Let $X$ be a random variable, and let $P ( X = x )$ denote the probability that $X$ takes the value $x$. Suppose that the points $( x , P ( X = x ) ) , x = 0,1,2,3,4$, lie on a fixed straight line in the $x y$-plane, and $P ( X = x ) = 0$ for all $x \in \mathbb { R } - \{ 0,1,2,3,4 \}$. If the mean of $X$ is $\frac { 5 } { 2 }$, and the variance of $X$ is $\alpha$, then the value of $24 \alpha$ is $\_\_\_\_$ .