Question

Average height and variance of 5 students in a class is $150$ and $18$ respectively. A new student whose height is $156$ cm is added to the group. Find new variance.

• A. $20$
• B. $22$
• C. $16$
• D. $14$

Answer 1

The correct option is A

$20$

Explanation for the correct option:

Step 1: Find the sum of the height of five students:

Find the sum using the given mean.

Given: Mean$=150$cm

$\begin{array}{rcl}\frac{s}{5}& =& 150\\ s& =& 750\end{array}$

Step 2: Find the new mean:

Find the new mean using the obtained sum.

$\begin{array}{rcl}\overline{x}& =& \frac{750+156}{6}\\ & =& 151\end{array}$

Step 3: Find the sum of squares of the height of five students:

Use the given variance to find the sum of squares of height.

Variance $=18{\text{cm}}^2$

Let $l$ be the sum of squares.

$\begin{array}{rcl}\frac{l}{5}-{\left(150\right)}^2& =& 18\\ l& =& 5\left(22500+18\right)\\ l& =& 112590\end{array}$

Step 4: Find the new variance:

Find the new variance by adding the square of the new height with the sum of the square of all heights.

$\begin{array}{rcl}{\sigma }^2& =& \frac{l+{\left(156\right)}^2}{6}-{\left(151\right)}^2\\ & =& \frac{112590+24336}{6}-22801\\ & =& 22821-22801\\ & =& 20\end{array}$

Hence, option (B) is correct.