Question

In a series where $n$ observations is $3a$, other $n$ observations is $-2a$ and the rest $n$ observations is $-a$. If the variance of this set of observations is $42$, then absolute value of $a$ is

Answer 1

Mean of the total observations is,
$\overline{x}=\frac{n\left(3a\right)+n(-2a)+n(-a)}{n+n+n}=0$
Variance
$=\frac{1}{3n}\left(\sum _{i=1}^{3n}\right(x_i-\overline{x}{)}^2)\Rightarrow 42=\frac{1}{3n}\left(\sum _{i=1}^{3n}\right(x_i{)}^2)\Rightarrow \frac{1}{3n}\left(n\right(3a{)}^2+n(-2a{)}^2+n(-a{)}^2)=42\Rightarrow \frac{14}{3}{a}^2=42\Rightarrow a=\pm 3\Rightarrow |a|=3$