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

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