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
Solution:
(b) is decreased by 8
Let the numbers be $x_{1}, x_{2} \ldots x_{n} .$
Hence, mean $=\frac{x_{1}+x_{2}+\ldots+x_{n}}{n}$
Now the new numbers after decreasing every number by $8:\left(x_{1}-8\right),\left(x_{2}-8\right) \ldots,\left(x_{n}-8\right)$
New Mean $=\frac{\left(x_{1}-8\right)+\left(x_{2}-8\right)+\ldots+\left(x_{n}-8\right)}{n}$
$=\frac{x_{1}+x_{2}+\ldots .+x_{n}-8 n}{n}$
$=\frac{x_{1}+x_{2}+\ldots .+x_{n}}{n}-8$
$\therefore$ New mean $=$ mean $-8$
Hence, mean is decreased by 8.