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Suppose $X$ is a random variable with finite variance. Let $m$ denote the median of $X$ and $\mu$ the mean of $X$, i.e. $\mu=\mathbb{E}(X)$. Show $$(m-\mu)^2\leq\text{var}(X)$$

Intuitively this is definitely correct, i.e. the median and mean can not be too far. How to show it? I was thinking of using Jensen's inequality on $\text{var}(X)=\mathbb{E}(X-\mu)^2$, but it does not work.

Richard Hardy
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Tan
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    Here is a [proof](https://en.wikipedia.org/wiki/Median#Inequality_relating_means_and_medians) from Wikipedia. – mhdadk Apr 06 '21 at 16:07

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