[Math] Can you really tell the relationship between mean and median in a skewed graph

Question

In two old statistics textbooks, I found the following pictures:-

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.]

Answer 1

This claim is false. Here's an interesting article on the subject: Mean, Median, and Skew: Correcting a Textbook Rule

Also note that you shouldn't confuse the terms skew with nonparametric skew, for which your claim is true by definition.