Question

Calculate variance for the following distribution.

Class	30-40	40-50	50-60	60-70	70-80	80-90	90-100
Frequency	3	7	12	15	8	3	2

Answer 1

Answer: 201

Let the assumed mean $A=65.$ Here, $h=10$.
We obtain the following table from the given data.

Class	Fre quency $f_i$	Mid -point $x_i$	$y_i=\frac{x_i-65}{10}$	$y_i^2$	$f_i{y}_i$	$f_i{y}_i^2$
30-40	3	35	-3	9	-9	27
40-50	7	45	-2	4	-14	28
50-60	12	55	-1	1	-12	12
60-70	15	65	0	0	0	0
70-80	8	75	1	1	8	8
80-90	3	85	2	4	6	12
90-100	2	95	3	9	6	18
total	N=50				-15	105

Therefore $\bar{x}=A+\frac{\sum f_i{y}_i}{50}\times h=65-\frac{15}{50}\times 10=62$
Variance ${\sigma }^2=\frac{h^2}{N^2}\left[N\Sigma f_i{y}_i^2-{\left(\Sigma f_i{y}_i\right)}^2\right]$
$=\frac{(10{)}^2}{(50{)}^2}\left[50\times 105-(-15{)}^2\right]$
$=\frac{1}{25}[5250-225]=201$