# 1.From the data given below 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

The group having a higher coefficient of variation will be more variable.

Let the assumed mean, A = 45 and h = 10

For Group A

 Marks Group A $f_i$ Midpoint $x_i$ $\dpi{100} y_i = \frac{x_i-A}{h}$ $\dpi{80} = \frac{x_i-45}{10}$ $y_i^2$ $f_iy_i$ $f_iy_i^2$ 10-20 9 15 -3 9 -27 81 20-30 17 25 -2 4 -34 68 30-40 32 35 -1 1 -32 32 40-50 33 45 0 0 0 0 50-60 40 55 1 1 40 40 60-70 10 65 2 4 20 40 70-80 9 75 3 9 27 81 $\sum{f_i}$ =N = 150 $\sum f_iy_i$ = -6 $\sum f_iy_i ^2$ =342

Mean,

$\overline{x} = A + \frac{1}{N}\sum_{i=1}^{n}f_iy_i\times h =45 + \frac{-6}{150}\times10 = 44.6$

We know, Variance, $\sigma^2 = \frac{1}{N^2}\left [N\sum f_iy_i^2 - (\sum f_iy_i)^2 \right ]\times h^2$

$\\ \implies \sigma^2 = \frac{1}{(150)^2}\left [150(342) - (-6)^2 \right ]\times10^2 \\ = \frac{1}{15^2}\left [51264 \right ] \\ =227.84$

We know,  Standard Deviation = $\sigma = \sqrt{Variance}$

$\therefore \sigma = \sqrt{227.84} = 15.09$

Coefficient of variation = $\frac{\sigma}{\overline x}\times100$

C.V.(A) = $\frac{15.09}{44.6}\times100 = 33.83$

Similarly,

For Group B

 Marks Group A $f_i$ Midpoint $x_i$ $\dpi{100} y_i = \frac{x_i-A}{h}$ $\dpi{80} = \frac{x_i-45}{10}$ $y_i^2$ $f_iy_i$ $f_iy_i^2$ 10-20 10 15 -3 9 -30 90 20-30 20 25 -2 4 -40 80 30-40 30 35 -1 1 -30 30 40-50 25 45 0 0 0 0 50-60 43 55 1 1 43 43 60-70 15 65 2 4 30 60 70-80 7 75 3 9 21 72 $\sum{f_i}$ =N = 150 $\sum f_iy_i$ = -6 $\sum f_iy_i ^2$ =375

Mean,

$\overline{x} = A + \frac{1}{N}\sum_{i=1}^{n}f_iy_i\times h =45 + \frac{-6}{150}\times10 = 44.6$

We know, Variance, $\sigma^2 = \frac{1}{N^2}\left [N\sum f_iy_i^2 - (\sum f_iy_i)^2 \right ]\times h^2$

$\\ \implies \sigma^2 = \frac{1}{(150)^2}\left [150(375) - (-6)^2 \right ]\times10^2 \\ = \frac{1}{15^2}\left [56214 \right ] \\ =249.84$

We know,  Standard Deviation = $\sigma = \sqrt{Variance}$

$\therefore \sigma = \sqrt{249.84} = 15.80$

Coefficient of variation = $\frac{\sigma}{\overline x}\times100$

C.V.(B) = $\frac{15.80}{44.6}\times100 = 35.42$

Since C.V.(B) > C.V.(A)

Therefore, Group B is more variable.

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