[Normal Probability Curve and Correlation] definition of ...
Transcript of [Normal Probability Curve and Correlation] definition of ...
99
[Normal Probability Curve and Correlation]
(Meaning and
definition of Normal Probability Curve)
(Characteristics of Normal
Probability Curve)
(Uses of Normal Probability
Curve)
(Meaning and definition of
correlation)
(Types of correlation)
(Utility, need
and Importance of correlation)
(Limitation of correlation)
(Measurement of correlation)
(Meaning and definition
of coefficient of correlation)
(Interpretation of coefficient of
correlation)
(Calculation of coefficient of
correlation)
100
(Spearmen rank order method)
(Characteristics of rank order
method)
(Limitation of rank order method)
(Pearson’s product moment
method)
(Characteristics of
Pearson’s product moment method)
(Limitation of
Pearson’s product moment method)
(Meaning of standard score)
(Z - score)
(T - score)
(Percentile)
(Calculation of
Percentile from ungrouped data)
(Calculation of
Percentile from grouped data)
101
[Normal Probability Curve and Correlation]
tc fdlh ijh{k.k ij fdlh lewg ds izkIrkad e/;eku ls nksuksa vksj ,dne
leku :Ik ¼vuqikr½ esa forfjr gksrs gSa rks izkIrkadksa ds bl izdkj ds forj.k dks
lkekU; forj.k dgrs gSaA
(Meaning and
definition of Normal Probability Curve)
fdlh Hkh lewg ds fdlh Hkh pj ij izkIr izkIrkad (Score) izk;% vkSlr
¼e/;eku½ dh vksj >qds gq, gksrs gSaA tc bu izkIrkadksa dk e/;eku ds nksuksa vksj
forj.k ,dne leku gksrk gS rks izkIrkadksa ds bl izdkj ds forj.k dks lkekU;
forj.k (Normal Distribution) dgrs gSa vkSj bl izdkj ds izkIrkadksa ds vkjs[k
(Graph) dks lkekU; forj.k oØ (Normal Distribution Curve) dgrs gSaA
okLrfodrk ;g gS fd izkIrkadksa dk bl izdkj dk lkekU; forj.k O;kogkfjd :Ik
esa dHkh ugha gksrk] blds yxHkx gh gksrk gS] vkSj bl yxHkx gksus vk/kkj ij gh
lEHkkouk dh tkrh gSA ;gh dkj.k gS fd bl izdkj ds lEHkkfor izkIrkadksa ds
vkjs[k dks lkekU; oØ (Normal Curve) u dgdj lkekU; lEHkkouk oØ
(Normal Probability Curve, NPC) dgrs gSaA
lkekU; lEHkkouk oØ dk fopkj loZizFke 1933 esa ÝkUl ds Mh0 eksboj
(De Movire) ds efLr’d esa vk;k FkkA mUgksaus bl oØ dh xf.krh; lehdj.k
102
Hkh izLrqr dh FkhA dqN fo}ku bl oØ dks muds uke ds vk/kkj ij Mh eksboj
oØ (De Moiver Curve) dgrs gSa vkSj bl oØ dk xf.krh; iz;ksx loZizFke
teZuh ds [kxksy'kkL=h xkWl (Carl Freidrich Gausss) us fd;k FkkA blfy,
dqN fo}ku bls xkWfl;u oØ (Gaussian Curve) dgrs gSa] ij lkekU; iz;ksx esa
bls lkekU; lEHkkouk oØ (Normal Probability Curve, NPC) gh dgrs gSaA
vc ;fn ge bls ifjHkkf"kr djuk pkgsa rks bl izdkj dj ldrs gSa &
lkekU; lEHkkouk oØ ,d ,slk lS)kfUrd] vkn'kZ ,oa xf.krh; oØ gS
ftlds izkIrkad e/;eku ds nksuksa vksj ,dne leku :Ik ls forfjr gksrs gSaA
(Characteristics of Normal
Probability Curve)
1- lkekU; lEHkkouk oØ e/; okys Hkkx rFkk nksuksa vksj lhekUrksa ij lefer
(Symmetrical) gksrk gSA bldh vkd`fr ?kaVkdkj (Bell Shaped) gksrh
gSA
2- lkekU; lEHkkouk oØ dh js[kk,¡ nksuksa vfUre Nksjksa (Ends) ij] X- v{k
dks u rks Li'kZ djrh gSa vkSj u gh X- v{k ds lekUrj gksrh gSaA ;g
/kkj.kk bl oØ esa vuUr rd cuh jgrh gSA
3- lkekU; lEHkkouk oØ ds e/; okys Hkkx esa vf/kdre vko`fÙk;k¡ gksrh gSaA
e/; ls tSls&tSls nkbZa o ckbZa vksj c<+rs gSa rks vko`fÙk;ksa dk vkdkj ,oa
foLrkj ,d fuf'pr Øe ls /khjs&/khjs de gksrk pyk tkrk gS vkSj fljksa
ij U;wure gksrk gSA
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4- lkekU; lEHkkouk oØ esa dsUnzh; izo`fÙk;ksa e/;eku (M), e/;kad (Mdn)
,oa cgqykad (Mo) ds eku leku gksrs gSa rFkk oØ ds e/; fcUnq ij fLFkr
gksrs gSaA
5- lkekU; lEHkkouk oØ u rks cgqr piVk gksrk gS vkSj u gh cgqr uqdhyk
gksrk gSA vkSlr Å¡pkbZ okys bl oØ dk oØrk xq.kkad (Coefficient of
Kurtosis) .263 gksrk gS vFkkZr~ Ku = .263
6- lkekU; lEHkkouk oØ lefer (Symmetrical) gksrk gS blfy, bldk
fo"kerk xq.kkad (Coefficient of Skewness) 'kwU; gksrk gS vFkkZr~ Sk = 0
7- lkekU; lEHkkouk oØ ds X- v{k dks ekud fopyu ( ) ds vk/kkj ij N%
Hkkxksa esa foHkkftr fd;k tkrk gSA e/;eku fcUnq ls rhu Hkkx nkbZa vksj vkSj
3 Hkkx ckbZa vksj gksrs gSaA nkbZa vksj ds rhu Hkkx e/;eku fcUnq ls Øe'k%
+1 , + 2 + 3 nwjh ij gksrs gSa vkSj ckbZa vksj ds rhu Hkkx e/;eku ls
Øe'k% -1 , - 2 - 3 nwjh ij gksrs gSaA
8- lkekU; lEHkkouk oØ ds e/; ls nksuksa vksj ds Hkkxksa esa vko`fÙk;ksa dk
forj.k o foLrkj leku :Ik ls 50% o 50% gksrk gSA
9- lkekU; lEHkkouk oØ esa e/;eku fcUnq ij fLFkr dksfV dh Å¡pkbZ
vf/kdre gksrh gS rFkk ;g dqy vko`fÙk;ksa (N) dh .3989 gksrh gSA
10- lkekU; lEHkkouk oØ e/;eku ls ,d ekud fopyu Åij o uhps vFkkZr~
1 ij viuh fn'kk ifjofrZr djrk gSA ;g oØ +1 , ls -1 ds chp
vk/kkj js[kk dh vksj vory (Concave) gksrk gS tcfd +1 ds Åij o
(Tails) ds uhps vFkkZr~ nksuksa fljks ij vk/kkj js[kk dh vksj mÙky
(Convex) gksrk gSA
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11- lkekU; lEHkkouk oØ rFkk vk/kkj js[kk ds e/; dk {ks=Qy lkekU;
lEHkkouk oØ {ks=Qy dgykrk gS vkSj ;g dqy vko`fÙk;ksa dks izdV djrk
gSA lkekU; lEHkkouk oØ dh fdUgha nks dksfV;ksa ds chp dk {ks=Qy mu
dksfV;ksa ds lkis{k izkIrkadksa ds chp vad ikus okys Nk=ksa dh la[;k dks
iznf'kZr djrk gSA ;g dqy {ks=Qy dk ,d fuf'pr izfr'kr gksrk gSA M
ls +1 ds e/; 34.13% izkIrkad gksrs gSa M ls +2 ds e/; 47.72%
izkIrkad gksrs gSa rFkk M ls +3 ds e/; 49.87% izkIrkad gksrs gSaA M ds
nksuksa rjQ 50% o 50% izkIrkad gksrs gSaA
12- lkekU; lEHkkouk oØ esa e/;eku ls fopyu dh fofHkUu lkaf[;dh; ekiksa
(Different Measures of Deviations) dh x.kuk ljyrkiwoZd dh tk
ldrh gS( tSls& Q1 ¼izFke prqFkkZa'k½ rFkk Q3 ¼r`rh; prqFkkZa'k½ leku :Ik
ls ,d gh nwjh ij gksrs gSaA ;g nwjh 0.6745 gksrh gS rFkk e/;eku ls
nksuksa vksj vadksa dh /kukRed ,oa _.kkRed fn'kk esa ekud fopyu nwjh
( ) Hkh leku gksrh gSA
13- lkekU; lEHkkouk oØ esa prqFkkZa'k fopyu dk
eku ekud fopyu ds eku yxHkx
gksrk gSA lw= esa bls fuEukafdr
:Ik esa fy[krs gSa&
rFkk
14- lkekU; lEHkkouk oØ dh xf.krh; lehdj.k gS &
√
ftlesa & fcUnq ij fLFkr dksfV dh Å¡pkbZ
105
e/;ekuksa ls izkIrkad nwjh
izkIrkadksa dk ekud fopyu
dqy vko`fÙk
fLFkjkad ftldk eku gksrk gSA
fLFkjkad ftldk eku gksrk gSA
(Uses of Normal Probability
Curve)
lkekU; lEHkkouk oØ ds lEca/k esa igys dgk tk pqdk gS fd euksfoKku
vkSj f'k{kk'kkL= esa cgqr ls ,sls pj gSa ftudk forj.k lkekU; lEHkkouk oØ ds
vuq:Ik gksrk gSA ekiu vkSj ewY;kadu ds {ks= esa lkekU; lEHkkouk oØ dk fo'ks"k
mi;ksx vkSj egRo gSA cqf)] Le`fr] fpUrk vkfn ls lEcaf/kr ekiksa ;k izkIrkadksa dks
lkekU; lEHkkouk oØ ds }kjk iznf'kZr fd;k tkrk gSA lkekU; lEHkkouk oØ dh
lgk;rk ls fuEukafdr izdkj dh leL;kvksa dk lek/kku mnkgj.kksa dh lgk;rk ls
le>k;k x;k gSA
1- nh gqbZ lhekvksa ds e/; izkIrkadksa dk izfr'kr Kkr djukA
2- lkekU; forj.k esa fofHkUu izfr'krksa ds e/; ds izkIrkadksa dh lhek,¡ Kkr
djukA
3- nks vfrO;kih forj.kksa dh rqyuk djukA
4- ijh{k.k ds iz'uksa ds dfBukbZ Lrj dks Kkr djukA
106
5- lkekU; lEHkkouk ds vk/kkj ij ,d vad forj.k dks fofHkUu mi&lewgksa esa
foHkkftr djukA
6- ekud izkIrkadksa dh x.kuk djukA
(Meaning and definition of
correlation)
lkekU;r% lglEca/k dk vFkZ nks oLrqvksa] lewgksa vFkok ?kVukvksa ds vkilh
lEca/k ls fy;k tkrk gS ijUrq lkaf[;dh esa lglEca/k ls rkRi;Z fdlh oLrq] lewg
vFkok ?kVuk ds nks ;k nks ls vf/kd pjksa (Variables) ds chp ik, tkus okys
lEca/kksa ls gksrk gSA ySFkjkWi ds 'kCnksa esa &
^lglEca/k nks pjksa ds chp ik, tkus okys la;qDr lEca/k dks bafxr djrk
gSA*
fdlh oLrq] lewg vFkok ?kVuk esa vusd pj gks ldrs gaS ijUrq ,d le;
esa muesa ls fdUgha nks pjksa ds chp lEca/k dk gh v/;;u fd;k tk ldrk gS
blfy, ySFkjkWi us nks pjksa dh gh ckr dh gSA bu nks pjksa esa Hkh igys pj esa
ifjorZu gksus ls ftl izdkj dk izHkko nwljs pj ij iM+rk gS Bhd mlh izdkj
dk izHkko nwljs pj esa ifjorZu gksus ls igys pj ij iM+rk gS blfy, ySFkjkWi us
la;qDr lEca/k dh ckr dgh gS ij bl ifjHkk"kk esa pjksa ds vk/kkj ¼oLrq] lewg
vFkok ?kVuk½ dks ugha fy;k x;k gS blfy, ;g ifjHkk"kk dqN v/kwjh lh yxrh
gSA gekjh n`f"V ls lglEca/k dks fuEufyf[kr :Ik esa ifjHkkf"kr djuk pkfg,&
^lg&lEca/k ls rkRi;Z fdlh oLrq] lewg vFkok ?kVuk ds ,d pj esa gksus
okys ifjorZu ls nwljs pj esa gksus okys ifjorZu ls gksrk gSA*
107
(Types of correlation)
lglEca/k rhu izdkj dk gksrk gS &
tc fdlh oLrq] lewg vFkok ?kVuk ds fdlh ,d
pj ds eku esa o`f) gksus ls nwljs lkgp;Z pj ds eku esa o`f) gksrh gS
vFkok mlds eku esa deh gksus ls nwljs lkgp;Z pj ds eku esa deh gksrh
gS rks eku nksuksa pjksa ds chp ik, tkus okys bl vuq:Ik lEca/k dks
/kukRed lglEca/k djrs gSaA mnkgj.kkFkZ fdlh xSl dk leku nkc ij
rkiØe c<+us ls mldk vk;ru c<+uk vFkok rkiØe de gksus ls mldk
vk;ru de gksuk xSl ds nks pjkS& rkiØe vkSj vk;ru ds chp /kukRed
lglEca/k gSA
tc fdlh oLrq] lewg vFkok ?kVuk ds fdlh ,d
pj ds eku esa o`f) gksus ls nwljs lkgp;Z pj ds eku esa deh vkrh gS
vFkok mlds eku esa deh gksus ls nwljs lkgp;Z pj ds eku esa o`f) gksrh
gS rks bu nksuksa pjksa ds chp ik, tkus okys bl izfrdwy lEca/k dks
_.kkRed lglEca/k dgrs gSaA mnkgj.kkFkZ fdlh xSl dk leku rkiØe ij
nkc c<+us ls mldk vk;ru de gksuk vFkok nkc de gksus ls mldk
vk;ru c<+uk] xSl ds nks pjksa&nkc vkSj vk;ru ds chp _.kkRed
lglEca/k gSA
tc fdlh oLrq ] lewg vFkok ?kVuk ds fdlh ,d pj
esa ifjorZu gksus ls nwljs lkgp;Z pj ij dksbZ izHkko ugha iM+rk rks bu
nksuksa pjksa ds chp ds lEca/k dks 'kwU; lglEca/k dgrs gSaA mnkgj.kkFkZ fdlh
xSl ds vk;ru ds c<+us vFkok ?kVus ls mlds jklk;fud lw= esa dksbZ
108
vUrj u gksuk xSl ds nks pjksa& vk;ru vkSj jklk;fud lw= ds chp 'kwU;
lglEca/k gSA
(Utility, need
and Importance of correlation)
foKku dk ewy vk/kkj dk;Z&dkj.k lEca/k (Cause and Effect
Relationship) gSA bl lEca/k dh tkudkjh ds vk/kkj ij fdlh ,d {ks= esa
gksus okys ifjorZu ls fdlh nwljs {ks= esa gksus okys ifjorZu dh Hkfo";ok.kh dh
tk ldrh gSA bl izdkj foKku ds {ks= eas rks lglEca/k dh lcls vf/kd
mi;ksfxrk gS] mldh lcls vf/kd vko';drk gS vkSj mldk lcls vf/kd egRo
gSA bl ;qx esa euksoSKkfudksa us Hkh ekuo O;ogkj ds dkjdksa dk irk yxkdj
dk;Z&dkj.k lEca/kksa dh LFkkiuk dh gSA vkt ekuoh; O;ogkj esa dk;Z&dkj.k
lEca/kksa dks le>us ds fy, lglEca/k izfof/k;ksa (Correlation Techniques) dk
iz;ksx fd;k tkrk gSA bl izdkj vkt euksfoKku vkSj f'k{kk ds {ks= esa Hkh
lglEca/k dh cM+h mi;ksfxrk gS] bldh cM+h vko';drk gS vkSj bldk cM+k
egRo gSA ;gk¡ f'k{kk ds {ks= esa lglEca/k dh mi;ksfxrk] vko';drk ,oa egRo
dk la{ksi esa o.kZu izLrqr gSA
1- nks fo"k;ksa ds lglEca/kksa dh lgk;rk ls fdlh Nk= dh ,d fo"k; dh
;ksX;rk ds vk/kkj ij mldh nwljs fo"k; dh ;ksX;rk dk vuqeku yxk;k
tk ldrk gSA
2- nks fo"k;ksa ds lglEca/k dh lgk;rk ls ;fn mijksDr vuqeku lgh u
fudys rks ;g funku djuk vko';d gks tkrk gS fd mldk dkj.k D;k
gSA funku djus ds ckn mipkjkRed f'k{k.k dh O;oLFkk dh tkrh gSA
109
3- v/;;u fo"k;ksa ds lglEca/k dh lgk;rk ls Nk=ksa dks 'kSf{kd ,oa
O;kolkf;d funsZ'ku nsus esa lgk;rk feyrh gSA
4- fØ;kRed vuqlU/kku esa lglEca/k dk iz;ksx fo'ks"k :Ik ls fd;k tkrk gSA
(Limitation of correlation)
1- fdUgha nks fo"k;ksa ds lglEca/k ls muds chp lglEca/kksa ds ewy dkj.kksa dk
Kku ugha gksrkA
2- fdUgha nks fo"k;ksa ds lglEca/k Nk=ksa dh la[;k ij fuHkZj djrs gSa] NksVs
lewg ls izkIr lglEca/k dh vis{kk cM+s lewg ls izkIr lglEca/k vf/kd
fo'oluh; gksrk gSA
3- fdUgha nks fo"k;ksa ds chp dk lglEca/k fo"k;ksa dh izd`fr ds lkFk&lkFk
Nk=ksa dh izd`fr ¼;ksX;rk] :fp vkSj vfHk:fp½ ij Hkh fuHkZj djrk gSA vr%
,d fun'kZ ls izkIr lglEca/k nwljs fun'kZ ij mlh :Ik esa ykxw ugha fd;k
tk ldrkA
4- lglEca/k xq.kkad dk vFkkZiu ifjfLFkfr;ksa ij fuHkZj djrk gS blfy,
mldh O;k[;k djuk FkksM+k dfBu dk;Z gSA
(Measurement of correlation)
lkekU; n`f"V ls fdUgha nks pjksa ds chp lglEca/k ds fo"k; esa dsoy bruk
gh dgk tk ldrk gS fd buesa /kukRed lglEca/k gS vFkok _.kkRed lglEca/k
gS vFkok 'kwU; lglEca/k gSA dHkh&dHkh ;g dFku djuk Hkh lEHko gksrk gS fd
mPp vFkok fuEu dksfV dk /kukRed vFkok _.kkRed lEca/k gS] ijUrq ;g ugha
dgk tk ldrk fd fdruk /kukRed vFkok _.kkRed lEca/k gSA lglEca/k dh
110
ek=k dk ekiu djus ds fy, fo}kuksa us lglEca/k xq.kkad (Coefficient of
Correlation) dk fodkl fd;k gSA ;gk¡ lglEca/k xq.kkad dk vFkZ ,oa mldh
x.kuk fof/k dk o.kZu izLrqr gSA
(Meaning and definition
of coefficient of correlation)
lglEca/k xq.kkad ,d ,slh vuqikfrd la[;k gksrh gS tks nks pjksa ds chp
lglEca/k dh izd`fr ¼/kukRed] _.kkRed vFkok 'kwU;½ vkSj mldh ek=k nksuksa dk
Li"V cks/k djkrh gSA fxyQksMZ ds 'kCnksa esa &
^lglEca/k xq.kkad og la[;k gS tks gesa ;g crkrh gS fd nks phtsa (pj]
Variables) vkil esa fdl lhek rd lEcaf/kr gSa vkSj muesa ls fdlh ,d esa
ifjorZu gksus ls nwljs esa fdl lhek rd ifjorZu gksrs gSaA*
(Interpretation of coefficient of
correlation)
lglEca/k xq.kkad ,d vuqikfrd la[;k gksrh gS ftldk eku &1 ls $1
rd gksrk gSA &fpUg _.kkRed lglEca/k vkSj $ fpUg /kukRed lglEca/k izdV
djrk gS vkSj &1 ls $1 ds chp dh la[;k,¡ lglEca/k dh ek=k dk cks/k djkrh
gSaA lglEca/k xq.kkad dh O;k[;k fuEukafdr lkfj.kh ds vk/kkj ij dh tkrh gSA
111
$ 1-00
$ -91 ls $ -99 rd
$ -71 ls $ -99 rd
$ -41 ;s $ -70 rd
$ -21 ls $ -40 rd
$ -01 ls $ -20 rd
00
& -01 ls & -20 rd
& -21 ls & -40 rd
& -41 ls & -40 rd
& -71 ls & -90 rd
& -91 ls & -99 rd
& 1-00
iw.kZ (Perfect) /kukRed lglEca/k
vR;Ur mPp (Very High) /kukRed
lglEca/k
mPp (High) /kukRed lglEca/k
lkekU; (Moderate) /kukRed
lglEca/k
fuEu (Low) /kukRed lglEca/k
vR;Ur fuEu (Very Low) /kukRed
lglEca/k
fuEu (Low) _.kkRed lglEca/k
lkekU; (Moderate) _.kkRed
lglEca/k
mPp (High) _.kkRed lglEca/k
vR;Ur mPp (Very High) _.kkRed
lglEca/k
iw.kZ (Perfect) _.kkRed lglEca/k
(Calculation of coefficient of
correlation)
lglEca/k xq.kkad Kkr dh djus dh dbZ fof/k;ksa dk fodkl fd;k x;k gS
ftuesa nks fof/k;ksa dk iz;ksx vf/kd fd;k tkrk gS& ,d Lih;jeSu dh jSad vUrj
112
fof/k (Spearman’s Rank Difference Method) vkSj nwljh ih;jlu dh
izksMDV eqeSUV fof/k (Pearson’s Product Moment Method) A ;gk¡ bu nksuksa
fof/k;ksa ds iz;ksx dks mnkgj.k }kjk Li"V fd;k tk jgk gSA
(Spearmen rank order method)
bl fof/k dk fodkl pkYlZ Lih;jeSu (Charls Spearman) us fd;k gS
blfy, bls mUgha ds uke ij Lih;jeSu dh jSad vUrj fof/k dgk tkrk gSA bl
fof/k dk iz;ksx rc fd;k tkrk gS tc lewg ¼fun'kZ] izfrn'kZ] Sample½ dk
vkdkj NksVk ¼30 ls de½ gksrk gS vkSj izkIrkad ¼vk¡dM+s] Data½ vadksa (Figures)
vFkok Js.kh Øe (Rank Order) esa fn, gksrs gSaA nksuksa ifjfLFkfr;ksa esa vxzkafdr
lw= dk iz;ksx fd;k tkrk gS&
∑
ftlesa& ¼jksg] Rho½ lglEca/k xq.kkad
∑ ;ksx
Øeksa (Ranks) ds vUrj ds oxZ (Square)
lewg ds lnL;ksa dh la[;k
10 Nk=ksa us 100&100 vad dh xf.kr vkSj foKku dh ijh{kkvksa esa
fuEukafdr vad izkIr fd, gSaA jSad vUrj fof/k ls bu izkIrkadka ds chp lglEca/k
xq.kkad Kkr dhft,A
113
Nk= A B C D E F G H I J
xf.kr esa izkIrkad 50 35 55 47 28 60 52 30 20 65
foKku esa izkIrkad 48 50 60 55 35 53 58 40 25 56
(Calculation)
loZizFke fuEufyf[kr rkfydk cuk,¡xs vkSj mlesa izFke dkye
esa Nk=] f}rh; dkWye esa xf.kr ds izkIrkad vkSj r`rh; dkWye esa foKku ds
izkIrkad p<+k,¡xsA
vc izFke pj ¼xf.kr ds izkIrkadksa½ dh jSad (R1) fu/kkZfjr
djsaxsA ftl Nk= ds xf.kr esa lcls vf/kd izkIrkad gS ml izkIrkad dks jSad 1]
blls de izkIrkad dks jSad 2 vkSj blh Øe ls jSad nsrs gq, lcls de izkIrkad
dks jSad 10 nsaxsA ;s jSad pkSFks dkWye esa vafdr dh tk,¡xhA Bhd blh izdkj nwljs
pj ¼foKku ds izkIrkadksa½ dh jSad (R2) fu/kkZfjr djsaxsA ;s jSad ik¡posa dkWye esa
p<+kbZ tk,¡xhA
nksuksa pjksa ds jSad (R1) rFkk (R2) fu/kkZfjr djus ds ckn
buds vUrj (R – R2 = D) Kkr djsaxs vkSj mls NBs dkWye esa p<+k,¡xsA
blds ckn lHkh D dks D2 esa ifjofrZr djds lkrosa dkWye esa
p<+k,¡xsA
blds ckn lHkh ∑D vkSj ∑ dh x.kuk djsaxs vkSj mUgsa
Øe'k% mUgha dkWyeksa ¼NVs vkSj lkrosa½ ds Bhd uhps vafdr djsaxsA ns[ksa rkfyd&
114
∑D dk iz;ksx lw= esa ugha fd;k tkrk gS ijUrq bldh x.kuk blfy, dh tkrh
gS fd blls iwoZ x.kukvksa esa gksus okyh Hkwy dk irk py tkrk gSA ∑D dk eku
ges'kk 'kwU; (Zero, 0) vkuk pkfg,A ;fn ;g 'kwU; ugha vkrk rks bldk vFkZ gS
fd jSad yxkus vFkok jSadksa dk vUrj fudkyus esa dgha Hkwy gks xbZ gSA ml fLFkfr
esa jSadksa vkSj jSadksa ds vUrj dks pSd djds Hkwy dk lq/kkj dj 'kq) x.kuk dh
tkrh gSA
R1
R2
(R1 –
R2 - D) D2
A
B
C
D
E
F
G
H
I
J
50
35
55
47
28
60
52
30
29
65
48
50
60
55
35
53
58
40
25
56
5
7
3
6
9
2
4
8
10
1
7
6
1
4
9
5
2
8
10
3
-2
1
2
2
0
-3
2
0
0
-2
4
1
4
4
0
9
4
0
0
4
N = 10 ∑ ∑
115
lw= esa ∑ vkSj N dk eku j[kdj x.kuk djsaxsA
∑
vUr esa lglEca/k xq.kkad .82 dh O;k[;k djsaxsA .82 Nk=ksa
ds xf.kr vkSj foKku ds izkIrkadksa esa mPp (High) /kukRed lglEca/k dks izdV
djrk gSA
(Tied Scores) dh fLFkfr esa jSdksa dk Øe fu/kkZj.k &
jSad vUrj fof/k ls lglEca/k xq.kkad Kkr djs le; dHkh&dHkh ,slh fLFkfr vkrh
gS fd nks ;k nks ls vf/kd Nk=ksa ds izkIrkad leku gksrs gSaA ,slh fLFkfr esa ge
izkIrkadksa dks Øekuqlkj jSad nsrs tkrs gSa vkSj tc leku izkIrkadksa dks jSad nsus dk
iz'u mBrk gS rks mUgsa ,d ds vkxs nwljk jSad nsrs gSa vkSj muds vkSlr dks fQj
lHkh izkIrkadksa dk jSad nsrs gSaA
116
(Characteristics of rank order
method)
¼1½ bl fof/k ls lglEca/k xq.kkad dh x.kuk ljyrk ls dh tk ldrh gSA
¼2z½ fo"ketkrh; vk¡dM+ksa ¼iznÙkksa] Data½ dh fLFkfr esa Hkh ;g fof/k mi;ksxh
gksrh gSA
(Limitation of rank order method)
¼1½ bl fof/k dk iz;ksx dsoy NksVs lewgksa ¼30 ls de½ ij gh fd;k tk ldrk
gSA
¼2z½ bl fof/k }kjk fudkyk x;k lglEca/k xq.kkad vis{kkd`r de fo'oluh;
gksrk gSA
(Pearson’s product moment
method)
bl fof/k dk fodkl ih;jlu (Karl Pearson) us fd;k gS blfy, bls
mUgha ds uke ij ih;jlu dh izksMDV eqeSUV fof/k dgrs gSaA bl fof/k dk iz;ksx
NksVs&cM+s lHkh lewgksa ij fd;k tk ldrk gSA ih;jlu fof/k ls lglEca/k xq.kkad
fudkyus ds fy, dbZ lw=ksa dk fodkl fd;k x;k gSA ;gk¡ ge NksVs lewg vkSj
cM+s lewgksa ds fy, lokZf/kd iz;ksx fd, tkus okys lw=ksa dh lgk;rk ls lglEca/k
xq.kkad fudkyus ds mnkgj.k izLrqr dj jgs gSaA
tc lewg ¼funFkZ] U;k;n'kZ] Sample½ NksVk ¼30 ls de½ gksrk gS rks
lglEca/k xq.kkad fudkyus ds fy, fuEukafdr lw= dk iz;ksx fd;k tkrk gSA
117
∑ ∑ ∑
√[ ∑ ∑ ][ ∑ ∑ ]
ftlesa& r = lglEca/k xq.kkad
N = lewg ds lnL;ksa dh la[;k
∑ = ;ksx
x = izFke pj (X) ds ekus gq, e/;eku ls fopyu
y = f}rh; pj (Y) ds ekus gq, e/;eku ls fopyu
(Characteristics of
Pearson’s product moment method)
¼1½ bl fof/k ls cM+s lewgksa ¼fun'kZ] Samples½ ds pjksa esa lglEca/k xq.kkad
fudkyk tk ldrk gSA
¼2½ 'kks/k dk;ksZa esa fun'kZ ges'kk cM+s gksrs gSa vr% ftu 'kks/kksa esa lglEca/k
fudkyk tkrk gS ogk¡ blh fof/k dk iz;ksx fd;k tkrk gSA
(Limitation of
Pearson’s product moment method)
¼1½ NksVs lewg (30 ls de) ij bl fof/k dk iz;ksx djus ls x.kuk esa le;
vf/kd yxrk gSA
118
¼2½ bl fof/k ds ftrus Hkh lw= gSa] os dkQh yEcs gSaA fQj mudh ykWftd D;k
gS] bldh tkudkjh lkaf[;dh ds lkekU; Kku esa ugha djkbZ tkrh gS]
blfy, bldk Lej.k djuk vkSj bldk iz;ksx djuk] nksuksa gh FkksM+s dfBu
dk;Z gSaA
10 Nk=ksa us 50&50 vadksa dh laLd`r vkSj fgUnh dh ijh{kkvksa esa fuEukafdr
vad izkIr fd,A ih;jlu fof/k ls bu izkIrkadksa ds chp lglEca/k Kkr dhftg,A
Nk= A B C D E F G H I J
laLd`r esa izkIrkad 30 40 38 35 28 40 35 20 34 45
fgUnh esa izkIrkad 26 38 35 30 30 36 30 20 28 40
loZ izFke 8 dkWyeksa dh rkfydk&4 [khaph vkSj mlds izFke
dkWye esa Nk=] f}rh; dkWye esa laLd`r ds izkIrkad vkSj rhljs dkWye esa fgUnh
ds izkIrkad vafdr fd,A
vc laLd`r esa izkIrkadksa dk e/;eku ekuus ds fy, mlds
lokZf/kd vSj U;wure izkIrkadksa dk vkSlr fudkysaxs tks
vk;sxkA vc rkfydk esa laLd`r ds izkIrkadksa esa 32-5 ds fudVre izkIrkad dk irk
yxk,¡xsA ;g 34 gSA blh dks laLd`r ds izkIrkadksa dk e/;eku (AM) ekusaxsA
119
blh izdkj fgUnh ds izkIrkadksa dk e/;eku gksxk
vr%
30 izkIrkad dks fgUnh ds izkIrkadksa dk e/;eku (AM) ekusaxsA
vc izFke pj laLd`r ds izkIrkadksa dk fopyu eku (X -
AMx) = x Kkr fd;k vkSj Bhd mlh izdkj fgUnh ds izkIrkadksa dk fopyu eku
(Y - AMy) = y Kkr fd;k vkSj mUgsa Øe'k% pkSFks vkSj ik¡pos dkWyeksa esa vafdru
fd;kA
vc Øe'k% x2,y2 vkSj xy dk eku fudkydj mUgsa Øe'k%
NBs] lkrosa vkSj vkBosa dkWyeksa esa vafdr fd;kA
blds ckn N, ∑ ∑ ∑ ∑ vkSj ∑ dk eku
fudkydj ;Fkk dkWyeksa esa vafdr djsaxsA
(X) (Y)
= x =y
x2 y2 xy
A
B
C
D
E
30
40
38
35
28
26
38
35
30
30AMy
-4
6
4
1
-6
-4
8
5
0
0
16
36
16
1
36
16
64
25
0
0
16
48
20
0
0
120
F
G
H
I
J
40
35
20
34AMx
45
36
30
20
28
40
6
1
-14
0
11
6
0
-10
-2
10
36
1
196
0
121
36
0
100
0
100
36
0
140
0
110
∑
∑
∑
∑
∑
lw= dk iz;ksx djus ij
∑ ∑ ∑
√ ∑ ∑ [ ∑ ∑ ]
√[ ][ ]
√[ ][ ]
√
√
121
vR;Ur mPp (Very High) /kukRed lglEca/kA
(Meaning of standard score)
vijks{k izd`fr ds pjksa ds ekiu ls izkIr izkIrkadksa dks vFkZ nsuk rFkk mudh
rqyuk djuk vius vki esa ,d leL;k gks tkrk gSA bl rjg dh ifjfLFkfr esa
izkIrkadksa dks vFkZ;qDr cukus ds fy,] izkIrkadksa dks ekud izkIRkkadksa (Standard
Scores) esa ifjofrZr fd;k tkr gSA ekud izkIrkad vius vki esa vFkZ;qDr gksrs
gS rFkk muds mi;ksx ds }kjk fofHkUu O;fDr;ksa dh ljyrk ls rqyuk dh tk
ldrh gSA ekud izkIrkad okLro esa fdlh lUnHkZ lewg (Reference Group) ds
fy, fofHkUu izkIrkadksa dh lkisf{kd fLFkfr dks izdV djrs gSaA blds fy, ewy
izkIrkadksa dks fdlh fuf'pr e/;eku o ekud fopyu okys rFkk Kkr izd`fr ds
forj.k ds :Ik esa ifjofrZr dj fn;k tkrk gSA ^^ekud izkIrkad** 'kCn ;qXe esa
ekud 'kCn dsoy bl ckr dk |ksrd gS fd bu izkIrkadksa dk e/;eku o ekud
fopyu iwoZ Kkr rFkk fuf'pr gS rFkk buds forj.k dh izd`fr Hkh iwoZ Kkr rFkk
fuf'pr gksrh gSA e/;eku rFkk ekud fopyu ds fHkUu&fHkUu gks ldus ds dkj.k
ekud izkIrkad vusd izdkj ds gks ldrs gSA tsM izkIrkad (Z-Scores), Vh
izkIrkad (T-Scores), lh izkIrkad (C-Scores), rFkk uoekud (Stanines) dqN
122
lokZf/kd izpfyr ekud izkIrkad gS rFkk bUgha ekud izkIrkadksa dh ppkZ vkxs dh
xbZ gSA
(Z - score)
tsM izkIrkad crkrs gSa fd izR;sd ewy izkIrkad ekud fopyu dh fdruh
bdkbZ;k¡ e/;eku ls vf/kd ;k de gSA nwljs 'kCnksa esa dgk tk ldrk gS fd tsM
izkIrkad] fdlh izkIrkad dh fLFkfr dks] e/;eku ds lUnHkZ esa rFkk ekud fopyu
dks ekiu dh bdkbZ ds :Ik esa ysrs gq,] Li"V djrk gSA tsM izkIrkad dk
/kukRed fpUg crkrk gS fd ewy izkIrkad e/;eku ls vf/kd gS rFkk _.kkRed
fpUg crkrk gS fd ewy izkIrkad e/;eku ls de gSA
tsM izkIrkadksa ds forj.k dk e/;eku 'kwU; rFkk ekud fopyu ,d cjkcj
gksrk gSA tsM izkIrkadksa dk eku izk;% -3Z ls +3Z ds chp gksrk gSA
ewy izkIrkadksa dks tsM izkIrkadksa esa rFkk tsM izkIrkadksa dks ewy izkIrkadksa esa
ljyrk ls ifjofrZr fd;k tk ldrk gSA ewy izkIrkadksa dks vaxzsth o.kZekyk ds
cM+s v{kj X ls iznf'kZr fd;k tkrk gS tcfd tsM izkIrkadksa dks v{kj Z ls fy[kk
tkrk gSA ;fn e/;eku ds fy, M rFkk ekud fopyu ds fy, ladsrk{kj
iz;qDr fd;s tk;s rks ewy izkIrkadksa ls tsM izkIrkad Kkr djus dk lw= fuEuor~
gksxk&
tcfd Z izkIrkadksa ls ewy izkIrkad fuEu lw= ls Kkr fd;s tk;saxs&
X = M + (Z )
123
(T - score)
tsM izkIrkadksa ds iz;ksx djus ij vkus okyh dfBukb;ksa dks nwj djus ds
fy, vU; vusd izdkj ds ekud izkIrkadksa dk lq>ko fo}kuksa us fn;k gSA buesa ls
T izkIrkad dkQh izpfyr gSaA T izkIrkad okLro esa tsM izkIrkadksa dk bl izdkj ls
js[kh; :ikUr (Linear Transformation) gS fd ifjofrZr izkIrkadksa dk e/;eku
50 rFkk ekud fopyu 10 gks tkrk gSA Li"Vr% T izkIrkad ,sls ekudhd`r
izkIrkad gSa ftudk e/;eku 50 rFkk ekud fopyu 10 gksrk gSA T izkIrkadksa dk
eku izk;% 20 ls 80 ds chp gksrk gSA
T izkIrkad (T-Scores) 'kCn dk iz;ksx 'kSf{kd ekiu ds eq[; izorZd
bZ0,y0 FkksuZMkbd (E.L. thorndike) ds lEeku esa iz;qDr fd;k x;k gSA T
v{kj buds miuke (Sirname) dks bafxr djrk gSA
tc e/;eku dks 50 rFkk ekud fopyu dks 10 djus ds fy, tsM
izkIrkadksa dk js[kh; :ikUrj.k djds T izkIrkad izkIr djus gksrs gS rc mijksDr
lehdj.k dk :Ik fuEuo~r gks tk;sxk&
T T = 50 + 10Z
;fn ewy izkIrkadksa ls lh/ks x.kuk djuh gksrh gS rc
T T = 50 + 10
(Percentile)
124
izkIrkad gSa tks vkdkj ds vuq:Ik Øec) izkIrkadksa
dh Js.kh dks lkS cjkcj Hkkxksa esa foHkkftr djrs gSaA vr% 'krkad O;ofLFkr
izkIrkadksa dh Ja[kyk (Series) dks lkS cjkcj Hkkxksa esa ck¡Vus okys foHkktd fcUnq
gksrs gSaA D;ksafd fdlh js[kk dks 99 fcUnqvksa ds }kjk lkS Hkkxksa esa ck¡Vk tk ldrk
gS blfy, 'krkadks dh la[;k 99 gksrh gSA nwljs 'kCnksa esa dgk tk ldrk gS fd
'krkad os izkIrkad gSa ftuls de vad izkIr djus okys Nk=ksa dh la[;k izfr'kr esa
nh xbZ gksrh gSA 'krkadksa dks Pk ladsrk{kjksa ls fy[kk tkrk gS tgk¡ k ml 'krkad
ls de vad izkIr djus okys Nk=ksa dh izfr'kr la[;k gksrk gSA tSls P10 og
izkIrkad gS ftlds uhps 10% Nk= vad izkIr djrs gSa] P15 og izkIrkad gS ftlds
uhps 15% Nk= vad izkIr djrs gSaA vr% 'krk¡d os izkIrkad gaS ftuds uhps
izkIrkadksa dk ,d fn;k x;k izfr'kr fLFkr gksrk gSA
(Calculation of
Percentile from ungrouped data)
voxhZd`r leadksa ls 'krk¡d Kkr djus ds fy, igys lHkh izkIrkadksa dks
vkjksgh Øe esa O;ofLFkr dj ysrs gSa rFkk fQj ml Js.kh dk og fcUnq Kkr dj
ysrs gSa tks izFke NK/100 izkIrkadksa dks 'ks"k izkIrkadksa ls vyx djrk gSA vkjksgh
Øe esa O;ofLFkr izkIrkadksa dk (NK/100+.5) ok¡ izkIrkad gh izFke (NK/100)
izkIrkadksa dks 'ks"k izkIrkadksa ls vyx djrk gSA vr% (NK/100+.5) ok¡ izkIrkad gh
ok¡fNkr 'krk¡d gksxkA
(
) oka izkIrkad
125
'krkad Kkr djrs le; dHkh&dHkh (NK/100+.5) dk eku iw.kZ la[;k esa
u vkdj n'keyo la[;k esa ;k fHkUu esa vkrk gSA rc (NK/100+.5) ok¡ izkIrkad
Kkr djus ds fy, bl la[;k ds iw.kZ vad okys izkIrkad esa mlls vxys izkIrkad
o ml izkIrkad ds vUrj dks n'keyo vadksa ls xq.kk djds tksM+ nsrs gSaA tSls&
5.7 ok¡ izkIrkad =5 ok¡ izkIrkad +.7 ¼6 ok¡ izkIrkad & 5 ok¡ izkIrkad½
nl Nk=ksa ds fuEu izkIrkadksa ds fy, P20, P25, P40, P57 o P85 Kkr djksA
48, 10, 20, 25, 11, 16, 31, 35, 12, 40
izkIrkad dks vkjksgh Øe esa O;ofLFkr djus ij
10, 11, 12, 16, 20, 25, 31, 35, 40, 48
1 2 3 4 5 6 7 8 9 10
(i) P20 ds fy,
vr% 2+ .5 vFkkZr 2.5 ok¡ izkIrkad gh P20 gksxkA
2.5 ok¡ izkIrkad = 2 ok¡ izkIrkad + .5 ¼3 ok¡ izkIrkad & 2 ok¡ izkIrkad½
= 11 + .5 (12 – 11)
= 11.25 vr% P20 = 11.25
126
(ii) P25 ds fy,
vr% 2.5+ .5 vFkkZr 3 ok¡ izkIrkad gh P25 gksxkA
vr% P25 = 12
(iii) P40 ds fy,
vr% 4+ .5 vFkkZr 4.5 ok¡ izkIrkad gh P40 gksxkA
4 ok¡ izkIrkad + 5 ok¡ izkIrkad
4.5 ok¡ izkIrkad =
2
= 18 vr% P40 = 18
(iv) P57 ds fy,
vr% 5.7+ .5 vFkkZr 6.2 ok¡ izkIrkad gh P57 gksxkA
6.2 ok¡ izkIrkad = 6 ok¡ izkIrkad +.2 ¼7 ok¡ izkIrkad & 6 ok¡ izkIrkad½
= 25 + .2 (31 – 25)
= 26.2 vr% P57 = 26.2
(v) P85 ds fy,
vr% 8.5+ .5 vFkkZr 9 ok¡ izkIrkad gh P85 gksxkA
9 ok¡ izkIrkad = 40 vr% P85 = 40
127
(Calculation of
Percentile from grouped data)
vko`fÙk forj.k ds :Ik esa O;ofLFkr leadksa ls 'krk¡dksa dh x.kuk djus dh
fof/k ewyr% ogh gS tks e/;k¡d Kkr djus dh gSA 'krk¡d Kkr djus dk lw=
fuEukuqlkj gS &
K ok¡ 'krkad]
tgk¡ L = 'krk¡d oxZ ¼ftlesa NK/100 oha lap;h vko`fÙk fLFkr gksrh gS½ dh
okLrfod fuEu lhek
cfB = 'krk¡d oxZ ls uhps okys oxZ dh lap;h vko`fÙk
f = 'krk¡d oxZ dh vko`fÙk
i = oxZ foLrkj
N = dqy izkIrkadksa dh la[;k
K = izkIrkadksa dk og izfr'kr tks Kkr fd;s tkus okys 'krk¡d ds uhps
fLFkr gSA
fuEu lkj.kh esa fn;s x;s vko`fÙk forj.k ds fy, P10, P65, P90 dh
x.kuk djksA
128
f cf
90 – 99 80 – 89 70 – 79 60 – 69 50 – 59 40 – 49 30 – 39 20 – 29 10 – 19
3 8
13 20 35 10 5 4 2
100 97 89 76 56 21 11 6 2
I = 10 N = 100
k ok¡ 'krk¡d Kkr djus dk lw= gS %
'krk¡d]
tgk¡ L = 'krkad oxZ dh fuEu lhek
cfB = 'krkad oxZ ls uhps lap;h vko`fÙk
f = 'krkad oxZ dh vko`fÙk
i = oxZ vUrjky
(i) P10 ds fy,
D;ksafd 10 oha lap;h vko`fÙk 30-39 okys oxZ esa fLFkr gS blfy, 30-39
okyk oxZ P10 oxZ gksxkA vr% lkj.kh esa&
P90 oxZ
P65 oxZ
P10 oxZ
129
L = 29.5 cfB = 6 f = 5 i = 10
lw= esa eku j[kus ij
vr% P10 = 37.5
(ii) P65 ds fy,
D;ksafd 65 oha lap;h vko`fÙk 60-69 okys oxZ esa fLFkr gS] blfy, 60-69
okyk gh oxZ P65 gksxkA vr% lkj.kh ls&
L = 59.5 cfB = 56 f = 20 i = 10
lw= esa eku j[kus ij
vr% P35 = 64.0
(iii) P90 ds fy,
D;ksafd 94 oha lap;h vko`fÙk 80-89 okys oxZ esa fLFkr gS] blfy, 80-89
okyk gh oxZ P90 gksxkA vr% lkj.kh ls&
L = 79.5 cfB = 89 f = 8 i = 10
lw= esa eku j[kus ij
130
vr% P70 = 80.75
(Steps)
oxhZd`r leadksa ls 'krk¡d Kkr djus ds lksikuksa dks fuEuor fy[kk tk
ldrk gS &
(i) vko`fÙk forj.k esa lap;h vko`fÙk;k¡ (cf) Kkr djksA
(ii) NK/100 dk eku Kkr djds ns[kksa fd NK/100 oha lap;h vko`fÙk
fdl oxZ esa fLFkr gSaA bl oxZ dks ml 'krkad ds fy, 'krkadh; oxZ
dgksA
(iii) 'krk¡dh; oxZ dh okLrfod fuEu lhek (L) Kkr djksA
(iv) 'krk¡dh; oxZ dh vko`fÙk (f) Kkr djksA
(v) 'krk¡dh; oxZ ls Bhd uhps ds oxZ dh lap;h vko`fÙk (cfB) Kkr
djksA
(vi) oxZ vUrjky (f) Kkr djksA
(vii) lw= esa lHkh eku j[kdj 'krk¡d ds eku dh x.kuk djksA
131
A B C D E F G H I J
40 44 35 42 30 38 50 45 25 46
35 32 30 34 25 36 45 40 28 26
132
1- Garrett, H.E. (1973) Statistics in psychology and education
Bombay :.
2- Popham, W.J. (1993). Educational evaluation. Boston : Allyn
and Bacon
3- Popham, W.J. (1993). Modern educational measurement :
Englewood Cliffs, NJ. : Prentice Hall.
4- Burke, K. (2005), How to assess authentic learning
Thousand Oaks, CA : Corwin.
5- 'kekZ] vkj- ,-] f'k{kk vuqla/kku] mPp f'k{kk v/;;u laLFkku esjB
¼fo'ofo|ky;] esjB ¼;w0ih0½
6- JhokLro] MkW- Mh- ,u-] lk¡f[;dh ,oa ekiu] fouksn iqLrd eafnj
vkxjk&2
7- xqIrk] MkW- ,l- ih- lka¡f[;dh; fof/k;k¡ ¼O;ogkijd foKkuksa esa½] 'kkjnk
iqLrd Hkou] bykgkckn