Question bank/JEE Main/Mathematics

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

A20B22C16D18

Show answer and solution▾

Let 5 students are x_1, x_2, x_3, x_4, x_5
Given $x = \frac{\Sigma x_{i}}{5} = 150$ $\Rightarrow$
$\sum_{i = 1}^{5}\mspace{2mu} x_{i} = 750$....(1)
$\frac{\Sigma x_{i}^{2}}{5} - (150)^{2} = 18$
$\Rightarrow$ $\Sigma x_{i}^{2} = (22500 + 18) \times 5$ $\Rightarrow$
$\sum_{i = 1}^{5}\mspace{2mu} x_{i}^{2} = 112590$.....(2)
Height of new student = 156 (Let x_6)
now x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 750 + 156
$\overline{x_{new}} = \frac{x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6}}{6} = \frac{906}{6} = 151$.......(3)
Variance (new) [IMAGE] $= \frac{x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} + x_{6}^{2}}{6} - (151)^{2}$
From equation (2) and (3)
 var (new) = $\frac{112590 + (156)^{2}}{6}$ - (151)^2 = 22821 - 22801
= 20

Source: Grass-G / M-22 Mathematics Paper 19 Dec Eng.docx