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

The mean and standard deviation of $15$ observations were found to be $12$ and $3$ respectively. On rechecking it was found that an observation was read as $10$ in place of $12$ . If $\mu$ and $\sigma^2$ denote the mean and variance of the correct observations respectively, then $15\left(\mu+\mu^2+\sigma^2\right)$ is equal to$……………….$

Find the mean and variance for the data $6,7,10,12,13,4,8,12$

If the variance of the frequency distribution is $3$ then $\alpha$ is ……