Question

If the mean of observation $x_1,x_2,....,x_n\mathrm{is}\overline{x}$, then the mean of x₁ + a, x₂ + a, ....., x_n + a is

(a) $a\overline{x}$
(b) $\overline{x}-a$
(c) $\overline{x}+a$
(d) $\frac{\overline{x}}{a}$

Answer 1

The mean of $x_1,x_2,...,x_n\mathrm{is}\overline{x}$.

$$\begin{gathered} \therefore \frac{x_1+x_2+x_3+...+x_n}{n}=\overline{x} \\ \Rightarrow x_1+x_2+x_3+...+x_n=n\overline{x} \end{gathered}$$

Mean of x₁ + a, x₂ + a, ... , x_n + a

$$\begin{gathered} =\frac{\left(x_1+a\right)+\left(x_2+a\right)+\left(x_3+a\right)+...+\left(x_n+a\right)}{n} \\ =\frac{\left(x_1+x_2+x_3+...+x_n\right)+\left(a+a+a+...+a\right)}{n} \\ =\frac{n\overline{x}+na}{n} \\ =\overline{x}+a \end{gathered}$$

Hence, the correct option is (c).