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

103

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

104

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