Consider three observations


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 $\mathrm{d}$, then which of the following is true?

  1. (1) $b^{2}=3\left(a^{2}+c^{2}\right)+9 d^{2}$

  2. (2) $b^{2}=a^{2}+c^{2}+3 d^{2}$

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

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

Correct Option: , 4


For $\mathrm{a}, \mathrm{b}, \mathrm{c}$

mean $=\frac{a+b+c}{3}(=\bar{x})$


$\Rightarrow \quad \bar{x}=\frac{2 b}{3}$

S.D. $(a+2, b+2, c+2)=S . D \cdot(a, b, c)=d$

$\Rightarrow \quad d^{2}=\frac{a^{2}+b^{2}+c^{2}}{3}-(\bar{x})^{2}$

$\Rightarrow \quad d^{2}=\frac{a^{2}+b^{2}+c^{2}}{3}-\frac{4 b^{2}}{9}$

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

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

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