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STATISTICS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSTATISTICS4 Marks
Question
From the following data, state which group is more variable, A or B?
Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Group A 9 17 32 33 40 10 9
Group B 10 20 30 25 43 15 7
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Answer

For group A,
Let assumed mean a = 45
Class interval Mid-point $(x_i)$ $u_i$ = $\frac{x_i - 45}{10}$ $u_i^2$ $f_i$ $f_iu_i$ $f_iu_i^2$
10 - 20 15 -3 9 9 -27 81
20 - 30 25 -2 4 17 -34 68
30 - 40 35 -1 1 32 -32 32
40 - 50 45 0 0 33 0 0
50 - 60 55 1 1 40 40 40
60 - 70 65 2 4 10 20 40
70 - 80 75 3 9 9 27 81
Total       150 -6 342
Here,
$\sum f_i$ = N = 150, $\sum f_{i} u_{i}$ = -6, and $\sum f_{i} u_{i}^{2}$ = 342, b = 10
$\therefore$ $\overline x$ = a + $\frac{\sum f_{i} u_{i}}{\sum f_{i}} \times$ b = 45 + $\frac{(-6)}{150}$ $\times$ 10 = 45 - 0.4 = 44.6
Variance, $\sigma_{A}=\frac{b^{2}}{N^{2}}\left[N \sum f_{i} u_{i}^{2}-\left(\sum f_{i} u_{i}\right)^{2}\right]$
= $\frac{100}{22500}$[150 $\times 342 - (-6)^2$]
= $\frac{1}{225}$ (51300 - 36) = $\frac{51264}{225}$ = 227.84
$\therefore$ Standard deviation, $\sigma _A$ = $\sqrt{227.84}$ = 15.09
Coefficient of variation (CV) = $\frac{\sigma_{{A}}}{\overline{x}} \times$ 100 = $\frac{15.09}{44.6}$ $\times$ 100 = 33.83
For group B,
Let the assumed mean a = 45
Class interval Mid-point $(x_i)$ $u_i$ = $\frac{x_i - 45}{10}$ $f_i$ $u_i^2$ $f_iu_i$ $f_iu_i^2$
10 - 20 15 -3 10 9 -30 90
20 - 30 25 -2 20 4 -40 80
30 - 40 35 -1 30 1 -30 30
40 - 50 45 0 25 0 0 0
50 - 60 55 1 43 1 43 43
60 - 70 65 2 15 4 30 60
70 - 80 75 3 7 9 21 63
Total     150   -6 366
$\sum F_i$ = N = 150,
$\sum f_{i} u_{i}$ = -6 and $\sum f_{i} u_{i}^{2}$ = 366
$\therefore$ $\overline{x}=a+\frac{\sum f_{i} u_{i}}{\sum f_{i}} \times b$
= 45 + $\frac{(-6)}{150}$ $\times$ 10 = 45 - 0.4 = 44.6
Variance, $\sigma_{B}^{2}=\frac{b^{2}}{N^{2}}\left[N \sum f_{i} u_{i}^{2}-\left(\sum f_{i} u_{i}\right)^{2}\right]$
= $\frac{100}{22500}$ [150 $\times 366 - (-6)^2$]
= $\frac{1}{225}$(54900 - 36) = $\frac{1}{225}$ $\times$ 54864 = 243.84
$\therefore$ Standard deviation, $\sigma _ B$ = $\sqrt{243.84}$ = 15.61
$\therefore$ Coefficient of variation (CV) = $\frac{\sigma_{B}}{\overline{x}} \times$ 100 = $\frac{15.61}{44.6}$ $\times$ 100 = 35
Since, CV(B) > CV(A)
So, group B is more variable than group A.

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