There is a commonly-encountered-but-wrong rule of thumb that says something like
If a probability distribution is positively skewed, its mean is greater than its median.
(You sometimes also see it phrased in terms of mean and mode). I'm looking for a nice counterexample to this rule. What do I mean by "nice"? Hard to say exactly, of course, but something like:
- It must be continuously differentiable.
- It must have only one maximum and at most two points of inflection.
- Its graph must be "obviously" positively skewed and the mean must be "obviously" less than the median.
I'm aware of the Weibull distribution, which can satisfy (1) and (2) - but for me the graph appears too close to being unskewed and the mean appears too close to the median for it to be an intuitively obvious counterexample. So I'm looking for something where the properties are more easily visible.
Alternatively: if such a distribution did exist it seems to me that it would be quite well-known. So it seems plausible that the rule of thumb might be almost true, in the sense that there might exist a true statement of the form
If a probability distribution is positively skewed, its mean can be at most __ less than its median.
I'm also very interested to see answers of this form.