Question

Find the mean and variance for the data

$X_i$	92	93	97	98	102	104	109
$f_i$	3	2	3	2	6	3	3

Answer 1

The data is obtained in tabular form as follows

$X_i$	$f_i$	$f_i-\overline{X}$	$X_i-\overline{X}$	${(X_i-\overline{X})}^2$	$f_i{(X_i-\overline{X})}^2$
92	3	276	-8	64	192
93	2	186	-7	49	98
97	3	291	-3	9	27
98	2	196	-2	4	8
102	6	612	2	4	24
104	3	312	4	16	48
109	3	327	9	81	243
109	3	327	9	81	243

22

2200

640

Here , $N=22$, $\sum _{i=1}^7{f}_i{X}_i=2200$
$\therefore \overline{X}=\frac{1}{N}\sum _{i=1}^7{f}_i{x}_i=\frac{1}{22}\times 2200=100$
and Variance $\left({\sigma }^2\right)=\frac{1}{N}\sum _{i=1}^7{f}_i{(X_i-\overline{X})}^2=\frac{1}{22}\times 640=29.09$