Question

In a class of $50$ students, the mean and variance of the marks scored in mathematics is $8.4$ and $20$ respectively. If there are $30$ boys in the class and their mean and variance of marks scored in mathematics is $8.25$ and $24$ respectively, then the variance of marks scored by the girls will be

• A. $13.69$
• B. $13.91$
• C. $13.96$
• D. $14.02$

Answer 1

The correct option is B $13.91$

Given
mean and variance of 50 students math marks $\overline{X}=8.4$
and ${\sigma }^2=20$
Also mean and variance of 30 boys math marks ${\overline{X}}_B=8.25$
and ${\sigma }_B^2=24$
Suppose mean and variance of 20 girls math marks ${\overline{X}}_G$ and ${\sigma }_G^2$
As we know
Combined mean $\overline{X}=\frac{n_1{\overline{X}}_B+n_2{\overline{X}}_G}{n}$
$\Rightarrow X_G=\frac{50\times 8.4-30\times 8.25}{20}=8.625$

So Combined variance ${\sigma }^2=\frac{n_1({\sigma }_B^2+d_B^2)+n_2({\sigma }_G^2+d_G^2)}{n_1+n_2}$
Where,$d_B=8.4-8.25=0.15$
$d_G=8.4-8.625=-0.225$
$\Rightarrow 20({\sigma }_G^2+(-0.225{)}^2)=50\times 20-30(24+\left(0.15{)}^2\right)$
$\Rightarrow {\sigma }_G^2=\frac{1000-720.675}{20}-0.050625\Rightarrow {\sigma }_G^2=13.96625-0.50625=13.91$