Question

The marks obtained by students $A$ and $B$ in $3$ examinations are given below

Marks of A	30	20	40
Marks of B	70	0	5

The ratio of the coefficient of variation of marks of $A$ and the coefficient of variation of marks of $B$ is

• A. 3: 1
• B. $5:8\sqrt{3}$
• C. 1: 3
• D. $5:3\sqrt{61}$

Answer 1

Answer: (D)

Marks of $A=30,20,40$.
${\bar{x}}_A=\frac{30+20+40}{3}=30$
${\sigma }_A=\sqrt{\frac{0^2+(-10{)}^2+(10{)}^2}{3}}=\sqrt{\frac{200}{3}}[\sigma =\sqrt{\frac{{\left(x_i-\bar{x}\right)}^2}{n}}$
$(CV{)}_A=\sqrt{\frac{200}{3}}\times \frac{1}{30}=\frac{10\sqrt{2}}{30\sqrt{3}}=\frac{\sqrt{2}}{3\sqrt{3}}\left[CV=\frac{\sigma }{\bar{x}}\right]$
Now, marks of $B=70,0,5$
${\bar{x}}_B=\frac{70+0+5}{3}=25$
${\sigma }_B=\sqrt{\frac{(70-25{)}^2+(0-25{)}^2+(5-25{)}^2}{3}}$
${\sigma }_B=\sqrt{\frac{3050}{3}}$
$(CV{)}_B=\sqrt{\frac{3050}{3}}\times \frac{1}{25}$
$=\frac{5}{25}\sqrt{\frac{122}{3}}=\frac{1}{5}\sqrt{\frac{122}{3}}$
$(CV{)}_A:(CV{)}_B=\frac{\sqrt{2}}{3\sqrt{3}}:\frac{1}{5}\sqrt{\frac{122}{3}}$
$\frac{\sqrt{2}}{3\sqrt{3}}:\frac{1}{5}\frac{\sqrt{2}\sqrt{61}}{\sqrt{3}}$
$5:3\sqrt{61}$