The $S.D.$ of $5$ scores $1, 2, 3, 4, 5$ is

If $\sum_{i=1}^{n}(x_{i}-a)=n$ and $\sum_{i=1}^{n}(x_{i}-a)^{2}=na$,where $n, a > 1$,then the standard deviation of $n$ observations $x_{1}, x_{2}, \ldots, x_{n}$ is

The variance of $20$ observations is $5$. If each observation is multiplied by $3$ and then $8$ is added to each number,then the variance of the resulting observations is:

Find the mean,variance,and standard deviation using the short-cut method for the following data:

Height in cms	$70-75$	$75-80$	$80-85$	$85-90$	$90-95$	$95-100$	$100-105$	$105-110$	$110-115$
No. of children	$3$	$4$	$7$	$7$	$15$	$9$	$6$	$6$	$3$

Determine the mean and standard deviation for the following distribution:
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Marks} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline \text{Frequency} & 1 & 6 & 6 & 8 & 8 & 2 & 2 & 3 & 0 & 2 & 1 & 0 & 0 & 0 & 1 \\ \hline \end{array}$

The mean and variance of six observations are $8$ and $16$ respectively. If each observation is multiplied by $3$,then the new variance of the resulting observations is: