The average of n numbers  $x_1,x_2,\dots ,x_n$ is $M.$ If $x_n$ is replaced by  $x\text{'}$ , then the new average is 

 We have, 

 $M=\frac{x_1+x_2+\dots +x_n}{n}$ 

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{x}_1+x_2+\dots +x_{n-1}+x_n=nM\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{x}_1+x_2+\dots +x_{n-1}=nM-x_n\end{array}$

Let M' be the average of $x_1,x_2,\dots ,x_{n-1},x^{\mathrm{\prime }} ...\left(i\right)$

$\begin{array}{l}{M}^{\mathrm{\prime }}=\frac{x_1+x_2+\dots +x_{n-1}+x^{\mathrm{\prime }}}{n}\\ \Rightarrow M^{\mathrm{\prime }}=\frac{nM-x_n+x^{\mathrm{\prime }}}{n}\end{array}$                                      [Using (i)]