evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

we¯Ívi cwigvc
Measures of Dispersion
6
f‚wgKv
Z_¨gvbmg~‡n A_ev †Kvb MYmsL¨v wb‡ek‡bi †ÿ‡Î Z_¨gvb¸‡jvi †K‡›`ªi w`‡K †K›`ªxf‚Z nIqvi cÖeYZv †hgb
_v‡K †Zgwb gvb¸‡jvi wewfbœ w`‡K cÖmvwiZ nIqvi cÖeYZvI _v‡K| A_©vr †Kvb Pj‡Ki gv‡bi †K›`ªxq cÖeYZvB
GKgvÎ ˆewkó¨ bq PjKwUi gv‡bi we¯ÍviI Ab¨ GKUv ˆewkó¨| Pj‡Ki gvb¸‡jvi wewfbœZv n‡jv we¯Ívi e‡j|
we¯Ív‡ii cwigvc hvi Øviv Kiv nq Zv‡K we¯Ívi cwigvcK ejv nq| G Aa¨v‡q wewfbœ cv‡V we¯Ív‡ii cwigvc,
cÖ‡qvRbxqZv, we¯Ívi cwigv‡ci myweav Amyweav BZ¨vw` m¤ú‡K© Av‡jvPbv Kiv n‡q‡Q|

BDwbU mgvwßi mgq BDwbU mgvwßi m‡e©v”P mgq 2 mßvn

G BDwb‡Ui cvVmg~n
cvV-6.1 : we¯Ívi I we¯Ívi cwigvc
cvV-6.2 : cwimi I cwimiv¼
cvV-6.3 : PZz_©K e¨eavb I PZz_©K e¨eavbv¼
cvV-6.4 : Mo e¨eavb I Mo e¨eavbv¼
cvV-6.5 : cwiwgZ e¨eavb I †f`vsK Ges cwiwgZ e¨eavbv¼ I we‡f`vsK

BDwbU 6 c„ôv 77
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

cvV-6.1 we¯Ívi I we¯Ívi cwigvc

Dispersion and Measures of Dispersion
D‡Ïk¨

G cvV †k‡l Avcwb-

• we¯Ív‡ii msÁv ej‡Z cvi‡eb;
• we¯Ív‡ii cwigvcK m¤ú‡K© ej‡Z cvi‡eb;
• we¯Ív‡ii cÖKvi‡f` m¤ú‡K© wjL‡Z cvi‡eb;
• we¯Ív‡ii cwigv‡ci cÖ‡qvRbxqZv m¤ú‡K© e¨vL¨v Ki‡Z cvi‡eb|

we¯Ívi (Dispersion)
Pj‡Ki †K›`ªxq cÖeYZvi cwigvcK †hgbÑ Mo, ga¨gv, cÖPyiK Gi mvnv‡h¨ Pj‡Ki gvbmg~‡ni ˆewkó¨ myôzfv‡e Rvbv
m¤¢e bq| D`vniY¯^iƒc †KD hw` g‡b K‡i MÖx®§Kv‡j †QvU GKUv b`xi cvwbi MfxiZv M‡o 2 dzU Ges mn‡R GUv
cvi nIqv hv‡e Ggb wm×všÍ wb‡j wec‡` co‡eb| KviY b`xi cvwbi MfxiZv †Kv_vq †Kgb we¯ÍvwiZfv‡e Zv‡K
Rvb‡Z n‡e| A_©vr cvwbi MfxiZvi e¨eavb †Kgb Rvb‡Z n‡e, †Kvb RvqMvq hw` 1 ev 2 dzU Avevi †Kvb RvqMvq
hw` 7 ev 8 dzU nq Z‡eB wec‡`i m¤§ywLb n‡Z nq| myZivs Z_¨ivwki e¨eavb ev we¯Ívi †Kvb Pj‡Ki wØZxq
ˆewkó¨|
we¯Ívi Øviv Pj‡Ki Z_¨ivwki e¨vwß wKsev wbw`©ó †Kvb gvb †_‡K ivwk¸‡jvi wePy¨wZ ev e¨eavb eySv‡bv n‡q _v‡K|

we¯Ívi cwigvc t Z_¨‡m‡Ui gvb¸‡jvi wfbœZv‡K wek¦vm Ges we¯Ív‡ii cwigvc †h gv‡bi Øviv Kiv nq Zv‡K we¯Ív‡ii
cwigvcK ejv nq| wKfv‡e Pj‡Ki Z_¨gvbmg~n wewÿß n‡q Av‡Q Zvi wewfbœ gvÎv we¯Ívi cwigvc‡Ki Øviv Rvbv
hvq|

we¯Ívi cwigv‡ci cÖKvi‡f`

we¯Ívi cwigvc `yB cÖKvi n‡Z cv‡i, †hgbÑ
1| cig we¯Ívi cwigvc (Absolute Measures of Dispersion)
2| Av‡cwÿK we¯Ívi cwigvc (Relative Measures of Dispersion)

1| cig we¯Ívi cwigvc

cig we¯Ívi cwigv‡ci †ÿ‡Î Pj‡Ki Z_¨gvb Ges we¯Ív‡ii GKK GKB _vK‡e| cig we¯Ívi Pvi ai‡bi| h_vÑ
K) cwimi (Range)
L) PZz_©K e¨eavb (Quartile deviation)
M) Mo e¨eavb (Mean deviation)
N) cwiwgZ e¨eavb I †f`vsK (Standard deviation and variance)

BDwbU 6 c„ôv 78
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2| Av‡cwÿK we¯Ívi cwigvc (Relative measures of dispersion)

†h cwigvc †Kv‡bv GKwU we¯Ívi cwigvc I †K›`ªxq cÖeYZvi cwigv‡ci mv‡_ Zzjbv K‡i wbY©q Kiv nq Zv‡K
Av‡cwÿK we¯Ívi cwigvc K‡j| GUv GKUv GKK wenxb msL¨v Ges G‡K kZKiv ev AbycvZ AvKv‡i cwigvc Kiv
nq| Av‡cwÿK we¯Ívi cwigvc Pvi cÖKvi, h_vÑ
K) cwimivsK (Co-efficient of Range)
L) PZz_©K e¨eavbvsK (Co-efficient of Quartile deviation)
M) Mo e¨eavbvsK (Co-efficient of Mean deviation)
N) cwiwgZ e¨eavbvsK I we‡f`vsK (Co-efficient of standard deviation and Co-efficient of Variation)

we¯Ívi cwigv‡ci cÖ‡qvRbxqZv

1| M‡oi wek¦vm‡hvM¨Zv wbY©q t Z_¨gvb mg~‡ni we¯Ívi cwigv‡ci Øviv M‡oi Ae¯’vb Ges mwVKZv wbY©q Kiv
hvq| we¯Ívi cwigvcK hw` Kg nq Z‡e eyS‡Z n‡e Pj‡Ki Z_¨ gvbmg~n Gi †K›`ªwe›`y ev M‡oi Lye KvQvKvwQ
Ae¯’vb Ki‡Q Ges G‡ÿ‡Î Mo wek¦vm‡hvM¨ A_©vr G gvb mKj gv‡bi cÖwZwbwaZ¡ K‡i| Avevi we¯Ívi
cwigv‡ci gvb hw` †ewk nq Z‡e eyS‡Z n‡e Z_¨gvbmg~n Mo †_‡K †ek `~‡i we¯Í„Z| G‡ÿ‡Î Mo Z_¨mg~‡ni
mKj gvb‡K cÖwZwbwaZ¡ K‡i bv|

2| `yB ev Z‡ZvwaK Pj‡Ki Zzjbvg~jK Av‡jvPbv t we¯Ívi cwigvc‡Ki Øviv `yB ev Z‡ZvwaK Pj‡Ki
Z_¨gvbmg~‡ni g‡a¨ Zzjbv Kiv hvq| †h Pj‡Ki we¯Ívi gvb Kg nq †mwUB fvj|

mvims‡ÿc:
Z_¨‡m‡Ui gvb¸‡jvi wewfbœZv n‡jv we¯Ívi|

BDwbU 6 c„ôv 79
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cvV-6.2 cwimi I cwimivsK

Range and Co-efficient of Range
D‡Ïk¨

G cvV †k‡l Avcwb-

• cwimi wKfv‡e wbY©q Ki‡Z nq ej‡Z cvi‡eb;
• cwimi Gi myweav I Amyweav mg~n ej‡Z cvi‡eb|

cwimi (Range)
cwimi nj we¯Ívi cwigv‡ci me‡P‡q mnR‡eva¨ I mnRfv‡e wbY©‡qi cwigvc| Pj‡Ki gvbmg~‡ni e„nËg I ÿz`Z ª g
gv‡bi ev msL¨vi cv_©K¨ ev e¨eavb‡K cwimi e‡j| A_©vr cwimi = e„nËg msL¨v Ñ ÿz`Zª g msL¨v|
MYmsL¨v wb‡ek‡bi †ÿ‡Î D”PZi †kÖwYi D”Pmxgv Ges wb¤œZi †kÖwYi wb¤œmxgvi e¨eavb‡K cwim‡ii cwigvY e‡j|
A_©vr cwimi = D”P‡kÖwYi D”Pmxgv Ñ wb¤œ‡kÖwYi wb¤œmxgv|
cwim‡ii Av‡cwÿK cwigvc n‡jv cwimivsK| Pj‡Ki Z_¨gvb mg~‡ni cwimi‡K e„nËg I ÿz`Z ª g gv‡bi †hvMdj
Øviv fvM Ki‡j cwimivsK cvIqv hvq| A_©vrÑ,
cwimi
cwimivsK = e„nËg msL¨v + ÿz`ªZg msL¨v × 100
K. A‡kÖwYK…Z Z‡_¨i †ÿ‡Î:
D`vniY-1 t 12 Rb e¨w³i D”PZv nj h_vµ‡g 62, 65, 68, 69, 71, 69, 67, 71, 66, 73, 72, 61 BwÂ|
cwimi Ges cwimivsK wbY©q Kiæb|

mgvavb t GLv‡b e„nËg msL¨v= 73

ÿz`ªZg msL¨v = 61
∴ cwimi = 73Ñ61
= 12 BwÂ|
12
cwimivsK = 73+61
12
= 134 × 100
= 8.96%
L. †kÖwYK…Z Z‡_¨i †ÿ‡Î:
D`vniY-2: wb‡¤œi MYmsL¨v wb‡ekb web¨vm n‡Z cwimi I cwimiv¼ wbY©q Kiæb|

‡kÖwY 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55
MYmsL¨v 7 11 14 19 27 48 43 21 13 9

BDwbU 6 c„ôv 80
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

mgvavb t cwimi = D”P‡kÖwYi D”Pmxgv Ñ wb¤œ‡kÖwYi wb¤œmxgv

= 55 Ñ 5
= 50
50
∴ cwimivsK = 55+5 × 100
50
= 60 × 100
= 83.33%

cwim‡ii myweav I Amyweav

myweav t
K) cwimi Lye mn‡R eySv hvq Ges mn‡R wbY©q Kiv hvq|
L) cwimi wbY©q Ki‡Z Lye Kg mgq jv‡M|

Amyweav t
K) cwimi ïaygvÎ Z_¨gvb mg~‡ni e„nËg I ÿz`Z ª g gv‡bi Dci wfwË K‡i Kiv nq| mKj Z_¨gv‡bi Dci wbf©i
K‡i Kiv nq bv e‡j GUv ZZUv wbf©i‡hvM¨ we¯Ívi cwigvc bq|
L) cÖvßgv‡bi cÖfve cwim‡ii Dci h‡_ó Av‡Q|
M) MvwYwZK cÖ‡qvR‡bi †ÿ‡Î GUv Dc‡hvMx bq|

mvims‡ÿc:
cwimi we¯Ívi cwigv‡ci me‡P‡q mnR fv‡e wbY©‡qi cwigvc|

BDwbU 6 c„ôv 81
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

cvV-6.3 Mo e¨eavb I Mo e¨eavbvsK

Mean deviation and Co-efficient of mean deviation
D‡Ïk¨

G cvV †k‡l Avcwb-

• Mo e¨eavb m¤ú‡K© ej‡Z cvi‡eb;
• Mo e¨eavb wbiƒcY Ki‡Z cvi‡eb;
• Mo e¨eavbvsK m¤ú‡K© ej‡Z cvi‡eb;
• Mo e¨eavbvsK wbY©q Ki‡Z cvi‡eb|

Mo e¨eavb I Mo e¨eavbvsK
Mo e¨eav‡bi †ÿ‡Î Z_¨‡m‡Ui cÖwZwU gvb n‡Z MvwYwZK Mo A_ev ga¨gv Gi e¨eavb †bqv nq| mvaviYZ
MvwYwZK Mo †_‡K cÖwZwU gv‡bi e¨eav‡bi †hvMdj k~b¨ weavq GB e¨eavb¸‡jv ïaygvÎ abvZ¥K a‡i †bIqv nq
A_©vr cig (absolute) gvb †bqv nq| Zvici H e¨eavb¸‡jvi MvwYwZK Mo wbY©q K‡i Mo e¨eavb cvIqv hvq|
K. A‡kÖwYK…Z Z‡_¨i †ÿ‡Î (mivmwi c×wZ)
hw` n msL¨K Z_¨gvb wewkó †Kvb Pj‡Ki gvb x1, x2, ................ xn nq
_ 1 n
Ges x = n ∑ xi
i=1
n
_
hw` H Pj‡Ki gvbmg~‡ni Mo x nq Z‡e Mo e¨eavb n‡eÑ
1 _
MD = n n |Xi- x | i = 1, 2, ..........n.
∑

i=1

BDwbU 6 c„ôv 82
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

Mo e¨eavbvsK nj
Mo e¨eavb
Mo e¨eavbvsK = MvwYwZK Mo × 100
D`vniY-1 t
GKwU K‡j‡Ri `k Rb Qv‡Îi eqm wb‡¤œ †`Iqv Av‡Q| Mo e¨eavb I Mo e¨eavbvsK †ei Kiæb|
eqm (erm‡i) t 16, 15, 17, 18, 14, 19, 21, 16, 20, 23

mgvavb t
_ ΣXi
G‡ÿ‡Î MvwYwZK Mo X = n
179
= 10
= 17.9
Mo e¨eavb †ei Ki‡Z n‡j wb‡¤œi mviwY e¨envi Ki‡Z n‡e|
Mo e¨eavb I Mo e¨eavbvsK wbY©q mviYx
eqm Xi Xi − X Xi − X
16 -1.9 1.9
15 -2.9 2.9
17 -0.9 0.9
18 0.1 0.1
14 -3.9 3.9
19 1.1 1.1
21 3.1 3.1
16 -1.9 1.9
20 2.1 2.1
23 5.1 5.1
_
Σ|Xi- X | = 23
10

∑ | xi − x |
∴ Mo e¨eavb MD = i =1

n
1
= 10 × 23
= 2.3
2.3
Ges Mo e¨eavbvsK = 17.9 × 100
= 12.849%

BDwbU 6 c„ôv 83
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

L. †kÖwYK…Z Z‡_¨i †ÿ‡Î (mivmwi c×wZ)

hw` n msL¨K gvb‡K k †kÖwY wewkó MYmsL¨v wb‡ek‡b cwiYZ Kiv hvq Ges Xi (i=1,2,........n) i Zg †kÖwYi ga¨gvb
nq Ges ƒi hw` D³ †kÖwYi MYmsL¨v nq Z‡eÑ
k  _
Mo e¨eavb (MD) = ∑ƒi xi- x
i=1
_ Σƒi Xi n
GLv‡b, x = N ; [N =
∑ ƒi ]
i=1
Mo e¨eavb
Mo e¨eavbvsK = MvwYwZK Mo × 100

†kYxK…Z Z‡_¨i †ÿ‡Î (mivmwi c×wZ)

D`vniY-2: wb‡¤œ mviwY †_‡K Mo e¨eavb I Mo e¨eav¼ wbY©q Kiæb|
†kÖwY 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
MYmsL¨v 2 3 15 37 23 13 5 2
mgvavb t
Mo e¨eavb I Mo e¨eavbvsK wbY©q mviwY
†kÖwY ga¨gvb MYmsL¨v ƒi ƒ Xi _ _
Xi i ƒi (Xi- X ) ƒi |Xi- X |
0-10 5 2 10 -69.0 69.0
10-20 15 3 45 -73.5 73.5
20-30 25 15 375 -217.5 217.5
30-40 35 37 1295 -166.5 166.5
40-50 45 23 1035 126.5 126.5
50-60 55 13 795 201.5 201.5
60-70 65 5 325 127.5 127.5
70-80 75 2 150 71.00 71.00
N=Σƒi =100 8 _
∑ƒi Xi =3950 Σƒi |Xi- X |
i=1 = 1053
_ 3950
X = 100 = 39.5

∑ fi | xi − x |
AZGe, Mo e¨eavb MD = i =1
N

BDwbU 6 c„ôv 84
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

1053
= 100 , n= 100
= 10.53
10.53
AZGe, Mo e¨eavbvsK = 39.5 ×100
= 26.658

Mo e¨eav‡bi myweav I Amyweav

myweav t
K) Mo e¨eavb mn‡R cwigvc Kiv hvq|
L) MvwYwZK wnmve mn‡R Kiv hvq|
M) `yB ev Z‡ZvwaK wb‡ek‡bi Zzjbv Kivi Rb¨ GUv GKUv wbf©i‡hvM¨ cwigvcK|
Amyweav t
K) Mo e¨eav‡bi †ÿ‡Î FbvZ¥K gvb AMÖvn¨ Ki‡Z nq d‡j G‡Z cieZx©‡Z exRMvwYwZK cÖwµqv Av‡ivc Kiv hvq
bv|

mvims‡ÿc:
M‡o e¨eavb wbY©‡qi †ÿ‡Î cÖwZwU Z_¨gvb n‡Z MvwYwZK Mo ga¨gvq cig e¨eavb †bqv nq|

BDwbU 6 c„ôv 85
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

cvV-6.4 PZz_©K e¨eavb I PZz_©K e¨eavbvsK

Quartile deviation and Co-efficient of Quartile deviation
D‡Ïk¨

G cvV †k‡l Avcwb-

• PZz_©K e¨eavb m¤ú‡K© ej‡Z cvi‡eb;
• PZz_©K e¨eavb wbY©q Ki‡Z cvi‡eb;
• PZz_©K e¨eavbvsK m¤^‡Ü ej‡Z cvi‡eb;
• PZz_©K e¨eavbvsK wbY©q Ki‡Z cvi‡eb;
• PZz_©K e¨eav‡bi myweav I Amyweav m¤ú‡K© ej‡Z cvi‡eb|

PZz_©K e¨eavb (Quartile deviation)

c~‡e©i cv‡V †`Lv †M‡Q †h, cwimi ïaygvÎ Pj‡Ki ÿz`Z ª g gvb Ges e„nËg gv‡bi Dci wbf©ikxj| GLb PZz_K
©
e¨eavb Pj‡Ki cÖ_g I Z…Zxq PZz_©‡Ki Dci wbf©ikxj| GwU we¯Ív‡ii wØZxq cwigvc Ges Gi gvb nj 3q I 1g
PZz_©K gv‡bi e¨eav‡bi A‡a©K| A_©vr Q1 hw` 1g PZz_K
© Ges Q3 hw` 3q PZz_K© nq Zvn‡jÑ
Q3-Q1
PZz_©K e¨eavb QD = 2
BDwbU-6 Gi cvV 6.4 G Q1, Q2, Q3 wKfv‡e wbY©q Ki‡Z nq Gi m~Î we¯ÍvwiZfv‡e †`qv Av‡Q|
PZz_©K e¨eavb cwigvc‡Ki GKK Av‡Q| PZz_K © e¨eav‡bi Av‡cwÿK cwigvcK nj PZz_K © e¨eavbvsK| PZz_K
©
e¨eavb‡K 3q I 1g PZz_©‡Ki MvwYwZK Mo Øviv fvM Ki‡j PZz_K
© e¨eavbvsK cvIqv hvq| GUv GKwU GKK wenxb
msL¨v|
(Q3-Q1)/2
PZz_©K e¨eavbvsK = (Q +Q )/2 × 100
3 1
Q3-Q1
= Q +Q × 100
3 1
A‡kÖYxK…Z Z‡_¨i †ÿ‡Î
D`vniY-1
wb‡¤œ `ywU Z‡_¨i gvbmg~n †`qv Av‡Q| cÖ_g PZz_K
© (Q1) Ges 3q PZz_K
© (Q3) wbY©q Kiæb|
Z_¨-1t 7, 4, 11, 15, 12, 21, 19, 16, 9, 8, 10, 14, 18, 13, 17|
Z_¨-2t 16, 15, 21, 22, 26, 24, 29, 25, 17, 30, 25, 23, 28, 31, 34, 33|

BDwbU 6 c„ôv 86
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

mgvavb t n hLb we‡Rvo:

cÖ_gwU‡Z 15wU gvb Av‡Q| G‡`i‡K DaŸ©µgvbymv‡i mvRv‡j cvIqv hvq|
4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21

15+1
∴ Q1 =
4
16
= 4 Zg gvb|
= 4_© gvb|
= 9|
3(15+1)
∴ Q3 = Zg gvb
4
= 12 Zg gvb
= 17|
Q3-Q1
GLb, PZz_©K e¨eavb QD = 2
47.826-31.351
= 2
=
Q3-Q1
Ges PZz_©K e¨eavbvsK = Q +Q × 100
3 1
47.826 - 31.351
= 47.826+31.351 × 100
16.475
= 79.187 × 100
=
n hLb †Rvo:
wØZxq Z‡_¨ 16wU msL¨vgvb Av‡Q hv‡`i‡K µgvbymv‡i mvRv‡j cvIqv hvq|
15, 16, 17, 21, 22, 23, 24, 25, 25, 26, 28, 29, 30, 31, 33, 34|
16 16 + 4
∴ Q1 =
4 Zg I Zg gv‡bi Mo
4
21 + 22
=
2
43
= 2
= 21.5
3 × 16 3 × 16 + 4
∴ Q3 = Zg I Zg
4 4
= 12Zg gvb I 13Zg gv‡bi MvwYwZK Mo

BDwbU 6 c„ôv 87
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29+30
= 2
= 29.5
Q3-Q1
GLb, PZz_©K e¨eavb QD = 2
47.826-31.351
= 2
=
Q3-Q1
Ges PZz_©K e¨eavbvsK = Q +Q × 100
3 1
47.826 - 31.351
= 47.826+31.351 × 100
16.475
= 79.187 × 100

†kÖYxK…Z Z‡_¨i †ÿ‡Î

D`vniY-2 : BDwbU 5 Gi cvV 5.4 Gi D`vniY-2 G ewY©Z Z_¨m~n †_‡K PZz_K © e¨eavb I PZz_K
© e¨eavsK wbY©q
Kiæb|
wb¤œwjwLZ MYmsL¨v wb‡ek‡bi †ÿ‡Î PZz_©K e¨eavb I PZz_K
© e¨eavbvsK wbY©q Kiæb|
‡kÖwY 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
MYmsL¨v 2 3 15 37 23 13 5 2

mgvavb t
PZz_K
© e¨eavb wbY©q mviwY
†kÖwY MYmsL¨v µg‡hvwRZ MYmsL¨v
0-10 2 2
10-20 3 5
20-30 15 20=Fc1
30-40 37 57= Fc3
40-50 23 80
50-60 13 93
60-70 5 98
70-80 2 100

n 100
GLv‡b 4 = 4 = 25 GLv‡b Q1 Ae¯’vb K‡i 30-40 †kÖwY‡Z|

BDwbU 6 c„ôv 88
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n
8 - F1
Q1 = L1 + fQ × C
1
GLv‡b L1 = 30, n= 100, Fc1 = 20, fQ1=37, C=10
100
4 - 20
Q1 = 30 + 37 × 10
5
= 30 + 37 × 10 = 31.351
3n 300
Avevi 4 = 4 = 75 hvi †P‡q eo µg‡hvwRZ MYmsL¨v n‡”Q 80, A_©vr 40-50 †kÖwY‡Z Q3 Aew¯’Z|
3n
4 - Fc3
Q3 = L3 + fQ ×C
3
GLv‡b L3 = 40, Fc3 = 57, fQ3=23, C=10

300
4 - 57
myZivs Q3 = 40 + 23 × 10
18
= 40+23 × 10
= 47.826
Q3-Q1
GLb, PZz_©K e¨eavb QD = 2
47.826-31.351
= 2
= 16.475
Q3-Q1
Ges PZz_©K e¨eavbvsK = Q +Q × 100
3 1
47.826 - 31.351
= 47.826+31.351 × 100
16.475
= 79.187 × 100
= 20.805%
PZz_©K e¨eav‡bi myweav I Amyweav:
K) we¯Ívi cwigv‡ci Rb¨ PZz_©K e¨eavb cwim‡ii †P‡q †ewk DËg|
L) PZz_K© e¨eavb mn‡R wbY©q Kiv hvq Ges GUv †ek mnR‡eva¨|
M) GUv cÖvšÍxq gvb Øviv cÖfvweZ nq bv|

mvims‡ÿc:
PZz_©K e¨eav‡bi Av‡cwÿK cwigvc nj PZz_K
© e¨eavbvsK

BDwbU 6 c„ôv 89
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

cvV-6.5 cwiwgZ e¨eavb I †f`vsK Ges cwiwgZ e¨eavbvsK I we‡f`vsK

Standard Deviation & Variance and Co-efficient of Standard
Deviation & Co-efficient of Variation
D‡Ïk¨

G cvV †k‡l Avcwb-

• cwiwgZ e¨eavb I †f`vsK m¤ú‡K© ej‡Z cvi‡eb;
• cwiwgZ e¨eavb I †f`vsK wbY©q Ki‡Z cvi‡eb;
• we‡f`vsK m¤ú‡K© e¨vL¨v Ges GUv wbY©q Ki‡Z cvi‡eb|

cwiwgZ e¨eavb I †f`vsK

(Standard deviation and Variance)
cwiwgZ e¨eavb GKwU ¸iæZ¡c~Y© I eûj cÖPwjZ we¯Ívi cwigvc| Mo e¨eav‡bi c×wZ MÖnY Kiv nq Ges msL¨vgvb
†_‡K M‡oi e¨eavb¸‡jv‡K G‡ÿ‡Î eM© Kiv nq d‡j mg¯Í eM©dj¸‡jvB abvZ¥K n‡q hvq|
†Kvb PjK ev wb‡ek‡bi mKj Z_¨gvb †_‡K G‡`i MvwYwZK M‡oi e¨eavbmg~‡ni e‡M©i Mo nj †f`vsK Ges
†f`vsK Gi abvZ¥K eM©g~j nj cwiwgZ e¨eavb|

K. A‡kÖwYK…Z Z‡_¨i †ÿ‡Î :

_ 1 n
hw` n msL¨K Z_¨gvb wewkó Pj‡Ki gvb X1, X2, ............Xn nq Ges Gi MvwYwZK Mo X = n ∑ Xi nq, Z‡e
i=1
1  _ 2
†f`vsK S2 = n Σ Xi - X
n 2
∑Xi2 - (ΣXi)
n
i=1
= n

n
(ΣXi)2
∑ Xi2 - n
i=1
Ges cwiwgZ e¨eavb S = n

L. †kÖwYK…Z Z‡_¨i †ÿ‡Î :

MYmsL¨v wb‡ek‡bi †ÿ‡Î †hLv‡b X1, X2, .......... Xn nj ga¨gvb Ges f1, f2, ............. fn n‡jv h_vµ‡g Zv‡`i
MYmsL¨v|

BDwbU 6 c„ôv 90
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

_
Σƒi (Xi-X)2
†f`vsK S2 = n
n 2
∑fixi2 - (Σfixi)
N
i=1
= N

n 2
∑fixi2 - (Σfixi)
N
i=1
Ges cwiwgwZ e¨eavb S = N

cwiwgZ e¨eavbvsK I we‡f`vsK : cwiwgZ e¨eavb I M‡oi AbycvZ‡K 100 Øviv ¸Y Ki‡j †h gvb cvIqv hvq
Zv‡KB cwiwgZ e¨eavbvsK e‡j|A_©vr,
cwiwgZ e¨eavb S
cwiwgZ e¨eavbvsK = Mo × 100 ev, × 100
x
†f`vsK σ
∴ we‡f`vsK = Mo ×100 ev ×100
x

D`vniY-1
GB BDwb‡Ui cvV-6.3 G D`vniY-1 G ewY©Z 10 Rb Qv‡Îi eq‡mi cwiwgZ e¨eavb, †f`vsK Ges cwiwgZ
e¨eavbvsK I we‡f`vsK †ei Kiæb|
eqm (erm‡i) t 16, 15, 17, 18, 14, 19, 21, 16, 20, 23
mgvavb t
GLv‡b n = 10
10 2
∑ (SXi)
Xi2 - n
i=1
∴S= N

10
GLb, ∑ Xi2 = 162+152+172+ ....... +232 = 3277
i=1
10
Ges ∑ Xi =16+15+17+ ............ + 23 = 179
i=1

BDwbU 6 c„ôv 91
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

(179)2
3277 - 10
∴S = 10

1
= 10 × 72.9
= 2.7
∴ myZivs cwiwgZ e¨eavb = 2.7
∴ †f`vsK = σ = (2.7)2 = 7.29
S
cwiwgZ e¨eavbvsK = × 100
X
2.7
= 17.9 × 100
= 15.084%
σ 7.29
∴ we‡f`vsK = × 100 =17.9 = 40.72%
X

D`vniY-2 t cvV 6.3 G D`vniY-2 G ewY©Z MYmsL¨v wb‡ek‡bi Rb¨ cwiwgZ e¨eavb I †f`vsK Ges cwiwgZ
e¨eavbvsK I we‡f`vsK wbY©q Kiæb|
wb‡¤œ mviYx †_‡K Mo e¨eavb I Mo e¨eav¼ wbY©q Kiæb|
†kÖwY 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
MYmsL¨v 2 3 15 37 23 13 5 2

mgvavb t
cwiwgZ e¨eavb wbY©q mviwY
‡kÖwY ga¨gvb Xi MYmsL¨v ƒi ƒi Xi ƒi Xi2
0-10 5 2 10 50
10-20 15 3 45 675
20-30 25 15 375 9375
30-40 35 37 1295 45325
40-50 45 23 1035 46575
50-60 55 13 715 39325
60-70 65 5 325 21125
70-80 75 2 150 11250
Σƒi = 100 Σƒi Xi= 3950 Σƒi Xi2=
173700

BDwbU 6 c„ôv 92
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

100
(Σfixi)2
∑ fixi2 - 100
i=1
cwiwgZ e¨eavb S= 100

(3950)2
173700 - 100
= 100

17675
= 100
= 176.75
= 13.295
∴ †f`vsK = (13.295)2 = 176.5
S
Ges cwiwgZ e¨eavbvsK = X × 100
13.295
=
39.5 × 100
= 33.657
σ
Ges we‡f`vsK = X × 100
176.75
=
39.5 × 100
= 447.48
cwiwgZ e¨eav‡bi myweav
K) cwiwgZ e¨eavb we¯Ívi cwigv‡ci me‡P‡q ¸iæZ¡c~Y© I eûj e¨eüZ cwigvc|
L) cwiwgZ e¨eav‡b MvwYwZK msÁv ¯úó Ges GUv mg¯Í Z_¨gv‡bi Dci wfwË K‡i wbY©q Kiv nq|
M) `yB ev †ewk MÖæ‡ci Z_¨gv‡bi Rb¨ mshy³ cwiwgZ e¨eavb †ei Kiv hvq wKšÍz Ab¨ cwigv‡ci †ÿ‡Î m¤¢e bq|
N) `yB ev Z‡ZvwaK web¨v‡mi Zzjbv Kivi Rb¨ cwiwgZ we‡f`vsK me‡P‡q †ewk Dc‡hvMx|

cwiwgZ e¨eav‡bi Amyweav

K) Ab¨vb¨ cwigv‡ci †P‡q GUv wbY©q Kiv GKUz KwVb|
L) cwiwgZ e¨eavb cÖvšÍxq gvb Øviv cÖfvweZ nq|

mvims‡ÿc:
‡Kvb PjK ev wb‡ek‡bi mKj Z_¨gvb †_‡K G‡`i MvwYwZK M‡oi e¨eavbmgy‡ni e‡M©i Mo nj †f`vsK Ges
†f`vs‡Ki eM©g~j nj cwiwgZ e¨eavb|

BDwbU 6 c„ôv 93
evsjv‡`k Dš§y³ wek¦we`¨vjq e¨emvq cwimsL¨vb

BDwbU g~j¨vqb:

1| we¯Ívi I we¯Ívi cwigvc ej‡Z Kx eySvq D`vniYmn wjLyb|

2| wewfbœ we¯Ívi cwigvc¸‡jv wK wK? eY©bv Kiæb| †Kvb cwigvcwU fvj, hyw³ mnKv‡i D‡jøL Kiæb|
3| Av‡cwÿK we¯Ívi cwigvc I cig we¯Ívi cwigvc ej‡Z Kx eySvq? G‡`i g‡a¨ cv_©K¨ Kx?
4| we¯Ívi cwigv‡ci cÖ‡qvRbxqZv m¤ú‡K© Av‡jvPbv Kiæb|
5| cwimi ej‡Z Kx †evSvq? GUv wKfv‡e wbY©q Ki‡Z nq? cwimivsK Kx? cwim‡ii myweav I Amyweavmg~n
wjLyb| wb¤œwjwLZ Z_¨gvb †_‡K cwimi Ges cwimivsK †ei Kiæb|
19, 16, 15, 14, 13, 17, 21, 22, 28, 15, 16, 19
6| PZz_©K e¨eavb I PZz_©K e¨eavbvsK ej‡Z Kx eySvq? G¸‡jv wbY©q Kivi c×wZ eY©bv Kiæb| PZz_K
©
e¨eav‡bi myweav I Amyweavmg~n wjLyb|
7| wb¤œwjwLZ MYmsL¨v wb‡ekb †_‡K PZz_K © e¨eavb I PZz_K
© e¨eavbvsK wbY©q Kiæb|
X 10-20 20-30 30-40 40-50 50-60 60-70
f 12 19 5 10 9 6
8| cwiwgZ e¨eavb I cwiwgZ we‡f`vsK ej‡Z Kx eySvq? wKfv‡e G¸‡jv wbY©q Ki‡Z nq? Gi myweav I
Amyweavmg~n wjLyb| cwiwgZ e¨eavb I Mo e¨eav‡bi g‡a¨ cv_©K¨ Kx?
9| wb‡¤œ cÖ`Ë wKQymsL¨K †jv‡Ki D”PZvi wb‡ek‡bi mviwY †_‡K cwiwgZ e¨eavb I cwiwgZ we‡f`vsK wbY©q
Kiæb|
D”PZv (Bw‡Z) †jv‡Ki msL¨v
60-65 20
65-70 180
70-75 31
75-80 09
10| `ywU msL¨vi MvwYwZK Mo 6 Ges †f`vsK 9| msL¨v `ywU wK wK?

BDwbU 6 c„ôv 94