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

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

1. A

$22$

2. B

$20$

3. C

$16$

4. D

$18$

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### Solution:

Let  be the heights  of five students of the class. It is given that

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

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,

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

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

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