Consider three observations $a, b,$ and $c$ such that $b = a + c$. If the standard deviation of $a + 2, b + 2, c + 2$ is $d$,then which of the following holds true?

The $S.D.$ of a variate $x$ is $\sigma$. The $S.D.$ of the variate $\frac{ax + b}{c}$,where $a, b, c$ are constants,is

The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results:
Number of observations $= 25$,mean $= 18.2 \text{ seconds}$,standard deviation $= 3.25 \text{ s}$.
Further,another set of $15$ observations $x_{1}, x_{2}, \ldots, x_{15}$,also in seconds,is now available and we have $\sum_{i=1}^{15} x_{i} = 279$ and $\sum_{i=1}^{15} x_{i}^{2} = 5524$. Calculate the standard deviation based on all $40$ observations.

$A$ scientist weighs $30$ fish. Their mean weight is $30 \text{ g}$ and the standard deviation is $2 \text{ g}$. Later,it is discovered that the weighing scale was not calibrated correctly and every fish's weight was recorded $2 \text{ g}$ less than the actual weight. What are the correct mean and standard deviation (in grams) of the fish weights,respectively?

Find the coefficient of variation of the first $n$ natural numbers.

The variance of the observations $2, 3, 5, 7, 11, 13, 17, 22$ is