Question

If each observation of a data is increased by 5, then their mean
(a) remains the same
(b) becomes 5 times the original mean
(c) is decreased by 5
(d) is increased by 5

Answer 1

(d) is increased by 5

$\mathrm{Let}\mathrm{the}\mathrm{numbers}\mathrm{be}x$₁, x_2,...x_n.
$\mathrm{Hence},\mathrm{mean}=\frac{x_1+x_2+....+x_n}{n}$

Now the new numbers after increasing every number by 5 : (x₁+5) , (x₂+5)...,(x_n+5)

$\mathrm{New}\mathrm{Mean}=\frac{\left(x_1+5\right)+\left(x_2+5\right)+....+\left(x_n+5\right)}{n}$

$=\frac{x_1+x_2+....+x_n+5n}{n}$

$$\begin{gathered} =\frac{x_1+x_2+....+x_n}{n}+5 \\ \therefore \mathrm{New}\mathrm{mean}=\mathrm{mean}+5 \end{gathered}$$

Hence, mean is increased by 5