JEE PYQ: Statistics Question 1
Question 1 - 2021 (16 Mar Shift 1)
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?
(1) $b^2 = 3(a^2 + c^2) + 9d^2$
(2) $b^2 = a^2 + c^2 + 3d^2$
(3) $b^2 = 3(a^2 + c^2 + d^2)$
(4) $b^2 = 3(a^2 + c^2) - 9d^2$
Show Answer
Answer: (4)
Solution
Mean $= \frac{a+b+c}{3} = \frac{2b}{3}$. S.D.$(a+2, b+2, c+2)$ = S.D.$(a, b, c) = d$. So $d^2 = \frac{a^2+b^2+c^2}{3} - \frac{4b^2}{9}$. Then $9d^2 = 3(a^2+b^2+c^2) - 4b^2$, giving $b^2 = 3(a^2+c^2) - 9d^2$.
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