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

 

Answers (1)

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}

= \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}

= \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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