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

$x_i$	$f_i$	$u_i=\frac{x_i-A}{h}$	$u_i^2$	$f_i{u}_i$	$f_i{u}_i^2$
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
Total	100			0	286

Here, $N=100,h=1$

Let assumed mean $A=64$

Mean $\overline{x}=A+\frac{{\sum }_{i=1}^9{f}_i{u}_i}{N}\times h$

$=64+\frac{0}{100}\times 1=64+0=64$

Variance ${\sigma }^2=\frac{h^2}{N^2}[N\sum _{i=l}^9{f}_i{u}_i^2-{\left(\sum _{i=l}^9{f}_i{u}_i\right)}^2]$
$=\frac{1}{{100}^2}[100\times 286-0]$
$=2.86$
$\therefore$ standard deviation $\left(\sigma \right)=\sqrt{2.86}=1.69$