# Bounding the standard deviation of a bounded random variable

Suppose $X$ is a random variable (discrete or continuous) whose values lie in the segment $[0,1]$. Is it safe to say that the standard deviation of $X$ is between $0$ and $\frac{1}{2}$?

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Yes. $(X-1/2)^2 \le 1/4$, so $$\text{Var}(X) = \text{Var}(X-1/2) = E[(X-1/2)^2] - (E[X]-1/2)^2 \le 1/4$$