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Mean and standard deviation of a set of observations are $20$ and $2$ respectively. If $p (\neq 0)$ is multiplied to each observation and then $q (\neq 0)$ is subtracted, then the new mean and standard deviation are half of the original values. The the value of $q,$ is

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Correct option is B

Let $x_{1}, x_{2}, x_{3}, \dots, x_{n}$ be $n$ values of a variable $X .$ It is given that $X = 20$ and $σ X = 2 .$ Let $y_{i} = p x_{i} - q; i = 1, 2, \dots, n$ . Then,

$\bar{Y} = p \bar{X} - q$ and $σ_{Y} = | p | σ_{X}$

$\Rightarrow \frac{\bar{X}}{2} = p \bar{X} - q and \frac{1}{2} σ_{X} = | p | σ_{X} [∵ \bar{Y} = \frac{\bar{X}}{2} and σ_{Y} = \frac{1}{2} σ_{X}]$

$\Rightarrow | p | = \frac{1}{2}$ and $\frac{\bar{X}}{2} = p \bar{X} - q$

$\Rightarrow p = \pm \frac{1}{2}$ and $\frac{\bar{X}}{2} = p \bar{X} - q$

If $p = \frac{1}{2},$ then $\frac{\bar{X}}{2} = p \bar{X} - q$ gives $q = 0$ , which is a contradiction.

$∴ p = - \frac{1}{2}$

Substituting $p = - \frac{1}{2}$ in $\frac{\bar{X}}{2} = p \bar{X} - q$ , we obtain

$\bar{X} = - q \Rightarrow q = - 20$ $[∵ \bar{X} = 20]$

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Mean and standard deviation of a set of observations are 20 and 2 respectively. If p ( ≠ 0) is multiplied to each observation and then q(≠0) is subtracted, then the new mean and standard deviation are half of the original values. The the value of q, is