Mathematics - Statistics Question with Solution | TestHub

Let remaining two observations are $a$ and $b$

$\frac{{\displaystyle \Sigma x_i}}{n}=\frac{5+7+12+10+15+14+a+b}{8}=10$

$\Rightarrow$ $63+a+b\mathit{=}80$

$\Rightarrow$ $a+b=17$

${\sigma }^2=\frac{{\displaystyle \Sigma x_i^2}}{n}-{\left(\frac{{\displaystyle \Sigma x_i}}{n}\right)}^2$

$\Rightarrow 13.5=\frac{25+49+144+100+225+196+a^2+b^2}{8}-100$

$\Rightarrow 908=a^2+b^2+739$

$\Rightarrow a^2+b^2=169$

$\Rightarrow {\left(a+b\right)}^2-2ab=169$

$\Rightarrow 289-169=2ab$

$\Rightarrow$ $ab=60$

$\therefore$ ${\left|a-b\right|}^2={\left(a+b\right)}^2-4ab=289-240=49$

$\therefore$ $\left|a-b\right|=7$.