Question

The arithmetic mean of a set of observations is $X$. If each observation is divided by $\alpha$ and then is increased by $10$, then the mean of the new series is

• A. $\frac{\overline{X}}{\alpha }$
• B. $\frac{\overline{X}+10}{\alpha }$
• C. $\frac{\overline{X}+10\alpha }{\alpha }$
• D. $\alpha \overline{X}+10$

Answer 1

The correct option is C $\frac{\overline{X}+10\alpha }{\alpha }$

Let $x_1,x_2,x_3,........x_n$ be the given observation,
$\therefore$ New observation is $\frac{x_i}{\alpha }+10,\frac{x_2}{\alpha }+10,\frac{x_3}{\alpha }+10,............\frac{x_n}{\alpha }+10$
Hence New observations mean $=\frac{\frac{x_1}{\alpha }+10+\frac{x_2}{\alpha }+10+\frac{x_3}{\alpha }+10+.................+\frac{x_n}{\alpha }+10}{n}$
$=\frac{\sum \frac{x_i}{\alpha }+10n}{n}=\frac{\overline{X}}{\alpha }+10=\frac{\overline{X}+10\alpha }{\alpha }$