Question

The mean and standard deviation 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 becomes half of their original values, then the value of $pq$ is

Answer 1

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 $\overline{x}=20$
$\Rightarrow 10=20p-q...\left(1\right)$

If each observation is multiplied by $p$ and then $q$ is subtracted,
then new variance $({\sigma }^{\prime }{)}^2$ is $p^2$ times of the initial variance $(\sigma {)}^2.$
$({\sigma }^{\prime }{)}^2=p^2(\sigma {)}^2$
$\Rightarrow 1=p^2\times 4\Rightarrow p=\pm \frac{1}{2}$
If $p=\frac{1}{2},q=0$ $($from $\left(1\right))$
If $p=-\frac{1}{2},q=-20.$ $($from $\left(1\right))$
Since $q\ne 0$
$\therefore p=-\frac{1}{2},q=-20$
Clearly, $pq=10$