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?

b^{2} = a^{2} + c^{2} + 3 d^{2}

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

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

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

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Solution

The correct option is B $b^{2} = 3 (a^{2} + c^{2}) - 9 d^{2}$
For $a, b, c$
Mean $(¯ x) = \frac{a + b + c}{3}$
$¯ x = \frac{2 b}{3}$
S.D. of $a, b, c = d$
$d^{2} = \frac{a^{2} + b^{2} + c^{2}}{3} - \frac{4 b^{2}}{9} b^{2} = 3 a^{2} + 3 c^{2} - 9 d^{2}$

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