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Tavi XIII. korelacia_______________________________________________________________________Tavi XIII

korelacia

__________________________________________________________wertilTa gabnevis diagrama

dadebiTi kavSiriuaryofiTi kavSiri

pirsonis korelaciis koeficientikorelaciis sazomis miReba: jvaredina namravlebi: -is Sedareba qulebis saerTo cvalebadobasTan

pirsonis korelaciis koeficientis alternatiuli formulebi-is standartuli qulebis formula-is gamoTvliTi formula

r -is formulebi _ Sejamebar -is Tvisebebi

kavSiris mimarTulebakavSiris xarisxidiapazonis SezRudva da -is mniSvnelobadeterminaciis koeficientikorelacia da mizezobrioba

pirsonis r koeficientis statistikuri mniSvnelovnobis Semowmebapopulaciis korelacia, aranulovani korelaciis Semowmeba populaciaSi

korelaciuri kvlevis Sedegebis warmodgenaspirmenis ranguli korelaciis koeficienti

-is gamoyenebis magaliTispirmenis ranguli korelaciis koeficientis statistikuri

mniSvneloba-is Tvisebebi

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Tavi XIII. korelacia_______________________________________________________________________

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 cvladi) da depresiis qulebi (aRniSnuli roogrc cvladi) warmodgenilia cxrilSi #13.1. cvladis -iT an -iT aRniSvna pirobiTia. Tumca zogadad mklevari svams kiTxvas: rogoria -is, rogorc -is funqciis mniSvneloba?

es debuleba gamoxatavs, rom mkvlevari dainteresebulia, erTi cvladis ( ) mniSvnelobis mixedviT iwinaswarmetyvelos meore cvladis ( ) mniSvneloba. cvladi, romelsac prediqtorad viyenebT, aRiniSneba -iT, xolo nawinaswarmetyvelebi cvladi - -iT. maSasadame, rodesac mkvlevari svams kiTxvas: `ras udris depresiis qula martoobis garkveuli qulis SemTxvevaSi?~, is cdilobs, depresiis done ( ) iwinaswarmetyvelos martoobis donis ( ) mixedviT.

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Tavi XIII. korelacia_______________________________________________________________________erTi qulis winaswarmetyvelebisaTvis meore cvladis gamoyenebis detalebi warmodgenilia me-14 TavSi.

qulaTa wyvilebi cxrilSi # 13.1 warmoadgens qulebis organzomilebian (bivariaciul) ganawilebas.

organzomilebiani ganawileba (Bivariate Distribution): ganawileba, romelSic TiToeuli cdis pirisagan miiReba ori monacemi (qula).

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 # 13.110 cdis piris hipoTeturi qulebi martoobisa da depresiis skalebze

cdis piri martoobisqula:

depresiisqula:

12345678910

427187301218231912

16373323343224292626

wertilTa gabnevis diagramebi

wertilTa gabnevis diagrama (Scatterplot): organzomilebiani ganawilebis grafiki, romelSic cvladi gadazomilia horizontalur RerZze da cvladi gadazomilia vertikalur RerZze.

cxrilis #13.1 monacemebis wertilTa gabnevis diagrama warmodgenilia suraTze #13.1. wertilTa gabnevis diagramaze -iT aRniSnuli cvladi (Cvens magaliTSi martoobis qula) gadazomilia horizontalur RerZze (abscisaze), xolo 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

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Tavi XIII. korelacia_______________________________________________________________________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), cvladis mniSvnelobis zrdasTan erTad izrdeba cvladis mniSvnelobac.

suraTi #13.110 cdis piris martoobisa da depresiis qulebis wertilTa gabnevis diagrama. wertilebis gverdiT miTiTebuli ricxvebi aRniSnavs cdis pirebs. qulebi aRebulia 13.1 cxrilidan.

dadebiTi kavSiri (positive relationship): or cvlads Soris kavSiri, romlis drosac erTi cvladis mniSvnelobis zrdasTan erTad izrdeba meore cvladis mniSvnelobac.

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.

wrfivi kavSiri (linear relationship): or cvlads Soris kavSiri, romelic SeiZleba aRiweros wrfiT.

sruli dadebiTi kavSiris SemTxvevaSi, rogoric gamosaxulia suraTze #13.2, -is TiToeul mniSvnelobas Seesabameba -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

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Tavi XIII. korelacia_______________________________________________________________________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 #13.210 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: cvladis mniSvnelobis matebasTan erTad 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: -is dabali mniSvnelobebi dakavSirebulia -is maRal mniSvnelobebTan da -is maRali mniSvnelobebi dakavSirebulia -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 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 #13.310 cdis piris martoobisa da depresiis qulebs Soris uaryofiTi kavSiri.

suraTi #13.410 cdis piris martoobisa da depresiis qulebs Soris sruli uaryofiTi kavSiri.

SeamowmeT Tqveni codna XIII-1

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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 -iT da martoobis qula -iT) da Semdeg am wertilebisaTvis aageT suraTze #13.1 gamosaxulis msgavsi wertilTa gabnevis diagrama. cvladis -iT an -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 4 192 12 203 28 334 18 325 14 266 6 197 22 288 23 369 15 3010 18 27

cxrili #2


1 5 292 11 313 23 234 17 175 28 166 13 217 12 268 4 289 21 1410 16 25

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4. qvemoT mocemulia 10 cdis piris SerCevis sxeulis wonisa (funtebSi) da TviTpativiscemis monacemebi

cdis piri sxeulis wona KTviTpativiscema

1 100 392 111 473 117 544 124 235 136 356 139 307 143 488 151 209 155 2810 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 -iT. am statistikas mieniWa karl pirsonis (1857-1936), im britaneli statistikosis saxeli, romelmac am koeficientis gamosaTvleli formula Seqmna. pirsonis korelaciis koeficienti ganisazRvreba, rogorc:

pirsonos korelaciis koeficienti =

sadac aris cdis piris cvladis Sesabamisi qula aris cvladis Sesabamisi yvela qulis saSualo Y aris cdis piris Y cvladis Sesabamisi qula aris Y cvladis Sesabamisi yvela qulis saSualo

pirsonis korelaciis koeficienti (Pearson correlation coefficient): statistika, romelic aRiniSneba -iT da asaxavs intervalebis an Sefardebis skalaze or

7

Tavi XIII. korelacia_______________________________________________________________________cvlads Soris wrfivi kavSiris xarisxs.

es formula SeiZleba Caiweros ufro martivi aRniSvnebis gamoyenebiT, rogorc:

sadac aris , anu aris –sa da –is jvaredina namravli ( da indeqsebi niSnavs, rom jvaredina namravlebi gamoTvlilia da cvladebisTvis)

aris monacemebis kvadratebis jami (anu ) aris monacemebis kvadratebis jami (anu )

ganmartebiTi formula Cveulebriv –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 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, an . magaliTad, cxrilze #13.1 martoobis qulebis saSualoa 17. me-2 cdis pirs aqvs martoobis 27 qula. Tu gamovTvliT , 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 ; 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.

qulebi SeiZleba gamoisaxos msgavsadve am qulebis saSualosTan, -Tan, mimarTebiT. magaliTad, depresiis qulebis saSualoa 28. me-2 cdis pirma miiRo depresiis 37 qula. am qulis saSualosgan gansxvavebaSi gadayvaniT viRebT: = 37-28 = +9. meores mxriv, me-4 cdis pirma miiRo depresiis 23 qula. saSualosgan gansxvavebis saxiT gamosaxuli, es qula udris: = 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

8

Tavi XIII. korelacia_______________________________________________________________________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-sa da -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

1 4 -13 16 -12 +1562 27 +10 37 +9 +903 18 +1 33 +5 +54 7 -10 23 -5 +505 30 +13 34 +6 +786 12 -5 32 +4 -207 18 +1 24 -4 -48 23 +6 29 +1 +69 19 +2 26 -2 -410 12 -5 26 -2 +10

17 28 jami +367

qulebis korelaciis siZlieris sazomi miiReba TiToeuli cdis piris qulebis cvladis saSualosgan sxvaobisa da 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, ( )() = (-10) (-5) = +50. am svetSi namravlebis jami aris +367.

sidides ewodeba -isa da -is jvaredina namravli, rac aRiniSneba, rogorc da is warmoadgens –is formulis mricxvels. 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 –sa da –s Soris, miT ufro izrdeba –is mniSvneloba. uaryofiTad dakavSirebuli monacemebisaTvis uaryofiTia da misi mniSvneloba miT ufro didia, rac ufro Zlieria kavSiri.

-isa da -is jvaredina namravlebi (Cross products of and ): mniSvneloba da cvladebisTvis

cxrilze #13.3a naCvenebia –is mniSvneloba suraTis #13.3 Sesabamisi –isa da –is uaryofiTi kavSirisaTvis. am monacemebisaTvis aris -399.

cxrili #13.3

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Tavi XIII. korelacia_______________________________________________________________________a) -isa da –is jvaredina namravlebi martoobisa da depresiis qulebisaTvis.

Mmartooba depresia

cdis piri

1 5 -10 32 +7 -702 24 +9 23 -2 -183 20 +5 17 -8 -404 12 -3 32 +7 -215 14 -1 21 -4 +46 20 +5 24 -1 -57 28 +13 14 -11 -1438 13 -2 25 0 09 6 -9 36 +11 -9910 8 -7 26 +1 -7

15 25 jami -399

b) -isa da –is jvaredina namravlebi im SemTxvevisaTvis, roca martoobisa da depresiis qulebs Soris kavSiris ar aris.

martooba depresia

cdis piri

1 24 +7 14 -12 -842 16 -1 36 +10 -103 4 -13 24 -2 +264 12 -5 16 -10 +505 22 +5 28 +2 +106 30 +13 26 0 07 10 -7 34 +8 -568 28 +11 30 +4 +449 14 -3 28 +2 -610 10 -7 24 -2 +14

17 26 jami _12

10


rodesac da qulebi erTmaneTTan kavSirSi ar aris, nuls uaxlovdeba. im SemTxvevisaTvis, roca da ar aris kavSirSi, naCvenebia cxrilze #13.3b. suraTze #13.5 am qulebis wertilTa gabnevis diagramaa naCvenebi. yuradReba miaqcieT imas, rom da qulebi ar korelirebs: -is maRali maCveneblebi SeTanxmebulad ar aris dakavSirebuli -is maRal an dabal qulebTan. msgavsadve, -is dabali qulebi ar aris SeTanxmebulad dakavSirebuli -is maRal an dabal qulebTan. cxrilze #13.3b gamoTvlili = -12 Seesabameba kavSiris ararsebobas.

suraTi #13.510 cdis piris martoobisa da depresiis qulebs Soris kavSiri ar arsebobs. qulebi aRebulia 13.3b cxrilidan.

: -is Sedareba monacemebis saerTo cvalebadobis sazomTan

gamoxatavs or cvlads Soris kavSiris mimarTulebas, magram –is ricxviTi mniSvneloba damokidebulia –isa da –is gazomvis erTeulze da Cvens xelT arsebuli qulebis raodenobaze. qulebis or gansxvavebul simravles SeiZleba hqondes erTnairi siZlieris kavSiri, magram -is gansxvavebuli mniSvnelobebi, radgan am qulebiT gazomilia gansxvavebuli erTeulebi an qulebis erTi simravle ufro didi SerCevidanaa miRebuli, vidre meore. imisaTvis, rom , rogorc -isa da –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). cvladiT aRniSnuli monacemebisaTvis = da cvladiT aRniSnuli monacemebisaTvis = . imisaTvis, rom gadaviyvanoT jvaredina namravlis erTeulebSi, ToToeuli -dan viRebT kvadratul fesvs. Semdeg -is kvadratuli fesvis gamravlebiT –is kvadratul fesvze viRebT monacemTa ori simravlis saerTo cvalebadobis sazoms, . pirsonis korelaciis koeficienti, , aris –is Sefardeba saerTo cvalebadobis am sazomTan. Sesabamisad, Cven vxedavT, rom Semdegi ori formula -is misaRebad erTsa da imave ricxviT mniSvnelobebs iyenebs:

da

.

yuradReba miaqcieT imas, rom –is formulis mniSvneli yovelTvis dadebiT mniSvnelobas iRebs, radgan sxvaobis kvadrats Seicavs. maSasadame, korelaciis koeficinetis dadebiT (+) Tu uaryofiT (_) niSans mricxvelis niSani

11

Tavi XIII. korelacia_______________________________________________________________________gansazRvravs. cxrilSi #13.4 naCvenebia cxrilis #13.1 10 qulisaTvis ganmartebiTi formulis gamoyenebiT –is gamoTvla. TiToeuli cdis pirisaTvis:

sveti a aris martoobis qula sveti b aris cdis piris martoobis qulisa da martoobis qulebis saSualos

Soris sxvaoba (anu ( )) 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 ( )) sveti v aris e svetis mniSvnelobis kvadrati sveti z warmoadgens b da e svetebis namravls (( )( )), anu -

s.

miiReba g svetis monacemebis dajamebiT, _ v svetis monacemebis dajamebiT da _ z svetis monacemebis dajamebiT. Semdeg xdeba am jamebis formulaSi Casma –is ricxviTi mniSvnelobis misaRebad, romelsac ewodeba (ix. cxrilis #13.4 me-2 safexuri). cxrilis #13.1 monacemebisaTvis,

= +0,78. sanam interpretacias ganvixilavde, warmogidgenT or alternatiul formulas, romlebic xSirad gamoiyeneba korelaciis koeficientis gamosaTvlelad.

cxrili #13.4–is gamoTvla cxrilis #13.1 martoobisa da depresiis qulebisaTvis ganmartebiTi

formulis gamoyenebiT.

safexuri 1. viTvliT -s, -s da -s.

(a) (b) (g) (d) (e) (v) (z)

martooba depresiacdis piri1 4 -13 169 16 -12 144 +1562 27 +10 100 37 +9 81 +903 18 +1 1 33 +5 25 +54 7 -10 100 23 -5 25 +505 30 +13 169 34 +6 36 +786 12 -5 25 32 +4 16 -207 18 +1 1 24 -4 16 -48 23 +6 36 29 +1 1 +69 19 +2 4 26 -2 4 -4

10 12 -5 25 26 -2 4 +10

17 630 28 352

+367,0

safexuri 2. viTvliT -s:

12


Dda

an

ricxviTi mniSvnelobebis CasmiT:

pirsonis korelaciis koeficientis alternatiuli formulebi

-is standartuli qulebis formula

-is ganmartebiTi formula SeiZleba Caiweros mniSvnelis dayofiT mis Semadgenel kvadratebis or jamad:

an

yuradReba miaqcieT, rom ( ) / Zalian hgavs standartul qulas, z = ()/S, romelic me-6 TavSia warmodgenili. imisaTvis, rom es gamosaxuleba standartul monacemad gardavqmnaT da miviRoT , monacmebis SerCevis standartuli gadaxra, saWiroa gavyoT N-ze, anu –is Semadgeneli monacemebis raodenobaze. igive msjeloba exeba ( ) / –sac. Sesabamisad, -is standartuli qulis formulaa:

,

sadac: aris ( )/ aris ( ) / aris monacemebis wyvilebis raodenoba

-s, qulebis SerCevis standartul gadaxras, Semdegnairad vpoulobT:

13


da -s, monacemebis SerCevis standartul gadaxras, Semdegnairad:

cxrili #13.5 gviCvenebs, rogor xdeba am formulis gamoyeneba gamosaTvlelad cxrilis #13.1 monacemebisaTvis. pirveli safexuria z qulebis gamosaTvlelad saWiro –is da –is miReba. am safexurze mocemuli

= 630 da = 352 aRebulia cxrilidan #13.4. am gamoTvlebiT Cven vigebT, rom qulebis standartul qulebad gardasaqmnelad = ( - 17,0)/7,937. msgavsadve, monacemebisaTvis = ( - 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 qula. b svetSi warmodgenilia sxvaoba TiToeuli cdis pirisaTvis da g svetSi - TiToeuli cdis piris z qula. d svetSi warmodgenilia TiToeuli cdis piris qula, e svetSi warmodgenilia sxvaoba da v svetSi _ z qula ToToeuli –Tvis. z svetSi gamravlebulia ToToeuli cdis piris

da qulebi da am svetis monacemebis jami, +7,794, gvaZlevs sidides, rac saWiroa –is formulis mricxvelisaTvis.

cxrilis me-4 safexuri gviCvenebs, Tu rogor xdeba –isa da mniSvnelobebis Casma formulaSi. miRebulia = +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–is gamoTvla cxrilis #13.1 martoobisa da depresiis qulebisaTvis standartuli qulis formulis

mixedviT safexuri I. -isa da –is miReba:

,

cxrilis #13.4 g svetidan cxrilis #13.4 v svetidan

=10 =10

safexuri II. -is, -sa da -is miReba:

qulebis standartul qulebSi gadasayvanad

qulebis standartul qulebSi gadasayvanad

14

Tavi XIII. korelacia_______________________________________________________________________ maSasadame,

maSasadame,

safexuri III. nedli qulebis qulebSi gadayvana

(a) (b) (g) (d) (e) (v) (z)martooba depresia

cdis piri1 4 -13,0 -1,638 16 -12,0 -2,023 +3,31

42 27 +10,0 +1,260 37 +9,0 +1,517 +1,91

13 18 +1,0 +0,126 33 +5,0 +0,843 +0,10

64 7 -10,0 -1,260 23 -5,0 -0,843 +1,06

25 30 +13,0 +1,638 34 +6,0 +1,011 +1,65

66 12 -5,0 -0,630 32 +4,0 +0,674 -0,4257 18 +1,0 +0,126 24 -4,0 -0,674 -0,0858 23 +6,0 +0,756 29 +1,0 +0,168 +0,12

89 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

17,0 28,0 jami +7,794

safexuri IV. -is gamoTvla:

(ixileT z sveti),=10,

.

ricxviTi mniSvnelobebis CasmiT,

cxrili #13.6cxrilis #13.1 martoobisa da depresiis qulebisaTvis gamoTvliTi formulis mixedviT –is gamoTvla

safexuri I. -is, -is, -is, -sa da -is gamoTvla:

(a) (b) (g) (d) (e)cdis piri1 4 16 16 256 642 27 37 729 1369 9993 18 33 324 1089 5944 7 23 49 529 161

15


5 30 34 900 1156 10206 12 32 144 1024 3847 18 24 324 576 4328 23 29 529 841 6679 19 26 361 676 494

10 12 26 144 676 312jamebi 170,0 280,0 3520,0 8192,0 5127,0

safexuri II. -is gamoTvla:

ricxviTi mniSvnelobebis CasmiT miviRebT:

r-is gamoTvliTi formula

r-is mniSvnelobis gamoTvliTi formulaa:

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 ( ) da depresiis ( ) qulebs. am svetebis monacemebis jamebi, = 170 da = 280, gvaZlevs formulisaTvis saWiro –sa –s. g sveti gviCvenebs ToToeul –s da am svetis monacemebis jami, 3520, gvaZlevs -s. d sveti warmoadgens TiToeul -s da am svetis monacemebis jami, 8192, -ia. e svetSi miRebulia -is mniSvneloba TiToeuli cdis pirisaTvis da am svetis monacemebis jami, 5127,

-is mniSvnelobaa. cxrilis #13.6 me-2 safexuri gviCvenebs –is formulaSi ricxviTi

mniSvnelobebis Casmas. = +0,78. yuradReba miaqcieT imas, rom

16

Tavi XIII. korelacia_______________________________________________________________________miuxedavad imisa, Tu ra formula iqna gamoyenebuli (ganmartebiTi, z qulis, Tu gamoTvliTi), cxrilis #13.1 monacemebisaTvis = + 0,78.

aq saWiroa kidev erT mniSvnelovan sakiTxze gamaxvildes yuradReba. rogorc aRvniSneT, cvladebis -iT an -iT aRniSvna absoluturad pirobiTia. Cvens magaliTSi martoobis qula aRiniSna -iT da depresiis qula -iT. Tumca, aRniSvna rom Segvebrunebina da depresiis qula aRgveniSna -iT da martoobis qula - -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 -iT da romeli - -iT.

r - is formulebi - Sejameba

-is 4 sxvadasxva formulis Sejameba mocemulia cxrilSi #13.7. yvela formula erTsa da imave mniSvnelobas iZleva, roca maT qulebis erTi simravlisTvis

viyenebT.

cxrili #13.7-is formulebis Sejameba

ganaszRvrebiTi formula

-iT, -Ta da -iT Cawerili ganmartebiTi formula

standartuli qulis formula

gamoTvliTi formula

SeamowmeT Tqveni codna XIII-2

1. gansazRvreT: jvaredina namravli, , , ,

17


2. axseniT, Tu ratom aris jvaredina namravlebi monacemTa or simravles Soris korelaciis sazomi.

3. mocemulia Semdegi monacemebi:


1 4 192 12 203 28 334 18 325 14 266 6 197 22 288 23 369 15 3010 18 27

a. am qulebisaTvis ganmartebiTi formulis gamoyenebiT gamoTvaleT .

b. am kiTxvis qulebi gadmotanilia davalebidan SeamowmeT Tqveni codna XIII-1, kiTxva #3, cxrili #1. SeadareT im kiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT, raTa TvalsaCinod warmoidginoT mimarTeba, romelsac aRwers.

4. mocemulia Semdegi monacemebi:


1 5 292 11 313 23 234 17 175 28 166 13 217 12 268 4 289 21 1410 16 25

a. am monacemebisaTvis standartuli qulis formulis gamoyenebiT gamoTvaleT .

am kiTxvis monacemebi gadmotanilia davalebidan SeamowmeT Tqveni codna XIII_1, kiTxva #3, cxrili #2. SeadareT im kiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT, raTa TvalsaCinod warmoidginoT mimarTeba, romelsac aRwers.

b. mocemulia Semdegi monacemebi:

cdis pirisxeulis wona

TviTpativiscema

18


1 100 392 111 473 117 544 124 235 136 356 139 307 143 488 151 209 155 2810 164 46

a. am monacemebisaTvis ganmartebiTi formulis gamoyenebiT gamoTvaleT .

b. am kiTxvis monacemebi gadmotanilia davalebidan SeamowmeT Tqveni codna XIII_1, kiTxva #4. SeadareT im SekiTxvisaTvis agebul wertilTa gabnevis diagramas. es Sedareba gamogadgebaT imisaTvis, raTa TvalsaConod warmoidginoT mimarTeba, romelsac 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 -is absoluturi mniSvneloba, sidide, romelic SeiZleba iRebdes 0-dan 1-mde mniSvnelobebs. niSnisa (+ an _) da ricxviTi mniSvnelobis gaTvlasiwinebiT 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 (mag., = +0,86), asaxavs –sa da

–s Soris dadebiT kavSirs; cvladis qulebis zrdasTan erTad izrdeba cvladis qulebic. qulebi 13.1 da 13.2 cxrilebze asaxavs dadebiT kavSirebs; cxrilisTvis #13.1 = +0,78 da cxrilisTvis #13.2 = +1,00.

uaryofiTi (magaliTad, = _0,86) aRniSnavs cvladebs Soris Sebrunebul kavSirs: roca cvladis qulebi izrdeba, cvladis qulebi klebulobs. uaryofiTi kavSiri naCvenebia cxrilebze #13.3 da 13.4. cxrilisTvis #13.3 udris _0,83-s da cxrilisTvis #13.4 udris _1,00-s.

kavSiris xarisxi

-is ricxviTi mniSvneloba gamoxatavs, imas, Tu ramdenad kargad aRwers wrfe da cvladebs Soris mimarTebas. 1,00-is toli niSnavs, rom kavSiri idealurad

wrfivia; wertilTa gabnevis diagramaze yvela wertili wrfeze Zevs. suraTi #13.2 gviCvenebs sruli dadebiTi kavSiris amsaxvel wertilTa gabnevis diagramas,

19

Tavi XIII. korelacia_______________________________________________________________________= +1,00 martoobisa da depresiis qulebisaTvis. sruli uaryofiTi kavSiri, = _1,00 warmodgenilia suraTze #13.4. rodesac udris plus an minus 1,00-s, -is yoveli mniSvnelobisTvis arsebobs misi Sesabamisi –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, qulebis mixedviT zustad viwinaswarmetyveloT . 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 daaxloebiT 0-is tolia, erTi qulis mniSvneloba ar aris kanonzomierad dakvSirebuli meore qulis mniSvnelobasTan. aseTi magaliTi naCvenebia suraTze #13.5, sadac = _0,2.

korelaciis koeficientis 0-dan +1,00-mde mniSvnelobebi gamoxatavs, rom –sa da –s Soris arsebobs kavSiri, magram ara sruli kavSiri.

TvalsaCinoebisTvis aRvniSnavT, rom suraTze #13.1, romlisTvisac r = +0,78, da 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 = _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 –is mniSvneloba, monacemebi wertilTa gabnevis diagramaze rom yofiliyo warmodgenili. es SesaZlebeli xdeba mxolod –is mniSvnelobebis im qulebis wertilTa gabnevis diagramasTan SedarebiT, romlebisTvisac korelacia gamovTvaleT. suraTi #13.6 warmoadgens –is gansxvavebuli mniSvnelobebis Sesabamis wertilTa gabnevis diagramebs. a-dan e-mde grafikebi asaxavs 1,00-dan 0,00-mde korelaciebs. yuradReba miaqcieT imas, rom –is klebasTan erTad –sa da –s Soris wrfivi kavSiri naklebad Zlieri xdeba, sanam = 0,00, roca kavSiri aRar arsebobs.

pirsonis aris da cvladebs Soris wrfivi kavSiris sazomi. grafiki #13.6v da cvladebs Soris arawrfiv, U-s formis kavSirs warmoadgens. am kavSiris dros –is mcire mniSvnelobebi dakavSirebulia –is did mniSvnelobebTan. garkveul wertilamde -is matebasTan erTad mcirdeba; –is Semdgomi mateba ki dakavSirebulia –is matebasTan. –sa da –s Soris aseTi kavSiris wrfiT warmodgena SeuZlebelia da misi aRwerisaTvis -is gamoyeneba ar SeiZleba.

suraTi #13.6wertilTa gabnevis diagramebi gamoxatavs da cvladis monacemebs Soris sxvadasva korelaciebs 10 cdis piris qulebisaTvis. qulebis nakrebebs Soris pirsonis korelaciis koeficientis mniSvneloba mocemulia -iT.(a) =+1,0 (b) =-1,0 (g) =-0,49

20

Tavi XIII. korelacia_______________________________________________________________________(d) =+0,25 (e) =0,00 (v) arawrfivi

diapazonis SezRudva da r –is mniSvneloba

-is mniSvneloba mgrZnobiarea mniSvnelobaTa im diapazonisadmi, rac –ma da –ma SeiZleba miiRos. erT-erTi an orive cvladis mniSvnelobaTa diapazonis SemcirebasTan erTad –is mniSvnelobac mcirdeba. aseTi kavSiris sailustraciod ganvixiloT suraTis #13.1 monacemebi, romelTaTvisac = +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 aris +0,69, rac ufro naklebia, vidre +0,78-is toli , 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): -is mniSvneloba, romelic da cvladebis saerTo cvalebadobas asaxavs.

me-9 TavSi warmovadgine (eta kvadrati), rogorc eqsperimentSi damoukidebeli cvladis damokidebul cvladze gavlenis siZlieris sazomi. korelaciur kvlevebSi determinaciis koeficienti, ( -is mniSvnelobis kvadrati), warmoadgens –sa da –s Soris kavSiris siZlieris analogiur sazoms.

-is interpretacia –is interpretaciis msgavsia. kerZod, gamoxatavs, qulebis erTi simravlis cvalebadobis ra proporciaa dakavSrebuli qulebis meore simravlis cvalebadobasTan. magaliTad, cxrilis #13.1 monacemebisaTvis, = +0,78; maSasadame, -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 da cvladebs Soris korelaciis arseboba ar niSnavs, rom erTi iwvevs meores. Tu da cvladebi korelaciaSia, maSin am cvladebs Soris mizez-Sedegobrivi kavSiris sul mcire sami varianti arsebobs:

iwvevs –s.

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iwvevs –s arc da arc mizezobriv gavlenas ar axdens erTmaneTze, orive

gamowveulia mesame, Z cvladiT. ganvixiloT magaliTi martoobisa ( cvladi) da depresiis ( 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 ( cvladi) da stresis done ( 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 ( 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 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 ( cvladi) da dRis ganmavlobaSi energiis dones ( 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 #13.8ori cvladis, martoobisa ( ) da depresiis ( ), korelaciis Sesabamisi SesaZlo mimzez-Sedegobrivi kavSirebi.

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a) miRebuli korelacia:

b) miRebuli korelaciis amxsneli SesaZlo mizez-Sedegobrivi kavSirebi:pirveli SesaZlebloba

cvladi iwvevs cvladspirovnebis martosuloba ( ) iwvevs mis depresiulobas ( ).

meore SesaZleblobacvladi iwvevs cvlads

pirovnebis depresiuloba ( ) iwvevs mis martosulobas ( ).

mesame SesaZleblobacvladi iwvevs da cvladebs

saTanado socialuri unarebis nakleboba ( cvladi) iwvevs pirovnebis martosulobasa ( ) da depresiulobas ( ).

SeamowmeT Tqveni codna XIII-31. ras gveubneba –is plus an minus miSani X da Y cvladebis kavSiris

Sesaxeb? 2. ras gveubneba –is ricxviTi mniSvneloba X da Y cvladebis kavSiris

Sesaxeb? 3. ra gavlena SeiZleba hqondes X an Y cvladis monacemebis

diapazonis SezRudvas –is mniSvnelobaze? 4. gamoTvaleT determinaciis koeficienti Semdegi

mniSvnelobebisaTvis:

a. +0,12 g. _0,74 e. _0,59b. _0,36 d. +0,23 v. +0,19

miRebulia da cvladebs Soris dadebiTi korelacia

adamianebs, romelTac aqvT martoobis maRali qula ( ), aqvT depresiis maRali qula ( ).adamianebs, romelTac aqvT martoobis dabali qula ( ), aqvT depresiis dabali qula ( ).

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5. Tqven akvirdebiT kavSirs da 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 Sorisb. dadebiTi korelacia yavis smasa da sigaretis mowevas Sorisg. dadebiTi korelacia bavSvebis mier televiziiT Zaladobis

yurebasa da maT agresiul qcevas Soris.

pirsonis r koeficientis statistikuri mniSvnelovnobis Semowmeba

populaciis korelacia, ρ

pirsonis koeficienti aRwers SerCevis da qulebs Soris wrfiv kavSirs. rogorc qulebis nebismieri simravlis SemTxvevaSi, SerCeva miRebulia qulebis ufro didi populaciidan da SeiZleba gvsurdes, SevawmoT, arsebobs Tu ara da monacemebs Soris korelacia populaciaSi. populaciaSi da monacemebs Soris wrfiv kavSirs warmoadgens parametri ρ (berZnuli anbanis me-17 aso, ro ). martoobisa da depresiis qulebis korelaciis Sesaxeb magaliTSi miRebuli = +0,78, gvaZlevs ρ–s savaraudo mniSvnelobas. Tumca, SerCevis Secdomis gamo, ρ–s toli ar iqneba. radgan SerCevis safuZvelzea miRebuli, sxvadasxva SerCevisaTvis gamoTvlili , 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 axasiaTebs garkveuli ρ–s mqone populaciidan aRebul SerCevas. Tumca, SesaZlebelia SevamowmoT hopoTeza ρ–s nebismier mniSvnelobasTan dakavSirebiT, yvelaze 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 populaciaSi nulovani korelaciisagan martivia. saWiroa hopoTezebis statistikuri Semowmebis nacnobi safexurebis gavla:

vaxdenT nulovani hopoTezis, -isa, da alternatiuli hipoTezis, -is, formulirebas;

vadgenT -is SerCeviT ganawilebas im varaudiT, rom WeSmaritia. es ganawileba mocemulia cxrilSi #A.5;

virCevT mniSvnelovnobis dones; cxrilSi #A.5 mocemuli –is SerCeviTi ganawilebis mixedviT vpoulobT -

is kritikul mniSvnelobas, romelic aRiniSneba, rogorc ; -is SerCeviT ganawilebaSi ganvsazRvravT uaryofis ares.

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: ro, populaciis korelaciis koeficientis aRmniSvneli simbolo.


SerCevis monacemebis mixedviT viTvliT korelaciis koeficients, romelic aRiniSneba ;

imis mixedviT, xvdeba Tu ara uaryofis areSi, viRebT gadawyvetilebas –is miRebis an uaryofis Sesaxeb.

statistikuri hipoTeza

nulovani hopoTeza. nulovani hopoTeza aseTia: : ρ = 0

es hipoTeza ambobs, rom populaciaSi X da Y cvladebs Soris korelacia nulis tolia.

alternatiuli hipoTeza. alternatiuli hipoTeza Semdegnairia: : ρ ≠ 0

es hipoTeza ambobs, rom populaciaSi X da Y cvladebs Soris korelacia ar aris nulis toli.

r -is SerCeviTi ganawileba da kritikuli mniSvnelobebiradgan –s SeuZlia 0-dan 1-mde mniSvnelobebi miiRos, misi SerCeviTi ganawileba : ρ = 0–sTvis SeiZleba advilad ganisazRvros. Tumca, rogorc sxva statistikuri kriteriumebis SemTxvevaSi, -is SerCeviTi ganawileba damokidebulia mis Tavisuflebis xarisxze. –saTvis df udris – 2, sadac

aris im qulebis wyvilebis raodenoba, saidanac gamoTvlilia . –is kritikuli mniSvnelobebi, anu , –is SerCeviTi ganawilebebidan im SemTxvevebisTvis, roca df meryeobs 1-dan 100-mde, da roca = 0,05 da 0,01-s, warmoadgenilia danarTis cxrilSi #A.5.

uaryofis zonis dadgenaTu WeSmaritia, mosalodnelia, rom –is iseT mniSvnelobebs miviRebT, romlebic aRemateba nuls. Sesabamisad, uaryofis zona aris –is is mniSvneloba, romelic udris an metia . roca absoluturi mniSvnelobiT (anu –is + an _ niSans yuradRebas Tu ar mivaqcevT,) tolia an masze metia, is xvdeba uaryofis areSi. Tu WeSmaritia, aseTi albaToba alfas mniSvnelobis tolia an masze naklebia. gadawyvetilebis miRebisaTvis saWiro wesebistatistikuri hipoTezis Sesaxeb gadawyvetilebis miRebis wesebi nacnobi nimuSis mixedviT xdeba:

Tu uaryofis areSia, maSin:uaryaviT : ρ = 0miiReT : ρ ≠ 0

Tu ar xvdeba uaryofis areSi, maSinar uaryoT ar miiRoT

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am process giCvenebT qvemoT mocemuli magaliTi. cxrili #13.9 ajamebs im gadawyvetilebebsa da daskvnebs, rasac –is mixedviT viRebT.

cxrili #13.9Sejameba: statistikuri hipoTezebis –is mixedviT Semowmebisas gadawyvetilebebis miReba da daskvnebis gakeTeba. gamoyenebulia mniSvnelovnobis 0,05 done.

Tu xvdeba uaryofis areSi, roca , maSin:

Tu ar xvdeba uaryofis areSi, roca , maSin:

albaToba 0,05-ze naklebia an misi tolia, anu

albaToba 0,05-ze metia, anu > 0,05.

uarvyofT : ρ = 0-s uarvyofT -s

viRebT : ρ ≠ 0-s. gviCvenebs, rom populaciis nulisagan gansxvavebulia.

uarvyofT -s. gviCvenebs, rom populaciis nulisagan gansxvavebulia.

statistikurad mniSvnelovania 0,05 doneze.

ar aris statistikurad mniSvnelovani 0,05 doneze.

viRebT gadawyvetilebas, rom im populaciaSi, saidanac SerCeva iyo aRebuli, da cvladebi korelaciaSia.

viRebT gadawyvetilebas, rom im populaciaSi, saidanac SerCeva iyo aRebuli, da cvladebi ar aris korelaciaSi.

daSvebebi r -is statistikuri mniSvnelovnobis SemowmebisaTvis-is statistikuri mniSvnelovnobis Semowmebisas igulisxmeba, rom populaciaSi da 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): da

cvladebis iseTi ganawileba, romlisTvisac (1) qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (2) qulebi normalurad aris ganawilebuli im populaciaSi, saidanac aRebulia SerCeva; (3) TiToeuli qulisaTvis qulebis ganawileba populaciaSi normaluria da (4) TiToeuli qulisaTvis qulebis ganawileba populaciaSi normaluria.

26


ã magaliTi #13.1

martoobasa da depresias Soris statistikuri mniSvnelovnobis Semowmeba

amocanacxrilis #13.1 martoobisa da depresiis qulebisTvis pirsonis korelaciis koeficientia

= +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.

saWiro statistika: = +0,78

daSveba gamoyenebisaTvis: martoobisa da depresiis monacemebs organzomilebiani normaluri ganawileba aqvs im populaciaSi, saidanac aRebulia SerCeva.

statistikuri hipoTezebi: : ρ = 0: ρ ≠ 0

mniSvnelovnobis done:

Tavisuflebis xarisxi (df): df = _ 2 = 10 _ 2 = 8.

r -is kritikuli mniSvneloba: cxrilis #A.5 mixedviT, (8) = 0,632.

uaryofis zonebi: is mniSvnelobebi, romelTa absoluturi mniSvneloba udris an metia 0,632-ze.

gadawyvetileba: = +0,78-is absoluturi mniSvneloba metia, vidre = 0,632. aqedan gamomdinare, is xvdeba uaryofis areSi. maSasadame,

uarvyofT : ρ = 0-sviRebT : ρ ≠ 0-s

daskvna: populaciaSi martoobisa da depresiis qulebs Soris mniSvnelovani korelacia arsebobs. = +0,78 mxolod SemTxveviT ar gansxvavdeba populaciaSi WeSmariti nulovani korelaciisgan. populaciis korelaciis arsebuli saukeTeso Sefaseba aris = +0,78. martosuli adamianebi depresiulebic arian. determinaciis koeficienti, = (+0,78)² = 0,61, gamoxatavs, rom martoobisa da depresiis qulebs cvalebadobis 61 procenti saerTo aqvT.

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ã magaliTi #13.2

dakavSirebulia Tu ara iumoris grZnoba fizikur janmrTelobasTan?

amocanaerT-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 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: fizikuri janmrTeloba:

1 36 92 68 33 61 124 47 125 35 156 48 177 42 108 30 179 56 1110 39 1711 51 1412 54 813 60 314 29 1315 65 716 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.

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Tavi XIII. korelacia_______________________________________________________________________saWiro statistika: r.

daSveba gamoyenebisaTvis: iumoris kiTxvarisa da janmrTelobis monacemebs organzomilebiani normaluri ganawileba aqvs im populaciaSi, saidanac aRebulia SerCeva.

statistikuri hipoTezebi: : ρ = 0: ρ ≠ 0

mniSvnelovnobis done:

Tavisuflebis xarisxi (df): df = _ 2 = 16 _ 2 = 14.

r -is kritikuli mniSvneloba: cxrilis #A.5 mixedviT, (14) = 0,497.

uaryofis zonebi: is mniSvnelobebi, romelTa absoluturi mniSvneloba udris an metia 0,497–ze.

gamoTvla: gamosaTvlelad gamoviyenebT gamoTvliT formulas:

am formulisaTvis saWiro ricxviTi mniSvnelobebia:

= 16= 7908= 768= 176

= 39,092= 2242

= 589.824= 30.976.

am formulaSi ricxviTi mniSvnelobebis CasmiT miviRebT:

= (_0,65) =0,42

gadawyvetileba: = _0,65-is absoluturi mniSvneloba metia, vidre = +0,497 da maSasadame, xvdeba uaryofis areSi. gadawyvetilebebi –sa da -Tan dakavSirebiT aseTia:

29


uarvyofT : ρ = 0-sviRebT : 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. determinaciis koeficienti aRniSnavs, rom iumoris kiTxvarisa da janmrTelobis monacemebs saerTo aqvT cvalebadobis 42 procenti.

korelaciuri kvlevis Sedegebis warmodgena

korelaciuri kvlevis angariSi unda Seicavdes mniSvnelobas, Tavisuflebis xarisxs (df) , gamoyenebul mniSvnelovnobis dones da informacias imis Sesaxeb, iyo Tu ara 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 koeficienti Sesaferisia maSin, roca im cvladebis Sesabamisi qulebi, romelTa korelaciis dadgenasac vapirebT intervalebis an Sefardebis skalazea. Tumca, zogjer qulebi da 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 –iT. spirmenis ranguli korlaciis koeficienti gamoiTvleba formuliT:

sadac D aris cdis piris monacemebis rangebs Soris gansxvaveba aris kvlevaSi rangebis wyvilebis raodenoba

rogorc pirsonis koeficientis SemTxveaSi, -is mniSvnelobebic SeiZleba iyos dadebiTi an uaryofiTi da iRebdes mniSvnelobas -1,00-dan nulis gavliT +1,00-mde.

-is gamoyenebis magaliTi

30


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 meTodi, rogoricaa TamaSi, devna, gaqceva, miwaze garTxma da ase Semdeg. davuSvaT, rom mkvlevars, romelic am meTods iyenebs, aqvs hopoTeza, rom katebis TamaSisa ( ) da devnis ( ) 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.

-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 ( ) gamoklebulia devnis rangi (Y). Semdeg TiToeuli D ayvanilia kvadratSi, rac naCvenebia d svetSi. am svetis monacemebis jami, 80,50, gvaZlevs -s mniSvnelobas. formulaSi vsvamT

-isa da mniSvnelobebs da viTvliT -s: = +0,51.

spirmenis ranguli korelaciis koeficientis statistikuri mniSvnelovnoba

-is statistikuri mniSvnelovnobis Semowmeba iseve xdeba, rogorc pirsonis r –sa. nulovani hopoTezaa, rom populaciaSi ranguli korelacia, ρs, nulis tolia, anu

: ρs = 0. alternatiuli hipoTezis mixedviT, populaciaSi korelacia ar aris nuli toli, anu : ρs ≠ 0. –is kritikuli mniSvnelobebi 0,05 da 0,01 doneebze warmodgenilia danarTis cxrilSi #A-9. kritikuli mniSvnelobebi damokidebulia

da ara Tavisuflebis xarisxze. Tu metia an tolia kritikuli mniSvnelobisa, maSin unda miviRoT –is uaryofisa da –is miRebis gadawyvetileba.

magaliTisaTvis, -is kritikuli mniSvneloba maSin, roca = 10 da , aris 0,648. radgan = +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.

-is Tvisebebi

spirmenis korelaciis koeficienti gviCvenebs ori cvladis rangebs Soris SeTanxmebas. Tu da cvladebs erTnairi rangebi aqvs miniWebuli, maSin iqneba +1,00. Tu cvladebi sapirispirod aris Sefasebuli ise, rom cvladis rangi

31

Tavi XIII. korelacia_______________________________________________________________________1 Seesabameba cvladis yvelaze dabal rangs da cvladis yvelaze dabali rangi Seesabameba cvladis rangs 1, maSin 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 #13.10spirmenis ranguli korelaciis koeficientis ( ) 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: devna:

123456789

10

268475

10319

6,584159

1023

6,5

-4,5-2432-401-22,5

20,254,00

16,009,004,00

16,000,001,004,006,25

jami 80,50-is formulaa

Tu CavsvamT da = 10 mniSvnelobebs,

SeamowmeT Tqveni codna XIII-41. gansazRvreT: organzomilebiani normaluri ganawileba, ρ.2. CamoayalibeT statistikuri hopoTeza, romelic saWiroa imisTvis, rom

SevamowmoT, mniSvnelovnad gansxvavdeba Tu ara nulisagan.

3. mocemuli mniSvnelobebidan TiToeulisaTvis gansazRvreT Tavisuflebis xarisxi (df) da Semdeg cxrilis #A.5 mixedviT ipoveT

, roca α = 0,05 da α = 0,01. =

a. 8 g. 15 e. 20b. 32 d. 43 v. 75

32


4. qvemoT mocemuli ToToeuli da CamoayalibeT nulovani da alternatiuli hipoTezebi; Semdeg ipoveT mniSvnelovnobis 0,05 doneze. gansazRvreT, xvdeba Tu ara uaryofis areSi da Semdeg miuTiTeT, Tu ra gadawyvetilebas iRebT statistikur hipoTezebTan dakavSirebiT.

rrealuri

a. b. g. d. e.v.

+0,580-0,346-0,682+0,204+0,249-0,731

12122024809

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 gacdenebiGgamocdis Sefasebebis

saSualo1 22 722 6 883 1 994 8 685 6 666 11 777 10 528 6 789 10 7610 0 9611 5 9112 10 86

a. am monacemebisaTvis gamoTvliTi formulis meSveobiT ipoveT mniSvneloba

b. SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara populaciaSi nulovani korelaciisagan.

g. ra daskvnas akeTebT gacdenebisa da gamocdebis saSualo maCveneblebis mimarTebis Sesaxeb?

33


6. -isa da TiToeuli mniSnelobisaTvis gansazRvreT statistikuri nulovani hipoTezebi; Semdeg ipoveT mniSvnelovnobis 0,05 doneze. miuTiTeT, xvdeba Tu ara miRebuli uaryofis areSi da Semdeg miuTiTeT, ra gadawyvetilebas iRebT statistikur hipoTezebTan dakavSirebiT.

a. b. g. d. e.v.

+0,401-0,620+0,541-0,366+0,544-0,377

171224102130

Sejameba

korelaciuri kvlevis mizania or an met cvlads Soris kavSiris xarisxis dadgena.

wertilTa gabnevis diagramaze, ori monacemidan erT-erTi ( cvladi) warmodgenilia horizontalur RerZze (abscisaze) da meore ( cvladi) gadazomilia vertikalur RerZze (ordinataze). cdis piris qulas oridan TiToeul cvladze wertilTa gabnevis diagramaze erTi wertili warmoadgens.

dadebiTi anu pirdapiri kavSiris SemTxvevaSi cvladis mniSvnelobis zrdasTan erTad izrdeba cvladis mniSvnelobac.

uaryofiTi, anu Sebrunebuli kavSiris SemTxvevaSi cvladis mniSvnelobis zrdasTan erTad cvladis mniSvneloba mcirdeba.

pirsonis korelaciis koeficienti ganisazRvreba Semdegnairad:

aris da cvladebis jvaredina namravli da aRiniSneba –iT. Sesabamisad, korelaciis koeficienti SeiZleba aseve gamoisaxos,

rogorc:

standartuli qulebis gamoyenebiT –is mniSvneloba mocemulia formuliT:

r -is mniSvneloba gamoTvliTi formulis gamoyenebiT Semdegia:

34


kavSiris mimarTulebas gamoxatavs –is dadebiTi an uaryofiTi niSani xarisxi imisa, Tu ramdenad qmnis wertilTa gabnevis diagramis wertilebi

wrfes, warmodgenilia –is absoluturi mniSvnelobiT, romelic SeiZleba icvlebodes 0,00-dan 1,00-mde.

populaciaSi da cvladebs Soris wrfivi kavSiri warmodgenilia –Ti. -is statistikuri mniSvnelovnobis SemowmebisaTvis gamoiyeneba cxrili

#A.5. determinaciis koeficienti gamoxatavs cvladis cvalebadobis im

proporcias, romelic dakavSirebulia cvladis cvalebadobasTan. Tu da cvladebs Soris korelacia arsebobs, maSin cvladebs Soris

mizez-Sedegobrivi mimarTebis, sul mcire, sami SesaZlebloba arsebobs: aris –is mizezi aris –is mizezi arc da arc ar axdens erTmaneTze mizez-Sedegobriv gavlenas

da orives mizezia mesame, cvladi. spirmenis ranguli korelaciis koeficienti gamoiTvleba formuliT:

-is statistikuri mniSvnelovnobis Semowmeba xdeba cxrilis #A.6 gamoyenebiT.

ZiriTadi terminebi da simboloebi

dadebiTi kavSiri determinaciis

koeficienti ( ) korelacia korelaciis

koeficienti korelaciuri kvleva organzomilebiani

ganawileba

organzomilebiani normaluri ganawileba

pirsonis korelaciis koeficienti ( )

spirmenis ranguli

korelaciis koeficienti ()

uaryofiTi kavSiri wertilTa gabnevis

diagrama wrfivi kavSiri -sa da -is

jvaredina namravlebi ()

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

35


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

12345678910111213141516

64303739172936344432374750414661

170132129144145122124145137115127148148144133158

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 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 . am ori sazomisaTvis gabrazebis donisa da sisxlis wnevis cvalebadobis ra proporciaa saerTo?

v. Tqveni kvlevis monacemebi warmoadgineT am Tavis Sesabamis ganyofilebaSi (korelaciuri kvlevis 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

36


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 simaRlesa da niSnebis furclis niSnebs Soris 53 mesameklasel gogonasTan da +0,32-is toli

42 mesameklasel biWTan. a. daadgineT, mniSvnelovnad gansxvavdeba Tu ara

TiToeuli korelaciis koeficienti populaciaSi nulovani korelaciisagan 0,05 doneze.

b. 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:

gamocdastudenti Sualeduri: saboloo:

12345678910

87899780738581748089

84919687669079808986

a. daxazeT wertilTa gabnevis diagrama Sualeduri (gadazomeT abscisaze) da saboloo (gadazomeT ordinataze) gamocdebisaTvis.

b. wertilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar vlindeba?

g. gamoTvleT 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

37


mixedviT. 12 gogonasa da 14 biWis Sefasebis Sedegad miiReT Semdegi rangebi:

gogonebifizikuri

mimzidveloba: popularoba:

362912178410511

173101138759612

biWebifizikuri

mimzidveloba: popularoba:

5113948121141061327

9110461251381421173

a. gamoTvaleT 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 biWebis monacemebisaTvis da SeamowmeT, mniSvnelovnad gansxvavdeba Tu ara is populaciaSi nulovani korelaciisagan 0,05 doneze.

38


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 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 (sT.):

siRrmis aRqmis

Secdoma (sm.):

12345678910111213

0,51,74,26,02,85,30,93,41,45,11,94,82,6

4,33,12,10,46,33,71,61,22,45,20,64,15,4

a. aageT wertilTa gabnevis diagrama kompiuterTan muSaobis xangrZlivobisa (gadazomeT abscisaze) da siRrmis aRqmis Secdomis (gadazomeT ordinataze) monacemebisaTvis.

39


b. werilTa gabnevis diagramis monacemebisaTvis kavSiri dadebiTia, uaryofiTi, Tu kavSiri ar vlindeba?

g. gamoTvliTi formulis gamoyenebiT gamoTvaleT 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

123456789101112131415

71481961017712176913225

341424152927143819212927181830

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 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?

40


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 =652; pirveli gamocdis niSnisaTvis =2094 da = _859.

a. am monacemebis gamoyenebiT ipoveT televizoris yurebasa da gamocdis niSans Soris.

b. 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 bavSvebis ranJireba moaxdina. davuSvaT, rom 16 baSvisaTvis miRebulia Semdegi rangebi:

bavSvi mowonebadobis done

CarTulobis done

12345678910111213141516

69121638410515127131411

81224141113561397101615

a. am qulebisaTvis ipoveT -is mniSvneloba.b. SeamowmeT da gansazRvreT, mniSvnelovnad

gansxvavdeba Tu ara miRebuli populaciaSi nulovani korelaciisagan.

g. ra daskvnas akeTebT mowonebadobasa da klasSi CarTulobas Soris kaSiris Sesaxeb?

41

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