Question

In a discrete data ${\frac{1}{4}}^{\text{th }}$ of the observations are equal to $a$, another ${\frac{1}{4}}^{th}$ of the observations are equal to $-a$. Out of the reamining, half of them are equal to $b$ and the rest are equal to $-b$. If the variance of all observations is $(ab)$, then

• A. $a^2=4b^2$
• B. $a=-2b$
• C. $a=b$
• D. $a=3b$

Answer 1

Answer: (C)

Given, $\frac{1}{4}$ th observation $=a$
another $\frac{1}{4}$ th observation $=-a$
$\frac{1}{4}$ th observation $=b$
$\Rightarrow \frac{1}{4}$ th observation $=-b$
$\therefore \bar{x}=\frac{a-a+b-b}{n}=0$
Variance $=ab$
$ab=\frac{\sum x_i^2}{n}-(\bar{x}{)}^2=\frac{\sum x_i^2}{n}$
$\Rightarrow ab=\frac{\frac{n}{4}\left(a^2+a^2\right)+\frac{n}{4}\left(b^2+b^2\right)}{n}$
$\Rightarrow ab=\frac{a^2+b^2}{2}$
$\Rightarrow a^2+b^2-2ab=0$
$\Rightarrow (a-b{)}^2=0$
$\Rightarrow a=b$