Question

The standard deviation of a variate x is $\sigma$. Then the standard deviation of the variate $\frac{ax+b}{c}$ where a, b, c are constants, 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

The correct option is B $\left|\frac{a}{c}\right|\sigma$

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