The mean of $100$ observations is $50$ and their standard deviation is $5$. Then,the sum of squares of all observations is

For a set of $50$ observations,the sum of their squares is $3050$ and their arithmetic mean is $6$. Find the standard deviation of these observations.

If $S_1$ and $S_2$ are the variances of the first $2k$ and $k$ $(k > 1)$ natural numbers respectively,then $(S_1 / S_2)$ lies in the interval

If each of given $n$ observations is multiplied by a certain positive number $k$,then for the new set of observations -

If $\sum_{i = 1}^9 (x_i - 5) = 9$ and $\sum_{i = 1}^9 (x_i - 5)^2 = 45$,then the standard deviation of the $9$ items $x_1, x_2, ..., x_9$ is:

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