Question

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

$x_i$	60	61	62	63	64	65	66	67	68
$f_i$	2	1	12	29	25	12	10	4	5

Answer 1

The data is obtained in tabular form as follows.

$x_i$	$f_i$	$fx_i=\frac{x_i-64}{1}$	$y_1^2$	$f_iy_i$	$f_iy_1^2$
60	2	-4	16	-8	32
61	1	-3	9	-3	9
62	12	-2	4	-24	48
63	29	-1	1	-29	29
64	25	0	0	0	0
65	12	1	1	12	12
66	10	2	4	20	40
67	4	3	9	12	36
68	5	4	16	20	80
	100	220		0	286

Mean, $\bar{x}=A\frac{\sum_{i=1}^9f_ix_i}{n}×h=64+\frac{0}{100}×1=64+0=64$ 

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

$=\frac{1}{100^2}[100×286-0]$

$=2.86$

$∴\,standard\,deviation\,(σ)=√2.86=1.69$