Is the median far away from the mean (for variables with finite variance)?

A few years ago, Ryan J. Tibshirani published “Fast computation of the median by successive binning” with a nice lemma, The Mean Is Within One Standard Deviation of Any Median And a rather nice and simple proof is given. More formally, If is a random variable with mean , variance , and median , then . Writethus, from Jensen’s inequality,and because the median minimizes the function ,and because, we can writeand if we use the concave version of Jensen’s inequality,i.e., Nice proof, isn’t it. … Conti …