Question

The mean of $n$ values of a distribution is $\overline{x}.$ If its first value is increased by $1$second by $2$ .... then the mean of new values will be

• A. $\overline{x}+n$
• B. $\overline{x}+\frac{n}{2}$
• C. $\overline{x}+\left(\frac{n+1}{2}\right)$
• D. None of these

Answer 1

The correct option is C $\overline{x}+\left(\frac{n+1}{2}\right)$

Let $x_1,x_2,...x_n$ be the $n$ values
Mean $\overline{x}=\frac{x_1+x_2+....+x_n}{n}$
$\Rightarrow x_1+x_2+...+x_n=n\overline{x}$

New mean = $\frac{(x_1+1)+(x_2+2)+.....+(x_n+n)}{n}$
$=\frac{x_1+x_2+.....+x_n}{n}+\frac{1+2+....+n}{n}$
$=\overline{x}+\frac{n+1}{2}$