Prove that $ES \leq \sigma$, with S being a random sample

Question

Assume we have a random sample S of (X1,...,Xn) from a population with finite variance $\sigma^2$. Prove that $ES \leq \sigma$

How would you prove this? Is there some way to do this using Jensen's inequality? If not, what other faster way is there?

Answer 1

Let $g(y) = y^{1/2}$. Since this is a concave function, we can apply Jensen's inequality if we flip the direction of the relationship: $E(g(y)) \leq g(E(y))$. Setting $y = s^2$ the result $E(s) \leq \sigma$ follows naturally.