Question

Find the mean and variance for the following frequency distrubution

Classes	0-10	10-20	20-30	30-40	40-50
Frequencies	5	8	15	16	6

Answer 1

Class interval	$f_i$	$x_i$	$u_i=\frac{x_i-A}{h}$	$u_i^2$	$f_i{u}_i$	$f_i{u}_i^2$
0-10	5	5	-2	4	-10	20
10-20	8	15	-1	1	-8	8
20-30	15	25	0	0	0	0
30-40	16	35	1	1	16	16
40-50	6	45	2	4	12	24
Total	50				10	68

We have $N=\sum f_i=50$

Width of class $h=10$

Assumed mean $A=25\left(say\right)$

Mean, $\overline{x}=A+\frac{{\sum }_{i=1}^5{f}_i{u}_i}{N}\times h$

$=25+\frac{10}{50}\times 10=25+2=27$

$\Rightarrow \overline{x}=27$

Variance ${\sigma }^2=h^2[\frac{{\sum }_{i=1}^5{f}_i{u}_i^2}{N}-{\left(\frac{{\sum }_{i=1}^5{f}_i{u}_i}{N}\right)}^2]$
${\sigma }^2={10}^2[\frac{68}{50}-{\left(\frac{10}{50}\right)}^2]$

$={10}^2\frac{[50\times 68-(10{)}^2]}{{\left(50\right)}^2}$
$=\frac{1}{25}[3400-100]$

$=\frac{3300}{25}$
$=132$