$5$ students of a class have an average height $150 cm$ and variance $18 cm2.$ A new student, whose height is $156 cm,$ joined them. The variance of the height of these six students is

Solution:

Let $x_1, x_2, x_3, x_4, x_5$ be the heights $(in cm)$ of five students of the class. It is given that

      $\frac{1}{5}\sum _{i=1}^5 x_i=150$   and    $18=\frac{1}{5}\sum _{i=1}^5 x_i^2-(150{)}^2$

$\Rightarrow \sum _{i=1}^5 x_i=750$  and   $\sum _{i=1}^5 x_i^2=112590$

Let the mean and variance of the heights of $6$ students be $\overline{X}$ and ${\sigma }^2$ respectively. Then,

   $\overline{X}=\frac{1}{6}\left\{\sum _{i=1}^5 x_i+156\right\}=\frac{1}{6}\{750+156\}=151$

   ${\sigma }^2=\frac{1}{6}\left\{\sum _{i=1}^5 x_i^2+{156}^2\right\}-(151{)}^2$

$\Rightarrow {\sigma }^2=\frac{1}{6}\left\{112590+{156}^2\right\}-{151}^2=20$