The variance of the first $20$ natural numbers is

For a frequency distribution,the standard deviation is computed by:

The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results:
Number of observations $= 25$,mean $= 18.2 \text{ seconds}$,standard deviation $= 3.25 \text{ s}$.
Further,another set of $15$ observations $x_{1}, x_{2}, \ldots, x_{15}$,also in seconds,is now available and we have $\sum_{i=1}^{15} x_{i} = 279$ and $\sum_{i=1}^{15} x_{i}^{2} = 5524$. Calculate the standard deviation based on all $40$ observations.

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	$3$	$4$	$7$	$7$	$15$	$9$	$6$	$6$	$3$

The variance for the first six prime numbers greater than $5$ is

Let ${x_1}, {x_2}, \ldots, {x_n}$ be $n$ observations,and let $\bar x$ be their arithmetic mean and ${\sigma ^2}$ be the variance.
Statement-$1$: The variance of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4{\sigma ^2}$.
Statement-$2$: The arithmetic mean of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4\bar x$.