Question

If the standard deviation of a variable X is $\sigma$ , then the standard deviation of variable $\frac{aX+b}{c}$ is

• A. $a\sigma$
• B. $\frac{a}{c}\sigma$
• C. $|\frac{a}{c}|\sigma$
• D. $\frac{a\sigma +b}{c}$

Answer 1

The correct option is C

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

Y = \frac{aX+b}{c} \)

$\overline{Y}=\frac{\sum y_i}{n}=\frac{\frac{a\sum x+nb}{c}}{n}$

$\frac{a\sum x}{n}+\frac{nb}{nc}$

$\frac{a\overline{X}}{c}+\frac{b}{c}$

Var (X) $=\frac{\sum (x_i-\overline{X}{)}^2}{n}={\sigma }^2$

Var(Y) $=\frac{\sum (y_i-\overline{Y}{)}^2}{n}$

$=\frac{\sum {(\frac{aX}{c}+\frac{b}{c}-\frac{a}{c}\times \frac{b}{c})}^2}{n}$

$=\frac{\sum {(\frac{aX}{c}-\frac{a}{c}\overline{X})}^2}{n}$