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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:
Correct Option: , 2
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 $=\frac{\sum_{i=1}^{6} x_{i}^{2}}{6}-\left(\bar{x}_{\text {new }}\right)^{2}$
$=\frac{112590+(156)^{2}}{6}-(151)^{2}$
$=22821-22801=20$
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