grouped Data) (Calculation method of Q) (Mean Deviation) (Characteristics of mean Deviation)

[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) 70

(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) 71

[Measures of Variability] lkaf[;dh dk mís'; fdlh lewg dh fdlh pj ds lunhkz esa fo'ks"krkvksa dk o.kzu djuk gs! dsunzh; izo`fùk 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 lewgks a 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=ks a 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 72

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 izkirkadks a ds QSyko ;k ijlij fhkuurk dks Hkh tkuuk vko';d gsa os lhkh ekisa tks izkirkadks a 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 izkirkadks a ds forj.k ;k QSyko ls gs] ;g QSyko izkirkadks a dh dsunzh; izo`fùk ds pkjksa vksj gksrk gsa^^ fdlh lewg ds izkirkadks a 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] 73

(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 74 (When does range is used) 1- tc vad&forj.k ds izkirkad cgqr fc[kjs gq, gksa rfkk vu; fopyu'khyrk ds ekiks a 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 izkirkadks a dh la[;k de gks rc izlkj dh x.kuk ugha djuh pkfg,a

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 of quartile Deviation ) (Meaning and definition tc fdlh lewg ds lhkh izkirkadksa dks pkj cjkcj Hkkxks a esa ck Vk tkrk gs rks izr;sd Hkkx dks prqfkkza'k (Quartile) dgrs gsaa Ldsy ij uhps ls Åij dh vksj 75

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'kks a (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 Deviation) 1- Q dk vfkz Q3 vksj Q1 ds e/; dh nwjh gsa (Characteristics of quartile 2- tc vad forj.k ds lhekurks a 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 76

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 vadks a (Extreme scores) dks egro ugha fn;k tkrk gsa Ungrouped Data) Q (Calculation of Q from vo;oflfkr vad lkexzh ls Q dh x.kuk dk lw= fueufyf[kr gs & tcfd ( ) 77

{ } ;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, izkirkadks a dks Øe esa O;ofLFkr dj yhft,a mnkgj.k uhps fn;s gq, vo;oflfkr izkirkadks a 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 { } ( ) ( ) 78

Q dh x.kuk mnkgj.k uhps fn;s gq, vo;oflfkr izkirkadks a ls Q dh x.kuk dhft, & gy % 9] 12] 15] 11] 14] 10] 6] 7 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 { } ( ) 79

( ) Q dh x.kuk grouped Data) Q (Calculation of Q from 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 75 th percentile) Q1 = izfke prqfkkza'k vfkok og prqfkkza'k ftlds uhps 25% vko`fùk;k gksrh gsa (First quartile or 25 th quartile) Q3 fudkyus dk lw= & ( ) Q1 fudkyus dk lw= & ( ) 80

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;ks a (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 81

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;ks a dks lw= esa j[kus ij] ( ) 82

( ) Q3 dh x.kuk ( ) iz'u esa] L = 109.5, 3N/4 = 34.5, F = 34, f = 6, C.I. 5 bu ewy;ks a 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 izkirkadks a ds fopyuksa dk e/;eku gs tcfd /ku rfkk _.k fpugksa dks /;ku esa u j[kk x;k gksa** (1985) ds vuqlkj] ^^izkirkadks a ds e/;eku ls fhkuuk&fhkuu izkirkadks a 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 83

(Characteristics of mean Deviation) 1-,d vad&forj.k ds lhkh izkirkadks a 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] 84

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 izkirkadks a 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 fopyuks a 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 85

fd e/;eku ls fopyu dh x.kuk djrs le; /ku rfkk _.k fpugks a dks egro ugha fn;k tkrk gsa r`rh; pj.k esa e/;eku ls lhkh fopyu Kkr dj ysus ds ckn eku Kkr djrs gsaa vur esa Kkr dj ysrs gsaa dk vksj N ds ekuks a dks lw= esa j[kdj AD dk eku (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) 28 28 30 35 34 33 32 36 x M = d 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 izkirkadks a ds e/;eku dh x.kuk & 86

= 32 iz'u esa] N = 8, bu ewy;ks a dks lw= esa j[kus ij] (Calculation tkrh gs & of mean deviation from grouped Data) O;ofLFkr vad lkexzh ls e/;eku fopyu dh x.kuk fueu lw= }kjk dh Tkcfd d = e/;fcunq ds e/;eku ds fopyu (Deviation of Mid-point from the mean), e/;eku ds e/;fcunqvks a 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 87

(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/;fcunqvks a 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;ks a dks j[kdj AD dk eku izkir dj ysrs gsaa (Long method) AD 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 88

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;ks a dks lw= esa j[kus ij] (Standard deviation) (1968) ds vuqlkj] ^^fn;s gq, izkirkadks a 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 izkirkadks a dk e/;eku ls fopyu Kkr dj yhft, fqj bu fopyuksa dk oxz djds budk 89

;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 90

(When does Standard Deviation used) 1- tc lokzf/kd 'kq) vksj fo'oluh; fopyu eki dh vko';drk gksa 2- tc dsunzh; ekidks a esa e/;eku dh x.kuk dh xbz gksa 3- tc nks vad&forj.kks a dk rqyukred v/;;u djuk gksa 4- tc lg&leca/k xq.kkad rfkk e/;ekuks a 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 izkirkadks a (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 Standard Deviation) (Determinant elements of 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 91

2- vad&forj.k ds izkirkadks a dk ewy; izkekf.kd fopyu dks egroiw.kz <ax ls izhkkfor djrk gsa Deviation of Ungrouped Data) (Standard 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 = izkirkadks a 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 vadks a dk vksj N dk ewy; Kkr djds e/;eku dh x.kuk dhft,a 2. nwljs pj.k esa izkirkadks a 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 92

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 gy % lozizfke e/;eku fudkfy, 5] 7] 8] 9] 10] 11] 12 Scores x M = d d 2 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;ks a dks S. D. ds lw= esa j[kus ij] 93

(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 fopyuks a ds xq.kuqyks a dk ;ksx (Sum of the product of fequencies and deviations), fopyuksa ds oxz,oa vko`fùk;ks a ds xq.kuqy dk ;ksx (Sum of the product of the frequencies and deviation squares), 94

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 nksuks a 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 95

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 ekuks a 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 N = 29 +4 +6 +6 +5 021-4 -6-6 - 420 16 18 12 05 00 04 12 18 16 iz'u esa bu ewy;ks a dks lw= esa j[kus ij] izfke lw= ls x.kuk & rfkk ( ) 96

( ) (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 97

(Long Method) S. D. C. I. X Mid- Point 34 36 1 35 31 33 2 32 28 30 3 29 25 27 5 26 22 24 6 23 19 21 4 20 16 18 3 17 13 15 2 14 10 12 1 11 N = 27 35 64 87 130 138 80 51 28 11 X Md 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 S. D. 98

99

(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. 70 71 68 69 66 67 64 65 62 63 60 61 58 59 56 57 54-55 52-53 50-51 F 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 3 100

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 101