Question

Calculate the standard deviation for the following data:

Class:	0-30	30-60	60-90	90-120	120-150	150-180	180-210
Frequency:	9	17	43	82	81	44	24

Answer 1

Class	$f_i$	Midpoint$\left(x_i\right)$	$y_i=\frac{x_i-105}{30}$	${y_i}^2$	$f_i{y}_i$	$f_i{y_i}^2$
0−30	9	15	−3	9	−27	81
30−60	17	45	−2	4	−34	68
60−90	43	75	−1	1	−43	43
90−120	82	105	0	0	0	0
120−150	81	135	1	1	81	81
150−180	44	165	2	4	88	176
180−210	24	195	3	9	72	216
	$\sum f_i=N=300$				$\sum f_i{y}_i=137$	$\sum f_i{y_i}^2=665$

Mean, $\overline{x}=a+h\left(\frac{1}{N}\Sigma f_i{y}_i\right)=105+30\left(\frac{137}{300}\right)=118.7$

Variance:
$$\begin{gathered} {\sigma }^2=\frac{h^2}{N^2}\left[N\sum f_i{y_i}^2-{\left(\sum f_i{y}_i\right)}^2\right] \\ =\frac{900}{90000}\left[300\times 665-18769\right] \\ =\frac{1}{100}\left[199500-18769\right] \\ =\frac{180731}{100}=1807.31 \\ SD,\sigma =\sqrt{1807.31}=42.51 \end{gathered}$$