If the standard deviation of a variable X is σ, then the standard deviation of the variable aX+b/c is

Question

If the standard deviation of a variable X is σ, then the standard deviation of the variable \(\frac{aX+b}{c}\) is

A. aσ

B. \(\frac{a}{c}\sigma\) 

C. \(|\frac{a}{c}|\sigma\) 

D. \(\frac{a\sigma + b}{c}\) 

Answer 1

We have X = \(\frac{aX+b}{c}\) 

Mean (X) = \(\frac{\Sigma y_i}{n}\) 

We can write as: Mean (X) 

=   

Mean (X) = \(\frac{a\Sigma \bar X}{nc} + \frac{nb}{nc}\)  

Var(X) = \(\Sigma \frac{(x_i-\bar X)^2}{n}\) 

Now, Substitute the value of y_i and Y, then we get

Var(X)  = \(\Big(\frac{a}{c}\Big)^2\sigma ^2\) 

SD (\(\sigma\)) = \(\sqrt{\Big(\frac{a}{c}\Big)^2\sigma ^2}\) 

(xσ) = \(|\frac{a}{c}|\sigma\) 

Hence, Proved