Question

The mean and standard deviation (s.d.) of $10$ observations are $20$ and $2$ respectively. Each of these $10$ observations is multiplied by $p$ and then reduced by $q$, where $p\ne 0$ and $q\ne 0$. If the new mean and standard deviation become half of their original values, then $q$ is equal to:

• A. $-20$
• B. $-5$
• C. $10$
• D. $-10$

Answer 1

The correct option is A $-20$

If each observation is multiplied by $p$ and then $q$ is subtracted,
then new mean ${\overline{x}}^{\prime }=p\overline{x}-q$
$\therefore {\overline{x}}^{\prime }=\frac{1}{2}\overline{x}$ and $\overline{x}=10$
$\Rightarrow 10=20p-q...\left(1\right)$

If each observation is multiplied by $p$ and then $q$ is subtracted,
then new standard deviation $\left({\sigma }^{\prime }\right)$ is $|p|$ times of the initial standard deviation $\left(\sigma \right).$
${\sigma }^{\prime }=|p|\sigma$
$\Rightarrow \frac{1}{2}\sigma =|p|\sigma \Rightarrow |p|=\frac{1}{2}$
If $p=\frac{1}{2},q=0$
If $p=-\frac{1}{2},q=-20.$