Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of $100$ consecutive positive integers $a _1, a _2, a _3, \ldots ., a _{100}$ is $25$. Then $S$ is

The mean and variance of the marks obtained by the students in a test are $10$ and $4$ respectively. Later, the marks of one of the students is increased from $8$ to $12$ . If the new mean of the marks is $10.2.$ then their new variance is equal to :

Let $x_1, x_2,........,x_n$ be $n$ observations such that $\sum {{x_i}^2 = 300} $ and $\sum {{x_i} = 60} $ on value of $n$ among the following is

The mean and variance of $10$ observations were calculated as $15$ and $15$ respectively by a student who took by mistake $25$ instead of $15$ for one observation. Then, the correct standard deviation is$.....$

Let sets $A$ and $B$ have $5$ elements each. Let the mean of the elements in sets $A$ and $B$ be $5$ and $8$ respectively and the variance of the elements in sets $A$ and $B$ be $12$ and $20$ respectively $A$ new set $C$ of $10$ elements is formed by subtracting $3$ from each element of $A$ and adding 2 to each element of B. Then the sum of the mean and variance of the elements of $C$ is $.......$.

The variance of first $50$ even natural numbers is