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

For $a, b, c$

mean $= \dfrac{a + b + c}{3}\left(= \overset{¯}{x}\right)$

$b = a + c$

$⇒ \overset{¯}{x} = \dfrac{2b}{3}$ $. . . ( 1 )$

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

$⇒ {d}^{2} = \dfrac{{a}^{2} + {b}^{2} + {c}^{2}}{3} - {\left(\overset{¯}{x}\right)}^{2}$

$⇒ {d}^{2} = \dfrac{{a}^{2} + {b}^{2} + {c}^{2}}{3} - \dfrac{4{b}^{2}}{9}$

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

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