Question

Consider a finite sequence of random values $X=\{x_1,x_2,x_3.....x_n\}\text{Let}{\mu }_x\text{be the mean and}{\sigma }_x$ be the standard deviation of X. Let another finite sequence Y of equal length be derived from this $y_i=a.x_i+b,$ where a and b are positive constants. Let ${\mu }_y\text{be the mean and}{\sigma }_y$ be the standard deviation of this sequence. Which one of the following statements is incorrect?

• A. Index position of mode of x in X is the same as the index position of mode of y in Y
• B. Index position of median of x in X is the same as the index position of median of y in Y
• C. ${\mu }_y=a{\mu }_x+b$
• D. ${\sigma }_y=a{\sigma }_x+b$

Answer 1

The correct option is D ${\sigma }_y=a{\sigma }_x+b$

y = ax + b
$\Rightarrow$ var (y) = Var(ax) + Var(b)

$=a^{2}Var\left(x\right)+0$
$[\because \text{Var (const) = 0}]$

or ${\sigma }_y=a^2{\sigma }_x$

Alternatively:
mean, median and mode are linear function over a random variable.
So, multiplying by constants or adding constants won't change their relative position.
Standard deviation is not a linear function over a random variable.
So, option (d) is correct.