If the mean and standard deviation of 10 observations $x_1,x_2,\dots ,x_{10}$ are 2 and 3 respectively, then mean of ${\left(x_1+1\right)}^2,{\left(x_2+1\right)}^2,\dots ,{\left(x_{10}+1\right)}^2$ is equal to

We have, Mean = 2 and S. D. = 3

$\Rightarrow \frac{1}{10}\sum _{i=10}^{10} x_i=2$ and $\frac{1}{10}\sum _{i=1}^{10} x_i^2-2^2=3^2$

$\Rightarrow \sum _{i=1}^{10} x_i=20$ and $\sum _{i=1}^{10} x_i^2=130$

Let $\overline{X}$ be the mean of ${\left(x_1+1\right)}^2,{\left(x_2+1\right)}^2,\dots ,{\left(x_{10}+1\right)}^2$. Then,

$\begin{array}{l}\overline{X}=\frac{1}{10}\sum _{i=1}^{10} {\left(x_i+1\right)}^2\\ \Rightarrow \overline{X}=\frac{1}{10}\left(\sum _{i=1}^{10} x_i^2\right)+\frac{2}{10}\left(\sum _{i=1}^{10} x_i\right)+\frac{1}{10}\sum _{i=1}^{10} 1\\ \Rightarrow \overline{X}=\frac{1}{10}\times 130+\frac{2}{10}\times 20+\frac{10}{10}=18\end{array}$