Question

Find the mean and variance for the following frequency distribution

Classes	0-30	30-60	60-90	90-120	120-150	150-180	180-210
Frequencies	2	3	5	10	3	5	2

Answer 1

Class	$f_i$	$x_i$	$u_i=\frac{x_i-A}{h}$	$u_i^2$	$f_i{u}_i$	$f_i{u}_i^2$
0-30	2	15	-3	9	-6	18
30-60	3	45	-2	4	-6	12
60-90	5	75	-1	1	-5	5
90-120	10	105	0	0	0	0
120-150	3	135	1	1	3	3
150-180	5	165	2	4	10	20
180-210	2	195	3	9	6	18
Total	30				2	76

Here, $N=30,h=30$

Mean, $\overline{x}=A+\frac{{\sum }_{i=1}^7{f}_i{u}_i}{N}\times h$

$=105+\frac{2}{30}\times 30$

$=105+2=107$

Variance $\left({\sigma }^2\right)=\frac{h^2}{N^2}[N\sum _{i=l}^7{f}_i{u}_i^2-{\left(\sum _{i=l}^7{f}_i{u}_i\right)}^2]$
=$\frac{{\left(30\right)}^2}{{\left(30\right)}^2}[30\times 76-(2{)}^2]$
$=2280-4$
$=2276$