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	Frequency \(f_i\)	\(mid-point\,x_i\)	\(y_i=\frac{x_i-105}{30}\)	\(f_iy_i\)	\(y_i^2\)	\(f_iy_1^2\)
0-30	2	15	-3	-6	9	18
30-60	3	45	-2	-6	4	12
60-90	5	75	-1	-5	1	5
90-120	10	105	0	0	0	0
120-150	3	135	1	3	1	3
150-180	5	165	2	10	4	20
180-210	2	195	3	6	9	18
	30			2		76

Mean,  \(\bar{x}=A\frac{\sum_{i=1}^7f_ix_i}{n}×h=105+\frac{2}{30}×30=105+2=107\)

Variance(σ²) = \(\frac{h^2}{N^2}[N\sum_{i=1}^7f_iy_i^2-(\sum_{i=1}^7f_iy_i)^2]\)

\(=\frac{(30)^2}{(30)^2}[30×76-(2)^2]\)

\(=2280-4\)

\(=2276\)