If each observation of a distribution with standard deviation $\sigma$ is increased by $\lambda$,find the variance of the new observations.

Given that $\bar{x}$ is the mean and $\sigma^{2}$ is the variance of $n$ observations $x_{1}, x_{2}, \ldots, x_{n}$,prove that the mean and variance of the observations $a x_{1}, a x_{2}, \ldots, a x_{n}$ are $a \bar{x}$ and $a^{2} \sigma^{2}$ respectively,where $a \neq 0$.

Statement $1$: The variance of the first $n$ odd natural numbers is $\frac{n^2 - 1}{3}$.
Statement $2$: The sum of the first $n$ odd natural numbers is $n^2$ and the sum of the squares of the first $n$ odd natural numbers is $\frac{n(4n^2 - 1)}{3}$.

The standard deviation of the numbers $31, 32, 33, \ldots, 46, 47$ is

Find the mean and variance for the following frequency distribution.

Classes	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
Frequencies	$5$	$8$	$15$	$16$	$6$

The mean of the numbers $a, b, 8, 5, 10$ is $6$ and the variance is $6.80$. Then which one of the following gives possible values of $a$ and $b$?