Question

Consider three observations $a,b$ and $c$ such that $b=a+c$. If the standard deviation of $a+2,b+2,c+2$ is $d$, then which of the following is true?

• A. $b^2=a^2+c^2+3d^2$
• B. $b^2=3(a^2+c^2)-9d^2$
• C. $b^2=3(a^2+c^2)+9d^2$
• D. $b^2=3(a^2+c^2+d^2)$

Answer 1

The correct option is B $b^2=3(a^2+c^2)-9d^2$

For $a,b,c$
Mean $\left(\overline{x}\right)=\frac{a+b+c}{3}$
$\overline{x}=\frac{2b}{3}$
S.D. of $a,b,c=d$
$d^2=\frac{a^2+b^2+c^2}{3}-\frac{4b^2}{9}{b}^2=3a^2+3c^2-9d^2$