The mean and the 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 new s.d. become half of their original values, then q is equal to:
A. $ - 10$
B. $ - 5$
C. $ - 20$
D. $10$

Hint: In order to provide a solution for this problem, we need to assume the initial mean as ($\bar x$) and standard deviation as (${\sigma _1}$​). When we are multiplying each observation by p and then reducing it by q, this causes a change in the original mean, and thus, the new mean $ = p.\bar x - q$. Whereas, it should be noted that subtraction of q from each observation will not affect the new standard deviation, and thus, the new standard deviation $ = \left| p \right|.\sigma {}_1$.

Complete step-by-step answer:
According to given information, we have
Initial mean ($\bar x$) and standard deviation (${\sigma _1}$) of $10$ observations are $20$ and $2$ respectively.

Let’s assume the new mean as ${\bar x_1}$and new standard deviation as ${\sigma _2}$.
And we know, ${\bar x_1} = 10$and ${\sigma _2} = 1$.
Now, we are multiplying each observation by p and then reducing it by q.
Henceforth, this causes a change in the original mean, and thus,
New mean $ = {\bar x_1} = p.\bar x - q$
$ \Rightarrow {\bar x_1} = \dfrac{1}{2}\bar x = \dfrac{1}{2} \times 20 = 10$
$ \Rightarrow 20p - q = 10$ $...(1)$
Further, we will calculate the new standard deviation,
New standard deviation is given by,
${\sigma _2}$$ = \left| p \right|.\sigma {}_1$ $...(2)$
It should be noted that subtraction of q from each observation will not affect the new standard deviation.
${\sigma _2}$$ = \dfrac{1}{2} \times 2 = 1$
On substituting the obtained value in equation $...(2)$, we get
$1$$ = \left| p \right|.2$
$ \Rightarrow \left| p \right| = \dfrac{1}{2}$
Therefore, we can conclude that the value of $p$is either $ + \dfrac{1}{2}$or $ - \dfrac{1}{2}$.
Firstly, considering $p = + \dfrac{1}{2}$ and substituting in the equation $...(1)$, we get
$ \Rightarrow 20 \times \dfrac{1}{2} - 10 = q$
$ \Rightarrow q = 0$
As we know according to the given data, $q \ne 0$.
Therefore, $p \ne + \dfrac{1}{2}$.
So, we are left with $p = - \dfrac{1}{2}$. On substituting in the equation $...(1)$, we get
$ \Rightarrow q = 20 \times ( - \dfrac{1}{2}) - 10$
$ \Rightarrow q = - 20$
Hence, Option C is the correct option.

Note: To solve this problem it is very much important to have a basic understanding of how an implication of arithmetic operations causes change in the original mean and standard deviation. Multiplication of each observation by p and then reducing it by q causes a change in the original mean. It should be noted that subtraction of q from each observation will not affect the new standard deviation.