Question

Find the mean and variance for the data

$X_i$	6	10	14	18	24	28	30
$f_i$	2	4	7	12	8	4	3

Answer 1

The data is obtained in tabular form as follows

$X_i$	$f_i$	$f_i{X}_i$	$X_i-\overline{X}$	${(X_i-\overline{X})}^2$	$f_i{(X_i-\overline{X})}^2$
6	2	12	-13	169	338
10	4	40	-9	81	324
14	7	98	-5	25	175
18	12	216	-1	1	12
24	8	192	5	25	200
28	4	112	9	81	324
30	3	90	11	121	363
	40	760			1736

Here, $N=40$, $\sum _{i=1}^7{f}_i{x}_i=760$
$\therefore \overline{x}=\frac{{\sum }_{i=1}^7{f}_i{x}_i}{N}=\frac{760}{40}=19$
and Variance $\left({\sigma }^2\right)=\frac{1}{N}\sum _{i=1}^7{f}_i(x_i-\overline{X}{)}^2=\frac{1}{40}\times 1736=43.4$