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,

\[\bar{x} = a + h\left( \frac{1}{N}\Sigma f_i y_i \right) = 105 + 30\left( \frac{137}{300} \right) = 118 . 7\]

Variance:

\[\sigma^2 = \frac{h^2}{N^2}\left[ N\sum f_i {y_i}^2 - \left( \sum f_i y_i \right)^2 \right]\]

\[ = \frac{900}{90000}\left[ 300 \times 665 - 18769 \right]\]

\[ = \frac{1}{100}\left[ 199500 - 18769 \right]\]

\[ = \frac{180731}{100} = 1807 . 31\]

\[SD, \sigma = \sqrt{1807 . 31} = 42 . 51\]