Consider three observations

For $\mathrm{a}, \mathrm{b}, \mathrm{c}$

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

$\mathrm{b}=\mathrm{a}+\mathrm{c}$

$\Rightarrow \quad \bar{x}=\frac{2 b}{3}$

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

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

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

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

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