Let X₁, X₂, ......., X₁₈ be eighteen observations such
that $\sum\limits_{i = 1}^{18} {({X_i} - } \alpha ) = 36$ and $\sum\limits_{i = 1}^{18} {({X_i} - } \beta {)^2} = 90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is 1, then the value of | $\alpha$ $-$ $\beta$ | is ____________.

Given, $\sum\limits_{i = 1}^{18} {({x_1} - \alpha ) = 36}$<br><br>$\Rightarrow \sum {{x_i} - 18\alpha = 36}$<br><br>$\Rightarrow \sum {{x_i} = 18(\alpha + 2)}$ .... (1)<br><br>Also, $\sum\limits_{i = 1}^{18} {{{({x_1} - \beta )}^2} = 90}$<br><br>$\Rightarrow \sum {x_i^2 + 18{\beta ^2} - 2\beta \sum {{x_i} = 90} }$<br><br>$\Rightarrow \sum {x_i^2 + 18{\beta ^2} + 2\beta \times 18(\alpha + 2) = 90}$ (using equation (1))<br><br>$\Rightarrow \sum {x_i^2 = 90} - 18{\beta ^2} + 36\beta (\alpha + 2)$<br><br>Given, $${\sigma ^2} = 1 \Rightarrow {1 \over {18}}{\sum {x_i^2 - \left( {{{\sum {{x_i}} } \over {18}}} \right)} ^2} = 1$$<br><br>$$ = {1 \over {18}}(90 - 18{\beta ^2} + 36\alpha \beta + 72\beta ) - {\left( {{{18(\alpha + 2)} \over {18}}} \right)^2} = 1$$<br><br>$$ \Rightarrow 90 - 18{\beta ^2} + 36\alpha \beta + 72\beta - 18{(\alpha + 2)^2} = 18$$<br><br>$\Rightarrow 5 - {\beta ^2} + 2\alpha \beta + 4\beta - {(\alpha + 2)^2} = 1$<br><br>$$ \Rightarrow 5 - {\beta ^2} + 2\alpha \beta + 4\beta - {\alpha ^2} - 4 - 4\alpha = 1$$<br><br>$\Rightarrow {\alpha ^2} - {\beta ^2} + 2\alpha \beta + 4\beta - 4\alpha = 0$<br><br>$\Rightarrow (\alpha - \beta )(\alpha - \beta + 4) = 0$<br><br>$\Rightarrow \alpha - \beta = - 4$<br><br>$\therefore$ $|\alpha - \beta |\, = 4$ $(\alpha \ne \beta )$