geodl-07 block-04 [measures of central tendency: …...37 [measures of central tendency: mean,...
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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.
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