Question

The S.D. of a variate x is σ. The S.D. of the variate$\frac{ax + b}{c}$ where a, b, c are constant, is

• A. $\left( \frac{a}{c} \right)\sigma$
• B. $\left| \frac{a}{c} \right|\sigma$
• C. $\left( \frac{a^{2}}{c^{2}} \right)\sigma$
• D. None

Answer 1

Answer: (B)

$\left| \frac{a}{c} \right|\sigma$

Answer 2

Let $y = \frac{ax + b}{c}$ i.e., $y = \frac{a}{c}x + \frac{b}{c}$ i.e. $y = Ax + B$, where

$A = \frac{a}{c}$, $B = \frac{b}{c}$

∴ $\bar{y} = A\bar{x} + B$

∴ $y - \bar{y} = A(x - \bar{x})$ ⇒ $(y - \bar{y})^{2} = A^{2}(x - \bar{x})^{2}$ ⇒

$\sum(y - \bar{y})^{2} = A^{2}\sum(x - \bar{x})^{2}$ ⇒ $n.\sigma_{y}^{2} = A^{2}.n\sigma_{x}^{2}$ ⇒ $\sigma_{y}^{2} = A^{2}\sigma_{x}^{2}$

⇒ $\sigma_{y} = |A|\sigma_{x}$ ⇒ $\sigma_{y} = \left| \frac{a}{c} \right|\sigma_{x}$

Thus, new S.D.$= \left| \frac{a}{c} \right|\sigma$.