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(C) The mean of $a, b, c$ is $\bar{x} = \frac{a+b+c}{3}$.Since $b = a + c$,we have $\bar{x} = \frac{b+b}{3} = \frac{2b}{3}$.The standard deviation of

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

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(C) The mean of $a, b, c$ is $\bar{x} = \frac{a+b+c}{3}$.Since $b = a + c$,we have $\bar{x} = \frac{b+b}{3} = \frac{2b}{3}$.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,$d^2 = \frac{a^2+b^2+c^2}{3} - (\bar{x})^2$.$d^2 = \frac{a^2+b^2+c^2}{3} - \left(\frac{2b}{3}\right)^2$.$d^2 = \frac{a^2+b^2+c^2}{3} - \frac{4b^2}{9}$.$d^2 = \frac{3(a^2+b^2+c^2) - 4b^2}{9}$.$9d^2 = 3(a^2+c^2+b^2) - 4b^2$.$9d^2 = 3(a^2+c^2) + 3b^2 - 4b^2$.$9d^2 = 3(a^2+c^2) - b^2$.Therefore,$b^2 = 3(a^2+c^2) - 9d^2$.

$X$	$5$	$6$	$7$	$8$	$10$
Frequency	$3$	$7$	$4$	$2$	$4$

Classes	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
Frequencies	$5$	$8$	$15$	$16$	$6$

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 holds true?

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