Question

Calculate the A.M. and S.D. for the following distribution:

Class:	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency:	18	16	15	12	10	5	2	1

Answer 1

Class	$f_i$	Midpoint$\left(x_i\right)$	$u_i=\frac{x_i-35}{10}$	$f_i{u}_i$	$f_i{u_i}^2$
0−10	18	5	$-$3	$-$54	162
10−20	16	15	$-$2	$-$32	64
20−30	15	25	$-$1	$-$15	15
30−40	12	35	0	0	0
40−50	10	45	1	10	10
50−60	5	55	2	10	20
60−70	2	65	3	6	18
70−80	1	75	4	4	16
	$\sum f_i=79$			$\sum f_i{u}_i=-71$	$\sum f_i{u_i}^2=305$

​$\overline{X}=a+h\left(\frac{\sum f_i{u}_i}{N}\right)=35+10\left(\frac{-71}{79}\right)=26.01$

AM = 26.01

$$\begin{gathered} {\sigma }^2=h^2\left[\frac{\sum f_i{u_i}^2}{N}-{\left(\frac{\sum f_i{u}_i}{N}\right)}^2\right]=100\left[\frac{305}{79}-\frac{5041}{6241}\right]=305.20 \\ \sigma =\sqrt{305.20}=17.47 \end{gathered}$$