Question

If each observation of the data is increased by 8, then their mean
(a) remains the same
(b) is decreased by 8
(c) is increased by 5
(d) becomes 8 times the original mean

Answer 1

(b) is decreased by 8

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

Now the new numbers after decreasing every number by 8 : (x₁−8) , (x₂−8)...,(x_n−8)

$\mathrm{New}\mathrm{Mean}=\frac{\left(x_1-8\right)+\left(x_2-8\right)+....+\left(x_n-8\right)}{n}$

$=\frac{x_1+x_2+....+x_n-8n}{n}$

$$\begin{gathered} =\frac{x_1+x_2+....+x_n}{n}-8 \\ \therefore \mathrm{New}\mathrm{mean}=\mathrm{mean}-8 \end{gathered}$$

Hence, mean is decreased by 8.