35

GEODL-07

Block-04

[Measures of Central Tendency: Mean, Median and Mode]

(Measures of Central Tendency)

(Types of measures of central

tendency)

(Meaning and Definition of

mean)

(Mathods of calculating mean from ungroup data)

(Mathods

of calculating mean from group data)

(Calculating mean by

long mathod)

(Calculating mean by

short mathod)

(Characteristics of mean)

(Limitations of mean)

(Uses

and importance of mean in field of education)

36

(Meaning and defination of

median)

(Mathods of calculating median from ungroup data)

(Mathods of calculating median from group data)

(Characteristics of median)

(Limitations of median)

(Uses and

importance of median in field of education)

(Meaning and defination of

mode)

(Mathods of calculating mode from ungroup data)

(Mathods

of calculating mode from group data)

(Method of

calculating crude mode)

(Method of calculating

true mode)

(Charactristics of mode)

(Limitations of mode)

(Uses and

importance of mode in field of education)

37

[Measures of Central Tendency: Mean, Median and Mode]

(Measures of Central Tendency)

fdlh lewg dh dsUnzh; izo`fÙk dks la{ksi esa izdV djus ds fy, ftu ekuksa

dk iz;ksx fd;k tkrk gS] mUgsa dsUnzh; eku dgrs gSaA fdlh lewg dk dsUnzh; eku

ml lewg dk izfrfuf/k izkIrkad gksrk gS] lewg ds vf/kdrj izkIrkad mlds

b/kj&m/kj gksrs gSa( dqN mlls dqN de gksrs gSa vkSj dqN mlls vf/kd gksrs gSaA

bl izdkj ;s ,sls izkIrkad gksrs gSa tks ml iwjs lewg ds izkIrkadksa dk izfrfuf/kRo

djrs gSa] ml iwjs lewg dh fo” ks"krkvksa dks iznf'kZr djrs gSaA bls ge

fuEufyf[kr :Ik ls ifjHkkf"kr dj ldrs gSa &

dsUnzorhZ eku fdlh lEiw.kZ lweg dk og izfrfuf/k izkIrkad gksrk gS ftlds

vkl&ikl ml lewg ds vf/kdrj izkIrkad dsfUnzr gksrs gSaA

(Types of measures of central

tendency)

dsUnzorhZ eku rhu izdkj ds gksrs gSa& e/;eku] e/;kad vkSj cgqykadA bl

v/;k; esa bu rhuksa ds vFkZ] mi;ksx vkSj x.kuk dh fof/k;ksa dh ppkZ dh xbZ gSA

(Meaning and Definition of

mean)

vadxf.kr dh Hkk"kk esa ftls vkSlr eku dgrs gSa] lkaf[;dh dh Hkk"kk esa

mlh dks e/;eku vFkok ek/; dgrs gSaA ;g fdlh lewg ds izkIrkadksa dh dsUnzh;

izo`fÙk dks Li"V djrk gS vkSj mlds leLr izkIrkadksa dk izfrfuf/kRo djrk gSA

bldh lcls cM+h [kwch ;g gksrh gS fd blds nksuksa vksj ¼blls vf/kd izkIrkadksa

38

dh vksj vkSj blls de izkIrkadksa dh vksj½ ds izkIrkadksa ds fopyuksa dk ;ksx 'kwU;

gksrk gSA ,d mnkgj.k }kjk ge bls Li"V fd, nsrs gSA

eku yhft, 50 vadksa dh ,d ijh{kk esa 5 Nk=ksa ds izkIrkad bl izdkj gSa &

15 37 23 28 vkSj 17

izkIrkadksa dk vkSlr

izkIrkadksa dks vkjksgh Øe esa j[kus ij vkSlr eku ls mudk fopyu

37

28

23

17

15

24

+ 13

+ 4

-1

-7 -17

-9

0

rc e/;eku dks fuEufyf[kr :Ik esa ifjHkkf"kr fd;k tk ldrk gS &

e/;eku fdlh lewg ds izkIrkadksa dk vkSlr eku gksrk gS ftlds nksuksa vksj

ds izkIrkadksa dk fopyu leku gksrk gSA

(Mathods of calculating mean from ungroup data)

17

39

vkSlr fudkyus dh fof/k ls vki lc ifjfpr gSa] cl mlh fof/k ls

e/;eku fudkyk tkrk gSA lkaf[;dh esa blds fy, fuEufyf[kr lw= gSa &

e/;eku] ∑

tgk¡ e/;eku dk ladsr gS] ∑ ;ksx dk ladsr gS] izkIrkadksa dk ladsr

gS vkSj Nk=ksa dh la[;k dk ladsr gSA

50 vadksa dh ,d ijh{kk esa 10 Nk=ksa ds izkIrkad Øe'k% 23] 15] 8] 30] 31]

21] 19] 24] 26 vkSj 33 gSa] e/;eku Kkr dhft,A

e/;eku ∑

(Note)

bl izdkj ls e/;eku ¼vkSlr½ fudkyuk ml fLFkfr esa FkksM+k dfBu gks

tkrk gS tc Nk=ksa dh la[;k dqN vf/kd gksrh gSA ml fLFkfr esa ge vko`fÙk

forj.k rkfydk cukdj e/;eku Kkr djrs gSaA fuEu mnkgj.k ls ge bl fof/k

dks Li’V dj jgs gSaA

40

50 vadksa dh fgUnh dh ,d ijh{kk esa 20 Nk=ksa us fuEufyf[kr vad izkIr

fd,A

8] 21] 17] 17] 13] 32] 28] 24] 17] 22] 24] 28] 20] 17] 31] 27] 20] 7]

36 vkSj 31

izkIrkadksa dks fLFkfr vkjksgh fLFkfr Øe (Rank Order) esa O;ofLFkr

djus ij &

7] 8] 13] 17] 17] 17] 17] 20] 20] 21] 22] 24] 24] 27] 28] 28] 31] 31]

32 vkSj 36

(x) (Scores)

(f) (Frequencies)

(Scores frequencies)

36 32 31 28 27 24 22 21 20 17 13 8 7

1 1 2 2 1 2 1 1 2 4 1 1 1

36 32 62 56 27 48 22 21 40 68 13 8 7

N = 20 ∑

41

e/;eku] ∑

bl iz'u dks vkSj Hkh vf/kd ljy fof/k ls fuEufyf[kr lw= ds }kjk

fudkyk tk ldrk gSA

e/;eku] ∑

ftlesa] e/;eku

dfYir e/;eku

∑ dfYir e/;eku ls fy, x, fopyuksa dk ;ksx

lEiw.kZ vko`fÙk;ksa dk ;ksx vFkkZr~ dqy izkIrkad

(Calculation)

loZizFke vko`fÙk forj.k rkfydk cukbZ tkrh gSA

blds ckn dfYir e/;eku fuf'pr fd;k tkr gSA rkfydk

dks ,d utj ls ns[kus ls gh e/;eku dk vuqeku yxk;k tk ldrk gSA

bl mnkgj.k esa geus izkIrkad 22 dks dfYir e/;eku ekuk gSA

blds ckn dfYir e/;eku ds vU; lc izkIrkadksa dk

fopyu (d) Kkr dj mls rkfydk esa vafdr fd;k tkrk gSA

42

blds ckn vko`fÙk;ksa dh fopyuksa ls xq.kk djds mu lcdk

;ksx ∑ Kkr fd;k tkrk gSA

vUr esa lw= dk iz;ksx dj lgh e/;eku Kkr fd;k tkrk

gSA

(x) (f) (d)

(x)

(fd)

36

32

31

28

27

24

22

21

20

17

13

8

7

1

1

2

2

1

2

1 A.M. 22

1

2

4

1

1

1

+ 14

+ 10

+ 9

+ 6

+ 5

+ 2

+ 0

- 1

- 2

- 5

- 9

- 14

- 15

+ 14

+ 10

+ 18

+ 12

+ 05

+ 04

- 1

- 4

- 20

- 9

- 14

- 15

N = 20 ∑

+63

-63

43

;gk¡ ij geus dfYir e/;eku 22 fy;k gSA

e/;eku] ∑

(1) bl fof/k ds }kjk x.kuk vf/kd yEch ugha djuh iM+rh gS vr% tc

vko`fÙk;k¡ vf/kd gksrh gSa rks gesa blh fof/k dk iz;ksx djuk pkfg,A

(Mathods

of calculating mean from group data)

tc Nk=ksa dh la[;k vkSj Hkh vf/kd gksrh gS rks mi;qZDr nksuksa esa ls fdlh

Hkh fof/k ls e/;eku fudkyuk cgqr dfBu gksrk gSA ml fLFkfr esa ge loZizFke

izkIrkadksa ds oxZ cukrs gSa vkSj fQj mldh vko`fÙk forj.k rkfydk cukdj mldh

lgk;rk ls e/;eku fudkyrs gSaA budh Hkh nks fof/k;k¡ gSa& ,d yEch fof/k

(Long Method) vkSj nwljh ljy fof/k (Short Method) ,d mnkgj.k }kjk

ge nksuksa fof/k;ksa dks Li"V fd, nsrs gSaA

44

xf.kr dh ,d ijh{kk esa 30 Nk=ksa us fuEufyf[kr vad izkIr fd,A bu

vadksa dks vko`fÙk forj.k rkfydk ds :Ik esa ifjofrZr dhft,A oxkZUrj vFkok oxZ

dk vkdkj 5 gksuk pkfg,A bl rkfydk dh lgk;rk ls yEch vkSj ljy nksuksa

fof/k;ksa ls e/;eku fudkfy, &

48 12 20 32 33 25 34 22 24 36

18 25 15 27 22 32 26 29 32 31

31 44 14 45 47 41 28 38 30 26

(Calculating mean by

long mathod)

bl fof/k esa leLr izkIrkadksa dks mfpr oxkZUrj ysdj mfpr oxksZa esa

O;ofLFkr fd;k tkrk gS vkSj fQj izR;sd oxZ dh vko`fÙk;k¡ Kkr dh tkrh gSA

blesa lHkh izkIrkadksa vkSj mudh vko`fÙk;ksa ds xq.kk ds LFkku ij oxksZa ds e/;

fcUnqvksa vkSj mudh vko`fÙk;ksa dh xq.kk dh tkrh gSA ;gk¡ ge ;Fkk rkfydk

cukdj bl fof/k ls e/;eku fudky jgs gSaA

bl lewg esa vf/kdre izkIrkad 48 gS U;wure izkIrkad 12

izkIrkad foLrkj ( )

45

oxZ foLrkj 5 gh ysuk Bhd jgsxkA ml fLFkfr esa 8 oxZ cusaxsA U;wure

izkIrkad 12 gS] blfy, izFke oxZ izFke oxZ 10 ls 'kq: fd;k tk,xkA

(Classes) (Frequencies)

(f)

(Mid-

Points of

Classes) (x)

(Frequencies

Mid-Points)

45 – 49

40 – 44

35 – 39

30 – 34

25 – 29

20 – 24

15 – 19

10 – 14

= 3

= 2

= 2

= 8

= 7

= 4

= 2

= 2

47

42

37

32

27

22

17

12

141

84

74

256

189

88

34

24

Ci = 5 N = 30 ∑

46

e/;eku] ∑

(Calculating mean by

short mathod)

yEch fof/k ls e/;eku fudkyus esa dkQh xq.kk vkSj tksM+ dh fØ;k djuh

iM+rh gS vkSj blfy, blesa le; cgqr yxrk gSA le; vkSj 'kfDr ds lnqi;ksx

dh n`f"V ls e/;eku fudkyus dh ,d ljy fof/k fudkyh xbZ gSA bl fof/k esa

,d dfYir e/;eku (Assumed Mean) ekudj lgh e/;eku fudkyk tkrk

gSA blfy, bls dfYir e/;eku fof/k (Assumed Mean Method) Hkh dgrs

gSaA blds fy, fuEukadr lw= gS &

e/;eku] ∑

47

tgk¡ A.M. dfYir e/;eku dk ladsr gS] ;ksx dk] fd vko`fÙk vkSj

dfYir e/;eku ls fopyu ds xq.kk dk] N Nk=ksa dh la[;k dk vkSj Ci oxkZUrj

ds vkdkj dkA

(Classes) (Frequencies)

(f)

(deviation

from A.M.) (d)

(Frequencies

Deviation from

A.M.) (fd)

45 – 49

40 – 44

35 – 39

30 – 34

25 – 29

20 – 24

15 – 19

10 – 14

= 3

= 2

= 2

= 8

= 7

= 4

= 2

= 2

3

2

1

(A.M. = 32)0

-1

-2

-3

-4

9

4

2

0

-7

-8

-6

-8

Ci = 5 N = 30 ∑

e/;eku (

)

-29

+17

48

(

)

(

)

(Precaution)

(1) dfYir e/;eku chp ds fdlh ,sls oxZ (Class) esa ekuuk pkfg,

ftlesa lcls vf/kd vko`fÙk;k¡ gksA

(2) ;Fkk oxZ dk e/; fcUnq dfYir e/;eku gksrk gSA

(3) vkjksgh Js.kh esa dfYir e/;eku ds Åij okys oxksZa dk fopyu

/kukRed vkSj uhps okyksa dk _.kkRed gksrk gSA

(Note)

(1) Åij okys mnkgj.k esa nksuksa fof/k;ksa ls e/;eku 29.67 vk;k gSA

vU; izkIrkadksa (vkdM+ksa] data) ds lkFk ;g FkksM+k fHkUu Hkh gks

ldrk gS( ij ;g vUrj ux.; gksrk gSA

49

(2) Ikgyh fof/k ls e/;eku fudkyus esa xq.kk&Hkkx cgqr djuk iM+rk gS

vkSj le; Hkh cgqr vf/kd yxrk gS vr% nwljh fof/k ls gh e/;eku

fudkyuk pkfg,A

(Charactristic of mean)

(1) e/;eku izkIrkadksa dk vkSlr eku gksrk gS] bls xf.krh; fof/k;ksa ls

ljyrk ls Kkr fd;k tk ldrk gSA

(2) e/;eku ,d ,slk dsUnzorhZ eku gksrk gS ftlds nksuksa vksj dk

xq:Roh; eku ¼fopyu½ leku gksrk gSA

(3) e/;eku cgqr laosnu'khy gksrk gS] lewg ds izkIrkdksa esa ls fdlh Hkh

izkIrkad ds eku esa ifjorZu gksus ls e/;eku dk eku ifjofrZr gks

tkrk gS blfy, ;g fdlh lewg ds izkIrkdksa dk lgh izfrfuf/k eku

tkrk gSA

(4) e/;eku vU; dsUnzorhZ ekuksa & e/;kad ,oa cgqykad dh vis{kk vf/kd

fo'oluh; gksrk gSA

(Limitations of mean)

(1) e/;eku fdlh lewg ds izkIrkadksa dk izfrfuf/kRo mlh fLFkfr esa

djrk gS tc izkIrkadksa dk forj.k lkekU; gksrk gSA

50

(2) ;fn fdlh lewg ds dqN gh izkIrkadksa dk eku vU; izkIrkadksa ds eku

dh rqyuk esa cgqr vf/kd vFkok cgqr de gksrk gS] rks ml fLFkfr esa

Hkh e/;eku lewg dk lgh izfrfuf/kRo ugha djrkA

(Uses

and importance of mean in field of education)

f'k{kk ds {ks= esa lcls vf/kd iz;ksx e/;eku dk gksrk gSA bldk iz;ksx

eq[; :Ik ls ml le; fd;k tkrk gS &

(1) tc fdlh lewg ds izkIrkad ds xq:Ro dsUnz dks tkuuk gksrk gSA

(2) tc fdlh lewg ds lHkh izkIrkadksa dks muds eku ds vuqlkj egRo

nsuk gksrk gSA

(3) tc nks ;k nks ls vf/kd lewgksa ds izkIrkadksa dh rqyuk djuh gksrh

gSA

(4) tc izkIrkadksa ds fopyu eku (Deviation) vkSj lg&lEca/k xq.kkad

(Coefficient) dh x.kuk djuh gksrh gSA

(5) tc 'kSf{kd 'kks/kksa esa bldh vko';drk gksrh gSA

(Meaning and defination of

median)

51

e/;kad og dsUnzorhZ eku gksrk gS tks fdlh lewg ds izkIrkadksa dks ,sls nks

cjkcj Hkkxksa esa ck¡Vrk gS fd ,d Hkkx ds lHkh izkIrkad mlls vf/kd gksrs gSa vkSj

nwljs Hkkx ds lHkh izkIrkad mlls de gksrs gSaA ,d mnkgj.k }kjk ge bls Li"V

fd, nsrs gSaA eku yhft, 50 vadksa dh ,d ijh{kk esa 5 Nk=ksa ds izkIrkad bl

izdkj gSa &

17 40 38 21 vkSj 41

Rkc izkIrkadksa dks fLFkfr Øe esa j[kus ij&

17 21 38 40 vkSj 41

vc vki tjk bu ik¡pksa izkIrkadksa ij n`f"V Mkfy,A buesa 38 ,d ,slk

izkIrkad gS ftlls de Hkh nks gh izkIrkad gSa vkSj vf/kd Hkh nks gh izkIrkad gSaA

blfy, bl lewg dk e/;kad 38 gSA

rc e/;kad dks fuEufyf[kr :Ik esa ifjHkkf"kr fd;k tk ldrk gS &

e/;kad fdlh lewg dk og izkIrkad gksrk gS tks mudks fLFkfr Øe esa j[kus

ij muds Bhd e/; esa iM+rk gS vkSj ftlds nksuksa vksj lewg ds vk/ks&vk/ks

¼50&50 izfr'kr½ izkIrkad forfjr gksrs gSaA

(Mathods of calculating median from ungroup data)

e/;kad eku fudkyus dh lcls lh/kh fof/k ;g gS fd ;Fkk lewg ds Nk=ksa

ds izkIrkadksa dks fLFkfr Øe (Rank Order) vFkkZr~ U;wure ls vf/kdre dh

52

fn'kk esa O;ofLFkr dj fy;k tk, vkSj fQj mlesa ls chp ds izkIrkad dks ys fy;k

tk,A

50 vadksa dh ,d ijh{kk esa 7 Nk=ksa ds izkIrkad Øe'k% 15] 8] 30] 21] 19]

vkSj 33 gSaA e/;kad Kkr dhft,A

(Calculation)

izkIrkadksa dks fLFkfr Øe esa O;fLFkr djus ij &

izkIrkadksa = 8] 15] 19] 21] 24] 30] 33

blesa chp dk izkIrkad e/;kad (mdn) 21 gS] ftlds Åij Hkh rhu izkIrkad

gSa vkSj uhps Hkh rhu izkIrkad gSaA

(Note)

ij tc Nk=ksa dh la[;k le gksrh gS rc ge chp ds nks izkIrkadksa dk

vkSlr ys ysrs gSaA

50 vadksa dh ,d ijh{kk esa 8 Nk=ksa us Øe'k% 23] 15] 8] 12] 24] 26] 33

vkSj 31 vad izkIr fd,A e/;kad Kkr dhft,A

(Calculation)

53

izkIrkadksa dks fLFkfr Øe esa yxkus ij &

8] 12] 15] 23] 24] 26] 31] 33

buesa chp ds izkIrkad 23 vkSj 24 nksuksa gSa] buds b/kj&m/kj 3&3 izkIrkad gSaA

e/;eku

(Note)

Åij ds nksuksa mnkgj.kksa dks nsf[k,A buesa e/;kad eku chp dk izkIrkad

vFkok chp ds nks izkIrkadksa ds ;ksx dk vk/kk gSA bls ;fn ge lw= :Ik esa izxV

djuk pkgsa rks bl izdkj dj ldrs gSa&

e/;kad]

ok¡ izkIrkad

tgk¡ e/;kad dk ladrs gS vkSj Nk=ksa dh la[;k dk ladsr gSA

igys mnkgj.k esa

e/;kad]

ok¡ izkIrkad

nwljs mnkgj.k esa

54

e/;kad]

ok¡ izkIrkad

ok¡ izkIrkad

vFkkZr~ pkSFks vkSj ik¡posa izkIrkad dk vkSlr

(Mathods of calculating median from group data)

tc Nk=ksa dh la[;k cgqr vf/kd gksrh gS rks muds izkIrkadksa dks O;ofLFkr

djuk FkksM+k dfBu gks tkrk gSA ml fLFkfr esa ge vko`fÙk forj.k rkfydk

cukdj mldh lgk;rk ls e/;kad eku fudkyrs gSaA

blesa lcls igys vko`fÙk forj.k rkfydk esa lap;h vko`fÙk;ksa dh x.kuk

djrs gSa vkSj fQj mldh lgk;rk ls ml oxZ dk irk yxkrs gSa ftlesa

ok¡

izkIrkad gksrk gSA blds ckn fuEufyf[kr lw=ksa dh lgk;rk ls ;Fkk izkIrkad dk

e/;kad fudkyk tkrk gSA

(i) e/;kad] (

)

ftlesa &

L e/;kad okys oxZ dk Bhd uhps dk lhekad (lower limit) dk ladsr gSA

55

N Nk=ksa dh dqy la[;k dk ladsr gSA

Fb e/;kad okys oxZ ls igys dh vko`fÙk;ksa ds ;ksx dk ladsr gSA

f e/;kad okys oxZ dh vko`fÙk dk ladsr gS] vkSj

Ci oxkZUrj vFkkZr~ oxksZa ds vkdkj dk ladsr gSA

vFkok

(ii) e/;kad] (

)

U e/;kad okys oxZ dk Bhd Åij dk lhekad (Upper limit) dk ladsr gSA

N Nk=ksa dh la[;k dk ladsr gSA

Fa e/;kad okys oxZ ds Åij dh lHkh vko`fÙk;ksa ds ;ksx dk ladsr gSA

f e/;kad okys oxZ dh vko`fÙk dk ladsr gS] vkSj

Ci oxkZUrj vFkkZr~ oxksZa ds vkdkj dk ladsr gSA

,d mnkgj.k }kjk ge bl fof/k dks Li"V fd, nsrs gSaA

vxzkafdr vko`fÙk forj.k rkfydk ls izkIrkdksa dk e/;eku Kkr dhft, &

(Classes) (Frequencies)

(Cumulative

Frequency) (cf)

56

(f)

45 – 49

40 – 44

35 – 39

30 – 34

25 – 29

20 – 24

15 – 19

10 – 14

3

2

5

11

8

5

4

2

40

37

35

30 (e/;kad eku oxZ)

19

11

6

2

Ci = 5 N = 40

(Calculation)

igys lw= ds vuqlkj

( )

bl lw= dk iz;ksx djus ls igys gesa lcls igys ;g irk yxkuk pkfg;s

fd e/;kad eku fdl oxZ esa iM+k gS! blds fy;s gesa lcls igys

dk eku

fudkyuk iM+rk gS vkSj ftl oxZ esa

vkrk gS ogh e/;kad oxZ gksrk gSA

e/;kad]

ok¡ izkIrkad

57

ok¡ izkIrkad

vc gesa ns[kuk gS fd 20 ok¡ izkIrkad fdl oxZ esa iM+ jgk gSA Åij dh

rkfydk nsf[k,A blesa 20 ok¡ izkIrkad 30 - 34 okys oxZ esa iM+ jgk gSA vr% gesa

bl oxZ dks e/;kad eku okyk oxZ ekuuk pkfg,A

rc

L vFkkZr~ bl oxZ dh uhps dh lhek (lower limit) = 29.5

N vFkkZr~ Nk=ksa dh la[;k = 40

Fb vFkkZr~ e/;kad okys oxZ ls igys dh vko`fÙk;ksa dk ;ksx = 19

f vFkkZr~ e/;kad eku okys oxZ dh vko`fÙk = 11

Ci vFkkZr~ oxZ dk vkdkj = 5

e/;eku] (

)

( )

58

nwljs lw= ds vuqlkj]

(

)

;gk¡

U vFkkZr~ e/;kad eku okys oxZ dh Åij dh lhek = 34.5

N vFkkZr~ Nk=ksa dh la[;k = 40

Fa vFkkZr~ e/;kad eku okys oxZ dh Åij dh vko`fÙk;k¡ = 11

f vFkkZr~ e/;kad eku okys oxZ dh vko`fÙk;k¡ = 5

vkSj Ci vFkkZr~ oxZ dk vkdkj = 5

e/;eku] (

)

( )

(Note)

59

;|fi e/;kad eku bu nksuksa lw=ksa ls fudkyk tk ldrk gS ijUrq iz;ksx esa

ge izk;% igyk lw= gh ykrs gSaA mÙkj dh tk¡p ds fy, ;fn ge nwljs lw= dk

Hkh iz;ksx dj ysa rks vPNk gSA

(Characteristics of median)

(1) e/;kad fdlh lewg ds izkIrkadksa dks fLFkfr Øe esa O;ofLFkr djus ds

ckn Bhd e/; dk eku gksrk gS blfy, ;g lewg ds vU; izkIrkadksa

dh rqyuk rFkk O;k[;k ds fy, mi;qDr gksrk gSA

(2) tc fdlh lewg esa izkIrkadksa dk fopyu vlkekU; gksrk gS vFkok

mlds dqN izkIrkadksa dk eku vU; izkIrkadksa dh vis{kk cgqr vf/kd

vFkok cgqr de gksrk gS rks ml fLFkfr esa e/;eku dh vis{kk e/;kad

vf/kd mi;ksxh gksrk gSA

(3) e/;kad fdlh lewg ds izkIrkadksa dh xq.kkRed O;k[;k ds fy, vU;

dsUnzorhZ ekuksa&e/;eku ,oa cgqykad dh vis{kk vf/kd mi;ksxh gksrk

gSA

(4) fdlh lewg ds izkIrkadksa ds xzkQ }kjk iznf'kZr e/;kad ls Hkh lewg ds

vU; izkIrkadksa dh O;k[;k dh tk ldrh gSA

(Limitations of median)

(1) e/;kad fdlh ,d gh lewg ds izkIrkadksa ds fy, Kkr fd;k tk

ldrk gS] nks lewgksa ds izkIrkadksa ds fy, ughaA

(2) e/;kad fdlh lewg ds izkIrkadksa dk fLFkfreku gksrk gS blfy,

e/;kad dks Kkr djus ds fy, lewg ds lHkh izkIrkadksa dh fLFkfr Øe

60

yxkuk vko';d gksrk gSA cM+k lewg gksus ij ;g lc djus esa le;

,oa 'kfDr vf/kd yxrh gSA

(Uses and

importance of median in field of education)

;w¡ f'k{kk ds {ks= esa lcls vf/kd mi;ksx e/;eku dk gksrk gS ijUrq dqN

ifjfLFkfr;k¡ ,slh gksrh gSa ftuesa e/;kad dk gh iz;ksx fd;k tkrk gS] tSls &

(1) tc fdlh lewg ds izkIrkadksa dk forj.k lkekU; ugha gksrk gSA

(2) tc fdlh lewg ds lHkh izkIrkad izkIr ugha gksrsA

(3) tc gesa fdlh lewg ds O;fDr fo'ks"k dh bl fLFkfr dk irk yxkuk

gksrk gS fd og izkIrkadksa dh n`f"V ls izFke 50% esa vkrk gS ;k vfUre

50% esa vkrk gSA

(4) tc e/;eku fudkyus dk le; ugha gksrkA

(5) tc 'kSf{kd 'kks/kksa ds fu"d"kZ izfriknu esa bldh vko';drk gksrh gSA

(Meaning and definition of

mode)

fdlh lewg ds izkIrkadksa esa ftl izkIrkad dh vko`fÙk lcls vf/kd gksrh gS]

mls ml lewg dk cgqykad dgrs gSaA mnkgj.k ds fy, fuEufyf[kr izkIrkadksa ij

n`f"V Mkfy, &

5 4 3 4 6 4 5 7 3 vkSj 6

buesa 5 dh vko`fÙk 2 gS

61

4 dh vko`fÙk 3 gS

3 dh vko`fÙk 2 gS

6 dh vko`fÙk 2 gS

vkSj 7 dh vko`fÙk 1 gS

blfy, bu izkIrkadksa dk cgqykad 4 gksxk D;ksafd lewg esa blh dh vko`fÙk

lcls vf/kd gSA

rc cgqykad dks fuEufyf[kr :Ik esa ifjHkkf"kr fd;k tk ldrk gS &

cgqykad fdlh lewg dk og izkIrkad gksrk gS ftldh ml lewg ds izkIrkadksa

esa lcls vf/kd vko`fÙk gksrh gSA

(Mathods of calculating mode from ungroup data)

cgqykad eku Kkr djus dh lcls lh/kh fof/k ;g gS fd ;Fkk lewg ds

Nk=ksa ds izkIrkadksa dks fLFkfr Øe vFkkZr~ U;wrue izkIrkad ls vf/kdre izkIrkad dh

fn'kk esa O;fLFkr fd;k tk, vkSj fQj izkIrkadksa dh vko`fÙk ns[k yh tk,A

vf/kdre vko`fÙk okyk izkIrkad gh cgqykad eku gksxkA

50 vadksa dh ,d ijh{kk esa 10 Nk=ksa us fuEufyf[kr vad izkIr fd;s gSaA

cgqykad fudkyks &40 11 17 18 8 17 24 37 18 17

(Calculation)

62

izkIrkadksa dks fLFkfr Øe esa yxkus ij

izkIrkadksa = 8 11 17 17 17 18 18 24 37 40

cgqykad = 17

(Note)

dHkh & dHkh ,slk gksrk gS fd nks ;k nks ls vf/kd izkIrkadksa dh vko`fÙk;k¡

leku gksrh gSa] ml fLFkfr esa cgqykad eku fudkyus ds fuEufyf[kr fu;e gSa &

(1) ftl lewg esa lehi ds nks izkIrkadksa dh vko`fÙk;k¡ lcls vf/kd vkSj

vkil esa leku gksrh gSa ml lewg esa bu nksuksa izkIrkadksa dk vkSlr

cgqykad gksrk gSA

mnkgj.k &

8 11 17 17 17 19 19 19 24 vkSj 37

cgqykad]

(2) ftl lewg esa nks ,sls izkIrkadksa dh vko`fÙk lcls vf/kd vkSj vkil esa

cjkcj gksrh gSa tks ,d nwljs ds fudV ugha gksrs rks msls lewg esa nksuksa

gh izkIrkad cgqykad gksrs gSaA ,sls lewgksa dks f}cgqykadh (Bi–Modal)

dgk tkrk gSA mnkgj.kkFkZ &

8 11 17 17 17 18 24 24 24 vkSj 26

cgqykad] vkSj

63

(3) ftl lewg esa vusd vFkok lHkh izkIrkadksa dh vko`fÙk;k¡ leku gksrh gSa

ml lewg esa cgqykad fudkyus dk iz'u gh ugha mBrkA mnkgj.kkFkZ &

11 11 11 16 16 16 17 17 17 19 19 19 23 23 esa dksbZ cgqykad ugha

gSA

(Mathods of

calculating mode from group data)

tc Nk=ksa dh la[;k cgqr vf/kd gksrh gS rks izkIrkadksa dks fLFkfr Øe esa

yxkuk dfBu gksrk gS vkSj blesa Hkwy dh Hkh lEHkkouk jgrh gSA ml fLFkfr esa

vko`fÙk forj.k rkfydk cukdj mldh lgk;rk ls cgqykad eku fudkyk tkrk

gSA bldh Hkh nks fof/k;k¡ gSa& ,d ls vuqekfur cgqykad fudkyk tkrk gS vkSj

nwljh ls lgh cgqykad fudkyk tkrk gSA

(Method of

calculating crude mode)

vuqekfur cgqykad fudkyus ds fy, lcls igys vko`fÙk forj.k rkfydk

cukrs gSaA fQj ftl oxZ esa izkIrkadksa dh vko`fÙk;k¡ lcls vf/kd gksrh gSa mldk

e/; fcUnq Kkr djrs gSaA ;g e/; fcUnq gh ml lewg dk vuqekfur cgqykad

gksrk gSA

fuEufyf[kr vko`fÙk forj.k rkfydk ls izkIrkadksa dk cgqykad Kkr dhft,A

64

(Classes) (Frequencies)

90 – 99

80 – 89

70 – 79

60 – 69

50 – 59

40 – 49

30 – 39

20 – 29

10 – 19

2

5

8

9

12

4

5

3

2

(Calculation)

vko`fÙk forj.k rkfydk dks /;ku ls nsf[k,A blesa izkIrkadksa dh lcls

vf/kd vko`fÙk 50 – 59 okys oxZ esa gSA

vuqekfur cgqykad]

(Method of calculating

true mode)

Lkgh cgqykad fudkyus ds fy, igys e/;eku fudkyk tkrk gS] fQj

e/;kad vkSj fQj e/;kad ds rhu xqus esa ls e/;eku dk nks xquk ?kVk fn;k tkrk

gSA bls lw= :Ik esa bl izdkj O;Dr fd;k tkrk gS &

65

e/;eku vkSj e/;kad fudkyus dh fof/k ge crk gh pqds gSaA ;gk¡ mi;qZDr

rkfydk dks iwjh djds mudh x.kuk dj cgqykad fudky jgs gSa&

(Classes) (Frequencies) (f) (Cumulative

Frequencies)(Deviaion

from A. M.) (d)

(Frequencies

Deviaton)

90 – 99 80 – 89 70 – 79 60 – 69 50 – 59 40 – 49 30 – 39 20 – 29 10 – 19

2 5 8 9

12 4 5 3 2

50 48 49 35 26 14 10 5 2

4 3 2 1

0 (A.M.) -1 -2 -3 -4

8 15 16 9 0 -4

-10 -9 -8

Ci = 10 N = 50 ∑ = 17

e/;eku] (∑ )

(

)

66

e/;kad] (

)

(

)

cgqykad]

(Note)

67

(1) bl vko`fÙk forj.k ls vuqekfur cgqykad 54.5 vk;k Fkk vkSj lgh

cgqykad 60.21 vk;k gSA bu nksuksa esa ;|fi FkksM+k gh vUrj gS ijUrq

bl vUrj ls gh ewY;kadu ds fu.kZ; fHkUu gks tkrs gSa] vr% tgk¡ rd

lEHko gks gesa lgh cgqykad gh fudkyuk pkfg,A

(2) tc forj.k yxHkx lkekU; lEHkkouk oØ cukrk gS rks e/;eku] e/;kad

eku vkSj cgqykad eku esa cgqr FkksM+k vUrj gksrk gSA Åij ds mnkgj.k

esa e/;eku 57.90 e/;kad eku 58.67 vkSj cgqykad eku 60.21 gSA

bu rhuksa esa cgqr de vUrj gSA

(3) cgqykad fudkyus ds fy, vyx ls Hkh lw= gSa ftuds }kjk fcuk

e/;eku rFkk e/;kad lh/kk cgqykad fudkyk tk ldrk gSA

(1) cgqykad]

(2) cgqykad]

ftuesa

L = cgqykad oxZ dh 'kq) fuEu lhek

cgqykad oxZ ls iwoZ ds oxZ dh vko`fÙk

cgqykad oxZ dh vko`fÙk

cgqykad oxZ ds rqjUr ckn okys oxZ dh vko`fÙk

blesa L = 49.5, 4 rFkk gS

cgqykad]

68

–

f}rh; lw= esa

cgqykad]

(Note)

(1) tc dsoy cgqykad gh Kkr djuk gksrk gS rks mijksDr lw= dk iz;ksx

djuk gh vf/kd mi;ksxh jgrk gS D;ksafd igyh fof/k esa ;fn Hkwyo'k

e/;eku ;k e/;kad esa ls dksbZ ,d xyr gks tk, rks cgqykad Lor%

xyr gks tk,xkA

69

(2) bl vko`fÙk forj.k ls vuqekfur cgqykad 54.5 vk;k rFkk izFke o

f}rh; fof/k ls 'kq) cgqykad Øe'k% 60.21 gS 56.21 rFkk 56.42

vk;kA ;g vUrj fofHkUu fof/k;ksa dk iz;ksx djus ds dkj.k gqvkA izk;%

'kq) cgqykad gh fudkyuk vf/kd mi;ksxh jgrk gSA

(Charactristics of mode)

(1) fdlh lewg ds izkIrkadksa dk cgqykad cgqr ljyrk ls Kkr fd;k tk

ldrk gS] vlkekU; izkIrkadksa dk HkhA

(2) fdlh lewg ds izkIrkadksa ds cgqykad dk eku izkIrkadksa ds xzkQ }kjk Hkh

Kkr fd;k tk ldrk gSA

(3) ;fn fdlh lewg ds dqN izkIrkadkas dk eku vU; izkIrkadksa ds eku dh

vis{kk cgqr vf/kd vFkok cgqr de gksrk gS rks ml fLFkfr esa Hkh ml

lewg ds izkIrkadksa dk cgqykad izHkkfor ugha gksrkA

(4) ;g fdlh lewg ds izkIrkadksa dh xq.kkRed O;k[;k ds fy, vf/kd

mi;ksxh gksrk gSA

(Limitations of mode)

(1) tc fdlh lewg esa dksbZ Hkh izkIrkad cgqykad ugha gksrk vFkok nks ls

vf/kd ,sls izkIrkad ftuesa cgqr vf/kd vUrj gksrk gS] cgqykad dh Js.kh

esa vkrs gSa rks ml fLFkfr esa cgqykad vFkZghu gksrk gSA

(2) cgqykad ,d vuqekfur dsUnzorhZ eku gksrk gS blfy, lkaf[;dh;

x.kukvksa ds fy, de mi;ksxh gksrk gSA

70

(Uses and

importance of mode in field of education)

;w¡ cgqykad vU; dsUnzorhZ eku&e/;eku ,oa e/;kad dh vis{kk de

fo'oluh; gksrk gS fQj Hkh f'k{kk ds {ks= esa dqN ,slh ifjfLFkfr;k¡ gksrh gSa tc

bldk iz;ksx fd;k tkrk gSA

(1) tc fdlh lewg dh dsUnzh; izo`fÙk dk vuqeku yxkuk gksrk gS rc

cgqykad eku dk mi;ksx fd;k tkrk gSA

(2) tc fdlh lewg dh lokZf/kd ilUnksa ;k miyfC/k;ksa dk irk yxkuk gks

rc Hkh cgqykad dk gh iz;ksx fd;k tkrk gSA

(3) tc fdlh lewg ds cgqpfpZr eku dk irk yxkuk gksrk gS rc Hkh bl

dsUnzorhZ eku cgqykad dk iz;ksx fd;k tkrk gSA

(4) tc e/;eku ;k e/;kad fudkyus ds fy, le; ugha gksrk rc ljyrk

ls cgqykad Kkr dj fy;k tkrk gS vkSj mlls dke pyk fy;k tkrk

gSA

(5) tc 'kSf{kd 'kks/kksa ds {ks= esa midYiukvksa (Hypothesis) dk fuekZ.k

djuk gksrk gS] rc dHkh & dHkh cgqykad dk iz;ksx fd;k tkrk gSA

71

(Central Tendency)

(Mean), (Median) (Mode)

(M), (Mdn)

(Mo)

C. I. 20 21 22 23 24 25 26

2 4 6 7 5 3 1

(M), (Mdn)

(Mo)

C. I.

80 – 84

75 – 79

70 – 74

65 – 69

60 – 64

55 – 54

45 – 49

40 - 44

2 4 6 8

10 9 7 4 1

N = 51

C. I.

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

1 3 4 5 6 4 2 2 1

N = 28

72

C.I. f 50 – 54 45 – 49 40 – 44 35 – 39 30 – 34 25 – 29 20 – 24 15 – 19 10 – 14

4 7 10 12 14 11 9 6 5

N = 78

C.I. 31 – 33 28 – 30 25 – 27 22 – 24 19 – 21 16 – 18 13 – 15 10 – 12 f 1 2 3 4 3 2 1 1

C.I. 40 – 44 35 – 39 30 – 34 25 – 29 20 – 24 15 – 19 10 – 45 5 – 9 0 – 4 f 1 1 2 3 4 3 2 1 1

73

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

vkxjk&2

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

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

3- xqIrk] MkW- ,l- ih-] xqIrk] MkW- vydk] mPprj f'k{kk euksfoKku] 'kkjnk

iqLrd Hkou] bykgkckn

4- Popham, W.J. (1993). Educational evaluation. Boston : Allyn

and Bacon

5- Popham, W.J. (1993). Modern educational measurement :

Englewood Cliffs, NJ. : Prentice Hall.

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

need to know New York : Prentice Hall.