Consider three observations $\mathrm{a}, \mathrm{b}$ and c such that $\mathrm{b}=\mathrm{a}+\mathrm{c}$. If the standard deviation of $a+2, b+2, c+2$ is $d$, then which of the following is true?

(A) $\mathrm{b}^{2}=3\left(\mathrm{a}^{2}+\mathrm{c}^{2}\right)-9 \mathrm{~d}^{2}$

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

(C) $\mathrm{b}^{2}=3\left(\mathrm{a}^{2}+\mathrm{c}^{2}+\mathrm{d}^{2}\right)$

(D) $b^{2}=3\left(a^{2}+c^{2}\right)+9 d^{2}$

JEE Main · 2021 · 16 Mar 2021 · Shift 1 · Statistics

Consider three observations

a, b

and c such that

b = a + c

. If the standard deviation of

a + 2, b + 2, c + 2

d

, then which of the following is true?

Answer: (A)

A shift leaves the standard deviation unchanged, so

d

is also the standard deviation of

a

b

c

. With

b = a + c

the mean is

\overset{x}{ˉ} = \frac{a + b + c}{3} = \frac{2 b}{3}

and therefore

d^{2} = \frac{a ^{2} + b ^{2} + c ^{2}}{3} - \frac{4 b ^{2}}{9}

Multiplying by

9

9 d^{2} = 3 (a^{2} + b^{2} + c^{2}) - 4 b^{2} = 3 (a^{2} + c^{2}) - b^{2}

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

JEE Main · 2021 · 16 Mar 2021 · Shift 1 · Single correct option