Question

The mean and the variance of five observations are $4$ and $5.20$, respectively. If three of the observations are $3,4$ and $4$; then the absolute value of the difference of the other two observations is

Answer 1

Answer: 7

Let two observations be $x_1$ and $x_2$, then
$\frac{x_1+x_2+3+4+4}{5}=4$
$\Rightarrow x_1+x_2=\mathrm{9...}\left(1\right)$
Variance $=\frac{\sum x_i^2}{N}-{\left(\bar{x}\right)}^2$
$\left(5.20\right)=\frac{9+16+16+x_1^2+x_2^2}{5}-16$
$26=41+x_1^2+x_2^2-80$
$x_1^2+x_2^2=\mathrm{65...}\left(2\right)$
From (1) and (2);
$x_1=8,x_2=1$
Hence, the required value of the difference of other two observations $=\left|x_1-x_2\right|=7$