Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

If the Standard Deviation of a Variable X is σ, Then the Standard Deviation of Variable a X + B C is - Mathematics

MCQ

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

•  $\frac{a}{c}\sigma$

• $\left| \frac{a}{c} \right| \sigma$

•  $\frac{a\sigma + b}{c}$

Solution

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

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

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

$= \frac{a\sum X}{nc} + \frac{nb}{nc}$

$= \frac{a \bar{X}}{c} + \frac{b}{c}$

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

$= \sigma^2$

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

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

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

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

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

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

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

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 32 Statistics
Q 12 | Page 51
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