Consider three observations a, b, and c such | vedclass

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

Marks	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$
Group $A$	$9$	$17$	$32$	$33$	$40$	$10$	$9$
Group $B$	$10$	$20$	$30$	$25$	$43$	$15$	$7$

Class	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$	$80-90$	$90-100$
$f_i$	$3$	$7$	$12$	$15$	$8$	$3$	$2$

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 holds true?

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