Question

The mean and variance of $8$ observations are $10$ and $13.5$ respectively. If $6$ of these observations are $5,7,10,12,14,15,$ then the absolute diffrence of the remaining two observations is

• A. $3$
• B. $9$
• C. $7$
• D. $5$

Answer 1

The correct option is C $7$

Let remaining two observations be $x$ and $y.$
Given: Mean $=10$
$\Rightarrow \frac{5+7+10+12+14+15+x+y}{8}=10$
$\Rightarrow x+y=17\cdots \left(1\right)$
Also,
variance $=13.5$
$\Rightarrow \frac{(5-10{)}^2+(7-10{)}^2+(10-10{)}^2+(12-10{)}^2+(14-10{)}^2+(15-10{)}^2+(x-10{)}^2+(y-10{)}^2}{8}=13.5$
$\Rightarrow 79+(x-10{)}^2+(y-10{)}^2=108$
$\Rightarrow x^2+y^2-20\left(17\right)+200=29$
$\Rightarrow x^2+y^2=169\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$ we get
$x=12,y=5$ or $x=5,y=12$
$\therefore |x-y|=7$