The range of the data $35, 12, 21, 24, 15, 7, 16, 12, 30, 32, 13, 17$ is

The mean and standard deviation of a set of $n_{1}$ observations are $\bar{x}_{1}$ and $s_{1},$ respectively,while the mean and standard deviation of another set of $n_{2}$ observations are $\bar{x}_{2}$ and $s_{2},$ respectively. Show that the standard deviation of the combined set of $(n_{1}+n_{2})$ observations is given by $SD = \sqrt{\frac{n_{1}(s_{1})^{2}+n_{2}(s_{2})^{2}}{n_{1}+n_{2}}+\frac{n_{1} n_{2}(\bar{x}_{1}-\bar{x}_{2})^{2}}{(n_{1}+n_{2})^{2}}}$.

The mean and standard deviation of $100$ observations were calculated as $40$ and $5.1$ respectively. Later on,it was found that one of the observations was taken as $50$ in place of $40$. If the wrong entry is replaced by the correct one,then the sum of the squares of all the observations is:

The mean of $n$ items is $\bar x$. If the first term is increased by $1$,the second by $2$,and so on,then the new mean is:

If the variance of the frequency distribution is $3$,then $\alpha$ is ......

$X_i$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Frequency $f_i$	$3$	$6$	$16$	$\alpha$	$9$	$5$	$6$

The mean of a data set comprising $16$ observations is $16$. If one observation with value $16$ is deleted and three new observations with values $3, 4,$ and $5$ are added to the data,then the mean of the resultant data is: