Consider a set of $3n$ numbers having variance $4$. In this set,the mean of the first $2n$ numbers is $6$ and the mean of the remaining $n$ numbers is $3$. $A$ new set is constructed by adding $1$ to each of the first $2n$ numbers and subtracting $1$ from each of the remaining $n$ numbers. If the variance of the new set is $k$,then $9k$ is equal to .... .

If $\bar{x}_1$ and $\bar{x}_2$ are the means of two distributions such that $\bar{x}_1 < \bar{x}_2$ and $\bar{x}$ is the mean of the combined distribution,then

The mean and standard deviation of a group of $100$ observations were found to be $20$ and $3$,respectively. Later on,it was found that three observations were incorrect,which were recorded as $21, 21,$ and $18$. Find the mean and standard deviation if the incorrect observations are omitted.

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 $20$ observations are found to be $10$ and $2$ respectively. On rechecking,it was found that an observation $8$ was incorrect. If the incorrect observation $8$ is replaced by $12$,calculate the correct mean and standard deviation.

The frequency distribution of the daily household expenditure of some families in a residential area is given below. If the mode of the distribution is $140$,what is the value of $b$?

Daily Expenditure (	$0-50$	$50-100$	$100-150$	$150-200$	$200-250$
Number of Families $(f)$	$24$	$33$	$37$	$b$	$25$