Question

If the standard deviation of a variable X is σ, then the standard deviation of variable \[\frac{a X + b}{c}\] is

 

Options

• a σ

•  \[\frac{a}{c}\sigma\]

   

• \[\left| \frac{a}{c} \right| \sigma\]

•  \[\frac{a\sigma + b}{c}\]

Answer 1

\[\left| \frac{a}{c} \right| \sigma\] 

\[Y = \frac{aX + b}{c}\]

\[Y = \frac{\sum y_i}{n} = \frac{\frac{a\sum X + nb}{c}}{n}\]

\[ = \frac{a\sum X}{nc} + \frac{nb}{nc}\]

\[ = \frac{a \bar{X}}{c} + \frac{b}{c}\]

\[Var\left( X \right) = \frac{\sum \left( x_i - \bar{X} \right)^2}{n}\]

\[ = \sigma^2 \]

\[Var\left( Y \right) = \frac{\sum \left( y_i - \bar{Y} \right)^2}{n}\]

\[ = \frac{\sum \left( \frac{aX}{c} + \frac{b}{c} - \frac{a}{c} \bar{X} - \frac{b}{c} \right)^2}{n}\]

\[ = \frac{\sum \left( \frac{aX}{c} - \frac{a}{c} \bar{X} \right)^2}{n}\]

\[ = \left( \frac{a}{c} \right)^2 \frac{\sum \left( x_i - \bar{X} \right)^2}{n}\]

\[ = \left( \frac{a}{c} \right)^2 \sigma^2 \]

\[SD \left( \sigma \right) = \sqrt{\left( \frac{a}{c} \right)^2 \sigma^2}\]

\[ = \left| \frac{a}{c} \right|\sigma\]