*I previously asked this on [here][1] on stats.stackexchange.com, but after not receiving an answer, was advised to post here on MO.* **Background** notation: RV= random variable, $\mu=$ mean $m=$ median Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV. If $f(x)$ strictly convex: $$\mu (f(x)) > f(\mu (x))\mathrm{\hspace{20mm}(1)}$$ Conversely if $-f(x)$ is strictly convex: $$\mu (f(x)) < f(\mu (x))$$ An analogous property of the median has been presented ([Merkle et al 2005][2], [pdf][3]). **Motivation** I have a nonlinear [function][4] [(pdf)][5] of positive random variables, too complex to post here, not directly pertinent to this question; I am looking for a more general answer. It is worth noting that it is, however, neither strictly concave nor convex. In practice, I find that the function of the medians provides a much better estimate of the median of the function than does the estimate of the mean of the function from the function of the means. I am interested in learning the conditions for which this is true. **Question** Under what conditions will the function of a median be closer to the median of a function than the mean of a function is to a function of the mean? Specifically for what types of $f(x)$ and $x$ is $$|\mu (f(x)) - f(\mu (x))| > |m (f(x)) - f(m (x))|$$ **References** [Merkle et al 2005 Jensen's inequality for medians. Statistics & Probability Letters, Volume 71, Issue 3, 1 March 2005, Pages 277-281][6] [1]: http://stats.stackexchange.com/q/4655/1381 [2]: http://dx.doi.org/10.1016/j.spl.2004.11.010 [3]: http://milanmerkle.com/documents/radovi/SPL-71.pdf [4]: http://www.esajournals.org/doi/full/10.1890/0012-9615%282001%29071%5B0557%3AAMFSVD%5D2.0.CO%3B2 [5]: http://www.oeb.harvard.edu/faculty/moorcroft/publications/publications/Moorcroft_etal_01.pdf [6]: http://dx.doi.org/10.1016/j.spl.2004.11.010