The marks obtained by students $A$ and $B$ in 3 examinations are given below The ratio of the coefficient of…

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
  1. 3 : 1
  2. $5: 8 \sqrt{3}$
  3. 1 : 3
  4. $5: 3 \sqrt{61}$

Solution

$\begin{aligned} & \text { Marks of } A=30,20,40 \\ & \bar{x}_A=\frac{30+20+40}{3}=30 \\ & \sigma_A=\sqrt{\frac{0^2+(-10)^2+(10)^2}{3}}=\sqrt{\frac{200}{3}} \\ & {\left[\sigma=\sqrt{\frac{\left(x_i-\bar{x}\right)^2}{n}}\right.}\end{aligned}$ $ (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} $

Asked in: AP EAMCET 2018 (23 Apr Shift 1)

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