Consider three observations $a, b$ and $c$ such that $b=a+c$. If the standard deviation of $\mathrm{a}+2,…

Consider three observations $a, b$ and $c$ such that $b=a+c$. If the standard deviation of $\mathrm{a}+2, \mathrm{~b}+2, \mathrm{c}+2$ is d, then what holds true.
  1. $\quad b^2=3\left(a^2+c^2+d^2\right)$
  2. $\mathrm{b}^2=\mathrm{a}^2+\mathrm{c}^2+3 \mathrm{~d}^2$
  3. $b^2=3\left(a^2+c^2\right)-9 d^2$
  4. $\mathrm{b}^2=3\left(\mathrm{a}^2+\mathrm{c}^2\right)+9 \mathrm{~d}^2$

Solution

Mean of a, b, c is $\begin{aligned} & \bar{x}=\frac{a+b+c}{3} \\ & \Rightarrow \bar{x}=\frac{2 b}{3} \end{aligned}$ $\therefore[\because b=a+c]$ S.D. of $a+2, b+2, c+2=S$. D. of $a, b, c$ $\begin{aligned} \therefore \quad d & =\sqrt{\frac{a^2+b^2+c^2}{3}-\left(\frac{2 b}{3}\right)^2} \\ & \Rightarrow d^2=\frac{a^2+b^2+c^2}{3}-\frac{4 b^2}{9} \\ & \Rightarrow d^2=\frac{3\left(a^2+b^2+c^2\right)-4 b^2}{9} \\ & \Rightarrow 9 d^2=3\left(a^2+c^2\right)-b^2 \\ & \Rightarrow b^2=3\left(a^2+c^2\right)-9 d^2 \end{aligned}$

Asked in: MHT CET 2024 (02 May Shift 2)

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