The mean and variance of a series of $5$ observations are $8$ and $24$ respectively. The mean and variance of another series of $3$ observations are $8$ and $24$ respectively. What is the variance of their combined series?

Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that $\sum_{i=1}^{10}(x_i-2)=30$,$\sum_{i=1}^{10}(x_i-\beta)^2=98$,$\beta > 2$ and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1-1)+4\beta, 2(x_2-1)+4\beta, \ldots, 2(x_{10}-1)+4\beta$,then $\frac{\beta\mu}{\sigma^2}$ is equal to:

The variance of the first $50$ even natural numbers is

$A$ data consists of $20$ observations $x_1, x_2, ..., x_{20}$. If $\sum_{i=1}^{20} (x_i + 5)^2 = 2500$ and $\sum_{i=1}^{20} (x_i - 5)^2 = 100$,then the ratio of mean to standard deviation of this data is:

The variance for the first six prime numbers greater than $5$ is

Variance of first $2n$ natural numbers is