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NCERT Class XI Mathematics - Statistics - Solutions

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Question : 23 of 34
Marks: +1, -0
From the data given below state which group is more variable, A or B?
MarksGroup A Group B
10 - 209 10
20 - 30 17 20
30 - 40 32 30
40 - 50 33 25
50 - 60 40 43
60 - 70 1015
70 - 809 7
Solution:  
For Group A :
Marks Mid values
xix_i		 fif_i uiu_i = xi−4510\frac{x_i-45}{10} fiuif_iu_i fiui2f_iu_i^2
10 - 25 15 9 - 3 - 27 81
20 - 30 25 17 - 2 - 34 68
30 - 40 35 32 - 1 - 32 32
40 - 50 45 33 0 0 0
50 - 60 55 40 1 40 40
60 - 70 65 10 2 20 40
70 - 80 75 9 3 27 91
150 - 6 342
Let assumed mean (A) = 45
Mean (x−)\left(\overset{-}{x}\right) = A + ΣfiuiN\frac{\Sigma f_iu_i}{N} × h = 45 - 6150\frac{6}{150} × 10 = 45 - 0.4 = 44.6
Standard deviation (σ1\sigma_1) = hNNΣfiui2−(Σfiui)2\frac{h}{N}\sqrt{N\Sigma f_iu_i^2-(\Sigma f_iu_i)^2} = 101580150×342−(−6)2\frac{10}{1580}\sqrt{150\times 342 - (-6)^2} = 11551300−36\frac{1}{15}\sqrt{51300-36}
= 115\frac{1}{15} × 226.41 = 15.09
For Group B :
Marks Mid values
xix_i		 fif_i uiu_i = xi−4510\frac{x_i-45}{10} fiuif_iu_i fiui2f_iu_i^2
10 - 25 15 10 - 3 - 30 90
20 - 30 25 20 - 2 - 40 80
30 - 40 35 30 - 1 - 30 30
40 - 50 45 25 0 0 0
50 - 60 55 43 1 43 43
60 - 70 65 15 2 30 60
70 - 80 75 7 3 21 63
150 - 6 366
Mean (x−)2\left(\overset{-}{x}\right)_2 = A + ΣfiuiN\frac{\Sigma f_iu_i}{N} × h = 45 - 6150\frac{6}{150} × 10 = 45 - 0.4 = 44.6
Standard deviation (σ2)(\sigma_2) = hNNΣfiui2−(Σfiui)2\frac{h}{N}\sqrt{N\Sigma f_iu_i^2-(\Sigma f_iu_i)^2} = 10150150×366−(−6)2\frac{10}{150}\sqrt{150\times 366-(-6)^2} = 11554900−36\frac{1}{15}\sqrt{54900-36}
= 115\frac{1}{15} × 234.23 = 15.61
The group which have greater S.D. is more variable. Thus group B is more variable.
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