# Consider three observations a,b and c such that b = a + c.

Question:

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

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

2. $b^{2}=a^{2}+c^{2}+3 d^{2}$

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

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

Correct Option: , 4

Solution:

For a, b, c

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

$b=a+c$

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

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

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

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

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

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