5 students of a class have an average height
Question:

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

1. 22

2. 20

3. 16

4. 18

Correct Option: , 2

Solution:

Given $\vec{x}=\frac{\Sigma x_{i}}{5}=150$

$\Rightarrow \sum_{i=1}^{5} x_{i}=750$   $\ldots \ldots(\mathrm{i})$

$\frac{\sum \mathrm{x}_{\mathrm{i}}^{2}}{5}-(\overrightarrow{\mathrm{x}})^{2}=18$

$\frac{\sum x_{i}^{2}}{5}-(150)^{2}=18$

$\Sigma \mathrm{x}_{\mathrm{i}}^{2}=112590$      ….(ii)

Given height of new student

$x_{6}=156$

Now, $\quad \vec{x}_{\text {new }}=\frac{\sum_{i=1}^{6} x_{i}}{6}=\frac{750+156}{6}=151$

Also, New variance $=\frac{\sum_{i=1}^{6} x_{i}^{2}}{6}-\left(\bar{x}_{\text {new }}\right)^{2}$

$=\frac{112590+(156)^{2}}{6}-(151)^{2}$

$=22821-22801=20$