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,
$h=5,a=27.5$
$\overline{X}=a+h\left(\frac{\sum f_i{u}_i}{N}\right)=27.5+5\left(\frac{8}{100}\right)=27.9$
${\sigma }^2=h^2\left[\frac{\sum f_i{u_i}^2}{N}-{\left(\frac{\sum f_i{u}_i}{N}\right)}^2\right]=25\left[\frac{150}{100}-\frac{64}{10000}\right]=37.34$

$\sigma =\sqrt{37.34}=6.11$

We know that
$$\begin{gathered} CV=\frac{\sigma }{\overline{X}}\times 100 \\ =\frac{6.11}{27.9}\times 100=21.9 \end{gathered}$$