If the mean of two samples of sizes $20$ and $30$ are $25$ and $10$ respectively,and their variances are $9$ and $16$ respectively,then their combined variance is:

If both mean and variance of $50$ observations $x_1, x_2, \ldots, x_{50}$ are equal to $16$ and $256$ respectively,then the mean of $(x_1-5)^2, (x_2-5)^2, \ldots, (x_{50}-5)^2$ is

The mean and variance of $8$ observations are $10$ and $13.5,$ respectively. If $6$ of these observations are $5, 7, 10, 12, 14, 15,$ then the absolute difference of the remaining two observations is

Based on the following statements,choose the correct option.
Statement-$I:$ The variance of the first $n$ even natural numbers is $\frac{n^2-1}{4}$.
Statement-$II:$ The difference between the variance of the first $20$ even natural numbers and their arithmetic mean is $112$.

Consider the frequency distribution of the given numbers. If the mean is known to be $3$,then the value of $f$ is:

Value	$1$	$2$	$3$	$4$
Frequency	$5$	$4$	$6$	$f$

The median of $100$ observations grouped in classes of equal width is $25$. If the median class interval is $20-30$ and the number of observations less than $20$ is $45$,then the frequency of the median class is: