Question

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

• A. $\frac{M-x_n+x^{\prime }}{n}$
• B. $\frac{(n-1)M+x^{\prime }}{n}$
• C. $\frac{nM-x_n+x^{\prime }}{n}$
• D. $M-x_n+x^{\prime }$

Answer 1

Answer: (C)

$\frac{nM-x_n+x^{\prime }}{n}$

We have,
$M=\frac{x_1+x_2+...+x_n}{n}$
$\Rightarrow x_1+x_2+...+x_{n-1}+x_n=nM$
$\Rightarrow x_1+x_2+...+x_{n-1}=nM-x_n$
Let $M^{\prime }$ be the average of $x_1,x_2,...,x_{n-1},x^{\prime }$. Then
$M^{\prime }=\frac{x_1+x_2+...+x_{n-1}+x^{\prime }}{n}\Rightarrow M^{\prime }=\frac{nM-x_n+x^{\prime }}{n}$