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(D) Given observations are $a, b, c$ with $b = a + c$.The mean $\bar{x} = \frac{a + b + c}{3}$.Substituting $a + c = b$,we get $\bar{x} = \frac{b + b}

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$b^{2} = 3(a^{2} + c^{2}) - 9d^{2}$

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(D) Given observations are $a, b, c$ with $b = a + c$.The mean $\bar{x} = \frac{a + b + c}{3}$.Substituting $a + c = b$,we get $\bar{x} = \frac{b + b}{3} = \frac{2b}{3}$.The standard deviation of $a + 2, b + 2, c + 2$ is the same as the standard deviation of $a, b, c$,which is $d$.Thus,$d^{2} = \frac{a^{2} + b^{2} + c^{2}}{3} - (\bar{x})^{2}$.Substituting $\bar{x} = \frac{2b}{3}$,we have $d^{2} = \frac{a^{2} + b^{2} + c^{2}}{3} - \frac{4b^{2}}{9}$.Multiplying by $9$,we get $9d^{2} = 3(a^{2} + b^{2} + c^{2}) - 4b^{2}$.$9d^{2} = 3(a^{2} + c^{2}) + 3b^{2} - 4b^{2}$.$9d^{2} = 3(a^{2} + c^{2}) - b^{2}$.Therefore,$b^{2} = 3(a^{2} + c^{2}) - 9d^{2}$.

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?

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