Question

Find the coefficient of variation for the following data:

Size (in cms):	10-15	15-20	20-25	25-30	30-35	35-40
No. of items:	2	8	20	35	20	15

Answer 1

Size (cm)	\[f_i\]	Midpoint  \[\left( x_i \right)\]	\[u_i = \frac{x_i - 27 . 5}{5}\]	\[f_i u_i\]	\[f_i {u_i}^2\]
10−15	2	12.5	- 3	- 6	18
15−20	8	17.5	- 2	- 16	32
20−25	20	22.5	- 1	- 20	20
25−30	35	27.5	0	0	0
30−35	20	32.5	1	20	20
35−40	15	37.5	2	30	60
	\[\sum f_i = N = 100\]			\[\sum f_i u_i  = 8 \]	\[\sum f_i {u_i}^2 = 150\]

Here,

\[h = 5, a = 27 . 5\]

\[\bar{X} = a + h\left( \frac{\sum f_i u_i}{N} \right) = 27 . 5 + 5\left( \frac{8}{100} \right) = 27 . 9\]

\[\sigma^2 = h^2 \left[ \frac{\sum f_i {u_i}^2}{N} - \left( \frac{\sum f_i u_i}{N} \right)^2 \right] = 25\left[ \frac{150}{100} - \frac{64}{10000} \right] = 37 . 34\]

\[\sigma = \sqrt{37 . 34} = 6 . 11\]

We know that

\[CV = \frac{\sigma}{\bar{X}} \times 100\]

\[ = \frac{6 . 11}{27 . 9} \times 100 = 21 . 9\]