Q. The mean of a set of observations is $\overline{x}$. If each observation is divided by, $\alpha \neq 0$ and then is increased by 10, then the mean of the new set is
A) $\frac{\overline{x}}{\alpha}$
B) $\frac{\overline{x} + 10}{\alpha}$
C) $\frac{\overline{x} + 10\alpha}{\alpha}$
D) $\alpha\overline{x} + 10$
Explanation
(c)
Let $x_{1},x_{2},x_{3},\ldots,x_{n}$ be $n$ observations. Then,
$\overline{x} = \frac{1}{n}\sum_{i = 1}^{n}x_{i}$
$\therefore\ \ New\ mean,\ \overline{X} = \frac{1}{n}\sum_{i = 1}^{n}\left( \frac{x_{i}}{\alpha} + 10 \right)$
$= \frac{1}{\alpha}\left( \frac{1}{n}\sum_{i = 1}^{n}x_{i} \right) + \frac{1}{n}.(10n)$
$= \frac{1}{\alpha}\overline{x} + 10 = \frac{\overline{x} + 10\alpha}{\alpha}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.