77

a{ d" { % fXg {yKg}Q gZ (collection), d(classification), ~sD (tabulation), [{ (analysis) yK %z p(interpretation) v{ fX.

dZ ~ZK %yg (Measures of Central tendency) y {yKg g$a{ ~_}r}Q dvyK. p} %yg (Measures of Dispersion) {yKg a{g{ %z }Q g{ U{p dR}}Q dvyK.

gg o d}g ~k{* {yKg gZ yK AK yp ~sD}Q y{.

* %dy g dy {yKg p, |Xd yK pg}Q dv{.

* {yKgg XK, kyzd k} yK p k}g}Q dv{.

o %|X{[ p} %yg{ }d k} (Standard deviation) yK }gd (co-efficient of variation) g g$ {.

o %|X}Q dy d dg ~sD p zXg}Q {.* %dy yK dy {yKg }d k}}Q dv{.* {yKg } gd}Q dv{ g Lpy yK %Lpy g$ {.* fX{ o %yg}Q I~ {yKg}Q %z{.

{yKg yp[} %g Oyg gyZd" { %y}Q ``p'' U}Qp.{yK fX p X{ UDppg p{ U{ {, p} %y{.

}X d[p p} %yg {p (i) XK (Range) (ii) kyzdk} (Quartile Deviation) (iii) p k} (Mean Deviation) yK (iv) }dk} (Standard Deviation).

XK , kyzd k} yK p k} g$ ogg k{M. o p}%yg[{, }d k} a{ A{ ~}{.

1. }d k} (Standard Deviation)XKg d_xg{ b, UyKp, yd, p}, ~d_[ g{ %dg EyX,

g % %g {yKgg. o fXg gx{ } gxd_xg}Q}X ``kp''g U{ dpyK.

o kpg Xg p yK pp{ %z %{{ {pk{p{. kpgg yK p gp Kp X}Q ``~Zp k}''U{ dpp.

`~Zpx k}' |}yWd g }d k}g{.

3 fX

78

2. %dy {yKg }d k} (Standard Deviation for anUngrouped Data)

{ 1 : Xg gxd" ~Zp k} yK }d k}g}Q d"kpv{.

X U{ kp, Xg gx p{ X : x1, x2 x3 .... xn

N : Xg yK

X : Xg gx{ p

D : p{ k} = (X X )Xg ~Zp k}}Q o d} yZ}Q I~ dv{.

~Zp k} = 2DN1

dd `~Zp k}' 22 DN1 = {. }d k}, ~Zp

k} |}yWd g Ap{{,

}d k} = Variance =

S.D = ND2

}d k}}Q Zd %d_p ~Z{.

}d k} p{ k}g gg yK{ |}yWd gAp{.

d :Kd Xg}Q dDg

1. p}Q "{.

2. ~Z Xd" yK p g Ep XyX}Q ~v{.

3. o XyX (k})g gg}Q "{.

4. k}g gg yK}Q Xg yK{ {.

5. E{ ~Zp k}g{.

6. ~Zpx k} |}yWd g }d k}g{.

~Zp k}

79

{ d"g1) d} yK ~Z~Kdg ~Zp k} yK }d k}}Q dv.

14, 16, 21, 9, 16, 17, 14, 12, 11 yK 20. y}Q .

d"kp{ ~sD

%dg p{ k} k}X (D = X X ) g D2

14 -1 116 +1 121 +6 369 -6 36

16 +1 117 +2 414 -1 112 -3 911 -4 1620 +5 25

X =150 2D = 130

~p : ~Z~Kdg fX N = 10(i) p}Q dv.

p = NXX =

= 10150

p = 15

p = 15

(ii) ~Zp k}}Q dv.

~Zp k} = 2DN1

2 = 101

(130) = 13 ~Zp k} = 13

(iii) `}d k}'}Q dv}d k} = Variance

= 13 = 3.6 }d k} = 3.6(Kd %dg %dgy{ p{ p 3.6 %dgD k} pyK.)

~Zp k}

80

2) aT Xb} Ur ~{Xg[ g{ p}g 35, 42, 23, 34, 39, 36, 32 yK 31A. (i) p p}Qg}Q yK }d k}}Q dv. (ii) y}Qg ?

~p : ~{Xg fX = N = 8

d"kp{ ~sDp}Qg p{ k} k}

X (D = X X ) g D2

35 +1 142 +8 6423 11 12134 0 039 +5 2536 +2 432 -2 431 -3 9

X = 272 2D = 228

i) pp}Q NXX ==

= 8272

= 34 p = 34

ii) }d k} ND

2=

= 8

228 = 5.28 = 5.34 }d k} = 5.34

Xb} g{ p}g p 34. p}Qg p{ I{ }d k}5.34 A{.

~ |}p ( X ) a{ ~d Ag{ Epg o |} %}{.y 1 : {{p a{ ~Z~Kd}Q %{m p (A) U{ ErDd\.y 2 : %{m p{ k} D = (X-A) g}Q dv.y 3 : k}g yK}Q D dv.y 4 : ~Z ~Z~Kd{ k}g gg}Q " yK o gg yK}Q 2D

dv.y 5 : o d} yZ}Q I~ }d k}}Q dv.

}d k} = 22

ND

ND

81

3) 10 {Xg a{ ~d_[ ~v{dv %dg o d}. 43, 48, 55, 57,42, 50, 47, 48, 58 yK 50. (i) p %dg}Q yK (ii) }d k}}Q dv.y}Q .

~p : {Xg fX = N = 10. %{m p A = 50 U{ ErDd\x.

d"kp{ ~sD%dg %{m p k}

X { k} g(D = X X ) D2

43 -7 4948 -2 455 +5 2557 +7 4942 -8 6450 0 047 -3 948 -2 458 +8 6450 0 0

X = 498 D = -2 2D = 268

i) p NXX ==

= 10498

= 49.8 p = 49.8

yK {Xg %dg p = 49.8

ii) }d k} 22

ND

ND

==

= 2

102

10268

= 2)2.0(8.26 = 04.08.26

= 76.26 = 5.173 }d k} = 5.173

(Kd %dg %dgy p{ p 5.173 %dgD k} pyK)

82

{ 2 : }Q Zx {yKg{Mg }d k}}Q dvd :X U{ kp yK f Aydp. Ag yp p{.

X x1 x2 x 3 ........... xn-1 x nf f1 f2 f3 ........... fn-1 fnN = f = Ay fXg yK

X = p = NfX

D = p{ k} = (X X )

Ag yp ~Zp k} = NfD2

~Zp k} = NfD22

=

yp }d k}

}d k} = NfD

2=

d ~Zp k} (Variance) yK }d k}}Q (Standard Deviation)}Q dvd

y 1: p X }Q NXX = %z N

fXX = yZ}Q I~ dv.

y 2: p{ k} D = (X X ) }Q ~Z ~Z~Kd X g ".y 3: D gg}Q dv{ D2 }Q ~v.

y 4: ~Z ~Z~Kd{ D2 }Q A Ay fX `f' { g, %g gxPgyK 2fd }Q dv.

y 5:NfD22

= }Q ". E{ {yK yp ~Zpx k}}Q dvyK{.

y 6: ~Zpx k} |}yWd g }Q yg{d\. NfD

2=

E{ {yK yp }d k}}Q dvyK{.

83

4) o dg dsDp yp (i) p yK (ii) }d k}}Q dv.

X 10 15 20 25 30 35

f 3 8 5 9 4 1

d"kp{ ~sD "pg AyfX p{ k} g f.D2

k} D2

X f fX D = (X X )

10 3 30 -11 121 363

15 8 120 -6 36 288

20 5 100 -1 1 5

25 9 225 +4 16 144

30 4 120 +9 81 324

35 1 35 +14 196 196

f =30 fX =630 f D2=1320

~p : yp AKg yK = N = f = 30

i) p = NfXX =

= 30630

= 21

p = 21yp p = 21

ii) }d k}NfD

2==

= 30

1320

= 44= 6.63

}d k} = 6.63

84

5) o dg dsDp yp (i) p (ii) ~Zp k} yK (iii) }d k}}Qdv .

X 2.5 3.5 4.5 5.5 6.5

f 4 3 5 10 3

d"kp{ ~sD"pg AyfX p{ k} g f.D2

k} D2

X f fX D = (X X )

2.5 4 10.0 -2.2 4.84 19.363.5 3 10.5 -1.2 1.44 4.324.5 5 22.5 -0.2 0.04 0.205.5 10 55.0 +0.8 0.64 6.406.5 3 19.5 +1.8 3.24 9.72

f =25 fX = 117.5 f D2=40.0

~p : yp AKg yK = N = f = 25

i) p NXX ==

= 255.117

= 4.5

p = 4.7

ii) ~Zp k} = NfD22

=

= 2540

= 1.6

~Zp k} = 1.6

iii) }d k} = Variance

= 6.1 = 1.26

}d k} = 1.26

~Zp k}

85

~ |} :y 1: {{p a{ ~Z~Kd X }Q %{m p (A) U{ ErDd\.

y 2: %{m p{ k} D = (X-A) g}Q dv.

y 3: ~Z k}}Q A Ay fX{ g, gxPg yK f D }Q~v.

y 4: k} g (D2) }Q p.

y 5: k} gg}Q dZ A Ay fX{ g, %g yK fD2~v.

y 6: o d} yZg}Q I~ yp p yK }d k}g}Qdv.

p = X = A + NfD

yK }d k} 22

NfD

NfD

==

6) o dg dsDp yp (i) p yK (ii) }d k}g}Q dv.

%dg 35 40 45 50 55

{Xg fX 2 4 8 5 1

~p : %{m p A = 45 U{ ErDd\x.

d"kp{ ~sD%dg AyfX %{m p k} g

{ k} D2 f.D f.D2

X f D = (X X )

35 2 -10 100 -20 20040 4 -5 25 -20 10045 8 0 0 0 050 5 +5 25 25 12555 1 +10 100 10 100

N = 20 fD = -5 fD2 = 525

86

i) p NfDAX +==

= 45 + 20)5(

= 45 0.25 p = 44.75

p %dg 44.75

ii) }d k} 22

NfD

NfD

=

2

205

20525

= 0625.025.26 =

1875.26= = 5.117 }d k} = 5.117

20 {Xg ~v{ %dg p 44.75. yK ~Z~Kdg p{ p5 %dgD k}}Q ~v.

3. dy {yKg }d k}%dy {yKg }d k} dv [ dy {yKg }d

k}}Q dv dv{.

E{}Q :1. gypg p %g |X {{ ~ZRvyK.2. p{ k}g}Q ~vyK.3. o k}g gg}Q A Ay fXg{ gyK.4. o gxPg yK}Q, Ay fXg yK{ {g y

~Zp k}.5. ~Zp k} |}yWd g }d k}.

7) o dg dsDp Ayd yp ~sD }d k}g}Q dv.

gyp Ay fX1-5 26-10 311-15 416-20 1

87

~p :y 1: ~Z gyp{ |X { (X) %}Q dv.

y 2: ~Z gypd" gxP f.X }Q dv.

gyp Ay fX |X{ p{f X fX k} D2 f.D2

D = (X X )

1-5 2 3 6 -7 49 986-10 3 8 24 -2 4 12

11-15 4 13 52 3 9 3616-20 1 18 18 8 64 64

N = 10 fX = 100 fD2 = 210

y 3: yp p}Q dv

p NfxX ==

= 10100

p = 10

y 4: D = X X }Q I~ %d gy{ p{ k} D }Q ~Zgypd" dv.

y 5: k}g gg}Q dv D2.

y 6: ~Z gypd" gxP fD2 %}Q dv g %g yK fD2%}Q dv.

y 7: yp }d k}}Q dv.

}d k} N

D.f2

==

10210

=

21= = 4.6 }d k} = 4.6

88

8) a{ ~Z{{[ p 20 m}p yp}Q (g[) o d}y}{.

b (g[) 30-34 35-39 40-44 45-49 50-54

m}p fX 2 5 6 5 2o yp (i) p b yK (ii) ~Zp k} g }d k}g}Qdv.

~p : d"kp{ ~sD

b Ay fX |X{ p{(g[) f X fX k} D2 f.D2

C.I. D = (X- X )

30-34 2 32 64 -10 100 20035-39 5 37 185 -5 25 12540-44 6 42 252 0 0 045-49 5 47 235 +5 25 12550-54 2 52 104 +10 100 200

20 840 650

g : N = f = AKg } yK = 20 fX = 840, f.D2 = 650

i) p = NfxX =

= 20840

= 42 p = 42

p b (g[) 42

ii) ~Zp k} = NfD22

=

= 20650

= 32.5 ~Zp k} = 32.5

iii) }d k} = Variance

= 5.32 = 5.7 }d k} = 5.7 yp }d k} (g[) = 5.7

~Zp k}

89

hrDk}dZ %z y-k}dZ1) {{p a{ gXK}Q A" %{p |X{ (X) }Q %{m p

(A) U{ ErDd\x.

2) %{m p{ ~Z hrD{ k}}Q d = iAX

, dv. E[ i U{

g XK }v} %yp{.

3) ~Z hrDk}}Q A Ay fX{, g yK gxP{ yK f.d.}Q ~v.

4) ~Z hrDk} g d2 }Q p.5) ~Z hrD k} g}Q A Ay fX{ g, %g gxPg

yK f.d2 }Q dv.6) o d} yZg}Q I~ yp p yK }d k}g}Q dv.

p ixNfdAX

+==

}d k} ixNfd

Nfd

22

==

9) 60 {Xg a{ p ~d_[ ~v{ %dg o d}.

%dg 5-15 15-25 25-35 35-45 45-55 55-65

{Xg fX 8 12 20 10 7 3

o Ayd yp ~sD (i) p yK (ii) }d k}g}Q dv.

~p :

hrD|X{ AK k}

%dg X f d=

i

AXd2 f.d f.d2

5-15 10 8 -2 4 -16 3215-25 20 12 -1 1 -12 1225-35 30 20 0 0 0 035-45 40 10 +1 1 10 1045-55 50 7 +2 4 14 2855-65 60 3 +3 9 9 27

N = 60 fd = 5 2fd =109

90

%{m p A = 30 U{ ErDd\xg : Aydg yK = N = 60

gXK }v} %yp = i = 10

f.d = 5 yK f.d2 = 109

i) p ixNfdAX

+==

10x60530

+=

6530+= = 30 + 0.83

= 30.83 p = 30.83

ii) }d k} ixNfd

Nfd

22

==

10x605

60109 2

=

( ) ( ) 10x08.0817.1 2=10x0064.0817.1 = 10x81.1=

= 1.345 x 10 = 13.45 }d k} = 13.4560 {Xg g{ %dg p 30.83yK %dg yp }d k} 13.45

4. } gd (Co-efficient of variation)} gd p} a{ ~d_ %y{. E{ {yK{ p X %}Q

yK }d k} ( ) }Q %{. } gd}Q o d} yZ}Q I~dv{.

} gd =

M.A

D.Sx 100

dd C.V =

X

x 100

}d k}p

91

}} :* } gd p} a{ ~d_ %y.* o gd}Q |px dv p~{[ g{.

* p ( X ) yK }d k} ( ) } gd}Q |yK.* E{ }g{ dK{ fX.* } gd Lpy %z %Lpy}Q |yK{.

10) %p yK py U ETp Zd Argpp 15 ~{Xg[ g{ p}Qg yKdZ 1050 yK 900 ApyK g }d k}g dZ 4.2 yK 3.0ApyK. p p}Qg[ p k+ g{p? p k+ Lpy}Q {Mp?

~p : ~{Xg yK = 15

%p g{ p}Qg p = 151050

= 70

py g{ p}Qg p = 15900

= 60

Argp p }d } gd

= 100xX

%p 70 4.2 0.6100x702.4

=

py 60 3.0 0.5100x600.3

=

%p g{ p}Qg p py g{ p}Qg p"y k+{. p %dg

{D{ %p IyK{ Argp}{M}.

py} } gd %p } gd"y d E{. A{{{ pyk+ Lpy\ Argp}{M}.

11) 20 {Xg a{ ~d_[ Ag[, gy yK uC} g[ p zX}Q ddv %g yK.

p }d k}

X Ag[ 56 5.75gy 73 6.25uC} 62 6.0

o {yK}Q I~ {[ o {Xg k+ Lp{U{}Q dv.

k}

X

92

(i) Ag[ : p X = 56 yK }d k} = 5.75

} gd 100xX

=

100x5675.5

=

Ag[ } gd = 10.27

(ii) gy : p X = 73 yK }d k} = 6.25

} gd 100xX

=

100x7325.6

=

gy{ } gd = 8.56

(iii) uC} : p X = 62 yK }d k} = 6.0

} gd 100xX

=

100x62

0.6

=

uC}{ } gd = 9.68

{g, gy{[ } gd d E{.

A{M{ gy{[ {Xg |} k+ Lpy {.

12) A yK } 5 p ~d_g[ gp %dg p d}.

p~d_fX 1 2 3 4 5

A 58 65 58 64 55

} 66 60 60 76 68

(i) Ep[ p +} zX\p? yK (ii) p +} Lpy\p{Mp?

93

~p :

d"kp{ ~sD

A }%dg p{ p{

X k} D2 X k} D2

D = (X X ) D = (X X )

58 -2 4 66 0 065 +5 25 60 -6 3658 -2 4 60 -6 3664 +4 16 76 +10 10055 -5 25 68 +2 4

300 74 330 176

(i) A :p~d_g yK N = 5

p NXX ==

5300

=

= 60 p = 60

}d k}ND

2==

574

= 5.14=

= 3.85 }d k} = 3.85

} gd 100xXCV

==

100x6085.3

=

= 6.42 A } gd = 6.42

94

(ii) } : p~d_g yK = N = 5

p NXX ==

5330

= p = 66

}d k}ND

2==

5176

= 2.35= }d k} = 5.93

} gd 100xXCV

=

100x6693.5

= } d gd = 8.98

Ay } k+ p %dg}Q ~vp{{ } IyK{.

A } gd, } } gd"y d Ep{{ A k+Lpy ~v{M.

13) o dg dsDp yp }d k} yK } gd}Q dv.

%dg (X) 10 20 30 40 50

{Xg fX (f) 4 3 6 5 2

~p : yp Aydg yK = N = 20

d"kp{ ~sD%dg Ay p{ k}

fX k} g f.D2

X f f.X. D = (X X ) D2

10 4 40 -19 361 144420 3 60 -9 81 24330 6 180 +1 1 640 5 200 +11 121 60550 2 100 +21 441 882

N = 20 fX = 580 fD2 = 3180

95

yp Aydg yK N = 20

i) p NfXX ==

20580

= = 29 p = 29

ii) }d k} N

D.f2

==

203180

=

159= = 12.61 }d k} = 12.61

iii) } gd 100xM.AD.S

=

100x29

61.12

= 29

1261= = 43.48 } gd = 43.48

14) 15 dgpp }g (p.g[) ~{}}Q o dg dsDp Aydyp ~sD[ dv{.

}g (p) 30-40 40-50 50-60 60-70 70-80

dgpp fX 2 3 5 3 2

o Ayd yp i) p ii) }d k} yK iii) } gdg}Qdv.

~p :d"kp{ ~sD

}g Ay |Xd p{ k}(p) fX f X fX k} g f.D2

C.I D = (X X ) D2

30-40 2 35 70 -20 400 80040-50 3 45 135 -10 100 30050-60 5 55 275 0 0 060-70 3 65 195 +10 100 30070-80 2 75 150 +20 400 800

15 fx = 825 2fd =2200

96

AKg yK = N = 15 f.X = 825 yK f.D2 = 2200

i) p NXX ==

=

15825

= 55 p = 55

}g yp }d k} (p.g[) = 55

ii) }d k}N

D.f2

==

152200

=

67.146= = 12.11 }d k} = 12.11

}g yp }d k} (p.g[) 12.11

iii) } gd 100xM.AD.S

=

100x55

11.12

=

=

551211

= 22.02 } gd = 22.02

%X : 31) o dg dsDp yK ~Z~Kdg }d k}}Q dv.

8, 9, 15, 23, 5, 11, 19, 8, 10 yK 12

2) aT Xb}} Ur Ebgbg[ "pg 48, 40, 36, 35, 46, 42, 36 yK 37A. o "pg (i) p (ii) ~Zp k} yK (iii) }d k}g}Qdvp.

3) o dg dsDp yp p yK }d k}g}Q dv.

X 5 15 25 35 45

f 5 8 15 16 6

97

4) 60 {Xg a{ gy p~d_[ gp %dg o d}.

%dg (X) 10 20 30 40 50 60

{Xg fX (f) 8 12 20 10 7 3

o %dg ~Zp k} yK }d k}g}Q dv.

5) o d} Ayd yp (i) p yK (ii) }d k}g}Q dv.

gyp Ay fX

20-25 8

25-30 3

30-35 15

35-40 12

40-45 8

45-50 4

N 50

6) a{ df} 40 ddp }g pg o ddv ~sD[ dv{.

}g p. 30-34 34-38 38-42 42-46 46-50 50-54

}dpp fX 4 7 9 11 6 3

o yp p, ~Zp k} yK }d k}g}Q dv gy}Q .

7) Xb} A g{ p}Qg p yK }d k}g dZ 64 yK 18ApyK yK Xb} B g{ p}Qg p yK }d k}g dZ43 yK 9 ApyK. A{p o ETp Xb}g zX yK Lpy dv{y}Q [.

8) a{ dgd ~Z{{[ }p A yK B U Upv df}g p{ py}g yK }d k}g}Q o dg dsD{.

df} p }d k} y} p. p.

A 34.5 6.21

B 28.5 4.56

df}, y} dv{p[ k+ %Lp{?

98

9) 20 m}p UyKp yK ydg p yK }d k}g}Q dg dsD{.

d_xg X UyKp (..) 175 3.5

yd (d.) 70 2.1

%p d_x k+ %Lp{?

10) o d} Ayd yp (i) p (ii) }d k} yK (iii) } gdg}Qdv.

gyp Ay fX

30-35 535-40 1040-45 1645-50 1550-55 4

arD 50

11) Ap Ebgbg[ ETp Xb}g g{ p}Qg p d}.

Xb} A 48 50 54 46 48 54

Xb} B 46 44 43 46 45 46

Ep[ 1) p IyK Argpp 2) p +} Lpy\p{Mp?

12) yK} ypg Upv gg A yK B a{ p~d_[ ~v{ %dg pd}{.

%dg {Xg fX {Xg fXg A g B

25-30 5 5

30-35 10 12

35-40 25 20

40-45 8 8

45-50 2 5

g{ y IyK{? yK g{ y k+ %Lp{?