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 $A=30,20,40$.
$\bar{x}_{A}=\frac{30+20+40}{3}=30$
$\sigma_{A}=\sqrt{\frac{0^{2}+(-10)^{2}+(10)^{2}}{3}}=\sqrt{\frac{200}{3}}\left[\sigma=\sqrt{\frac{\left(x_{i}-\bar{x}\right)^{2}}{n}}\right.$
$(C V)_{A}=\sqrt{\frac{200}{3}} \times \frac{1}{30}=\frac{10 \sqrt{2}}{30 \sqrt{3}}=\frac{\sqrt{2}}{3 \sqrt{3}}\left[C V=\frac{\sigma}{\bar{x}}\right]$
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}} $
$(C V)_{B} =\sqrt{\frac{3050}{3}} \times \frac{1}{25} $
$=\frac{5}{25} \sqrt{\frac{122}{3}}=\frac{1}{5} \sqrt{\frac{122}{3}} $
$(C V)_{A}:(C V)_{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} $