Question

$5$ students of a class have an average height $150cm$ and variance $18c{m}^2$. A new student, whose height is $156cm$, joined them. The variance (in $c{m}^2$) of the height of these six students is :

• A. 22
• B. 20
• C. 16
• D. 18

Answer 1

Answer: (B)

Given $\vec{x}=\frac{\sum x_i}{5}=150$
$\Rightarrow {\sum }_{i=1}^5{x}_i=750$ ....(i)
$\frac{\sum x_1^2}{5}-{\left(\vec{x}\right)}^2=18$
$\frac{\sum x_1^2}{5}{\left(150\right)}^2=18$
$\sum x_1^2=112590$ ....(ii)
Given height of new student
$x_6=156$
Now, ${\vec{x}}_{new}=\frac{{\sum }_{i=1}^6{x}_i}{6}=\frac{750+156}{6}=151$
Also, New variance $=\frac{{\sum }_{i=1}^6{x}_1^2}{6}-{\left({\bar{x}}_{new}\right)}^2$
$=\frac{112590+{\left(156\right)}^2}{6}-{\left(151\right)}^2$
$=22821-22801=20$