If each of the observation $x_{1}, x_{2}, \ldots ., x_{n}$ is increased by $'a'$ where $a$ is a negative or positive number, show that the variance remains unchanged.

Let $ \bar x , M$ and $\sigma^2$ be respectively the mean, mode and variance of $n$ observations $x_1 , x_2,…,x_n$ and $d_i\, = – x_i – a, i\, = 1, 2, …. , n$, where $a$ is any number.
Statement $I$: Variance of $d_1, d_2,…..d_n$ is $\sigma^2$.
Statement $II$ : Mean and mode of $d_1 , d_2, …. d_n$ are $-\bar x -a$ and $- M – a$, respectively

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

The variance $\sigma^2$ of the data is $ . . . . . .$

The mean and variance of $7$ observations are $8$ and $16,$ respectively. If five of the observations are $2,4,10,12,14 .$ Find the remaining two observations.