Question

$5$ students of a class have an average height $150\, cm$ and variance $18 \; cm^{2}$. A new student, whose height is $156\, cm$, joined them. The variance (in $cm^2$) 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^{5}_{i=1} 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^{6}_{i=1} x_{i}}{6} = \frac{750+156}{6} = 151 $
Also, New variance $ = \frac{\sum^{6}_{i=1}x_{1}^{2}}{6} -\left(\bar{x}_{new }\right)^{2} $
$ = \frac{112590+\left(156\right)^{2}}{6} - \left(151\right)^{2}$
$ = 22821- 22801 = 20$