Question

The mean of n observation is $\overline{x}$. If the first observation is increased by 1, the second by 2, the third by 3, and so on, then the new mean is

(a) $\overline{x}+\left(2n+1\right)$
(b) $\overline{x}+\frac{n+1}{2}$
(c) $\overline{x}+\left(n+1\right)$
(d) $\overline{x}-\frac{n+1}{2}$

Answer 1

Let $x_1,x_2,x_3,...,x_n$ be the n observations.

Mean$=\overline{x}=\frac{x_1+x_2+...+x_n}{n}$

$\Rightarrow x_1+x_2+x_3+...+x_n=n\overline{x}$

If the first item is increased by 1, the second by 2, the third by 3 and so on.

Then, the new observations are $x_1+1,x_2+2,x_3+3,...,x_n+n$.

New mean = $\frac{\left(x_1+1\right)+\left(x_2+2\right)+\left(x_3+3\right)+...+\left(x_n+n\right)}{n}$
$$\begin{gathered} =\frac{x_1+x_2+x_3+...+x_n+\left(1+2+3+...+n\right)}{n} \\ =\frac{n\overline{x}+{\displaystyle \frac{n\left(n+1\right)}{2}}}{n} \\ =\overline{x}+\frac{n+1}{2} \end{gathered}$$

Hence, the correct answer is (c).