If the mean and variance of the following data: $6, 10, 7, 13, a, 12, b, 12$ are $9$ and $\frac{37}{4}$ respectively,then $(a-b)^{2}$ is equal to:

Let $a_1, a_2, \ldots, a_{10}$ be $10$ observations such that $\sum_{k=1}^{10} a_k = 50$ and $\sum_{k < j} a_k a_j = 1100$. Then the standard deviation of $a_1, a_2, \ldots, a_{10}$ is equal to:

The variance of the ungrouped data $2, 12, 3, 11, 5, 10, 6, 7$ is

Consider $10$ observations $x_1, x_2, \ldots, x_{10}$ such that $\sum_{i=1}^{10}(x_i-\alpha)=2$ and $\sum_{i=1}^{10}(x_i-\beta)^2=40$,where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. The value of $\frac{\beta}{\alpha}$ is equal to :

Let in a series of $2n$ observations,half of them are equal to $a$ and the remaining half are equal to $-a$. Also,by adding a constant $b$ to each of these observations,the mean and standard deviation of the new set become $5$ and $20$,respectively. Then the value of $a^{2} + b^{2}$ is equal to ....... .

Suppose values taken by a variable $x$ are such that $a \le x_i \le b$,where $x_i$ denotes the value of $x$ in the $i^{th}$ case for $i = 1, 2, ..., n$. Then: