x f Class xm rf <cf >cf <cpf >cpf

Boundarie
s
26-32 2 25.5-32.5 29 2.86 2 70 6 140
33-39 4 32.5-39.5 36 5.71 6 68 18 136
40-46 7 39.5-46.5 43 10 13 64 34 128
47-53 13 46.5-53.5 50 18.57 26 57 58 114
54-60 15 53.5-60.5 57 21.43 41 44 88 88
61-67 10 60.5-67.5 64 14.29 51 29 114 58
68-74 8 67.5-74.5 71 11.43 59 19 134 38
75-81 6 74.5-81.5 78 8.57 65 11 130 22
82-88 5 81.5-88.5 85 7.14 70 5 140 10

x f xm Fxm X¹ Fx¹ X¹ Fx¹ cf

26-32 2 29 58 -4 -8 -3 -6 2
33-39 4 36 144 -3 -12 -2 -8 6
40-46 7 43 301 -2 -14 -1 -7 13
47-53 13 50 650 -1 -13 0 0 26
54-60 15 57 855 0 0 1 15 41
61-67 10 64 640 1 10 2 20 51
68-74 8 71 568 2 16 3 24 59
75-81 6 78 468 3 18 4 24 65
82-88 5 85 425 4 20 5 25 70

x f cf xm X̄ |xm- f(xm- ∑ f (xm− x̄ )²∑ f (xm− x̄ )³ ∑ f (xm− x̄ )⁴

X̄ | X̄ )
26- 2 2 29 58.7 29.7 59.4 1,764.18 52,396.15 1,556,165.66
32
33- 4 6 36 58.7 22.7 90.8 2,061.16 46,788.33 1,062,095.09
39
40- 7 13 43 58.7 15.7 109.9 1,725.43 27,093.96 425,375.17
46
47- 13 26 50 58.7 8.7 113.1 983.97 8,560.54 74,476.70
53
54- 15 41 57 58.7 1.7 25.5 43.35 73.70 125.29
60
61- 10 51 64 58.7 5.3 53 280.9 1,488.77 7,890.48
67
68- 8 59 71 58.7 12.3 98.4 1,210.32 14,886.94 183,109.36
74
75- 6 65 78 58.7 19.3 115.8 2,234.94 43,134.34 832,492.76
81
82- 5 70 85 58.7 26.3 131.5 3,458.45 90,957.24 2,392,175.41
88
141.7 797.4 13,762.7 285,379.97 6,533,905.92
 HISTOGRAPH
16 15
14 13
12
10
10
8
8 7
6
6 5
4
4
2
2

0
26-32 33-39 40-46 47-53 54-60 61-67 68-74 75-81 82-88

FREQUENCY POLYGON
f
16 15

14 13

12
10
10
8
8 7
6
6 5
4
4
2
2
0
2 6 -3 2 3 3 -3 9 4 0 -4 6 4 7 -5 3 5 4 -6 0 6 1 -6 7 6 8 -7 4 7 5 -8 1 8 2 -8 8

OGIVE
80
70 68
70 64 70
57 65
60 59
50 44 51

40 41
29
30
26 19
20
11
13 5
10
6
0 2
26-32 33-39 40-46 47-53 54-60 61-67 68-74 75-81 82-88

<cf >cf
 Mean Median Mode

( )
∑ fxm 4109 n Mo = 3(Mdn) – 2(x̄ )
x̄ ¿ = =58.7 −cf
n 70 Mdn = XLB+ 2 = 3(57.7) – 2(58.7)
i = 173.1 – 117.4
f
assume mean = 55.7
= XO + (
∑ fx 1
i ) = 53.5+ (
3 5−26
15 )
7

= 57 +
n
17
( )
7
= 53.5 + 4.2
= 57.7
Mo =XLB + (
D1
D 1+ D 2
i
)
( )
70 2
= 53.5+ 7
= 57 + 1.7 2+5
= 58.7 = 53.5 + 2
= 55.5
= XO + ( ∑ fxn 1 )i Modal class
= 50 + ( )7
87 D1 = 15-13 = 2
70 D2 = 15-10 = 5
= 50 + 8.7
= 58.7

Quartiles

( ) ( ) ( )
1 2 3
n−cf n−cf n−cf
Q1 = XLB + 4 Q2 = XLB + 4 Q3 = XLB + 4
i i i
f f f

( ) ( ) (
1 2 = XLB +
(7 0)−cf (7 0)−26

)
= XLB + 4 = 53.5 + 4 3
7 7 (7 0)−51
f 15 4
i
f
= 46.5 + (17.5−1 3
7 ) = 53.5 +( )
35−26
7
13
= 46.5 + 2.42
15
= 53.5 + 4.2
= 67.5 + ( 52.5−51
8 )7
= 48.92 = 57.7 = 67.5 + 1.31
= 68.81

Deciles

( ) ( )
1 4
n−cf n−cf
D2 = XLB + 5 D4 = XLB + 10
i i
f f

( ) ( )
1 4
( 7 0 )−6 ( 7 0 )−26
= 39.5 + 5 = 92.5 + 10 7
7
7 15
= 39.5 + 8 = 53.5 + 0.93
= 47.5 = 54.43

( ) ( )
6 7
n−cf n−cf
D6 = XLB + 10 D7 = XLB + 10
i i
f f

( ) ( )
6 7
(7 0)−41 ( 7 0 )−4 1
= 60.5 + 10 = 60.5 + 10
7 7
10 10
= 60.5 + 0.7 = 60.5 + 5.6
= 61.2 = 66.1
 Percentile

P96=
n [
100 ( P−X LB ) f
i
+ cf ]
=
70 [
100 ( 57−53.5 )
7
+13 ]
=1.43 (0.5 + 13)
= 1.43(13.5)
= 19.31

Quartile Deviation Mean Deviation

Q 3−Q1 ∑ f ( xm− x̄)
Qd= Md =
2 n
68.81−48.92 797.4
Qd = =
2 70
= 9.95 = 11.39

Variance Standard Deviation

√
∑ f ( xm− x̄)² ∑ f (xm− x̄)2
s² = s=
n−1 n−1
13,762.7 =√ 199.46
=
7 0−1 = 14.12
= 199.46

Coefficient of Variation Coefficient of Quartile Skewness

SD Q3 −Q1 3 ( x̄−Median )
CV = ×100 % CQD= ×100 % Sk=
x̄ Q 3+Q 1 SD
14.12 68.81−48.92 3 (58.7−57.7 )
¿ ×100 % ¿ ×100 % ¿
58.7 68.81+48.92 14.12
= 0.24× 100% 19. 89 3 (1 )
= 0.24 ¿ × 100 % ¿
117 . 73 14.12
= 0.10× 100% 3
¿
= 0.17 14.12
¿ 0.21(positively skewed)
Moments (4 Moments) Kurtosis
∑ f ( xm− x̄ ) ∑ f ( xm− x̄ )
4
1. K=
n n¿ ¿ ¿
797.4 6 ,533,905.92
¿ ¿
70 7 0 ( 19 9 . 46 )
2

= 11.39 6 ,533,905.92
∑ f ( xm− x̄ )
2 ¿
2. 2,784,900.41
n
13,762.7 = 2.35
¿
70
= 196.61
3
∑ f ( xm− x̄ )
3.
n
 285,379.97
¿
70
= 4,076.86
4
∑ f ( xm− x̄ )
4.
n
6 ,533,905.92
¿
70
= 93,341.51