Question

In a series of $2n$ observations, half of them equal to $a$ and remaining half equal $-a$. If the standard deviation of the observations is $2$, then $\left|a\right|$ equals

• A. $\frac{\sqrt{2}}{n}$
• B. $\sqrt{2}$
• C. $2$
• D. $\frac{1}{n}$

Answer 1

Answer: (C)

$2$

Here, $x_1=x_2=...x_n=a$
$x_{n+1}=x_{n+2}=...x_{2n}=-a$
$\therefore \overline{x}=0$
$2=S.D.=\sqrt{\frac{\sum (x-\overline{x}{)}^2}{2n}}=\sqrt{\frac{\sum x^2}{2n}-\overline{x^2}}$
$\sqrt{\frac{\sum x^2}{2n}}=\sqrt{\frac{2n{a}^2}{2n}}=|a|$
$\therefore |a|=2$