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. $b^2=3\left(a^2+c^2\right)-9d^2$
• C. $b^2=3\left(a^2+c^2\right)+9d^2$
• D. $b^2=3\left(a^2+c^2+d^2\right)$

Answer 1

The correct option is B

$b^2=3\left(a^2+c^2\right)-9d^2$

Explanation for correct answer:

Step-1: Finding mean.

Given, observation $a,b$and $c$ and relation between them is $b=a+c$

mean $\left(\overline{x}\right)=\frac{a+b+c}{3}$

$\Rightarrow$$\left(\overline{x}\right)=\frac{2b}{3}$

Step-2: Finding standard deviation.

By using formula for standard deviation ${\sigma }^2=\sum _{i=1}^n\frac{{x_i}^2}{N}-{\left(\frac{x_i}{N}\right)}^2$

Given, $\sigma =d$

$\Rightarrow$$d^2=\left(\frac{a^2+b^2+c^2}{3}\right)-{\left(\frac{2b}{3}\right)}^2$

$\Rightarrow$$d^2=\left(\frac{a^2+b^2+c^2}{3}\right)-\left(\frac{4b^2}{9}\right)$

$\Rightarrow$$9d^2=3\left(a^2+b^2+c^2\right)-4b^2$

$\Rightarrow$$b^2=3\left(a^2+c^2\right)-9d^2$

Hence, correct answer is option $\left(B\right)$.