Question

If the standard deviation of $n$ observations $x_1,x_2,...,x_n$ is $4$ and another set of $n$ observations $y_1,y_2,...,y_n$ is $3$. The standard deviation of $n$ observations $x_1-y_1,x_2-y_2,...,x_n-y_n$ is

• A. $1$
• B. $\frac{2}{\sqrt{3}}$
• C. $5$
• D. data insufficient

Answer 1

The correct option is D data insufficient

$S.D={\sigma }_{x-y}^2$
$\frac{1}{n}\sum _{i=1}^n{(x_i-y_i-\overline{x}+\overline{y})}^2$
$=\frac{1}{n}\sum _{i=1}^n{(x_i-\overline{x})}^2+\frac{1}{n}\sum _{i=1}^n(y_i-\overline{y})-\frac{2}{n}\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})$
$={\sigma }_x^2+{\sigma }_y^2-2cov(x,y)$
As $cov(x,y)$ is not known, therefore we cannot find ${\sigma }_{x-y}^2$ or ${\sigma }_{x-y}.$ Hence data is insufficient.