TITLE: Why is $\operatorname{Var}[X]≤(b−a)^2/4$ if $X$ is a random variable with values between $[a,b]$
QUESTION [6 upvotes]: I don't know how to start here. I tried to proof it with $$\operatorname{Var}(X) =  E(X^2) - [E(X)]^2$$ but this wasn't a good idea in the end.

REPLY [13 votes]: Let $Y$=$X-\frac{a+b}{2}$.  Since they differ by a constant, $\operatorname{Var}(Y)=\operatorname{Var}(X)$.  Since $|Y|$ is always less than or equal to $|\frac{b-a}{2}|$, we have $Y^2$ is always less than $\frac{(b-a)^2}{4}$.  Thus $\operatorname{Var}(X)=\operatorname{Var}(Y)=E(Y^2)-(EY)^2\le E(Y^2)\le \frac{(b-a)^2}{4}$.  QED