koeficienti.

Sρ populaciis spirmenis ranguli korelaciis koeficienti.

Sr spirmenis ranguli korelaciis koeficienti. r 2 determinaciis koeficienti. σ sigma, populaciis standartuli gadaxra.

σ 2 populaciis dispersia.

Xσ populaciis saSualos standartuli Secdoma.

21 AXAX −σ saSualoebs Soris sxvaobis standartuli Secdoma.

∑=

N

i 1

i=1-dan N-mde Sejamebis aRniSvna.

S SerCevis standartuli gadaxra. s populaciis standartuli gadaxris Sefaseba. SD standartuli gadaxra. simbolo gamoiyeneba publikaciebSi,

romlebic misdeven amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils.

2S SerCevis dispersia. 2s populaciis dispersiis Sefaseba. 2

ulikombinirebs populaciis or saSualos Soris kombinirebuli dispersiis

Sefaseba.

Xs saSualos standartuli Secdomis Sefaseba.

SE saSualos standartuli Secdoma. simbolo gamoiyeneba publikaciebSi, romlebic misdeven amerikis fsiqologTa asociaciis publikaciebis saxelmZRvanelos (1994) stils.

21 AXAXs−

saSualoebs Soris sxvaobis standartuli Secdomis Sefaseba.

YXs • standartuli Secdomis Sefaseba X-dan Y-is winaswarmetyvelebisas.

SIQR semikvartilTSorisi gabnevis diapazoni.

∑R rangebis jami. SS kvadratebis jami. dispersiul analizSi SS-s Cveulebriv

moyveba indeqsebi, rogoricaa SSA, SSB, SSAxB, SSSecdoma da SSsruli, raTa aRiniSnos, cvalebadobis romel wyaros asaxavs is.

SSnaSTi ∑ ′− 2)( YY mniSvneloba.

XSS korelaciaSi X cvladis kvadratebis jami.

YSS korelaciaSi Y cvladis kvadratebis jami.

t t statistika t kriteriumSi. kritikulit t-s kritikuli mniSvneloba.

elidamoukidebt damoukidebeli jgufebis t kriteriumisaTvis miRebuli t statistika.

realurit monacemebidan miRebuli t-s mniSvneloba.

lidamokidebut damokidebuli jgufebis t kriteriumisaTvis miRebuli t statistika.

T uilkoqsonis T kriteriumi.

kritikuliT T-s kritikuli mniSvneloba.

realuriT monacemebidan miRebuli T-s mniSvneloba.

U man-uitnis U kriteriumi. kritikuliU U-s kritikuli mniSvneloba.

realuriU monacemebidan miRebuli U-s ufro mcire mniSvneloba. X cdis piris qula im cvladis mixedviT, romelic

gansazRvrulia, rogorc X cvladi. gamoyenebuli sqemis mixedviT, qula SeiZleba aRniSnuli iyos Xi, Xij an Xijk.

imaqsimalurX yvelaze didi qula qulaTa ganawilebaSi.

minimaluriX yvelaze mcire qula qulaTa ganawilebaSi.

LX sixSireTa ganawilebaSi im intervalis dabali realuri sazRvari, romelic Seicavs X qulas.

PX ganawilebaSi procentilis P wertilze mdebare qula.

25X qula, romelic Seesabameba ganawilebis 25-e procentils.

50X qula, romelic Seesabameba ganawilebis medianas.

75X qula, romelic Seesabameba ganawilebis 75-e procentils.

X SerCevis saSualo.

1AX , 2AX ,

3AX A faqtoris doneebis mTavari efeqtebi. 1BX , 2BX ,

3BX B faqtoris doneebis mTavari efeqtebi. ABX ujredis saSualo orfaqtorian sqemaSi.

GX sruli saSualo.

SX cdis piris saSualo cdis pirTa Sida sqemaSi.

XX saSualoebis empiriuli SerCeviTi ganawilebis saSualo.

21 AXAXX − saSualoebs Soris sxvaobis empiriuli SerCeviTi ganawilebis saSualo.

Y cdis piris qula im cvladis mixedviT, romelic gansazRvrulia, rogorc Y cvladi.

Y SerCevis saSualo qulebisaTvis, romlebic gansazRvrulia, rogorc Y cvladi.

lY Y-is nawinaswarmetyvelebi mniSvneloba wrfiul regresiul analizSi.

z z gardaqmnis gamoyenebiT miRebuli qulis mniSvneloba.

danarTi E amocanebis pasuxebi aq mocemulia seqciebSi “SeamowmeT Tqveni codna” da “kiTxvebi” moyvanili amocanebis amoxsnebis umetesoba. formulebisa da axsnebisaTvis ixileT Sesabamisi seqciebi teqstSi. Tavi I SeamowmeT Tqveni codna I.1 2. b. SerCevis statistikis safuZvelze (52%) populaciis parametris (52%) Sesaxeb. SeamowmeT Tqveni codna I.2 2. a. micemuli ukukavSiris tipi. b. fexebis muxlebSi moxris xangrZlivoba.

g. kvleva moicavda ori ekvivalenturi jgufis Seqmnas cdis pirTa Soris sqemiT da damoukidebeli cvladis manipulirebas. d. imisaTvis, rom ganisazRvros, SemTxveviTze metia Tu ara fexebis moxris xangrZlivobebis gasxvaveba.

SeamowmeT Tqveni codna I.3 2. a. imis dadgena, aris Tu ara kavSirSi paranormaluri movlenebis

rwmena da arCeuli samecniero kursebis raodenoba. b. paranormaluri movlenebis rwmena; arCeuli samecniero kursebis

raodenoba. kiTxvebi 3. a. korelacia da regresia.

g. mrisxanebis done, sisxlis wneva. 4. a. aRwera da statistikuri hipoTezebis Semowmeba.

b. bavSvisaTvis daniSnuli mkurnalobis tipi. g. bavSvis sxeulis temperatura.

5. a. aRwera. b. procenti.

6. a. nedli monacemebi. b. daskvna. b. statistika. b. nedli monacemebi. b. daskvna. b. statistika.

Tavi II SeamowmeT Tqveni codna II.1

2. a. gamokiTxva. b. eqsperimenti.

g. saarqivo Canawerebi. d. bunebrivi dakvirveba.

SeamowmeT Tqveni codna II.2 3. adamianebi nawildebian imave kategoriaSi, Tumca maT ar aqvT

gazomvis qveS myofi cvladis igive raodenoba. 5. b. ara. SeamowmeT Tqveni codna II.3 2. a. diax, es intervalebis skalaa. b. ara.

g. ara. 3. a. diax, es Sefardebis skalaa. b. ki.

g. ki. SeamowmeT Tqveni codna II.4 2. a. uwyveti. b. uwyveti.

g. wyvetili. d. uwyveti. e. uwyveti. v. wyvetili. z. wyvetili. T wyvetili.

3. a. 152.35-dan 152.45-is CaTvliT sm. b. 17.25-dan 17.35- is CaTvliT wm.

g. 3.5-dan 4.5- is CaTvliT. d. 0.4365-dan 0.4375 is CaTvliT wm. e. 6.85-dan 6.95 is CaTvliT mm. v. 67.805-dan 67.815 is CaTvliT girvanqa.

4. a. 27.44. b. 119.028.

g. 1.44. d. 1263.7592. e. 0.35. v. 13.27.

kiTxvebi 1. a. gamokiTxva. b. saarqivo Canawerebi.

g. eqsperimenti. d. bunebrivi dakvirveba.

3. a. profesiul miRwevaze orientacia. b. rigis. 4. a. sxeulis wona. b. Sefardebis.

g. 134.55-dan 134.65-is CaTvliT girvanqa. d. 127.6 an 127.65 girvanqa, SeTanxmebis Tanaxmad.

5. a. emociuri stabiloba.

b. Sefasebis skala: rigisa da intervalebis skalebs Soris. g. 7.5-dan 8.5 is CaTvliT. d. ara, is unda damrgvaldes 5.8-mde an 5.83-mde, SeTanxmebis Tanaxmad.

6. a. Sefardebis.

b. 96.5-dan 97.5 is CaTvliT wuTi. g. 82.7 an 82.67 wuTi, SeTanxmebis Tanaxmad.

7. rigis. 8. saxeldebis. 9. a. wyvetili.

b. uwyveti. g. wyvetili. d. uwyveti.

10. a. 141.25-dan 141.35-is CaTvliT girvanqa.

b. 33.5-dan 34.5-is CaTvliT wT. g. 175.55-dan 175.65-is CaTvliT iardi. d. 98.15-dan 98.25-is CaTvliT gradusi farengeitiT. e. 108.5-dam 109.5-is CaTvliT mm vercxliswyali.

11. a. 42.26.

b. 1.16. g. 163.65. d. 27.43. e. 13.90. v. 24.62. z. 19.28.

13. a. rigis.

b. Sefardebis. Tavi III SeamowmeT Tqveni codna III.1

2. a.

b. minimaluri = 3; maqsimaluri = 19. g. 8.

SeamowmeT Tqveni codna III.2 2. 10-dan 20-mde. 3. 5. 4. 11. 6. a. 123, 125.

b. 129, 131. g. 122.5, 125.5. d. 128.5-dan 131.5-mde. e. 121. v. 3.

7. a. i = 10.

intervali

realuri sazRvrebi


Tvlis Sedegi

f

rf

%f

cf

crf

c%f zeda qveda 200-209 190-199 180-189 170-179 160-169 150-159 140-149 130-139 120-129 110-119

199.5 189.5 179.5 169.5 159.5 149.5 139.5 129.5 119.5 109.5

209.5 199.5 189.5 179.5 169.5 159.5 149.5 139.5 129.5 119.5

204.5 194.5 184.5 174.5 164.5 154.5 144.5 134.5 124.5 114.5

/ / / /

1 1 0 0 0 1 1 0 0 0

.02

.02

.00

.00

.00

.02

.02

.00

.00

.00

2 2 0 0 0 2 2 0 0 0

50 49 48 48 48 48 47 46 46 46

1.00 .98 .96 .96 .96 .96 .94 .92 .92 .92

100 98 96 96 96 96 94 92 92 92

qula Tvlis Sedegi

f rf %f

19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

/ / / / / / / // //// // // // /

1 0 0 0 1 1 1 1 1 1 2 4 0 2 2 2 1 0

.05

.00

.00

.00

.05

.05

.05

.05

.05

.05

.10 .20 .00 .10 .10 .10 .05 .00

5 0 0 0 5 5 5 5 5 5 10 20 0 10 10 10 5 0

100-109 90-99 80-89 70-79 60-69 50-59 40-49 30-39 20-29 10-19

99.5 89.5 79.5 69.5 59.5 49.5 39.5 29.5 19.5 9.5

109.5 99.5 89.5 79.5 69.5 59.5 49.5 39.5 29.5 19.5

104.5 94.5 84.5 74.5 64.5 54.5 44.5 34.5 24.5 14.5

// /// ////// /////////// //////// ////// /////// // /

0 2 3 6 11 8 6 7 2 1

.00

.04

.06 .12 .22 .16 .12 .14 .04 .02

0 4 6 12 22 16 12 14 4 2

46 46 44 41 35 24 16 10 3 1

.92

.92

.88

.82

.70

.48

.32

.20

.06

.02

92 92 88 82 70 48 32 20 6 2

b. distanciebis umetesoba iyo 99 metri an naklebi. yvelaze xSiri iyo distancia 60-dan 69 metramde. arc erTi distancia ar iyo 10 metrze naklebi an 209 metrze meti. g. P68 = 67. d. X50 = 60.4 metrs.

SeamowmeT Tqveni codna III.3 2. a. suraTebi unda iyos

b. ganawilebebi igivea, rac suraTebze, im gamonaklisiT, rom fardobiTi sixSireebi ordinatazea asaxuli. g. distanciebis umetesoba naklebia, vidre 104.5 metri. yvelaze xSiri iyo distancia intervalSi, romlis Suawertilia 64.5 metri. arc erTi distancia ar iyo 9.5 metrze naklebi an 209.5 metrze meti.

3. Rero foToli 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

5 0 4 0 11 133 022256 11234666778 04456789 237899 1377889 59 8

a. 18 metri. b. 205 metri. g. 66 da 72 metri. SeamowmeT Tqveni codna III.4 2. a. simetriuli, unimodaluri.

b. asimetriuli, dadebiTad asimetriuli, unimodaluri. g. asimetriuli, dadebiTad asimetriuli, bimodaluri. d. asimetriuli, uaryofiTad asimetriuli, unimodaluri. kiTxvebi 1. a. intervalis zoma = 20; intervalebis iwyeba dadgenili

sazRvridan 220, pirveli Suawertili 229.5; bolo intervalis zeda dadgenili sazRvari = 499.

b. 400-dan 419-is CaTvliT; Suawertili = 409.5; ara, is ufro moklea. g. mniSvneloba iqneba intervalSi 340-dan 359-is CaTvliT. d. X50 = 356.2. e. P431 = 89. 2. a. intervalis zoma = 5; intervalebis iwyeba dadgenili sazRvridan

0, pirveli Suawertili 2.0; bolo intervalis zeda dadgenili sazRvari = 64.

b. 0-dan 4-is CaTvliT. g. X50 = 6.2. d. P24 = 86. e.

Rero foToli *

6 . *

5 . *

4 . *

3 . *

2 . *

1 . *

0 .

3 8 1 67 7 66 114 5778 0124 5555566666678899 000000001111223333444444

minimaluri qula = 0, maqsimaluri qula = 63. yvelaze xSiri qula = 0.

4. a. gacdenili dReebis

raodenoba

dajildoebis politikis mqone

skola

dajildoebis politikis

armqone skola 25-ze meti

24-25 22-23 20-21 18-19 16-17 14-15 12-13 10-11 8-9 6-7 4-5 2-3

.02

.00

.02

.00

.01 .00 .02 .00 .03 .08 .04 .13 .23

.03

.03 .005 .04 .00 .02 .11 .05 .09 .10 .015 .07 .17

0-1 .42 .27

b. X50 = 2.2 5.2

g. P24 = 68 46 d. dajildoebis politikis mqone skolis SemTxvevaSi moswavleebi 78% acdens 5 dRes an naklebs, sadac gacdenis medianaa 2.2 dRe. dajildoebis politikis armqone skolis SemTxvevaSi moswavleebis mxolod 51% acdens 5 dRes an naklebs, xolo gacdenis mediana aris 5.2 dRe.

5. b. minimaluri qula = 5; maqsimaluri qula = 29. g. 20. 6. a. i = 30. b. 209.5; 239,5. g. 224.5. e. 340 wuTi. v. P264 = 53. 7. a. 40. g. P78 = 59. d. 79.3 wuTi. Tavi IV SeamowmeT Tqveni codna IV.1 3. a. 49. b. unimodaluri. 4. a. 27. b. unimodaluri. 5. a. 24,27,49. b. multimodaluri. 7. 46. 8. 42. 9. 46. SeamowmeT Tqveni codna IV.2

2. a. X = 88.0. g. 262. 3. a. 41.7. b. 39.5. g. 37.7. SeamowmeT Tqveni codna IV.3 1. a. mediana. b. saSualo. g. moda. d. saSualo. 2. igive, rac mediana. 3. dadebiTad asimetriuli. kiTxvebi 1. SesaZlebelia uaryofiTad asimetriuli, radgan saSualo medianaze

naklebia. 2. dadebiTad asimetriuli, radgan saSualo sagrZnoblad metia medianaze. 3. dadebiTad asimetriuli, radgan saSualo sagrZnoblad metia medianaze. 4. jgufi 1

b. moda = 38. Mdn = 38. X = 38.6. g. daaxloebiT simetriuli; centraluri tendencies yvela sazomi

daaxloebiT erTnairia. jgufi 2

b. moda = 28. Mdn = 29. X = 33.5. g. dadebiTad asimetriuli; saSualo medianaze metia.

jgufi 3

b. moda = 52. Mdn = 50. X = 46.7. g. uaryofiTad asimetriuli; saSualo medianaze naklebia. jgufi 4

b. moda = bimodaluri, 26, 52. Mdn = 40. X = 38.6. g. daaxloebiT simetriuli; bimodaluri.

d. radgan ganawileba bimodaluria, arc mediana da arc X ar waradgens kargad cdis piris tipiur qulas.

5.

X mediana moda

roberto dimitri kareni

64 58 71

65 65 67

90 ar aris 56

moda mediana saSualo

6. a. rogorc Cans, zogierTma cdis pirma alkoholis dalevis Zalian

maRali raodenoba miuTiTa kviraSi, ramac ganapiroba dadebiTad asimetriuli ganawileba.

b. kviraSi alkoholis dalevis raodenobis mediana. 7. a. moda an mediana. b. moda = 793. Mdn = 832. Tavi V SeamowmeT Tqveni codna V.1 2. a. diapazoni = 44, IQR = 18, SIQR = 9. 3. a. diapazoni = 287 miliwami, IQR = 85 miliwami, SIQR = 42.5 miliwami. SeamowmeT Tqveni codna V.2 2. a. da b.

g. jgufSi, romelsac maswavlebeli ukiTxavda. d. 10.23. 3. a.

kiTxulobdnen

TviTon

ukiTxavdaT

maswavlebeli

X s s2

59.0 5.54 30.69

57.0 10.23 104.65

mamrobiTi mdedrobiTi

X s s2

15.55 7.41 54.91

7.05 2.59 6.71

b. mamrobiTSi. g. 7.41 wuTi. d. gogonebisaTvis; s qalebisaTvis naklebia, vidre mamakacebisaTvis.

SeamowmeT Tqveni codna V.3 1. a. Mdn, SIQR. b. Mdn, SIQR. g. X , s. d. Mdn, SIQR. e. X , s. v. Mdn, SIQR. z. Mdn, SIQR. kiTxvebi

1. a. Mdn = 50, X = 49.60, IQR = 10, s = 7.26. b. orive erTnairad kargia.

2. a. Mdn = 50, X = 49.45, IQR = 10, s = 11.56. b. orive erTnairad kargia. g. IQR-ze ar axdens gavlenas qulebi ganawilebis kudebidan. 3. a. diapazoni. b. IQR. g. SIQR. d. s2. e. s. v. IQR. z. diapazoni. T. S2. i. s2. k. S. l. s. m. SIQR. n. σ. o. σ2. 4. a. 15.4 alkoholis miReba kviraSi. b. did cvalebadobaze.

g. didi standartuli gadaxrebis migviTiTebs, rom ganawilebebi dadebiTad iyo asimetriuli. zogma cdis pirma mogvca dalevis Zalian didi raodenoba kviraSi da gazarda s-is mniSvneloba. es Sedegi gviCvenebs, rom mediana am qulebisaTvis centraluri tendenciis ukeTesi sazomi iqneboda.

5. IQR da SIQR, radgan isini ar moiTxoven mecxre pacientis qulis ricxviTi mniSvnelobis codnas.

Tavi VI SeamowmeT Tqveni codna VI.1

4. a. .3413. b. .4772. g. .4987. d. .3413. e. .4772. v. .4987. z. .8400. T. .9544. i. .9974. k. .1359. l. .1574. m. .0215. SeamowmeT Tqveni codna VI.2 2.

z farTobi 0-dan z-mde farTobi z-is zemoT a. b. g. d. e. v.

+1.90 -0.52 +0.77 -1.83 +1.75 -1.26

.4713

.1985 .2794 .4664 .4599 .3962

.0287

..3015

.2206

.0336

.0401

.1038

3. a. .0401. b. .1465. g. .3345. d. daaxloebiT 58.8-dan 85.2-mde, rodesac z = + an -1.65. SeamowmeT Tqveni codna VI.3 3. .0014. 4.

z p a. b. g. d. e. v.

+1.90 -0.52 +0.77 -1.83 +1.75 -1.26

.0287

..3015

.2206

.0336

.0401

.1038

5. a. .0668. b. .1865. g. .2426. kiTxvebi 3. a. .1587. b. .0228. g. .0038. d. .4082. e. .4525. v. daaxloebiT 80.0-dan 119.2-mde, rodesac z = + an -1.28.

z. .6568. T. .0918. i. .0475. k. .0038. 4. a. .016. b. .018. 5. a. p = .06. b. p = .08. g. p = .54. 6. a. .0968. b. .0968. g. .8064. 7. a. .0668. b. .0401. g. .9544. d. .8664. e. .2266. v. .3085. 8. a.

videoTamaSebi semestruli niSnebis saSualo

X S

8.0 5.86

2.7 0.87

aleqsi lela ia levani daviTi ivane

-0.34 0.00 -1.37 +1.19 -0.85 +1.37

-0.23 -0.80 +1.38 +0.80 +0.46 -1.61

b. lela jgufis saSualoa videoTamaSebisaTvis daxarjuli drois mixedviT. g. ia aris 1.38 standartuli gadaxriT maRla jgufis semestruli niSnebis saSualosTan SedarebiT. d. adamianis semestruli niSnebis saSualo jgufis saSualoze qvemoT aris. e. adamianis mier videoTamaSebze daxarjuli dro jgufis saSualoze maRalia.

9.

literatura istoria z = +0.83 z = +1.00

istoriaSi, radganac Tqveni qula klasis saSualoze metia erTi standartuli gadaxriT. 10. a. p = .206. b. p = .045. 11. a. p = .458. b. p = .267. Tavi VII SeamowmeT Tqveni codna VII.1 2.

(a) proporcia (b) raodenoba generalizebuli SfoTviTi aSliloba agorafobia martivi fobia

.27 .23 .19

27 23 19

panikuri aSliloba obsesiur-kompulsiuri

.22

.09

22 9

SeamowmeT Tqveni codna VII.2 2. Zalmosili; ufro naklebad afasebs. 3. Caunacvlebeli. 4. Zalmosili; odnav naklebad afasebs. 7. a. .6826. b. .1587. g. .0228. d. 0475. e. .4525. 8. a. 9, 6, 3 wami, Sesabamisad.

b. N-is zrdasTan erTad mcirdeba Xs .

9. a. s = 10, Xs = 2.

s = 15, Xs = 3.

s = 20, Xs = 4.

b. s-is zrdasTan erTad aseve izrdeba Xs

10. populaciidan A2, radganac mas σ-is ufro didi mniSvneloba aqvs. SeamowmeT Tqveni codna VII.3 2. a. 73.0. b. 69.1-dan 76.9-mde. g. 67.8-dan 78.2-mde. d. 73.0. e. 71.0-dan 75.0-mde. v. 70.4-dan 75.6-mde.

z. Xσ -is mniSvneloba mcirdeba N-is zrdasTan erTad. kiTxvebi

1. a. σ. b. µ. g. 2σ . d. Xσ .

2. N = 24, radgan X aris Zalmosili Sefaseba. 3. stratificirebuli; axsnisaTvis ixileT teqsti. 5. a. 9.0 sigareti dReSi. b. 7.5-idan 10.5-mde sigareti dReSi. 6. a. 0.3. b. .0013. g. .1587. d. .9544. 8.

SerCeva a b g

a

b

g

d

e

2.00

1.33

1.00

0.80

0.40

.1587

.0668

.0228

.0062

<.0001

.3413

.4332

.4772

.5938

>.4999

Tavi VIII SeamowmeT Tqveni codna VIII.1

3. a. .100:0 =µH

.100:1 ≠µH

b. .4.4:0 =µH saaTi.

.4.4:1 ≠µH saaTi.

g. 0.423:0 =µH miliwami.

0.423:1 ≠µH miliwami. 4.

realuriz

a. b. g. d.

-2.13

-1.20

+2.25

+1.72

xvdeba uaryofis areSi; uarvyofT

oH ; viRebT 1H .

ar xvdeba uaryofis areSi; ar

uarvyofT oH ; ar viRebT 1H .

xvdeba uaryofis areSi; uarvyofT

oH ; viRebT 1H .

ar xvdeba uaryofis areSi; ar

uarvyofT oH ; ar viRebT 1H .

5. a. ( )XXz σµ /−=

b. .100:0 =µH .100:1 ≠µH

g. α = .05. d. kritikuliz =1.96.

e. realurisz ufro eqstremaluri mniSvnelobebi, vidre z = - 1.96 an + 1.96.

v. realuriz = + 2.50.

z. uarvyofT oH ; viRebT 1H .

T. SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis saSualosagan.

SeamowmeT Tqveni codna VIII.2 3. a. 7. b. 6. g. 6. 4. a. 9. b. 15. g. 6. d. 53. e. 119. v. 29. 5. ganawileba 13 df-iT. 6. rodesac df = ∞. SeamowmeT Tqveni codna VIII.3

1. a. 493:0 =µH ; .493:1 ≠µH

b. 1.36:0 =µH ; .1.36:1 ≠µH

g. 100:0 =µH ; .100:1 ≠µH

2. a. 2.447. b. 2.898. g. 2.069. d. 2.042. e. 2.042. v. 2.660. z. 2.000. T. 1.980. 3.

xvdeba uaryofis

areSi

gadawyvetileba

a. ki uarvyofT oH

viRebT 1H

b. ara ver uarvyofT oH

ver viRebT 1H

g. ki uarvyofT oH

viRebT 1H

d. ara ver uarvyofT oH

ver viRebT 1H

e. ki uarvyofT oH

viRebT 1H

v. ki uarvyofT oH

viRebT 1H

z. ara ver uarvyofT oH

ver viRebT 1H

T. ki uarvyofT oH

viRebT 1H

i. ki uarvyofT oH

viRebT 1H

k. ki uarvyofT oH

viRebT 1H

l. ki uarvyofT oH

viRebT 1H

5. a. ( )XsXt /µ−=

b. ixileT teqsti.

g. 0.7:0 =µH ; .0.7:1 ≠µH

d. α = .05. e. kritikulit (10) = 2.228.

v. realurist ufro eqstremaluri mniSvnelobebi, vidre –2.228 an +2.228.

z. realurit = -4 979.

T. uarvyofT oH , viRebT 1H .

i. SerCevis saSualo 5.8 wuTi mniSvnelovnad gansxvavdeba populaciis saSualosagan – 7.0 wuTi.

6. angariSi 1 a. 70.4 kg.

b. 7.6 kg. g. t erTi SerCevisaTvis. d. 75:0 =µH kg, 75:1 ≠µH kg.

e. 6. v. –1.48. z. 5. T. 2.571.

i. ver uarvyofT oH .

k. ver viRebT 1H . l. SerCevis saSualo mniSvnelovnad ar gansxvavdeba populaciis saSualosagan.

angariSi 2

a. 10.4 dRe. b. 2.3 dRe. g. t erTi SerCevisaTvis.

d. 7.8:0 =µH dRe; 7.8:1 ≠µH dRe.

e. 25. v. +3.70. z. 24. T. 2.064.

i. uarvyofT oH .

k. viRebT 1H . l. SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis saSualosagan; 10.4 dRe mniSvnelovnad ufro grZeli periodia, vidre hipoTeturi saSualo 8.7 dRe.

SeamowmeT Tqveni codna VIII.4

2. b. p = α g. pirveli gvaris Secdoma ver moxdeba, Tu oH ar aris WeSmariti.

d. ara. 4. 80.1-dan 113.9 wamamde. Tavis mimoxilva

5. a. uarvyofT oH ; viRebT 1H .

b. ver uarvyofT oH ; ver viRebT 1H .

7. a. SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi saSualosagan.

b. SerCevis saSualo mniSvnelovnad ar gansxvavdeba populaciis hipoTeturi saSualosagan.

8. uaryofis are Sedgeba realurisz ufro ukiduresi mniSvnelobebisagan,

vidre z = -2.58 an +2.58.

realuriz xvdeba uaryofis

areSi

gadawyvetileba

+3.71

-2.60

+1.74

-3.05

-1.96

+2.58

ki

ki

ara

ki

ara

ki

uarvyofT oH ; viRebT 1H .


ver uarvyofT oH ; ver

viRebT 1H .


ver uarvyofT oH ; ver

viRebT 1H .


9. realuriz = -1.52; aramniSvnelovani; ver uarvyofT oH .

11. a. SerCevis saSualo mniSvnelovnad gansxvavdeba populaciis hipoTeturi saSualosagan.

b. SerCevis saSualo mniSvnelovnad ar gansxvavdeba populaciis hipoTeturi saSualosagan.

xvdeba uaryofis

areSi

gadawyvetileba

a. ki uarvyofT oH

viRebT 1H

b. ara ver uarvyofT oH

ver viRebT 1H

g. ki uarvyofT oH

viRebT 1H

d. ki uarvyofT oH

viRebT 1H

e. ara ver uarvyofT oH

ver viRebT 1H

v. ara ver uarvyofT oH

ver viRebT 1H

13. a. realurit (8) = -2.924; aramniSvnelovani .05 doneze; uarvyofT oH ,

viRebT 1H . g. ki. d. ara.

e. 6.81-dan 7.59-mde saaTi; µ = 7.7 ar xvdeba am intervalSi.

14. a. realurit (11) = +3.139; mniSvnelovani .05 doneze; uarvyofT oH , viRebT

1H . b. ara. 15. a. pirveli gvaris Secdoma. b. meore gvaris Secdoma. g. swori gadawyvetileba. d. swori gadawyvetileba.

Tavi IX SeamowmeT Tqveni codna IX.1 6. a. damatebiTi cvladi = asaki; jgufebi sistematurad gansxvavdebian

saSualo asakisa da xmauris pirobis mixedviT. b. damatebiTi cvladi = eqsperimentatoris waxaliseba; jgufebi

sistematurad gansxvavdebian waxalisebisa da xmauris pirobis mixedviT.

g. damatebiTi cvladi = genderi; jgufebi sistematurad gansxvavdebian genderisa da xmauris pirobis mixedviT.

d. damatebiTi cvladi = kolejSi aRebuli wamyvani sagani; jgufebi sistematurad gansxvavdebian kolejSi aRebuli wamyvani sagnisa da xmauris pirobis mixedviT.

e. damatebiTi cvladi = dRis monakveTi, rodesac eqsperimenti Catarda; jgufebi sistematurad gansxvavdebian dRis monakveTisa da xmauris pirobis mixedviT.

SeamowmeT Tqveni codna IX.2 3. a. 3.77. b. 3.56. g. 3.81. d. 2.25. SeamowmeT Tqveni codna IX.3 6.

SerCeva df kritikulit xvdeba

uaryofis areSi

gadawyvetileba

1. 18 2.101 ki uarvyofT oH

viRebT 1H

2. 18 2.101 ara ver uarvyofT oH

ver viRebT 1H


viRebT 1H


viRebT 1H


viRebT 1H

6. 33 2.042 ara ver uarvyofT oH

ver viRebT 1H


viRebT 1H


viRebT 1H

9. amocana # 1

a. 210 : AAH µµ = ; 211 : AAH µµ ≠ .

b. realurit (34) = -3.656.

g. 34. d. 2.042.

e. realurist ufro ukiduresi mniSvnelobebi, vidre –2.042 an +2.042.

v. ki.


T. gansxvavebuli populaciebidan. i. ki. k. ki; sakontrolo jgufSi statistikurad mniSvnelovani ufro

mokle drois intervali gamovlinda, vidre eqsperimentul jgufSi. amocana # 2 a. 210 : AAH µµ = ; 211 : AAH µµ ≠ .

b. realurit (28) = -1.742.

g. 28. d. 2.048.

e. realurist ufro ukiduresi mniSvnelobebi, vidre –2.048 an +2.048.

v. ara.

z. ver uarvyofT oH ; ver viRebT 1H . T. ar arsebobs imis TvalsaCinoeba, rom isini erTi da imave populaciidan arian.

i. ara; aramniSvnelovania. k. ara; gansxvaveba jgufebis saSualoebs Soris unda ganvixiloT,

rogorc SemTxveviTi gansxvaveba. SeamowmeT Tqveni codna IX.4 2. a. jgufi 2. b. jgufi 2. g. jgufi 1.

3. mosalodnelia, rom 2021 == AA nn eqneba ufro mcire 21 AXAXs−

da

migviyvans ufro mZlavr statistikur kriteriumamde. 4. a. .42. b. .27. g. .19. d. .23. e. .08. 7. angariSi 1 a. 2. b. 62. g. 79.7; 73.0. d. 8.57; 5.85. e. 3.60. v. 2.000.

z. uarvyofT oH . 1H .

T. viRebT 1H . i. statistikurad mniSvnelovani. k. Carevis efeqts.

angariSi 2 a. 2. b. 30. g. 22.78; 24.33.

d. 5.23; 5.14. e. –0.82. v. 2.048.

z. ver uarvyofT oH . 1H .

T. ver viRebT 1H . i. aramniSvnelovani. k. SerCevis Secdomas. SeamowmeT Tqveni codna IX.5

2. a. 210 : AAH µµ = ; 211 : AAH µµ ≠ .

b. realurit (10) = -3.148.

g. 10. d. 2.228.

e. realurist ufro eqstremaluri mniSvnelobebi, vidre –2.228 an +2.228.

v. ki.

z. uarvyofT oH ; viRebT 1H . 1H .

T. gansxvavebuli populaciebidan. i. ki. k. ki; sakontrolo jgufma statistikurad mniSvnelovani reaqciis

droiT upasuxa, gansxvavebiT eqsperimentuli jgufisagan. l. .50. m. - 41.8-dan - 7.2 miliwamamde. kiTxvebi 1.

jgufi uxmauro xmauriani

X s

122.64 8.76

131.27 10.45

a. realurit (20) = -2.099, p<.05. uxmauro jgufis saSualo sisxlis wneva

mniSvnelovnad dabalia, vidre xmauriani jgufisa.

b. 18.2 =η ; damokidebuli cvladis dispersiis 18%. g. 0.05-dan 17.21-mde. 2.


X s

113.09 4.55

115.27 6.48

realurit (20) = -0.913, p>.05. ori jgufis saSualo sisxlis wneva

mniSvnelovnad ar gansxvavdeba erTmaneTisagan. 3.

sqesi mamr. mdedr.

X s

17.77 4.99

18.59 5.32

realurit (28) = -0.429, p>.05. mamakacebi da qalebi mniSvnelovnad ar

gansxvavdebian Sefasebuli ganmartoebis moTxovnilebis mixedviT. 4. a.

instruqciis piroba instruqciamde instruqciis

Semdeg

X s

2.45 .604

2.52 .677

b. alfas .05 doniTa da ormxrivi kriteriumiT, instruqciamde gakeTebuli Sefasebebis saSualo Secdoma (M = 2.45 mm, SD = .604 mm) mniSvnelovnad ar gansxvavdeboda instruqciis Semdeg gakeTebuli Sefasebebis saSualo Secdomisagan (M = 2.52 mm, SD = .677 mm), t(11) = - 1.261, p > .05.

5. a. realurit (88) = +3.170, p<.01. kontrolis lokusis saSualo qula

naSvilebi bavSvebis mSoblebis jgufSi mniSvnelovnad maRali iyo, vidre lokusis kontrolis saSualo qula biologiuri mSoblebis jgufSi.

b. 10.2 =η g. 1.29-dan 5.69-mde.

6. a. realurit (5398) = -25.003, p<.001; ori jgufis saSualo qulebis

mniSvnelovnad gansxvavdeba. wesrigis damrRvevi studentebis saSualo qula mniSvnelovnad dabalia, vidre im studentebisa, romlebic wesrigs ar arRvevdnen.

b. 10.2 =η g. 0.47-dan 0.55-mde.

7. realurit (32) = -0.818, p>.05. aramniSvnelovani t miuTiTebs, rom jgufis

saSualoebi mniSvnelovnad ar gansxvavdeba.05 doneze.

8. a. realurit (24) = - 3.744, p<.05. im mosaubris saSualo sandoobis

reitingi, romelic uSualo qcevas avlens, mniSvnelovnad metia, vidre im mosaubris reitingi, romelic ar aris uSualo qcevis pirobaSi.

b. η2 = .37. g. - 3.223-dan - 0.931-mde.

d. alfas .05 doniTa da ormxrivi kriteriumiT, saSualo sandoobis reitingi mosaubrisa, romelic uSualo qcevas avlens, (M = 6.9, SD = 1.41) mniSvnelovnad meti iyo vidre im mosaubris reitingi, romelic ar aris uSualo qcevis pirobaSi (M = 4.8, SD = 1.42), t(24) = - 1.261, p > .01, η2 = .37.

Tavi X SeamowmeT Tqveni codna X.1

6. a = 4, An = 11, N = 44. 11. 150.000 12. 25.00 13. 79. 14. 25.00. 15. 10.00.

16. An = 3, Secdomadf = 40, srulidf = 43.

17.

wyaro SS df MS F A faqtori Secdoma sul

24.00 16.00 40.00

1 4 5

24.00 4.00

6.00

18.


42.00 24.00 66.00

2 6 8

21.00 4.00

5.25

19. cxrili # 1


50.00 260.00 310.00

1 26 27

50.00 10.00

5.00

a. 2. b. 14. g. 28. cxrili # 2


18.00 152.00 170.00

3 76 79

6.00 2.00

3.00

a. 4. b. 20. g. 80. SeamowmeT Tqveni codna X.2 6. daaxloebiT 1.00 7. is 1.00-ze meti xdeba.

8. a. 210 : AAH µµ =

1H : Aµ -ebi ar aris toli.

b. 43210 : AAAAH µµµµ ===

1H : yvela Aµ ar aris erTmaneTis toli.

g. 6543210 : AAAAAAH µµµµµµ =====

1H : yvela Aµ ar aris erTmaneTis toli. 9. damoukidebel cvlads ar aqvs efeqti. 10. damoukidebel cvlads aqvs efeqti. 11. nuli. 12. zeda sazRvari ar aris. 13.

eqsperi-menti

kritikuliF xvdeba uaryofis

areSi

gadawyvetileba

1. 4.96 ki uarvyofT oH

viRebT 1H

2. 3.40 ara ver uarvyofT oH

ver viRebT 1H


viRebT 1H


viRebT 1H

5. 3.88 ki ver uarvyofT oH

ver viRebT 1H


ver viRebT 1H


ver viRebT 1H


ver viRebT 1H


viRebT 1H


ver viRebT 1H

SeamowmeT Tqveni codna X.3 1.

jgufi sakontrolo eqsperimentuli

X s

7.09 1.30

8.82 1.47

a. 210 : AAH µµ =

1H : Aµ -ebi ar aris toli.

b. realuriF (1,20) = 8.51.

g. Adf = 1, Secdomadf =20.

d. kritikuliF (1,20) = 4.35.

e. realurisF mniSvnelobebi ≥ 4.35. v. ki.


T. gansxvavebuli populaciebidan. i. ki. k. isini gansxvavdeba; sakontrolo jgufma ufro mokled drois

intervali gviCvena. l. ara, radganac icvleboda damoukidebeli cvladis mxolod ori

done.

m. 30.2 =η n. 30 procenti. 2.

klasi meore meoTxe meeqvse

X s

565.80 32.410

530.60 29.99

501.00 24.42

a. 3210 : AAAH µµµ ==

1H : yvela Aµ ar aris erTmaneTis toli.

b. realuriF (2, 39) = 17.76.

g. Adf = 2, Secdomadf = 39.

d. kritikuliF (2, 39) = 3.32. (kritikuli mniSvneloba 2, 30 df-isaTvis)

e. realurisF mniSvnelobebi, romlebis metia an tolia 3.32.

v. ki.


T. saSualoebs Soris aris sul mcire erTi mniSvnelovani sxvaoba.

i. ki, radganac realuriF statistikurad mniSvnelovania da

eqsperimentSi Sedardeba sami saSualo. k. CD = 26.86. l. TiToeuli jgufi mniSvnelovnad gansxvavdeba yvela sxva

jgufisagan. m. Zilis periodebi yvelaze grZelidan yvelaze moklesaken aris me-

2, me-4 da me-6 klasis moswavleebi.

n. η2 = .48. o. 48 procenti. 3.1. a. 2. b. 62. g. 14.15. d. 4.00

e. uarvyofT oH .

v. viRebT 1H . z. ki. T. nacnobi da ucnobi leqsis jgufebis saSualoebi. 3.2. a. 4. b. 40. g. 1.13. d. 2.92.

e. ver uarvyofT oH .

v. ver viRebT 1H . z. ara. T saSualoebs Soris ar aris mniSvnelovani gansxvaveba. 3. a. 3. b. 39. g. 33.94. d. 3.32.

e. uarvyofT oH .

v. viRebT 1H . z. ki. T. tukis CD gviCvenebs, rom mZime rokis saSualo (29.9)

mniSvnelovnad maRali iyo rogorc klasikuri musikis (18.5), aseve musikis gareSe sakontrolo jgufis (17.5) saSualoebze. klasikuri musikisa da musikis gareSe sakontrolo jgufebis saSualoebi ar gasxvavdeboda mniSvnelovnad.

kiTxvebi 1. a.

eqsperimentuli jgufi musikis gareSe

Zveli musika mZime roki

X s

36.0 3.70

35.6 3.50

22.9 3.60

realuriF (2,21) = 34.39, p < .01. tukis CD = 4.6. mZime rokis jgufma

mniSvnelovnad naklebi Secdomebi aRmoaCina, vidre danarCenma orma jgufma. musikis gareSe da Zveli musikis jgufebis saSualoebi ar gansxvavdeboda mniSvnelovnad.

b. η2 = .77. 2. a.

instruqciis piroba videofiris

gareSe fexburTis videofiri

srolis, daWeris

videofiri

X s

45.22 5.63

44.33 5.74

57.11 6.68

realuriF (2,24) = 12.57, p < .01. tukis CD = 7.1. srolis, daWeris

videofiris yurebis saSualo mniSvnelovnad gansxvavdeba rogorc videofiris gareSe pirobis, iseve fexburTis videofiris pirobis saSualoebisagan. videofiris gareSe pirobisa da fexburTis videofiris pirobis saSualoebi mniSvnelovnad ar gansxvavdeba.

b. η2 = .51. 3. a.


X s

122.64 8.76

131.27 10.45

realuriF (1,20) = 4.41, p < .05. uxmuro jgufis saSualo mniSvnelovnad

dabalia xmauriani jgufis saSualoze.

b. ki, t = 41.4=F = 2.10.

4. a. a = 6, An = 7, N = 42. b. Adf = 5, Secdomadf = 36, srulidf = 41.

5. cxrili # 1


50.00 270.00 320.00

2 27 29

25.00 10.00

2.50

a. 3. b. 10. g. 30.

d. kritikuliF (2,27) = 3.35; realuriF aramniSvnelovania.

e. ver uarvyofT oH ; ver viRebT 1H .

cxrili # 2


60.00 216.00 276.00

5 72 77

12.00 3.00

4.00

a. 6. b. 13. g. 78.

d. kritikuliF (5,72) = 2.37; realuriF statistikurad mniSvnelovania.

e. uarvyofT oH ; viRebT 1H .

cxrili # 3


45.00 320.00 365.00

3 64 67

15.00 5.00

3.00

a. 4.

b. 17. g. 68.

d. kritikuliF (3,64) = 2.76; realuriF statistikurad mniSvnelovania.


7. realuriF (1,18) = 4.00.

8. realurit (22) = 3.00.

10. statistikurad mniSvnelovani realuriF (3,64) = 21.64 gviCvenebs, rom

arsebobs sul mcire erTi mniSvnelovani gansxvaveba jgufebs Soris. Tu wyvilTaSorisi SedarebebisaTvis CD = 5.0 wuTs gamoviyenebT, aRmoCndeba, rom qcevis Secvlas da sigaretebis anonimur klubs mniSvnelovnad didi saSualo drois periodebi aqvT sigaretebis mowevas Soris, vidre rogorc sakotrolo, aseve mowevis Sewyvetis saReWi rezinis jgufs. qcevis Secvlisa da sigaretebis anonimuri klubis saSualoebi erTmaneTisagan mniSvnelovnad ar gansxvavdeba, xolo sakontrolo jgufis da mowevis Sewyvetis saReWi rezinis jgufis saSualoebic aseve erTmaneTisagan mniSvnelovnad ar gansxvavdeba.

Tavi XI SeamowmeT Tqveni codna XI.1 2.

damoukidebeli cvladebis raodenoba

doneebi saWiro cdis pirebis

raodenoba a. b. g. d. e. v. z. T.

2 2 2 2 2 3 3 3

A = 3; B = 3 A = 3; B = 2 A = 2; B = 3 A = 6; B = 2 A = 4; B = 4

A = 2; B = 2, C = 2 A = 2; B = 4, C = 3 A = 3; B = 3, C = 2

90 60 60 120 160 80 240 180

3. cxrili # 1 faqtori A

A1 A2


B1 faqtori B

B2

13.3 20.2 16.8 22.8

16.8

19.8 mTavari efeqtis

saSualoebi A faqtorisaTvis

15.1 21.5

3.18=GX

cxrili # 2

faqtori A

A1 A2


B1 faqtori B

B2

18.9 30.1 31.3 33.9

24.5

32.6 mTavari efeqtis

saSualoebi A faqtorisaTvis

25.1 32.0

6.28=GX

SeamowmeT Tqveni codna XI.2

8. 240.00. 9. 45.00. 10. 55.00. 11. 60.00

12. .35,30,2,2,1 ===== sruliSecdoma dfdfdfdfdf AxBBA

13. .77,71,2,1,2 ===== sruliSecdoma dfdfdfdfdf AxBBA

14. 79.

15. 00.860/00.480

;00.222/00.44;00.132/00.26;00.141/00.14==

======

SecdomaMSMSMSMS AxBBA

16. a. wyaro SS df MS F A faqtori B faqtori A x B Secdoma sul

108.00 0.00 12.00 32.00 152.00

1 1 1 8 11

108.00 0.00 12.00 4.00

27.00 0.00 3.00

b. B faqtoris orive mTavari efeqtis saSualo udris 32.0.

17. cxrili # 1

wyaro SS df MS F A faqtori B faqtori A x B Secdoma sul

32.00 24.00 24.00 480.00 560.00

2 1 2 60 65

16.00 24.00 12.00 8.00

2.00 3.00 1.50

a. 3 b. 2. g. 11. d. 66.

cxrili # 2

wyaro SS df MS F A faqtori B faqtori A x B Secdoma sul

36.00 24.00 160.00 840.00 1060.00

2 4 8 210 224

18.00 6.00 20.00 4.00

4.50 1.50 5.00

a. 3

b. 5. g. 15. d. 225.


6. realuriF 1.00-ze meti xdeba.

7. realuriF 1.00-ze meti xdeba.

9. a. F Sefardeba: statistikuri hipoTezebi

A faqtorisaTvis H0 : µ A1 = µ A2;

H1 : µ A.ebi ar aris toli

B faqtorisaTvis H0 : µ B1 = µ B2 = µ B3 = µ B4 ;

H1 : µ B-ebi ar aris toli

A x B H0 : yvela (µAB - µ A - µ B + µ G) = 0 H1 : yvela (µAB - µ A - µ B + µ G) ar udris 0

b. F Sefardeba: statistikuri hipoTezebi

A faqtorisaTvis H0 : µ A1 = µ A2 = µ A3;


B faqtorisaTvis H0 : µ B1 = µ B2;


A x B H0 : yvela (µAB - µ A - µ B + µ G) = 0 H1 : yvela (µAB - µ A - µ B + µ G) ar udris 0

g. F Sefardeba: statistikuri hipoTezebi

A faqtorisaTvis H0 : µ A1 = µ A2 = µ A3;


B faqtorisaTvis H0 : µ B1 = µ B2 = µ B3;


A x B H0 : yvela (µAB - µ A - µ B + µ G) = 0

H1 : yvela (µAB - µ A - µ B + µ G) ar udris 0

10. cxrili # 1

wyaro df realuriF kritikuliF realuriF

xvdeba uaryofis areSi

gadawyvetileba

A faqtori B faqtori A x B

1 3 3

4.63

2.11

3.97

4.08

2.84

2.84

ki

ara ki

uarvyofT oH

viRebT 1H

ver uarvyofT oH

ver viRebT 1H

uarvyofT oH

viRebT 1H

cxrili # 2



gadawyvetileba


3 2 6

1.42

3.51

2.37

2.68

3.07

2.17

ara ki ki

ver uarvyofT oH

ver viRebT 1H

uarvyofT oH

viRebT 1H

uarvyofT oH

viRebT 1H

cxrili # 3



gadawyvetileba


2 4 8

4.12

2.83

1.87

3.07

2.45

2.02

ki ki

ara

uarvyofT oH

viRebT 1H

uarvyofT oH

viRebT 1H

ver uarvyofT oH

ver viRebT 1H


suraTi # 11.2 a. ki. b. ara. g. ki.

d. statistikurad mniSvnelovani urTierTqmedebis gamo swavlebis meTodis tipis efeqti damokidebulia masalis tipze, B faqtorze. maTematikuri masalis SemTxvevaSi (B1) swavlebis kompiuterizebul meTods (A1) mivyavarT testis mniSvnelovnad maRal qulebamde, vidre arakompiuterirebul meTods (A2). socialuri Sinaarsis masalis SemTxvevaSi ki (B2) swavlebis meTodebis erTmaneTisagan ar gansxvavdebian. e. statistikurad mniSvnelovani urTierTqmedeba miuTiTebs, rom masalis tipis efeqti damokidebulia swavlebis meTodze, A faqtorze. swavlebis kompiuterizebuli meTodis SemTxvevaSi (A1) maTematikuri Sinaarsis masala (B1) iZleva testis mniSvnelovnad maRal qulebs socialuri Sinaarsis masalasTan SedarebiT (B2). swavlebis arakompiuterizebuli meTodis SemTxvevaSi (A2) masalis tipebs Soris gansxvaveba ar aris. v. ara; A faqtoris mTavari efeqti miuTiTebs swavlebis kompiuterizebuli meTodis sayovelTao upiratesobaze, magram swavlebis kompiuterizebuli meTodi testis mniSvnelovnad maRal qulebs iZleva mxolod maTematikuri Sinaarsis masalis SemTxvevaSi. amdenad, A faqtoris mTavari efeqti urTierTqmedebis artefaqtia; is ar iZleva mniSvnelovan informacias A faqtoris efeqtis Sesaxeb testis qulebze.

suraTi # 11.2 a. ara. b. ki. g. ki.

d. statistikurad mniSvnelovani urTierTqmedeba miuTiTebs, rom swavlebis meTodis efeqti damokidebulia masalis tipze, B faqtorze. maTematikuri Sinaarsis masalisaTvis (B1) swavlebis meTodebi mniSvnelovnad ar gansxvavdebian. socialuri Sinaarsis masalisaTvis (B2) swavlebis arakompiuterizebuli meTodi (A2) iZleva testis mniSvnelovnad maRal qulebs, vidre swavlebis kompiuterizebuli meTodi (A1). e. statistikurad mniSvnelovani urTierTqmedeba miuTiTebs, rom masalis tipis efeqti damokidebulia swavlebis meTodze, A faqtorze. swavlebis orive meTodi testis ufro maRal qulebs iZleva maTematikuri Sinaarsis masalisaTvis, magram gansxvaveba maTematikuri da socialuri Sinaarsis masalebs Soris ufro didia swavlebis kompiuterizebuli meTodis SemTxvevaSi, vidre swavlebis arakompiuterizebuli meTodis SemTxvevaSi. v. ki; B faqtoris mTavari efeqti miuTiTebs maTematikuri Sinaarsis masalis sayovelTao upiratesobaze da maTematikuri Sinaarsis masala testis mniSvnelovnad maRal qulebs iZleva swavlebis orive meTodisaTvis. Tumca, urTierTqmedeba migviTiTebs, rom

gansxvaveba maTematikuri da socialuri Sinaarsis masalebs Soris damokidebulia swavlebis meTodis tipze.

2. cxrili # 1 a. ki. b. ara. g. ara. d. swavlebis kompiuterizebuli meTodi (A1) testis mniSvnelovnad

maRal qulebs iZleva, vidre arakompiuterizebuli swavleba (A2). e. efeqti ar aris. v. ki, A da B faqtorebs Soris ar aris urTierTqmedeba. cxrili # 2 a. ara. b. ki. g. ara. d. efeqti ar aris. e. maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal

qulebs iZleva, vidre socialuri Sinaarsis masala (B2). v. ki, A da B faqtorebs Soris ar aris urTierTqmedeba. cxrili # 3 a. ki. b. ki. g. ara. d. swavlebis arakompiuterizebuli meTodi (A2) testis

mniSvnelovnad maRal qulebs iZleva, vidre kompiuterizebuli swavleba (A1).

e. maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal qulebs iZleva, vidre socialuri Sinaarsis masala (B2).

v. ki, A da B faqtorebs Soris ar aris urTierTqmedeba. cxrili # 4 a. ara. b. ara. g. ki. d. statistikurad mniSvnelovani urTierTqmedebis gamo swavlebis

meTodis efeqti damokidebulia masalis tipze, B faqtorze. maTematikuri Sinaarsis masalisaTvis (B1) swavlebis kompiuterizebuli meTodi (A1) testis mniSvnelovnad maRal qulebs iZleva, vidre swavlebis arakompiuterizebuli meTodi (A2). Tumca, socialuri Sinaarsis masalisaTvis (B2) arakompiuterizebuli meTodi (A2) testis mniSvnelovnad maRal qulebs iZleva, vidre kompiuterizebuli meTodi (A1).

e. statistikurad mniSvnelovani urTierTqmedebis gamo masalis tipis efeqti damokidebulia swavlebis meTodze, A faqtorze. v. ki, A da B faqtorebs Soris ar aris urTierTqmedeba. swavlebis kompiuterizebuli meTodisTvis (A1) maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal qulebs iZleva, vidre socialuri Sinaarsis masala. swavlebis arakompiuterizebuli meTodisTvis (A2)

socialuri Sinaarsis masala (B2) testis mniSvnelovnad maRal qulebs iZleva, vidre maTematikuri Sinaarsis masala. v. ara; TiToeuli damoukidebeli cvladis mTavari efeqti ar miuTiTebs damoukidebeli cvladis sayovelTao efeqtze. miuxedavad imisa, rom orive damoukidebel cvlads aqvs efeqti, TiToeuli damoukidebeli cvladis efeqti damokidebulia meore damoukidebeli cvladis doneze.

cxrili # 5 a. ki. b. ki. g. ki. d. statistikurad mniSvnelovani urTierTqmedebis gamo swavlebis

meTodis tipis efeqti damokidebulia masalis tipze. maTematikuri Sinaarsis masalisaTvis (B1) gansxvaveba kompiuterizebul (A1) da arakompiuterizebul (A2) swavlebis meTodebs Soris aramniSvnelovania. Tumca, socialuri Sinaarsis masalisaTvis arakompiuterizebuli swavlebis meTodi (A2) testis mniSvnelovnad maRal qulebs iZleva, vidre kompiuterizebuli swavlebis meTodi (A1).

e. statistikurad mniSvnelovani urTierTqmedebis gamo masalis tipis efeqti damokidebulia swavlebis meTodze. swavlebis rogorc kompiuterizebuli (A1), ise arakompiuterizebuli (A2) meTodebisaTvis maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal qulebs iZleva, vidre socialuri Sinaarsis masala (B2). Tumca, gansxvaveba maTematikuri da socialuri Sinaarsis masalebis qulebs Soris ufro didia kompiuterizebuli meTodis SemTxvevaSi (A1), vidre arakompiuterizebuli meTodis SemTxvevaSi (A2). v. A faqtoris - swavlebis meTodis - mTavari efeqti artefaqturia, radganac is miuTiTebs swavlebis arakompiuterizebuli meTodis sayovelTao upiratesobaze. magram mwavlebis arakompiuterizebuli meTodi mniSvnelovnad ukeTes Sesrulebas iZleva mxolod socialuri Sinaarsis masalis SemTxvevaSi. B faqtoris - masalis tipis - mTavari efeqti mniSvnelovania; maTematikuri Sinaarsis masala mniSvnelovnad ukeTes Sesrulebas iZleva rogorc kompiuterizebuli, iseve arakompiuterizebuli swavlebis meTodis SemTxvevaSi.

cxrili # 6 a. ki. b. ki. g. ki. d. statistikurad mniSvnelovani urTierTqmedebis gamo swavlebis

meTodis tipis efeqti damokidebulia masalis tipze. maTematikuri Sinaarsis masalisaTvis (B1) swavlebis kompiuterizebuli meTodi (A1) testis mniSvnelovnad maRal qulebs iZleva, vidre swavlebis arakompiuterizebuli meTodi (A2). xolo socialuri Sinaarsis masalisaTvis ar arsebobs swavlebis meTodis efeqti.

e. masalis tipis efeqti damokidebulia swavlebis meTodze. swavlebis kompiuterizebuli meTodisaTvis (A1) maTematikuri Sinaarsis masala (B1) testis mniSvnelovnad maRal qulebs iZleva, vidre socialuri Sinaarsis masala (B2). swavlebis arakompiuterizebuli meTodisaTvis (A2) gansxvaveba maTematikuri (B1) da socialuri (B2) Sinaarsis masalebis mier mocemul qulebs Soris aramniSvnelovania. v. rogorc A faqtoris - swavlebis meTodis, iseve B faqtoris - masalis tipis mTavari efeqtebi artefaqturia.

SeamowmeT Tqveni codna XI.5 1. dispersiuli analizis Sejameba: wyaro SS df MS F lokus-kontroli (A) instruqciebi (B) AxB Secdoma sul

704.167 1,472.667 46,112.666 12,233.000 60,522.500

1 1 1 20 23

704.167 1,472.667 46,112.666 611.650

1.15 2.41 75.39*

* p < .01 a. (1) arc erTi damoukidebeli cvladisaTvis ar arsebobs

mniSvnelovani mTavari efeqti. (2) urTierTqmedeba mniSvnelovania. b. (CD = 36.15) statistikurad mniSvnelovani urTierTqmedeba

miuTiTebs, rom instruqciebis efeqti damokidebulia kontrolis lokusze. kontrolis Sida lokusis mqone cdis pirebisaTvis unarebis instruqtirebam mogvca mniSvnelovnad meti dro, romelic daixarja davalebis Sesrulebaze, vidre SemTxveviTobis instruqtirebam. xolo kontrolis gare lokusis mqone cdis pirebisaTvis SemTxveviTobis instruqtirebam mogvca mniSvnelovnad meti dro, romelic daixarja davalebis Sesrulebaze, vidre unarebis instruqtirebam.

g. urTierTqmedebis gamo aramniSvnelovani mTavari efeqtebi ar gvawvdian gamosadeg informacias Sedegebis axsnisaTvis.

d. η2 = .76 A da B faqtorebis urTierTqmedebisaTvis. 2. dispersiuli analizis Sejameba: wyaro SS df MS F instruqcia (A) danaSauli (B) AxB Secdoma sul

191.361 1667.361 210.250 259.333 2328.305

1 1 1 32 35

191.361 1667.361 210.250 8.104

23.61* 205.75* 25.94*

* p < .01 a. (1) arsebobs mniSvnelovani mTavari efeqti rogorc A, iseve B

faqtorisaTvis. (2) urTierTqmedeba mniSvnelovania. b. (CD = 3.31) statistikurad mniSvnelovani urTierTqmedeba

miuTiTebs, rom instruqciebis tipis efeqti damokidebulia

danaSaulis tipze; gansxvaveba folis saxiT micemul da normalur instruqciebs Soris manqanis moparvisaTvis aramniSvnelovania, magram fizikuri Zaladobis gamoyenebiT TavdasxmisaTvis mniSvnelovnad grZeli patimrobis vada aris miCenili filmis saxiT instruqciis SemTxvevaSi, vidre normaluri instruqciis SemTxvevaSi.

g. statistikurad mniSvnelovani urTierTqmedeba miuTiTebs, rom danaSaulis tipis efeqti damokidebulia instruqciebis tipze. orive tipis instruqciisaTvis Tavdasxma fizikuri Zaladobis gamoyenebiT iZleva miCenili patimrobis mniSvnelovnad grZel vadas, vidre manqanis moparva. magram gansxvaveba patimrobis miCenil vadebSi manqanis moparvasa da Tavdasxmas Soris ufro didia filmis saxiT instruqtaJis SemTxvevaSi, vidre normaluri instruqtaJis SemTxvevaSi.

d. danaSaulis tipis statistikurad mniSvnelovani mTavari efeqti mniSvnelobis mqonea; mniSvnelovnad ufro grZeli patimrobebia misjili TavdaxsmisaTvis rogorc video, iseve normaluri instruqtaJis pirobebSi. instruqciis tipis mTavari efeqti artefqturia; ar arsebobs gansxvaveba video da normalur instruqciebs Soris manqanis moparvisaTvis; magram mniSvnelovnad ufro grZeli patimrobaa misjili Tavdasxmis videoinstruqtaJis SemTxvevaSi.

e. η2 = .72 danaSaulis tipisaTvis.

v. η2 = .09 A da B faqtorebis urTierTqmedebisaTvis. 3.1. a. 3. b. 2. g. 42. d.

kritikuliF

A B

AxB

3.32 4.17 73.32

e. ki. v. ki. z. ara. 3.2. a. 2. b. 4. g. 96. d.

kritikuliF

A B

AxB

4.00 2.76 2.76

e. ki. v. ara. z. ki. 3.3. a. 2. b. 2.

g. 60. d.

kritikuliF

A B

AxB

4.08 4.08 4.08

e. ara. v. ara. z. ki. kiTxvebi 1. dispersiuli analizis Sejameba. wyaro SS df MS F mxedvelobiTi kontaqti (A) emocia (B) AxB Secdoma sul

24.500 780.125 325.125 903.750 2033.500

1 1 1 28 31

24.500 780.125 325.125 32.277

0.76 24.17* 10.07*

* p < .01 a. ki; xdeba A da B faqtorebis mniSvnelovan urTierTqmedeba (tukis

CD = 7.1). dadebiTi emociis pirobisaTvis mxedvelobiTi kontaqtis ararsebobis saSualo mniSvnelovnad maRalia, vidre mxedvelobiTi kontaqtis damyarebis saSualo. uaryofiTi emociis pirobisaTvis mxedvelobiTi kontaqtis pirobis saSualoebi mniSvnelovnad ar gansxvavdeba.

b. mxedvelobiTi kontaqtis ararsebobis pirobisaTvis ar arsebobs mniSvnelovani gansxvaveba dadebiTi da uaryofiTi emociebis pirobebs Soris. xolo mxedvelobiTi kontaqtis damyarebis pirobisaTvis uaryofiTi emocia iZleva mniSvnelovnad ufro maRal guliscemis sixSires, vidre dadebiTi emocia.

g. emociis pirobis statistikurad mniSvnelovani mTavari efeqti artefaqturia; ar arsebobs mniSvnelovani gansxvaveba emociis pirobebs Soris mxedvelobiTi kontaqtis ararsebobis pirobaSi, magram mxedvelobiTi kontaqtis damyarebis pirobaSi uaryofiTi emociis ujredis saSualo mniSvnelovnad maRalia, vidre dadebiTi emociis ujredis saSualo.

d. η2 = .16 A da B faqtorebis urTierTqmedebisaTvis. 2. dispersiuli analizis Sejameba. wyaro SS df MS F pirovnebis tipi (A) xmauris foni (B) AxB Secdoma sul

36.750 30.083 27.000 245.833 339.666

1 1 1 44 47

36.750 30.083 27.000 5.587

6.58* 5.38* 4.83*

* p < .05 a. dispersiuli analizi avlens statistikurad mniSvnelovan

urTierTqmedebas; ase rom, saWiroa tukis CD martivi

efeqtebisaTvis. es CD 3,44 df-isaTvis aris 2.35. martivi efeqtebis analizi gviCvenebs, rom introvertebis saSualo xmamaRali fonis pirobaSi (M = 8.2 swor debulebas) mniSvnelovnad dabalia, vidre introvertebis saSualo xmadabali fonis pirobaSi (M = 11.3 swor debulebas) da aseve mniSvnelovnad dabalia, vidre eqstravertebis saSualo xmamaRali fonis pirobaSi (M = 11.4 swor debulebas). introvertebis saSualo xmadabali fonis pirobaSi mniSvnelovnad ar gansxvavdeboda eqstravertebis saSualosagan xmadabali fonis pirobaSi (M = 11.5 swor debulebas) da eqstravertebis saSualo xmadabali fonis pirobaSi mniSvnelovnad ar gansxvavdeboda eqstravertebis saSualosagan xmamaRali fonis pirobaSi.

b. η2 = .08. g. orive mTavari efeqti statistikurad mniSvnelovania, magram

artefaqturia da maTi mniSvnelovnad axsna SeuZlebelia. 3. dispersiuli analizis Sejameba. wyaro SS df MS F instruqciis meTodi (A) varjiSi (B) AxB Secdoma sul

436.594 60.168 69.777 219.332 785.871

2 1 2 48 53

436.594 60.168 43.888 4.569

47.78* 13.17* 7.64*

* p < .01

a. xdeba statistikurad mniSvnelovani urTierTqmedeba; amdenad, instruqciis meTodis efeqti damokidebulia varjiSis raodenobaze. tradiciuli meTodisTvis erTkviriani varjiSis saSualo (M = 13.1 swor sityvas) mniSvnelovnad naklebi iyo, vidre eqvskviriani varjiSis saSualo (M = 17.9 swor sityvas; CD = 2.9). miuxedavad amisa, ar aRmoCnda mniSvnelovani gansxvaveba erTkvirian da eqvskvirian varjiSis pirobebs Soris rogorc sityvebis yuTis (Sesabamisad, M = 18.7 swor sityvas; M = 21.0 swor sityvas), aseve sityvebis dajgufebis meTodisTvis (Sesabamisad, M = 22.8 swor sityvas; M = 22.0 swor sityvas). garda amisa, gansxvavebebi instruqciis meTodebs Soris damokidebulia varjiSis raodenobaze. erTkviriani varjiSis pirobisaTvis intruqciis meTodis yoveli jgufi sxva jgufebisagan mniSvnelovnad gansxvavdeboda; sityvebis dajgufebis meTodis cdis pirebma yvelaze maRali qulebi miiRes, sityvebis yuTis meTodis cdis pirebma - momdevno maRali qulebi da tradiciuli meTodis cdis pirebma - yvelaze dabali qulebi. eqvskviriani varjiSis pirobisaTvis, tradiciuli meTodis saSualo mniSvnelovnad naklebi iyo rogorc sityvebis yuTis, iseve sityvebis dajgufebis saSualoebze. Tumca, sityvebis yuTis da sityvebis dajgufebis saSualoebi mniSvnelovnad ar gansxvavdeboda.

b. η2 = .09. g. orive mTavari efeqti artefaqturia.

4. a = 2, b = 3, .47,42,2,2,1,48,8 ======= sruliSecdoma dfdfdfdfdfNn AxBBAAB

5. cxrili # 1

wyaro SS df MS F davalebis tipi (A) xmauris done (B) AxB Secdoma sul

15.00 16.00 36.00 120.00 187.00

3 2 6 60 71

5.00 8.00 6.00 2.00

2.50 4.00* 3.00*

* p < .05. a. 4. b. 3. g. 6. d. 72.

e. realuriF B faqtorisaTvis da A da B faqtorebis urTierTqmedebisaTvis statistikurad mniSvnelovania.

v.

A faqtori: B faqtori A x B

ver uarvyofT oH

ver viRebT 1H

uarvyofT oH

viRebT 1H

uarvyofT oH

viRebT 1H

cxrili # 2 wyaro SS df MS F treningis xangrZlivoba (A) unaris done (B) AxB Secdoma sul

40.00 120.00 960.00 3200.00 4320.00

4 3 12 160 179

10.00 40.00 80.00 20.00

0.50 2.00 4.00*

* p < .01. a. 5. b. 4. g. 9. d. 180.

e. realuriF A da B faqtorebis urTierTqmedebisaTvis statistikurad

mniSvnelovania. v.


ver uarvyofT oH

ver viRebT 1H

ver uarvyofT oH

ver viRebT 1H

uarvyofT oH

viRebT 1H

cxrili # 3 wyaro SS df MS F ukukavSiris tipi (A) davalebis sirTule (B) AxB Secdoma sul

200.00 160.00 240.00 2700.00 3300.00

4 2 8 135 149

50.00 80.00 30.00 20.00

2.50* 4.00* 1.50

* p < .05. a. 5. b. 3. g. 10. d. 150.

e. realuriF A faqtorisaTvis da B faqtorisaTvis statistikurad mniSvnelovania.

v.


uarvyofT oH

viRebT 1H

uarvyofT oH

viRebT 1H

ver uarvyofT oH

ver viRebT 1H

6. a. statistikurad mniSvnelovani urTierTqmedebis gamo xatis tipis efeqti damokidebulia damuSavebis doneze. damuSavebis dabali donisaTvis xatis Cveulebrivi tipi iwvevs aRdgenis mniSvnelovnad ukeTes Sedegebs, vidre xatis ucnauri tipi. xolo damuSavebis maRali donisaTvis xatis Cveulebrivi tipi iwvevs mniSvnelovnad uares aRdgenas, vidre xatis ucnauri tipi. xatis tipis mTavari efeqti artefaqturia.

b. aqac, statistikurad mniSvnelovani urTierTqmedebis gamo, damuSavebis donis efeqti damokidebulia xatebis tipze. Cveulebrivi xatebisaTvis dabali damuSaveba iwvevs mniSvnelovnad ukeTes aRdgenas, vidre maRali damuSaveba. ucnauri xatebisaTvis aris sawinaaRmdego kavSiri; dabali damuSaveba iwvevs mniSvnelovnad ufro dabal aRdgenas, vidre maRali damuSaveba.

7. a. statistikurad mniSvnelovani urTierTqmedeba migviTiTebs, rom

xelsawyos tipis efeqti damokidebulia Zilis deprivaciis raodenobaze. rodesac Zilis deprivacia ar xdeba, sxvaoba swor wakiTxvebs Soris analoguri da cifruli xelsawyoebisaTvis aramniSvnelovania. Zilis 24-saaTiani deprivaciisaTvis cifruli xelsawyo iZleva mniSvnelovnad met swor wakiTxvas, vidre analoguri xelsawyo. xelsawyos tipis mTavari efeqti artefaqturia.

b. aqac, statistikurad mniSvnelovani urTierTqmedeba migviTiTebs, rom Zilis deprivaciis raodenobis efeqti damokidebulia

xelsawyos tipze. rogorc analoguri, iseve cifruli xelsawyoebisaTvis Zilis 24-saaTiani deprivacia Sedegad iZleva ufro nakleb swor wakiTxvas, vidre Zilis deprivaciis ararseboba. Tumca, sxvaoba swori wakiTxvebis raodenobebs Soris deprivaciis ararsebobasa da 24-saaTiani deprivaciis pirobebisaTvis ufro didia analoguri xelsawyosaTvis, vidre cifruli xelsawyosaTvis. radganac Zilis 24-saaTiani deprivacia Sedegad iZleva ufro nakleb swor wakiTxvebs rogorc analoguri, iseve cifruli xelsawyosaTvis, Zilis deprivaciis raodenobis mTavari efeqti mniSvnelovania.

Tavi XII SeamowmeT Tqveni codna XII.1 4. 199.00. 5. 33.00. 6. 25.00. 7. 96.00.

8. .64,48,12,4 ==== srulidfdfdfdf AxSSA

9. 59.

10. .00.2,00.2,00.11 === AxSSA MSMSMS

11. cxrili # 1 wyaro SS df MS F A faqtori S faqtori AxS sul

48.00 27.00 72.00 147.00

4 9 36 49

12.00 3.00 2.00

6.00

a. 5. b. 10. g. 10. cxrili # 2 wyaro SS df MS F A faqtori S faqtori AxS sul

6.00 36.00 36.00 78.00

2 12 24 38

3.00 3.00 1.50

2.00

a. 3. b. 13. g. 13. SeamowmeT Tqveni codna XII.2

3. ;543210 AAAAAH µµµµµ =====

=1H yvela Aµ -ebi ar aris erTmaneTis toli.

SeamowmeT Tqveni codna XII.3 2.

cdis piri

qalaqi (A) dalasi tihuana tripoli

X s

30.8 4.14

21.6 3.15

22.6 3.55

a. ;3210 AAAH µµµ ===

=1H yvela Aµ -ebi ar aris erTmaneTis toli. b.

wyaro SS df MS F qalaqi (A) cdis pirebi (S) AxS sul

709.000 269.333 245.667 1224.000

2 13 26 41

354.500 20.718 9.449

37.52*

* p <.01. g. 2,26.

d. .37.3)26,2( =kritikuliF

e. realurisF mniSvnelobebi meti an toli 3.37-ze.

v. ki.


T. gansxvavebuli populaciebidan. i. ki.

k. ki, radganac realuriF statistikurad mniSvnelovania da icvleba

damoukidebeli cvladis sami done. l. tukis CD = 2.90. m. dadgenili ganedis saSualo dalasisaTvis mniSvnelovnad metia,

vidre saSualoebis tihuanasaTvis da tripolisTvis. tihuanas da tripolis saSualoebi mniSvnelovnad ar gansxvavdeba.

n. .74.2 =η 3. a. 2. b. 19. g. 4.76. d. 4.41.

e. uarvyofT oH .

v. viRebT 1H . z. ki. T. martivi da rTuli Carevis pirobebi. 4. a. 3. b. 12. g. 2.81. d. 3.44.

e. ver uarvyofT oH .

v. ver viRebT 1H .

z. ara. T. Carevis pirobebi mniSvnelovnad ar gansxvavdeba. kiTxvebi 1. a.

cdis piri

sityvis tipi bednieri neitraluri mowyenili

X s

4.5 1.41

3.0 1.20

2.5 1.20

wyaro SS df MS F sityvis tipi (A) cdis pirebi (S) AxS sul

17.333 16.667 17.333 51.333

2 7 14 23

8.666 2.381 1.238

7.00*

* p <.05.

tukis CD = 1.5. bednieri sityvebis saSualo mniSvnelovnad didia, vidre rogorc neitraluri, ise mowyenili sityvebis saSualoebi. neitraluri da mowyenili sityvebis saSualoebi erTmaneTisagan mniSvnloevnad ar gansxvavdeba.

b. .50.2 =η 2.

instruqciis piroba instruqciamde instruqciis

Semdeg

X s

2.45 0.60

2.52 0.68

a. wyaro SS df MS F instruqciebi (A) cdis pirebi (S) AxS sul

0.027 8.843 0.203 9.073

1 11 11 23

0.027 0.804 0.018

1.50

realuriF aramniSvnelovania .05 doneze. ar arsebobs imis

TvalsaCinoeba, rom instruqciebma gavlena moaxdina aRqmul iluziaze.

b. .255.150.1.261.1)11( === Ftkritikuli (mniSvnelobebis mcire gansxvaveba

t-sa da F-s Soris gamoTvlebSi damrgvalebebis Secdomis gamoa.) g. erTsa da imave daskvnebs.

3. a. ;543210 AAAAAH µµµµµ =====

=1H yvela Aµ -ebi ar aris erTmaneTis toli.

b. .59,44,11,4 ==== srulidfdfdfdf AxSSA

g. .61.2)44,4( =kritikuliF

d. uarvyofT oH ; viRebT 1H .

e. arsebobs sul mcire erTi mniSvnelovani sxvaoba saSualoebs Soris. tukis CD kriteriumi aucilebelia imisaTvis, raTa ganisazRvros specifikuri mniSvnelovani sxvaobebi.

4. cxrili # 1


10.668 69.834 27.332 107.834

1 11 11 23

10.668 6.349 2.485

4.29

a. 3. b. 24. g. 24. d. ara.

e. ver uarvyofT oH ; ar viRebT 1H .

cxrili # 2


60.00 133.00 380.00 573.00

5 19 95 119

12.0 7.0 4.0

3.00

a. 6. b. 20. g. 20. d. ki.


5. statistikurad mniSvnelovani realuriF (2,26) = 15.99 migviTiTebs, rom

arsebobs sul mcire erTi mniSvnelovani sxvaoba saSualoebs Soris. wyvilTaSorisi Sedarebebi tukis CD-s gamoyenebiT gviCvenebs, rom yoveli saSualo mniSvnelovnad gansxvavdeba yoveli sxva saSualosagan. aqedan gamomdinare, Cvilebi yvelaze met dros nacnob saxeebs akvirdebodnen. ucnob saxeze fiqsaciis xangrZlivoba mniSvnelovnad naklebi iyo, vidre nacnob saxeze, xolo cariel ovalebze fiqsaciis xangrZlivoba mniSvnelovnad naklebi iyo, vidre nacnob da ucnob saxeebze. Tavi XIII SeamowmeT Tqveni codna XIII.1 2. ara, or qulas Soris damokidebuleba ar icvleba. 3. cxrili # 1 a. dadebiTi kavSiri. b. srulyofilze naklebi. g. ki. martoobis qulebis codna depresiis ukeTesad

winaswarmetyvelebis saSualebas iZleva, vidre am qulebis arcodna. cxrili # 2 a. uaryofiTi kavSiri. b. srulyofilze naklebi. g. ki. 4. a. susti uaryofiTi kavSiri.

b. srulyofilze naklebi. g. ara, es ar gazrdis wianswarmetyvelebis sizustes. SeamowmeT Tqveni codna XIII.2

3. a. .870.+=realurir

4. a. .764.−=realurir


SeamowmeT Tqveni codna XIII.3 4.

2r a. b. g. d. e. v.

.0144

.5476 .3481 .1296 .0529 .0361


2. .0;0:0 ≠= ρρH

3. kritikuli mniSvneloba df .05 .01 a. b. g. d. e. v.

6 13 18 30 41 73

.707 .514 .444 .349 .304 .232

.834

.641

.561 .449 .393 .302

4.


kritikuli mniSvneloba

xvdeba uaryofis

areSi

gadawyvetileba

a.

0:0:

1

0

≠=

ρρ

HH

.576 ki

0H : uarvyofT

1H : viRebT

b.

0:0:

1

0

≠=

ρρ

HH

.576 ara

0H : ver uarvyofT

1H : ver viRebT

g.

0:0:

1

0

≠=

ρρ

HH

.444 ki

0H : uarvyofT

1H : viRebT

d.

0:0:

1

0

≠=

ρρ

HH

.423 ara

0H : ver uarvyofT

1H : ver viRebT

e.

0:0:

1

0

≠=

ρρ

HH

.232 ki

0H : uarvyofT

1H : viRebT

v.

0:0:

1

0

≠=

ρρ

HH

.666 ki

0H : uarvyofT

1H : viRebT

5. a. 550.−=realurir

b. realurikritikuli rr ;576.)10( = aramniSvnelovania.

g. ar dasturdeba hipoTeza, rom im populaciaSi, romlidanac aRebulia es qulebi, arsebobs kavSiri gacdenebsa da testis qulebs Soris.

6. statistikuri

hipoTezebi kritikuli mniSvneloba

xvdeba uaryofis

areSi

gadawyvetileba

a.

0:0:

1

0

≠=

SHH

ρρ

.490 ara

0H : ver uarvyofT

1H : ar viRebT

b.

0:0:

1

0

≠=

SHH

ρρ

.591 ki

0H : uarvyofT

1H : viRebT

g.

0:0:

1

0

≠=

SHH

ρρ

.409 ki

0H : uarvyofT

1H : viRebT

d.

0:0:

1

0

≠=

SHH

ρρ

.648 ara

0H : ver uarvyofT

1H : ver viRebT

e.

0:0:

1

0

≠=

SHH

ρρ

.438 ki

0H : uarvyofT

1H : viRebT

v.

0:0:

1

0

≠=

SHH

ρρ

.364 ki

0H : uarvyofT

1H : viRebT

kiTxvebi 1. b. dadebiTi.

g. realurikritikulirealuri rrr ;497.)14(;655. =+= statistikurad

mniSvnelovania. d. arsebobs dadebiTi kavSiri sibrazis donesa da sistolur

arteriul wnevas Soris. rac ufro maRalia sibrazis done, miT ufro maRalia sistoluri arteriuli wneva.

e. .43.2 =r

2. a. 12.−=realurir gogonebisaTvis; realurikritikuli rr ;273.)51( =

aramniSvnelovania. 32.+=realurir vaJebisaTvis;

realurikritikuli rr ;304.)40( = statistikurad mniSvnelovania.

b. 01.2 =r gogonebisaTvis. 10.2 =r vaJebisaTvis. g. gogonebisaTvis ar arsebobs kavSiri. vaJebisaTvis kavSiri dadebiTia; maRal biWebs aqvT tendencia, miiRon ufro maRali qulebi angariSis baraTebze.

3. b. dadebiTi.

g. realurikritikulirealuri rrr ;632.)8(;781. =+= statistikurad mniSvnelovania.

d. isini dadebiTad aris dakavSirebuli; rac ufro maRalia niSani semestris SuaSi, miT ufro maRali iqneba saboloo gamocdis niSani.

4. a. realurikritikulirealuri ss rrr ;545.)14(;937. =+= statistikurad

mniSvnelovania. b. popularoba da fizikuri mimzidveloba dadebiT kavSirSia im

gogonebis populaciaSi, saidanac aRebuli iyo SerCeva.

g. realurikritikulirealuri ss rrr ;545.)14(;134. =−= aramniSvnelovania.

d. ar arsebobs kavSiri vaJebis popularobasa da fizikur mimzidvelobas Soris im populaciaSi, saidanac aRebuli iyo SerCeva.

5. kolejis studentebis inteleqtis koeficienti moicavs cvladis SezRudul diapazons. inteleqtis koeficientis Zalian dabali da Zalian maRali qulebi ar Sedioda SerCevaSi. SesaZlebelia aseve, rom niSnebis saSualoebi msgavsad aris SezRuduli diapazonSi. 6. b. ar aris kavSiri.

g. realurikritikulirealuri ss rrr ;553.)11(;001. =−= aramniSvnelovania.

d. kompiuteris gamoyenebis saaTebis raodenoba da siRrmis aRqmis Secdoma ar aris dakavSirebuli gazomil SerCevaSi.

7. b. kavSiri, rogorc Cans, uaryofiTia.

g. realurikritikulirealuri ss rrr ;514.)13(;807. =−= statistikurad

mniSvnelovania. d. dRis uaryofiTi movlenebis raodenoba da skolisgan miRebuli

kmayofileba uaryofiTadaa dakavSirebuli. rac ufro metia yoveldRiuri uaryofiTi movlenebi, miT ufro dabalia skolisgan miRebuli kmayofileba.


b. realurikritikuli rr ;482.)15( = statistikurad mniSvnelovania.

g. uaryofiTi kavSiri gviCvenebs, rom X cvladis ufro maRali qulebi dakavSirebulia Y cvladis ufro dabal qulebTan. amdenad, Tqven ufro dabal niSnebs elodebiT im megobrisagan, romelic kviraSi 15 saaTis ganmavlobaSi uyurebs televizors, vidre megobrisagan, romelic 4 saaTs uyurebs televizors kviris manZilze.

9. a. .847.+=realurir

b. realurikritikuli rr ;507.)16( = statistikurad mniSvnelovania.

g. arsebobs dadebiTi korelacia mowonebis rangsa da monawileobis rangs Soris. rac ufro maRalia mowonebis rangi, miT ufro maRalia monawileobis rangi.

Tavi XIV SeamowmeT Tqveni codna XIV.1 4. a. b = +2; a = 0; Y = +2X.

b. b = -1; a = +2; Y = -1X+2. g. b = +1.5; a = +1; Y = +1.5X+1. d. b = -2; a = +4; Y = -2X+4. SeamowmeT Tqveni codna XIV.2

3. a. 98.0 ++= XY l b.

X lY 60 65 80 85 90

57 61 73 77 81

g. Y =69.8. regresiis wrfe gadis wertilebs X = 0, Y = 9; .8.69,76 == YX

4. a. 234.1 +−= XY l b.

X lY 5 8 11 14 17

16.0 11.8 7.6 3.4 -0.8

g. Y =7.6. regresiis wrfe gadis wertilebs X = 0, Y = 23; .6.7,11 == YX

SeamowmeT Tqveni codna XIV.3 5. 4.98. 6. 4.80. 7. 4.30. SeamowmeT Tqveni codna XIV.4 1. a. +11.507. b. 202.215.

g. .215.202507.11 ++= XY l d. regresiis wrfe gadis wertilebs X = 0, Y = 202.215;

.7.261,17.5 == YX e.

X lY 6 3 9 1 4 2 3

271.3 236.7 305.8 213.7 248.2 225.2 236.7

5 8 7 8 6

259.8 294.3 282.8 294.3 271.3

v.

cdis piri

naSTi

(Y- lY ) 1 2 3 4 5 6 7 8 9 10 11 12

-54.3 -6.7 +9.2 +10.3 +22.8 -27.2 +19.3 +3.2 +26.7 +6.2 -29.3 +19.7

z. 6824.43. T. 26.12.

i. ki; XYs • naklebia Ys -ze. 2. a. b = -0.43. b. a = 24.6. g. lY =-0.43X+24.6. d. 41 skanireba, lY = 7.0.

e. 18 skanireba, lY = 16.9. kiTxvebi 1. a. +0.884. b. +0.884.

g. lY =+0.884X + 10.96. d. 79.0. e. 5.52.

2. a. lY =-0.116X + 3.23. b. 2.88. g. 2.64.

3. a. lY =+-4.3X + 128, sadac X = dReSi moweuli sigaretebis raodenobas. g.

sigaretebi dReSi

nawinaswarmetyvelebi wona dabadebisas (oz)

0 1 5 8 10 13

128.0 123.7 106.5 93.6 85.0 72.1

20 42.0

4. a. lY =+0.786X + 107.16. b.

cdis piri

nawinaswarmetyvelebi arteriuli wneva

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

157.5 130.7 136.2 137.8 120.5 130.0 135.5 133.9 141.7 132.3 136.2 144.1 146.5 139.4 143.3 155.1

d. XYs • aris Ys -is .78. 5. a. –0.562. b. 28.33.

g. lY =-0.562X + 28.33. e. 2.90. 6.

aplikanti nawinaswarmetyvelebi Sesruleba

samsaxurSi: lY

diana rodrigo kolini maria rubeni luki holi bliTi

55.2 61.2 54.0 67.2 64.8 52.8 50.4 70.8

rodrigos, marias, rubenis da bliTis samsaxurSi Sesrulebis nawinaswarmetyvelebi qulebi 60-ze metia. amdenad, am aplikantebs samsaxurSi aiyvanen.

7. a. lY = -1.317X + 89.17.

b. lY 4 saaTisTvis = 83.9; lY 15 saaTisTvis = 69.4.

g. XYs • =8.01; XYs • aris Ys -is .70. Tavi XV

SeamowmeT Tqveni codna XV.1 7.

2realuriχ

df 2kritikuliχ xvdeba

uaryofis areSi

gadawyvetilebebi

7.26

8.33

14.19

26.94

2 4 7 26

5.99

9.49

14.1

31.4

ki

ara

ki

ara

oH : uarvyofT

1H : viRebT

oH : ver uarvyofT

1H : ar viRebT

oH : uarvyofT

1H : viRebT

oH : ver uarvyofT

1H : ar viRebT

8. TiToeul ujredSi mocemuli mniSvnelobebi mosalodneli sixSireebia. a.

sveti 1 2

1 12.6 31.4 striqoni

2 15.4 38.6

2realuriχ (1) = 0.073, p>.05;

aramniSvnelovani. b.

sveti 1 2


2 24.8 38.2

2realuriχ (1) = 9.256, p<.01;

statistikurad mniSvnelovani. g.

sveti 1 2


2 17.6 23.4

2realuriχ (1) = 12.442, p<.01;

statistikurad mniSvnelovani.

9. 2realuriχ (2) = 0.168. es 2

realuriχ aramniSvnelovania; am magaliTSi ar

arsebobs kavSiris kviris dResa da dRis periods Soris im manqanebisaTvis, romlebic gadian SeCerebis niSanTan.

10. 2realuriχ (1) = 70.914, p<.01; hipoTeturi kavSiri arsebobs.

SeamowmeT Tqveni codna XV.2 1.

qulebi, dalagebuli umciresidan udidesamde

7 10 14 15 16 19 23 26

jgufis identoba 2 1 2 1 2 1 1 2 rangi 1 2 3 4 5 6 7 8 raodenoba SemTxvevebisa, rodesac A1 qula win uswrebs A2 qulas

3 2 1 1

raodenoba SemTxvevebisa, rodesac A2 qula win uswrebs A1 qulas

4 3 2

UA1 = 7; UA2 = 9. a. UA1. b. ara. 2. UA1 = 7; UA2 = 9.

a. ki. 4. UA1 daaxloebiT unda utoldebodes UA2-s. 5. isini sakmaod unda gansxvavdebodnen. 6.

Ukritikuli xvdeba uaryofis areSi

gadawyvetileba

a. 0 ara oH : ver uarvyofT

1H : ar viRebT

b. 16 ki oH : uarvyofT

1H : viRebT

g. 27 ara oH : ver uarvyofT

1H : ar viRebT

d. 22 ki oH : uarvyofT

1H : viRebT

e. 56 ki oH : uarvyofT

1H : viRebT

v. 87 ara oH : ver uarvyofT

1H : ar viRebT

7. ufro mcire U = UA1 = 54.5; Ukritikuli = 41; UA1 ar xvdeba uaryofis areSi. ar dasturdeba, rom jgufebi gansxvavdeba SfoTvis SefasebebSi. SeamowmeT Tqveni codna XV.3

4. Tkritikuli xvdeba uaryofis

areSi gadawyvetileba

a. 4 ki oH : uarvyofT

1H : viRebT

b. 8 ara oH : ver uarvyofT

1H : ar viRebT

g. 17 ki oH : uarvyofT

1H : viRebT

d. 46 ki oH : uarvyofT

1H : viRebT

e. 90 ara oH : ver uarvyofT

1H : ar viRebT

5. Trealuri (14) = 27; Tkritikuli (14) = 21; Trealuri ar aris statistikurad mniSvnelovani. am SerCevaSi SfoTvis done ar Seicvala pirveli TamaSidan meore TamaSamde. kiTxvebi

1. a. 2realuriχ (1) = 1.235, p > .05. skolidan garicxvis sixSire ar aris

kavSirSi sqesTan.

b. 2realuriχ (2) = 28.990, p > .01. skolidan garicxvis sixSire kavSirSia

problemis seriozulobasTan. iricxeba saSualo problemebis mqone individebis mosalodnelze ufro meti raodenoba; aseve, iricxeba seriozuli problemebis mqone individebis mosalodnelze naklebi raodenoba.

2. ∑ − )( R = 50, ∑ + )( R = 5; amdenad, Trealuri (11) = 5, p < .01. uarvyofT oH ;

davalebis tipi gavlenas axdens mTqnarebaze. a. uintereso videofilmis pirobaSi qulebi dadebiTadaa

asimetriuli. b. damTqnarebebis raodenobis ganawilebebi gansxvavebulia. ufro

meti mTqnareba xdeba, rodesac stimuli uinteresoa, vidre rodesac stimuli sainteresoa.

3. .05.,5.382 >= pU A ganawilebebi mniSvnelovnad ar gansxvavdeba.

4. .05.,5.442 >= pU A ganawilebebi mniSvnelovnad ar gansxvavdeba.

5. realurikritikulirealuri TTT .3)8(;3 == xvdeba uaryofis areSi; ase rom

“ 0H :dakavSirebuli A1 da A2 qulebis populaciis ganawilebebi

identuria” uaryofilia.

danarTi F xSirad gamoyenebuli formulebi

sixSireTa ganawilebebi daujgufebel sixSireTa ganawilebebi

qulis rf = N

rfqulis

qulis %f = (qulis rf) x 100

dajgufebel sixSireTa ganawilebebi

klasis intervalis zoma = i = raodenoba isintervaleb

minimaluriimaqsimalur − XX

intervalSi qulebis rf = N

f qulebis intervalSi

intervalSi qulebis %f = N

f qulebis intervalSix 100

procentilis rangi

XP = ( )[ ]

−+

NfiiXXcf LL / x 100

procentilis wertili

pX = +LX ( )

−

i

L

fcfNP i

centraluri tendenciis sazomebi

mediana

Mdn = 50X = +LX ( )

−

i

L

fcfN50. I

saSualo

X = N

X∑


rangi

rangi = minimaluriimaqsimalur XX −

kvartilTSorisi rangi

IQR = 2575 XX −

semikvartilTSorisi rangi

ISQR = 2

IQR =2

2575 XX −

dispersia

populaciis dispersia

σ 2 = populaciaN

X∑ − )( 2µ

SerCevis dispersia

2S =N

XX∑ − 2)(

populaciis dispersiis Sefaseba

ganmartebiTi formula: 2s =1

)( 2

−

−∑N

XX

kvadratebis jamebis formula: 2s =1−N

SS

gamoTvliTi formula: 2s =[ ]

1/)( 22

−

−∑ ∑N

NXX

standartuli gadaxra populaciis standartuli gadaxra

σ = 2σ


S = 2S = N

XX∑ − 2)(


ganmartebiTi formula: s = 2s = 1

)( 2

−

−∑N

XX

kvadratebis jamis formula: s = 1−N

SS

gamoTvliTi formula: s = ( )[ ]

1/22

−

−∑ ∑N

NXX

saSualos standartuli Secdoma

σ cnobili

σ X = Nσ

σ ucnobi

s X = Ns

saSualoebs Soris sxvaobis standartuli Secdoma

σ cnobili

σ 21 AXAX − =

2

22

1

21

A

A

A

A

nnσσ

+

σ ucnobi

s 21 AXAX − =

( ) ( )

+

−+

−+−

2121

222

211 11

211

AAAA

AAAA

nnnnsnsn

albaToba xdomilebis albaToba (p)

p (xdomileba) = ipopulaciaS raodenoba isxdomilebeb iSesaZlebel

ipopulaciaS raodenoba xdomilebis

z qulebi da standartuli qulebi

erTeuli qula

z = σ

µ−X

standartuli qula

z = S

XX − an z =

( )∑ −

−

NXX

XX

/2

SerCevis saSualo

z = X

Xσ

µ−

t kriteriumebi t kriteriumi erTi SerCevisaTvis

t = σ

µ−X an t =

NsX/

µ−, df = N – 1

t kriteriumi damoukidebeli jgufebisaTvis

ganmartebiTi formula: elidamoukidebt = 21

21

AXAX

AA

sXX

−

−

gamoTvliTi formula: elidamoukidebt = ( ) ( )

+

−+

−+−

−

2121

222

211

21

112

11

AAAA

AAAA

AA

nnnnsnsn

XX,

df = N – 2 t kriteriumi damokidebuli jgufebisaTvis

lidamokidebut =

( )1/)( 2221

−

−

−

∑ ∑wyvilebiwyvilebi

wyvilebi

NNNDD

XX AA, df = 1−wyvilebiN

dispersiuli analizi cdis pirTa Soris erTfaqtoriani sqema wyaro SS dfa MS F A faqtori Secdoma sul

( )∑∑ −2

GA XX

( )∑∑ −2

AXX

( )∑∑ −2

GXX

a – 1 N – a N - 1

AA dfSS /

SecdomaSecdoma dfSS /

ar gamoiTvleba

SecdomaMSMS A /

a a = A faqtoris doneebis raodenoba, N = qulebis sruli raodenoba.

cdis pirTa Soris orfaqtoriani sqema wyaro SS dfa MS F A faqtori B faqtori A da B faqtorebis urTierTqme- deba Secdoma sul

ASS

BSS

AxBSS

SecdomaSS

sruliSS

a – 1 b – 1 (a – 1)( b – 1) ab( ABn -1) N - 1

A

A

dfSS

B

B

dfSS

AxB

AxB

dfSS

Secdoma

Secdoma

dfSS

ar

gamoiTvleba

SecdomaMSMSA

SecdomaMSMSA

SecdomaMSMS AxB

a a = A faqtoris doneebis raodenoba; b = B faqtoris doneebis raodenoba;

ABn = qulebis raodenoba TiToeul ujredSi; N = qulebis sruli

raodenoba. cdis pirTa SigniT erTfaqtoriani sqema wyaro SS dfa MS F A faqtori S faqtori AxS sul

ASS

SSS

AxSSS

sruliSS

a – 1

An – 1

(a – 1)( An – 1)

N - 1

A

A

dfSS

S

S

dfSS

AxS

AxS

dfSS

ar gamoiTvleba

AxSMSMS A

a a = A faqtoris doneebis raodenoba; An = qulebis raodenoba Carevis

pirobaSi, anu, Sesabamisad, cdis pirebis raodenoba; N = qulebis sruli raodenoba.

tukis HSD mravaljeradi Sedarebis kriteriumi cdis pirTa SigniT erTfaqtoriani sqemisaTvis

CD = qAn

SecdomaMS

cdis pirTa Soris erTfaqtoriani sqemisaTvis

CD = qA

AxS

nMS

ubralo efeqtebisaTvis cdis pirTa SigniT erTfaqtorian sqemaSi

CD = qABn

SecdomaMS

efeqtis siZlieris sazomebi eta kvadrati t kriteriumisaTvis

dftt

+= 2

22

realuri

realuriη

eta kvadrati cdis pirTa Soris erTfaqtoriani disperiuli analizisaTvis

sruliSSSS A=2η

an

( )( ) Secdomarealuri

realuri

dfFdfFdf

A

A

+=

))((2η

eta kvadrati cdis pirTa SigniT erTfaqtoriani disperiuli analizisaTvis

AxSA

A

SSSSSS+

=2η

eta kvadrati cdis pirTa Soris orfaqtoriani disperiuli analizisaTvis

sruliSSSS A=2η A faqtorisaTvis

sruliSSSSB=2η B faqtorisaTvis

sruliSSSS AxB=2η A da B faqtorebis urTierTqmedebisaTvis

korelacia pirsonis korelaciis koeficienti

ganmartebiTi formula: ( )( )

( )[ ] ( )[ ]∑∑∑

−−

−−=

22YYXX

YYXXr

jvaredini namravlis formula: ( )( )YX

XY

SSSSCPr =

gamoTvliTi formula:

( )( ) ( )( )( )( ) ( )[ ]( )( ) ( )[ ]2222 ∑∑∑∑

∑∑∑−−

−=

YYNXXN

YXXYNr

wyvilebiwyvilebi

wyvilebi

standartuli qulis formula: ( )wyvilebiN

zzr YX∑= ,

df = Nwyvilebi – 2 yvela formulisaTvis. determinaciis koeficienti r2 spirmenis ranguli korelaciis koeficienti

( )( )

−−= ∑

16

1 2

2

wyvilebiwyvilebi NND

rs

regresia

wrfis gantoleba

Y = bX + a

wrfis daxriloba

aSimniSvnelob isX cvlileba

aSimniSvnelob is cvlileba

−−

=Y

b = 12

12

XXYY

−−

umcires kvadratTa meTodiT miRebuli regresiis wrfis gantoleba

abXY +=′ umcires kvadratTa meTodiT miRebuli regresiis wrfis daxriloba

gadaxraTa namravlebis jamis formula: ( )( )

( )∑∑

−

−−= 2

XX

YYXXb

jvaredini namravlis formula: X

XY

SSCPb =

gamoTvliTi formula: ( )( ) ( )( )

( )( )( )22 ∑∑∑∑∑ −

=YXN

YXXYNb

wyvilebi

wyvilebi

korelaciuri formula:

=

Y

X

ssrb

umcires kvadratTa meTodiT miRebuli regresiis wrfis Y- TanakveTa

a = XbY − Sefasebis standartuli Secdoma

ganmartebiTi formula: 2−

=•wyvilebi

naSTi

NSSs XY

an

( )2

2

−

−= ∑

•wyvilebiN

YYs

l

XY

gamoTvliTi formula:

( ) ( ) ( ) ( )( )[ ]( )

−

−−−

−= ∑

∑∑∑∑∑∑• 22

222

21

XXN

YXXYNYYN

NNs XY

wyvilebi

wyvilebi

wyvilebi

wyvilebiwyvilebi

korelaciuri formula: ( )2121

realuri

wyvilebi

wyvilebi rNN

ss YXY −

−

−=•


xi-kvadrat damoukideblobis kriteriumi

( )∑∑ −

=rc

rcrc

EEO 2

2χ , df = (r - 1)(c - 1)

=

sixSire

imosalodnel

ujredis

= ( )( )

raodenoba sruli pasuxebis

sveti imarginalur isujredisaTvstriqoni imarginalur isujredisaTv

Tanxmobis xi-kvadrat kriteriumi

( )∑ −

=c

cc

EEO 2

2χ , df = c – 1

man-uitnis U kriteriumi

( ) ∑−+

+= 111

211 21

AAA

AAA RnnnnU

( ) ∑−

++= 2

22212 2

1A

AAAAA RnnnnU

an

1212 AAAA UnnU −= uilkoqsonis niSniani rangebis kriteriumi

T = ufro mcires ( )∑ − R da ( )∑ + R -dan

statistika socialur mecnierebebSi - edu.aris.ge · winasityvaoba . saqarTveloSi socialuri...

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Transcript of statistika socialur mecnierebebSi - edu.aris.ge · winasityvaoba . saqarTveloSi socialuri...