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 X is a random variable with mean \mu, variance \sigma^{2}, and median m, then m\in[\,\mu-\sigma,\;\mu+\sigma]. Write|\mu-m|=\bigl|\mathbb{E}(X-m)\bigr|thus, from Jensen’s inequality,\bigl|\mathbb{E}(X-m)\bigr|\leq\mathbb{E}\bigl|X-m\bigr|and because the median minimizes the function a\mapsto \mathbb{E}|X − a|,\mathbb{E}\bigl|X-m\bigr|\leq\mathbb{E}\bigl|X-\mu\bigr|and because|a|=\sqrt{a^2}, we can write\mathbb{E}\bigl|X-\mu\bigr|=\mathbb{E}\!\sqrt{(X-\mu)^{2}}and if we use the concave version of Jensen’s inequality,|\mu-m|\leq \sqrt{\mathbb{E}(X-\mu)^{2}}i.e.,|\mu-m|\leq\sigma

Nice proof, isn’t it.

Of course, the result is quite old, almost 100 years old… it seems that it first appeared in

Harold Hotelling & Leonard M. Solomons (1932) “The Limits of a Measure of Skewness” Annals of Mathematical Statistics. 3(2): 141-142

There were also a couple of references in the early 80’s,

Stephen A. Book & Lawrence Sher (1979) “How close are the mean and the median?” The Two-Year College Mathematics Journal, Vol. 10, No. 3, pp. 202-204

Warren Page & V. N. Murty (1982) “Nearness Relations Among Measures of Central Tendency and Dispersion: Part 1” The Two-Year College Mathematics Journal, Vol. 13, No 5, pp 315-327 

but then, in the 90’s, Colm O’Cinneide mentioned that old papers from Harold Hotelling and Leonard Solomons

Arthur Charpentier

Arthur Charpentier, professor in Montréal, in Actuarial Science. Former professor-assistant at ENSAE Paristech, associate professor at Ecole Polytechnique and assistant professor… Read more

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