The mean and the standard deviation of a data set of $8$ items are $25$ and $5$ respectively. If two items $15$ and $25$ are added to this data,then the variance of the new data is:

$\text{Assertion (A):}$ Variance of $4x_1, 4x_2, \ldots, 4x_n$ is $16$ times the variance of $x_1, x_2, \ldots, x_n$. $\text{Reason (R):}$ If $y = ax + b$,then variance of $y$ is $a(\text{variance of } x) + b$. The correct option among the following is

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 nine observations $x_1, x_2, \ldots, x_9$ is

Find the mean and variance for the first $n$ natural numbers.

The standard deviation of the numbers $22, 26, 28, 20, 24, 30$ is

The means of two groups of size $200$ and $300$ are $25$ and $10$ respectively. Their standard deviations are $3$ and $4$ respectively. What is the variance of the combined sample of size $500$?