Consider three observations a, b and c such that b = a + c . If t… - Vidyadip

MCQ

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 ?

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

✓

Answer

Correct option: D.

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

d
For $a, b, c$

mean $=\frac{a+b+c}{3}(=\bar{x})$

$b = a + c$

$\Rightarrow \quad \bar{x}=\frac{2 b}{3}$ $.....(1)$

S.D. $(a+2, b+2, c+2)=$ S.D. $(a, b, c)=d$

$\Rightarrow \quad d ^{2}=\frac{ a ^{2}+ b ^{2}+ c ^{2}}{3}-(\overline{ x })^{2}$

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

$\Rightarrow 9 d^{2}=3\left(a^{2}+b^{2}+c^{2}\right)-4 b^{2}$

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

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