Question

The S.D of a variable $x$ is $\sigma$. 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 of these

Answer 1

Answer: (B)

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

Let $y=\frac{ax+b}{c}i.e.,y=\frac{a}{c}x+\frac{b}{c}$
$i.e.,y=Ax+B,$ where $A=\frac{a}{c},B=\frac{b}{c}$
$\therefore \overline{y}=A\overline{x}+B$
$\therefore y-\overline{y}=A(x-\overline{x})\Rightarrow {(y-\overline{y})}^2=A^2{(x-\overline{x})}^2$
$\Rightarrow \sum {(y-\overline{y})}^2=A^2\sum {(x-\overline{x})}^2$
$\Rightarrow n.{\sigma }_y^2=A^2,n{\sigma }_x^2$
$\Rightarrow {\sigma }_y^2=A^2{\sigma }_x^2$
$\Rightarrow \sigma y=\left|A\right|{\sigma }_x\Rightarrow {\sigma }_y=\left|\frac{a}{c}\right|{\sigma }_x$
Thus, new $S.D.=\left|\frac{a}{c}\right|\sigma .$