Interview problem: distance between median and mean

Problem:
Provide a bound on the distance between median and mean.





Solution:
Let $X$ be a random variable and $M(X)$ be the median. Besides $\mathbb{E}(X)=\text{argmin}_c\mathbb{E}\left(X-c\right)^2$, we have \begin{equation}M(X)=\text{argmin}_c\mathbb{E}\left|X-c\right|\,.\end{equation}
As a result,  we can prove using Jensen's inequality: \begin{eqnarray} \left|\mathbb{E}(X)-M(X)\right| &\leq& \mathbb{E}\left|X-M(X)\right| \\ &\leq& \mathbb{E}\left|X-\mathbb{E}(X)\right| \\ &\leq& \sqrt{\mathbb{E}\left(X-\mathbb{E}(X)\right)^2}=\sqrt{\text{Var}(X)}\,.\end{eqnarray} 
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