Let $x_1, x_2, \dots, x_n$ be the values taken by a variable $X$ and $y_1, y_2, \dots, y_n$ be the values taken by a variable $Y$ such that $y_i = ax_i + b$,where $i = 1, 2, \dots, n$. Then...

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 $\frac{n(4n^2 - 1)}{3}$.

If the standard deviation of the numbers $-1, 0, 1, k$ is $\sqrt{5}$ where $k > 0$,then $k$ is equal to

If the variance of the first $n$ natural numbers is $10$ and the variance of the first $m$ even natural numbers is $16$,then $m + n$ is equal to

If the variance of the following frequency distribution is $50$,then $x$ is equal to:

Class	$10-20, 20-30, 30-40$
Frequency	$2, x, 2$

Let the observations $x_{i} (1 \leq i \leq 10)$ satisfy the equations $\sum_{i=1}^{10}(x_{i}-5)=10$ and $\sum_{i=1}^{10}(x_{i}-5)^{2}=40$. If $\mu$ and $\lambda$ are the mean and the variance of the observations $x_{1}-3, x_{2}-3, \dots, x_{10}-3$,then the ordered pair $(\mu, \lambda)$ is equal to: