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NCERT Class XI Mathematics - Statistics - Solutions

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Question : 21
Total: 34
Find the mean, variance and standard deviation using short-cut method.
Height in cms No. of children
70 - 75 3
75 - 80 4
80 - 85 7
85 - 90 7
90 - 95 15
95 - 100 9
100 - 105 6
105 - 110 6
110 - 115 3

Solution:  
Height in cms Mid values
xi		 fi ui =
xi−92.5
5
 fiui fiui2
70 - 75 72.5 3 - 4 - 12 48
75 - 80 77.5 4 - 3 - 12 36
80 - 85 82.5 7 - 2 - 14 28
85 - 90 87.5 7 - 1 - 7 7
90 - 95 92.5 15 0 0 0
95 - 100 97.5 9 1 9 9
100 - 105 102.5 6 2 12 24
105 - 110 107.5 6 3 18 54
110 - 115 112.5 3 4 12 48
60 6 254

Let assumed mean (A) = 92.5
Mean (
−
x
) = A +
Σfiui
N
 × h = 92.5 +
6
60
 × 5 = 92.5 + 0.5 = 93
Variance (σ2) =
h2
N2
[NΣfiui2−(Σfiui)2] =
(5)2
(60)2
 [60×254−(6)2]
=
25
3600
 × 15204 = 105.58
Standard deviation (σ) = √105.58 = 20.27
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