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**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.

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