Search any question & find its solution
Question:
Answered & Verified by Expert
The marks obtained by students $A$ and $B$ in 3 examinations are given below

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

The ratio of the coefficient of variation of marks of $A$ and the coefficient of variation of marks of $B$ is
Solution:
2183 Upvotes
Verified Answer
The correct answer is:
$5: 3 \sqrt{61}$
$\begin{aligned} & \text { Marks of } A=30,20,40 \\ & \bar{x}_A=\frac{30+20+40}{3}=30 \\ & \sigma_A=\sqrt{\frac{0^2+(-10)^2+(10)^2}{3}}=\sqrt{\frac{200}{3}} \\ & {\left[\sigma=\sqrt{\frac{\left(x_i-\bar{x}\right)^2}{n}}\right.}\end{aligned}$
$$
(C V)_A=\sqrt{\frac{200}{3}} \times \frac{1}{30}=\frac{10 \sqrt{2}}{30 \sqrt{3}}=\frac{\sqrt{2}}{3 \sqrt{3}} \quad\left[C V=\frac{\sigma}{\bar{x}}\right]
$$
Now, marks of $B=70,0,5$
$$
\begin{aligned}
\bar{x}_B & =\frac{70+0+5}{3}=25 \\
\sigma_B & =\sqrt{\frac{(70-25)^2+(0-25)^2+(5-25)^2}{3}} \\
\sigma_B & =\sqrt{\frac{3050}{3}} \\
(C V)_B & =\sqrt{\frac{3050}{3}} \times \frac{1}{25} \\
& =\frac{5}{25} \sqrt{\frac{122}{3}}=\frac{1}{5} \sqrt{\frac{122}{3}} \\
(C V)_A:(C V)_B & =\frac{\sqrt{2}}{3 \sqrt{3}}: \frac{1}{5} \sqrt{\frac{122}{3}} \\
& \frac{\sqrt{2}}{3 \sqrt{3}}: \frac{1}{5} \frac{\sqrt{2} \sqrt{61}}{\sqrt{3}}
\end{aligned}
$$
$$
(C V)_A=\sqrt{\frac{200}{3}} \times \frac{1}{30}=\frac{10 \sqrt{2}}{30 \sqrt{3}}=\frac{\sqrt{2}}{3 \sqrt{3}} \quad\left[C V=\frac{\sigma}{\bar{x}}\right]
$$
Now, marks of $B=70,0,5$
$$
\begin{aligned}
\bar{x}_B & =\frac{70+0+5}{3}=25 \\
\sigma_B & =\sqrt{\frac{(70-25)^2+(0-25)^2+(5-25)^2}{3}} \\
\sigma_B & =\sqrt{\frac{3050}{3}} \\
(C V)_B & =\sqrt{\frac{3050}{3}} \times \frac{1}{25} \\
& =\frac{5}{25} \sqrt{\frac{122}{3}}=\frac{1}{5} \sqrt{\frac{122}{3}} \\
(C V)_A:(C V)_B & =\frac{\sqrt{2}}{3 \sqrt{3}}: \frac{1}{5} \sqrt{\frac{122}{3}} \\
& \frac{\sqrt{2}}{3 \sqrt{3}}: \frac{1}{5} \frac{\sqrt{2} \sqrt{61}}{\sqrt{3}}
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.