The mean of four observations is $3$. If the sum of the squares of the observations is $48$, then their standard deviation is

The correct option is C

$\sqrt{3}$

Explanation for the correct option:

Find the standard deviation for the given observations.

It is given that, the mean of four observations is $3$ and the sum of the squares of the observations is $48$.

We know that, the standard deviation $\sigma$ is given by $\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^n}\left(x_i^2\right)}{n}-{\left(\overline{x}\right)}^2}$.

So, the standard deviation $\sigma$ for the given observation is as follows:

$$\begin{gathered} \sigma =\sqrt{\frac{48}{4}-3^2} \\ \Rightarrow \sigma =\sqrt{12-9} \\ \Rightarrow \sigma =\sqrt{3} \end{gathered}$$

Therefore, the standard deviation $\sigma$ for the given observation is $\sqrt{3}$.

Hence, option $C$ is the correct option.