Question

Find the mean, variance and standard deviation using short-cut method. 

Height in cms	70-75	75-80	80-85	85-90	90-95	95-100	100-105	105-110	110-115
No. of children	3	4	7	7	15	9	6	6	3

Answer 1

Class Interval	Frequency \(f_i\)	\(mid-point\,x_i\)	\(y_i=\frac{x_i-92.5}{5}\)	\(f_iy_i\)	\(f_i^2\)	\(f_iy_1^2\)
70-75	3	72.5	-4	-12	16	48
75-80	4	77.5	-3	-12	9	36
80-85	7	82.5	-2	-14	4	28
85-90	7	87.5	1	-7	1	7
90-95	15	92.5	0	0	0	0
95-100	9	97.5	1	9	9	9
100-105	6	102.5	2	12	12	24
105-110	6	107.5	3	18	18	54
110-115	3	112.5	4	12	12	48
	60			6	6	254

Mean, \(\bar{x}=A\frac{\sum_{i=1}^9f_ix_i}{n}×h=92.5+\frac{6}{60}×5=92.5+0.5=93\) 

Variance (σ²) = \(\frac{h^2}{N^2}[N\sum_{i=1}^9f_iy_i^2-(\sum_{i=1}^9f_iy_i)^2]\)

\(=\frac{(5)^2}{(60)^2}[60×254-(6)^2]\)

\(=\frac{25}{3600}(15204)=105.58\)

\(Standard\,\,deviation\.(σ) =√105.58=10.27\)