70

[Measures of Variability]

(Range)

(Characteristics of Range)

(When does range is used)

(Limitations of range)

(Calculation of range)

(Meaning and definition

of quartile Deviation )

(Characteristics of quartile

Deviation)

Q (When does quartile

Deviation calculated)

Q (Limitations of quartile Deviation)

Q (Calculation of Q from

Ungrouped Data)

Q (Calculation of Q from

grouped Data)

(Calculation method of Q)

(Mean Deviation)

(Characteristics of mean Deviation)

71

(Limitations of mean Deviation)

(Calculation of mean deviation from Ungrouped Data)

(Calculation

of mean deviation from grouped Data)

(Standard deviation)

(Characteristics of Standard

Deviation)

(When does

Standard Deviation used)

(Determinant elements of

Standard Deviation)

(Standard

Deviation of Ungrouped Data)

(Standard

Deviation of grouped Data)

72

[Measures of Variability]

lkaf[;dh dk mís'; fdlh lewg dh fdlh pj ds lUnHkZ esa fo'ks"krkvksa dk

o.kZu djuk gS! dsUnzh; izofÙk ds fofHkUu eku lewg ds ckjs esa dqN mi;ksxh

lwpuk,a rks iznku djrs gSaA ijUrq ;s lwpuk,a lexz lewg ds lEca/k esa dksbZ Li"V

/kkj.kk cukus ds fy, i;kZIr ugha gSaA mnkgj.kkFkZ ekuk fd pkj lewg rFkk izR;sd

esa N% N% Nk= gSa] ftuds fdlh ijh{k.k ij izkIrkad fuEufyf[kr gSa &

18] 18] 18] 18] 18] 18

17] 18] 18] 18] 18] 19

16] 17] 18] 18] 19] 20

16] 16] 17] 19] 20] 20

Li"V gS fd mijksDr pkjksa lewgksa dk e/;eku 18 gS] ijUrq D;k ;s pkjksa

lewg leku gSa\ ugha! dnkfi ugha! e/;keu dk 18 gksuk dsoy bruk crkrk gS

fd lHkh lewgksa ds izkIrkad 18 ds vklikl gSaA lewg v ds lHkh Nk= cjkcj vad

ikrs gSa] lewg c ds pkj ek= ,d vf/kd vad gSa] lewg l ds nks Nk=ksa ds vad

18 ds cjkcj gSa rFkk 'ks"k ds 18 ls Øe'k% 2 de] 1 de] 1 vf/kd o 2 vf/kd

izkIrkad gS tcfd lewg n ds nks Nk= 18 ls nks vad de] ,d Nk= 18 ls ,d

vad de] ,d 18 ls ,d vad vf/kd o nks Nk= 18 ls nks vad vf/kd ikrs gSA

Li"V gS fd pkjksa lewgksa dk e/;eku leku gksrs gq, Hkh lewgksa esa vUrj gSA lewgksa

73

esa ;g fHkUurk muds izkIrkadksa esa ijLij vUrj ;k QSyko dh otg ls gSA vr%

lewg ds ckjs esa dksbZ Li"V /kkj.kk cukus ds iwoZ mlds izkIrkadksa ds QSyko ;k

ijLij fHkUurk dks Hkh tkuuk vko';d gSA os lHkh ekisa tks izkIrkadksa ds QSyko

(Dispersion) vFkok oSfHkU;rk dks crkrh gSa fopyu'khyrk ds eku

(Measures of Variability) dgykrh gSaA

(1985) ds vuqlkj] ^^fopyu'khyrk dk rkRi;Z izkIrkadksa ds forj.k

;k QSyko ls gS] ;g QSyko izkIrkadksa dh dsUnzh; izo`fÙk ds pkjksa vksj gksrk gSA^^

fdlh lewg ds izkIrkadksa ds fopyu dks dbZ izdkj ls ekik tkrk gS] mUgsa

fopyu'khyrk ds eku dgrs gSaA fopyu 'khyrk ds eku pkj izdkj ds gksrs gSa &

1- foLrkj fopyu (Range)

2- prqFkkZa'k fopyu (Quartile Deviation)

3- e/;eku fopyu (Mean Deviation)

4- ekud fopyu (Standard Deviation)

(Range)

,d vad forj.k dh fopyu'khyrk dk ljyre eki izlkj (Range) gSA

izlkj dk vFkZ ml eku ls gS tks ,d vad&forj.k ds mPpre izkIrkad dks

U;wure izkIrkad esa ?kVkus ls izkIr gksrk gSA izlkj dk lw= fuEufyf[kr gS &

izlkj = mPpre izkIrkad - U;wure izkIrkad

[Range = Highest Score – Lowest Score]

74

(Characteristics of Range)

1- ,d vad&forj.k esa izlkj mPpre izkIrkad vkSj U;wure izkIrkad ds e/;

nwjh gS] fLFkfr (Location) ugha gSA

2- ,d vad&forj.k esa mPpre vkSj U;wure izkIrkadks ds e/; ftrus Hkh

izkIrkad gksrs gSa mudk izHkko izlkj ij ugha iM+rk gSA

3- izfrn'kZ (Sample) dk vkdkj izlkj ds eku dks izHkkfor djrk gSA cgq/kk

NksVs izfrn'kZ dk izlkj eku NksVk gksrk gS vkSj cM+s izfrn'kZ dk izlkj eku

cM+k gksrk gSA

4- vad&forj.k ds o.kZukRed Lrj rd gh izlkj dk mi;ksx fd;k tkrk gSA

bldk mi;ksx vad&forj.k ds lEca/k esa fu"d"kZ fudkyus esa ugha fd;k

tkrk gSA

(When does range is used)

1- tc vad&forj.k ds izkIrkad cgqr fc[kjs gq, gksa rFkk vU; fopyu'khyrk

ds ekiksa dk mi;ksx u fd;k tk ldsA

2- tc fdlh vad&forj.k dh fopyu'khyrk dks vfr'kh?kzrk ls Kkr djuk

gksA

3- tc vad&forj.k dk dqy QSyko Kkr djuk gksA

4- tc va'k&forj.k ds (Extreme Score) dks egRo nsuk gksA

5- tc fopyu'khyrk ds vf/kd 'kq) eku dh vko';drk u gksA

(Limitations of range)

1- tc izkIrkadksa dh la[;k de gks rc izlkj dh x.kuk ugha djuh pkfg,A

75

2- tc nks izfrn'kksZa ds N dk eku fHkUu&fHkUu gks rc izlkj dh x.kuk ugha

djuh pkfg,A

3- tc forj.k esa chp&chp esa [kkyh LFkku gks rc izlkj dh x.kuk ugha

djuh pkfg,A

4- izlkj ds vk/kkj ij nks lewgksa dh rqyuk ugha djuh pkfg, D;ksafd bl

izdkj dh rqyuk ls dsoy viw.kZ Kku izkIr gksrk gSA

5- izlkj ,d vfo'oluh; eki gSA fxyQksMZ (1958) us fy[kk gS] “The

total range is indicator of variability that is easiest and must

quickly ascertained but is also the most unreliable.”

(Calculation of range)

iz'u esa mPpre vad = 28 U;wure vad = 22

izlkj = mPpre vad – U;wure vad

= 28 – 22 = 6

(Meaning and definition

of quartile Deviation )

tc fdlh lewg ds lHkh izkIrkadksa dks pkj cjkcj Hkkxksa esa ck¡Vk tkrk gS rks

izR;sd Hkkx dks prqFkkZa'k (Quartile) dgrs gSaA Ldsy ij uhps ls Åij dh vksj

76

izFke prqFkkZa'k dks Q1 f}rh; prqFkkZa'k dks Q2 vkSj r`rh; prqFkkZa'k dks Q3 ls

O;Dr djrs gSaA

fdlh lewg ds e/;kad (Q2) ls mlds nksuksa vksj ds prqFkkZa'kksa (Q1 vkSj

Q3) ds fopyu ds vkSlr dks prqFkkZa'k fopyu dgrs gSaA

(1973) ds vuqlkj] ^^fdlh vko`fÙk forj.k esa 75osa izfr'krkad vkSj

25osa izfr'krkad ds chp dh vk/kh nwjh gksrh gSA^^

prqFkkZa'k fopyu dk nwljk uke v)Z&e/;kad&prqFkkZa'k (Semi-Inter-

Quartile Range) gSA vr% ;g blds nwljs uke ls Li"V gS fd ;g ,d izdkj

dk izlkj gSA bldk ladsr fpUg (Symbol) QD vFkok Q gSA

(Characteristics of quartile

Deviation)

1- Q dk vFkZ Q3 vkSj Q1 ds e/; dh nwjh gSA

2- tc vad forj.k ds lhekUrksa ij fopyu dh ek=k vf/kd gksrh gS rc

bldk mi;ksx fd;k tkrk gSA

3- tc vad forj.k fo"ke (Skewed) gks ;k eqDr Nksj (Open-ended) gks

rc Q dh x.kuk dh tkrh gSA

4- o.kZukRed (Descriptive) lkaf[;dh esa bldk mi;ksx cgqr vf/kd gSA

5- jpuk] xq.k vkSj x.kuk dh n`f"V ls Q dk lEca/k cgqr dqN e/;kad ls gSA

77

Q (When does quartile

Deviation calculated)

1- tc vad forj.k iw.kZ gks rc Q dh x.kuk djuh pkfg,A

2- tc SD dh x.kuk u dh tk lds vFkok nwf"kr ifj.kke izkIr gksus dh

lEHkkouk gksA

3- tc izfrn'kZ (Sample) NksVk gksA

4- tc e/;kad (Mdn) dh x.kuk dh xbZ gksA

5- tc vad forj.k lkekU; rFkk iw.kZ gks rc Q dh x.kuk djuh pkfg,A

Q (Limitations of quartile Deviation)

1- dsoy Q ds eku ds vk/kkj ij forj.k ds Lo:Ik dks ugha le>k tk

ldrk gSA

2- bldh x.kuk esa lhekUr vadksa (Extreme scores) dks egRo ugha fn;k

tkrk gSA

Q (Calculation of Q from

Ungrouped Data)

vO;ofLFkr vad lkexzh ls Q dh x.kuk dk lw= fuEufyf[kr gS &

tcfd (

)

78

{

}

;gk¡ N = izkIrkadksa dh la[;k (Total number of scores)

(Calculation)

1- nh gqbZ vO;ofLFkfr vad lkexzh ls igys Q1 fQj Q3 dh x.kuk dhft,

vUr esa lw= esa Q3 vkSj Q1 ds eku j[kdj Q dk eku Kkr dj yhft,A

2- Q1 vkSj Q3 dh x.kuk esa dsoy N dk eku Kkr gksuk vko';d gSA

3- Q1 vkSj Q3 Kkr djus ls igys fn;s gq, izkIrkadksa dks Øe esa O;ofLFkr

dj yhft,A

mnkgj.k uhps fn;s gq, vO;ofLFkr izkIrkadksa ls Q dh x.kuk dhft, &

12 13 11 14 13 18 17 16 15

gy % Øe esa O;ofLFkr izkIrkad 11 12 13 13 14 15 16 17 18

Q1 dh x.kuk (

)

(

)

(

)

Q3 dh x.kuk {

}

(

)

(

)

79

Q dh x.kuk

mnkgj.k uhps fn;s gq, vO;ofLFkr izkIrkadksa ls Q dh x.kuk dhft, &

9] 12] 15] 11] 14] 10] 6] 7

gy %

9 13 15 11 14 10 6 7

6 7 9

10 11 13 14 15

;gk¡ N = 8

Q1 dh x.kuk (

)

(

)

(

)

Q3 dh x.kuk {

}

(

)

80

(

)

Q dh x.kuk

Q (Calculation of Q from

grouped Data)

O;ofLFkr vad lkexzh ls Q dh x.kuk fuEu lw= }kjk dh tkrh gS &

Q fudkyus dk lw= &

tcfd] Q = prqFkkZa'k fopyu (Quartile deviation),

Q3 = r`rh; prqFkkZa'k vFkok og prqFkkZa'k ftlds uhps 75% vko`fÙk;k¡ gksrh gSaA

(Third quartile or 75th percentile)

Q1 = izFke prqFkkZa'k vFkok og prqFkkZa'k ftlds uhps 25% vko`fÙk;k¡ gksrh gSa

(First quartile or 25th quartile)

Q3 fudkyus dk lw= & (

)

Q1 fudkyus dk lw= & (

)

81

tcfd L = ml oxkZUrj dh fuEure 'kq) lhek ftlds uhps Q1 iM+rk gS ;k Q3

iM+rk gSA

(Exact lower limit of C.I. in which first quartile or third quartile lies),

F = ml oxkZUrj dh uhps dh lafpr vko`fÙk ftlesa Q1 ;k Q3 gS

(Cumulative frequency of the C.I. below Q1 or Q3),

f = ml oxkZUrj dh vko`fÙk ftlesa Q1 ;k Q3 gS

(Frequency of that C.I. containing Q1 or Q3),

N = vko`fÙk;ksa dk dqy ;ksx

(Total number of Frequencies),

C.I. = oxkZUrj dk vkdkj (Length of C.I.)

(Calculation method of Q)

1- fn;s gq, vad&forj.k dks vkjksgh Øe (Ascending order) esa fyf[k,

fQj nh gqbZ vko`fÙk;ksa dks lafpr vko`fÙk;ksa (Cumulative frequency F)

esa ifjofrZr dhft,A

2- loZizFke Q1 dh x.kuk dhft,A N/4 ds eku dh lgk;rk ls Q1 dks

fuf'pr dhft,A blh izdkj ls N/4 dh lgk;rk ls Q3 dks fuf'pr

djds Q3 dh x.kuk dhft,A

82

3- Q1 vkSj Q3 dh x.kuk e/;kad (Md) dh x.kuk ls feyrh&tqyrh gSSA

4- Q3 vkSj Q1 dk eku izkIr dj ysus ds ckn Q dh x.kuk fn;s gq, lw= dh

lgk;rk ls dhft,A

5- Q1, Q3 vkSj Q dh x.kuk,¡ n'keyo ds r`rh; LFkku rd dh ftlls fd

'kq) ifj.kke izkIr gks ldsaA

mnkgj.k fuEufyf[kr O;ofLFkr vad lkexzh ls Q dh x.kuk dhft, &

C. I. F

120 – 124 115 – 119 110 – 114 105 – 109 100 – 104

95 – 99 90 – 94 85 – 89 80 – 84

2 4 6 8 9 7 5 3 2

46 44 40 34 26 17 10 5 2

N = 46

Q1 dh x.kuk (

)

iz'u esa L = 94.5, N/4 = 11.5, F = 10, = 7

bu ewY;ksa dks lw= esa j[kus ij]

(

)

83

(

)

Q3 dh x.kuk (

)

iz'u esa] L = 109.5, 3N/4 = 34.5, F = 34, f = 6, C.I. 5

bu ewY;ksa dks lw= esa j[kus ij]

(

)

(

)

Q dh x.kuk

(Mean Deviation)

(1958) us e/;eku fopyu dks ifjHkkf"kr djrs gq, fy[kk gS fd]

^^e/;eku fopyu] e/;eku ls fHkUUk&fHkUUk izkIrkadksa ds fopyuksa dk e/;eku gS

tcfd /ku rFkk _.k fpUgksa dks /;ku esa u j[kk x;k gksA**

(1985) ds vuqlkj] ^^izkIrkadksa ds e/;eku ls fHkUUk&fHkUu izkIrkadksa dk

fopyu Kkr fd;k tk;s fQj /ku (+) rFkk _.k (-) fpUgksa dks /;ku fn;s fcuk

e/;eku Kkr fd;k tk;s rks izkIr la[;k e/;eku fopyu (AD) dgyk;sxhA

e/;eku ls fopyu dk ekiu e/;eku fopyu gSA bldk ladsr fpUg

AD ;k MD gSA

84

(Characteristics of mean Deviation)

1- ,d vad&forj.k ds lHkh izkIrkadksa dk izHkko e/;eku fopyu dh x.kuk

ij iM+rk gSA vr% ml vad forj.k dk iw.kZ izfrfuf/kRo djrk gSA

2- e/;eku fopyu dh izd`fr dks ljyrk ls le>k tk ldrk gSA

3- e/;eku fopyu dh x.kuk ij Extreme Score dk U;wrue izHkko iM+rk

gSA

1- tc izekf.kd fopyu (SD) dh x.kuk lEHko u gks vkSj vad forj.k ds

izkIrkad fc[kjs gq, gksa rc AD dh x.kuk djuh pkfg,A

2- e/;eku fopyu dh x.kuk ml le; Hkh dh tkrh gS tc 'kq)rk dh

vko';drk gksrh gSA

3- tc vad&forj.k ds izR;sd izkIrkad dks mlds vkdkj ds vuqlkj egRo

nsuk gks rc AD dh x.kuk dh tkrh gSA

4- e/;eku fopyu dh x.kuk ml le; Hkh dh tkrh gS tc e/;eku ds

nksuksa vksj ds izkIrkadksa dk fopyu Kkr djuk gksA

5- tc lk/kkj.k 'kq)rk ds fopyu eki dh vko';drk gks rc AD dh x.kuk

djuh pkfg,A

(Limitations of mean Deviation)

e/;eku fopyu dh x.kuk djrs le; /ku rFkk _.k fpUgksa dks egRo

fn;k tkrk gSA xf.krh; n`f"Vdks.k ls bl izdkj dh x.kuk =qfViw.kZ gSA

(Edward, W. Minium, 1970) us lekykspuk djrs gq, fy[kk gS fd]

85

“It is for example, of no use in statistical inference. You may run

across its use in either works, but it has seldom appeard in research

literature for the past 30 years.”

(Calculation of mean deviation from Ungrouped Data)

tcfd] d = e/;eku ls izkIrkadksa dk fopyu

d = d ds nksuksa vksj f[kaph js[kkvksa dk rkRIk;Z gS fd fopyu dk ;ksxQy fudkyrs

le; /ku rFkk _.k fpUgksa dks egRo ugha fn;k tkrk gSA (Bars embracing

the d indicate that signs are disregarded in arriving at the sum of

deviations).

N = izkIrkadksa dh la[;k (Number of scores)

e/;eku ls fopyuksa dk ;ksx (Sum of all the deviations taken

from mean ignoring + and - signs),

(Calculation mathod)

loZizFke nh gqbZ vO;ofLFkr vad lkexzh dk e/;eku (M) Kkr dhft,A

f}rh; pj.k esa izkIrkadksa dk e/;eku ls fopyu Kkr dhft,A blds fy, izR;sd

izkIrkad esa ls e/;eku dks ?kVkb;s vFkkZr~ d = X – MA ;gk¡ dsoy vUrj Kkr

djuk gSA e/;eku fopyu dh ifjHkk"kk esa ;g igys gh Li"V fd;k tk pqdk gS

86

fd e/;eku ls fopyu dh x.kuk djrs le; /ku rFkk _.k fpUgksa dks egRo

ugha fn;k tkrk gSA

r`rh; pj.k esa e/;eku ls lHkh fopyu Kkr dj ysus ds ckn dk

eku Kkr djrs gSaA vUr esa vkSj N ds ekuksa dks lw= esa j[kdj AD dk eku

Kkr dj ysrs gSaA

(AD)

gy % fn;s gq, izkIrkadksa dks ,d ykbu esa fy[kdj e/;eku fudkfy, vkSj

e/;eku ls fopyu fuEu izdkj Kkr dhft, &

y (Scores) x – M = d

28 28 30 35 34 33 32 36

28 – 32 = 4 28 – 32 = 4 30 – 32 = 2 35 – 32 = 3 34 – 32 = 2 33 – 32 = 1 32 – 32 = 0 36 – 32 = 4

= 20

uksV & ;gk¡ $ rFkk & fpUgksa dks fopyu Kkr djrs le; egRo ugha fn;k x;k

gSA

mi;qZDRk fn;s vO;ofLFkr izkIrkadksa ds e/;eku dh x.kuk &

87

= 32

iz'u esa] N = 8,

bu ewY;ksa dks lw= esa j[kus ij]

(Calculation

of mean deviation from grouped Data)

O;ofLFkr vad lkexzh ls e/;eku fopyu dh x.kuk fuEu lw= }kjk dh

tkrh gS &

Tkcfd d = e/;fcUnq ds e/;eku ds fopyu (Deviation of Mid-point from

the mean),

e/;eku ds e/;fcUnqvksa ds fopyuksa dk ;ksx tc lEcaf/kr vko`fÙk;ksa ls

xq.kk fd;k x;k gks rFkk $ rFkk & dk /;ku u j[kk x;k gks (Sum of all the

deviation s of mid-point mean when multiplied by their respective

frequencies and ignoring + and - signs),

N = vko`fÙk;ksa dk dqy ;ksx (Total number of frequencies),

vko`fÙk;k¡ (Frequecies) A

88

(1) loZizFke nh gqbZ O;ofLFkr vad lkexzh dk e/;eku Kkr fd;k tkrk gSA

e/;eku Kkr djus dh nks fof/k;k¡ gSa & ¼v½ nh?kZ fof/k (Long

method), ¼c½ laf{kIr fof/k (Short method) A nksuksa esa ls fdlh Hkh

fof/k }kjk e/;eku Kkr dj ldrs gSaA

(2) nwljs pj.k esa e/;fcUnq dk e/;eku ls fopyu Kkr fd;k tkrk gSA

e/;eku fopyu dh x.kuk esa pw¡fd $ vkSj & fpUgksa dks egRo ugha

fn;k tkrk gS vr% vko';d ugha gS fd e/;fcUnq esa ls gh e/;eku

?kVk;k tk;s cfYd e/;fcUnq vkSj e/;eku dk vUrj ekywe djuk

vko';d gSA mnkgj.k esa ;g d = x - M okys dkWye esa fn;k gqvk gSA

(3) e/;eku ls e/;fcUnqvksa dk fopyu Kkr djus ds i'pkr~ bu fopyuksa

dks lEcaf/kr oxkZUrjksa dh vko`fÙk;ksa ls xq.kk dhft,A mnkgj.k esa ;g fd

dkye esa x.kuk djds n'kkZ;k x;k gSA

(4) vUr esa dk eku Kkr djds AD ds lw= esa ewY;ksa dks j[kdj

AD dk eku izkIr dj ysrs gSaA

AD

(Long method)

60 – 64 55 – 59 50 – 54 45 – 49

1 2 3 4

62 57 52 47

62 104 156 188

19.13 14.13 9.13 4.13

19.13 28.26 27.39 16.52

89

40 – 44 35 – 39 30 – 34 25 – 29 20 - 24

6 3 2 1 1

42 37 32 27 22

252 111 64 27 22

.87 5.87

10.87 15.87 20.87

5.22 17.61 21.74 15.87 20.87

N = 23 = 986 = 172.61

loZizFke nh?kZ fof/k (Long method) ls e/;eku Kkr dhft, fQj AD

dh x.kuk dhft,A

e/;eku fopyu dh x.kuk

iz'u esa]

bu ewY;ksa dks lw= esa j[kus ij]

(Standard deviation)

(1968) ds vuqlkj] ^^fn;s gq, izkIrkadksa ds e/;eku ls izkIrkadksa ds

fopyuksa ds oxksZa ds e/;eku dk oxZewy }kjk izkIr eku gh izkekf.kd fopyu

gSA**

mi;qZDr ifjHkk"kk dks Li"V djrs gq, dgk tk ldrk gS fd fn;s gq,

izkIrkadksa dk e/;eku Kkr djds /ku o _.k fpUgksa dks /;ku fn;s fcuk izkIrkadksa

dk e/;eku ls fopyu Kkr dj yhft, fQj bu fopyuksa dk oxZ djds budk

90

;ksx Kkr dj yhft,A bl izdkj izkIr eku dks N ls Hkkx nsdj oxZewy Kkr

dhft,A bl izdkj izkIr eku gh izkef.kr fopyu gSA

lEiw.kZ vad&forj.k dh fopyu'khyrk (Variability) crkus okyk eki ghs

izkekf.kd fopyu gSA f'k{kk vkSj euksfoKku ds vuqla/kku dk;ksZa esa cgq/kk izkekf.kd

fopyu dh gh x.kuk dh tkrh gSA fopyu'khyrk dk ;g lcls 'kq) vkSj

fo'oluh; eki gSA bldk ladsr SD gSA

,d lkekU; vad forj.k esa lkekU;r% izkekf.kd fopyu dh 6 bdkb;k¡

gksrh gSaA vr% lkekU; vad&forj.k ds e/;eku ls /kukRed fn'kk esa 3 bdkb;k¡

gksrh gSa rFkk e/;eku ls _.kkRed fn'kk esa rhu bdkb;k¡ gksrh gSaA vr% lkekU;

vad&forj.k dk lEiw.kZ foLrkj ;k fopyu'khyrk M 3 SD dh bdkb;ksa ds

e/; gksrh gSA M 1 SD esa lEiw.kZ vad&forj.k dk 68.26% Hkkx vkrk gSA M

2 SD esa lEiw.kZ vad&forj.k dk 95.44% Hkkx vkrk gSA M 3 SD esa

lEiw.kZ vad&forj.k dk 99.73% Hkkx vkrk gSA

(Characteristics of Standard

Deviation)

1- vad forj.k ds izR;sd vad ls izkekf.kd fopyu izHkkfor gksrk gSA

2- izkekf.kd fopyu lkekU; lEHkkouk oØ dk eq[; vk/kkj gSA

3- fopyu'khyrk dk ;g loZ'kq) vkSj fo'oluh; eki gSA

4- vad&forj.k ds e/;eku dh fo'oluh;rk dk v/;;u izkekf.kd fopyu

ds vk/kkj ij fd;k tkrk gSA

5- SD = 1.483 Q rFkk SD = 1.253 AD ds cjkcj gksrh gSA

91

(When does

Standard Deviation used)

1- tc lokZf/kd 'kq) vkSj fo'oluh; fopyu eki dh vko';drk gksA

2- tc dsUnzh; ekidksa esa e/;eku dh x.kuk dh xbZ gksA

3- tc nks vad&forj.kksa dk rqyukRed v/;;u djuk gksA

4- tc lg&lEca/k xq.kkad rFkk e/;ekuksa ds vUrj dh lkFkZdrk dh tk¡p

djuh gksrh gS rc SD dh x.kuk vko';d gksrh gSA

5- izkekf.kd fopyu dh x.kuk dh vko';drk rc Hkh iM+rh gS tc ewy

izkIrkadksa dks izkekf.kd izkIrkadksa (Standard scores) esa cnyuk gksrk gSA

6- lkekU; lEHkkouk oØ (Normal probability curve) ds v/;;u esa Hkh

bldh x.kuk dh vko';drk iM+rh gSA

7- fopyu xq.kkad (CR) vkSj izkekf.kd =qfV (Standard error) ds v/;;u

esa bldh x.kuk dh vko';drk gksrh gSA

8- tc lhekUr izkIrkadksa (Exreme scores) dks egRo nsuk gksrk gS rc Hkh

SD dh x.kuk dh tkrh gSA

(Determinant elements of

Standard Deviation)

1- tc vad forj.k ysIVksdfVZd (Leptocurtic) gksrk gS rc SD dk eku

de gksrk gS rFkk vad&forj.k IysVksdfVZd (Platocurtic) gksrk gS rc

SD dk eku vf/kd gksrk gSA

92

2- vad&forj.k ds izkIrkadksa dk ewY; izkekf.kd fopyu dks egRoiw.kZ <ax ls

izHkkfor djrk gSA

(Standard

Deviation of Ungrouped Data)

vO;ofLFkr vad&lkexzh ls izkekf.kd fopyu dh x.kuk ds lw=

fuEufyf[kr gSaA lHkh lw=ksa ls S. D. dk leku eku izkIr gksrk gSA

√

tcfd] d = izkIrkadksa dk e/;eku ls fopyu (Deviation of scores form

mean),

e/;eku ls fy, x;s fopyuksa ds oxksZa dk ;ksx (Sum of the

squared deviations taken from the mean),

N = izkIrkadksa dh la[;k (Number of scores)

√

√

(Calculation mathod)

1. loZizFke nh gqbZ vad lkexzh dks Øec) dhft, fQj lHkh vadksa dk

vkSj N dk ewY; Kkr djds e/;eku dh x.kuk dhft,A

2. nwljs pj.k esa izkIrkadksa dk e/;eku ls fopyu Kkr fd;k tkrk gSA blds

fy, izR;sd izkIrkad esa ls e/;eku ?kVkrs gSaA mnkgj.k esa X – M = d okys

93

dkye esa bl izfØ;k dks n'kkZ;k x;k gSA ;gk¡ /ku vkSj _.k yxkus dh

vko';drk ugha gSA

3. rhljs pj.k esa fopyuksa dk oxZ djrs gSaA izR;sd fopyu dk oxZ djds

okys dkye esa j[krs gSaA vUr esa dk eku izkIr dj ysrs gSaA

4. vfUre pj.k esa vkSj N dk eku SD ds lw= esa j[krs gSa vkSj x.kuk

djds SD dk eku izkIr dj ysrs gSaA

5] 7] 8] 9] 10] 11] 12

gy % loZizFke e/;eku fudkfy,

Scores x – M = d d2

5 7 8 9

10 10 11 12

5 – 9 = - 4 7 – 9 = - 2 8 – 9 = - 1 9 – 9 = 0

10 – 9 = 1 10 – 9 = 1 11 – 9 = 1 12 – 9 = 3

16 4 1 0 1 1 4 9

iz'u esa]

bu ewY;ksa dks S. D. ds lw= esa j[kus ij]

94

√

√

√

(Standard

Deviation of grouped Data)

O;ofLFkr vad lkexzh ds S. D. dh x.kuk ds fuEu rhu lw= izpfyr gSa &

S. D.

√

(

)

√

tcfd SD = izkekf.kd fopyu (Standard deviation),

i = oxkZUrj dk vkdkj (Length of the C. I.),

vko`fÙk;ksa ,oa fopyuksa ds xq.kuQyksa dk ;ksx (Sum of

the product of fequencies and deviations),

fopyuksa ds oxZ ,oa vko`fÙk;ksa ds xq.kuQy dk ;ksx

(Sum of the product of the frequencies and deviation

squares),

95

N = izkIrkadksa dh la[;k (Number of Scores)

f}rh; lw= izFke lw= dk gh ljyhd`r :Ik gSA ;|fi izFke lw= SD

dh x.kuk ds fy, vf/kd yksdfiz; gS fQj Hkh x.kuk dh n`f"V ls f}rh; lw=

vf/kd ljy gSA ;gk¡ nksuksa gh lw=ksa ds mnkgj.k fn;s gq, gSa &

S. D.

tcfd] e/; fcUnqvksa dk e/;eku ls fopyu (Deviation of midpoint

from mean)

fopyuksa ds oxZ ,oa vko`fÙk;ksa ds xq.kuQy dk ;ksx (Sum of the

product of the frequencies and deviation squares),

N = izkIrkadksa dh la[;k (Numbe of seores)

(Short Method) S. D.

1. bl fof/k }kjk S. D. dh x.kuk djrs le; dqN x.kuk,¡ oSls gh djuh

iM+rh gSaA tSls e/;eku esa x.kuk,¡ djuh iM+rh gSaA

2. loZizFke ftl oxkZUrj dh vko`fÙk lokZf/kd gksrh gS ;k tks oxkZUrj e/; esa

gksrk gS mlesa dfYir e/;eku (AM) ekudj 'kwU; yxk nsrs gSa rFkk

96

vkSj dkye dks iwjk dj ysrs gSa A vUr esa dh

x.kuk djrs gSaA

3. x.kuk ds r`rh; pj.k esa rFkk vkSj ds eku izkIr

dj ysrs gSaA

4. x.kuk ds vfUre pj.k esa S. D. ds lw= esa lHkh ekuksa dks j[kdj S. D. dh

x.kuk dj S. D. dk eku izkIr dj ysrs gSaA

SD

C. I. D

26 – 27 24 – 25 22 – 23 20 – 21 18 – 19 16 – 17 14 – 15 12 – 13 10 – 11

1 2 3 5 8 4 3 2 1

+4 +3 +2 +1 0 -1 -2 -3 -4

+4 +6 +6 +5

021 -4 -6 -6

- 420

16 18 12 05 00 04 12 18 16

N = 29

iz'u esa rFkk

bu ewY;ksa dks lw= esa j[kus ij]

izFke lw= ls x.kuk &

√

(

)

97

√

(

)

√ √

√

√

√

√

(Long Method) S. D.

1. bl fof/k }kjk Hkh O;ofLFkr vad lkexzh dk S. D. Kkr fd;k tkrk gSA

bl fof/k }kjk S. D. Kkr djrs le; loZizFke nh?kZ fof/k }kjk e/;eku

(M) dh x.kuk dh tkrh gSA

2. f}rh; pj.k esa fopyu (d) dh x.kuk dh tkrh gSA ;g fopyu e/;eku

(M) vkSj e/;fcUnq (Mid-point) ds vUrj ds cjkcj gksrk gSA

3. r`rh; pj.k esa d dk eku Kkr gks tkus ds ckn vkSj dh x.kuk

dh tkrh gSA

4. vUr esa nh?kZ fof/k ds lw= esa ladsrksa dk eku j[kdj S. D. dk eku x.kuk

}kjk izkIr dj ysrs gSaA

98

(Long

Method) S. D.

C. I. X Mid-Point

X – Md

34 – 36 31 – 33 28 – 30 25 – 27 22 – 24 19 – 21 16 – 18 13 – 15 10 – 12

1 2 3 5 6 4 3 2 1

35 32 29 26 23 20 17 14 11

35 64 87 130 138 80 51 28 11

11.89 8.89 5.89 2.89 0.11 3.11 6.11 9.11 12.11

11.89 17.78 17.67 14.45 00.66 12.44 18.33 18.22 12.11

23.78 26.67 23.56 17.34 00.77 15.55 24.44 27.33 24.22

N = 27

S. D.

√

√

√

99

100

(Q)

(Q)

(SD)

(MD)

C.I. 10–14 15–19 20–24 25–29 30–34 35–39 40–44 45–49 50-54 F 1 2 3 4 5 3 3 2 2 N = 23

Q

C.I. F 70–71 68–69 66–67 64–65 62–63 60–61 58–59 56–57 54-55 52-53 50-51

2 2 3 4 6 7 5 4 2 3 1

N = 39

S.D.

C.I. 45–49 40–44 35–39 30–34 25–29 20–24 15–19 10–14 5--9 0-4 F 1 2 3 6 8 10 7 5 5 3

101

1- Agarwal, Y.P. (1990). Statistical methods : concepts,

applications and computations. New Delhi : Sterling

Publishers.

2- Garrett, H.E. (1973) Statistics in psychology and education

Bombay :.

3- Popham, W.J. (2010). Classroom assessment : What teachers

need to know New York : Prentice Hall.

4- xqIrk] MkW- ,l- ih-] lka¡f[;dh; fof/k;k¡ ¼O;ogkijd foKkuksa esa½] 'kkjnk

iqLrd Hkou] bykgkckn

5- JhokLro] MkW- Mh- ,u-] lk¡f[;dh ,oa ekiu] fouksn iqLrd eafnj

vkxjk&2

6- yky] jeu fcgkjh tks'kh] lqjs'k pUnz] f'k{kk euksfoKku ,oa ekiu]

jLrksxh ifCyds'ku 115, gjh uxj] esjB 'kgj