Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
Advertisement Remove all ads

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

  •  \[\frac{a}{c}\sigma\]

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

  •  \[\frac{a\sigma + b}{c}\]

Advertisement Remove all ads

Solution

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

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

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

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

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

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

\[ = \sigma^2 \]

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

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

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

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

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

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

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

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 32 Statistics
Q 12 | Page 51
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×