Can you really tell the relationship between mean and median in a skewed graph? In two old statistics textbooks, I found the following pictures:- !graphs of distributions Without any explanation, they both inferred the following:- When it is right skewed (as in fig. 3.2), mean > median. (And is the otherwise for the left skewed.) 1. I wonder if the claim is always true? 2. If it is, is there any simple proof? [By simple, I mean something like by inspection or simple logical reasoning but not deep into the statistical theory please.]

This proof isn't quite right. Step 3 is a new assumption that doesn't follow from anything before. The mean-median=mode inequality is true for "nice" distributions, but you have to define nice more carefully than this proof does. The author gives the Gaussian as an example of a nice distribution, but the Gaussian has no skew. You might think the Weibull is a nice distribution, but the inequality doesn't hold for the Weibull.

Thee following article has a good discussion of violations, with citations that clarify the conditions under which the inequality is true.

* Paul T. von Hippel. (2005). Mean, Median, and Skew: Correcting a Textbook Rule. _Journal of Statistics Education_ , Volume 13, Number 2, jse.amstat.org/v13n2/vonhippel.html