The Mean-Median-Mode Inequality: Counterexamples

Let x be a random variable whose first three moments exist. If the density of x is unimodal and positively skewed, then counterexamples are provided which show that the inequality mode ≤ median ≤ mean does not necessarily hold.I thank Andrey Vasnev for help with the graphs and Jan Magnus for various helpful discussions. I also thank Martin Bland, Paolo Paruolo, Peter Phillips, Michael Rockinger, and a referee for their comments. ESRC grant R000239538 is gratefully acknowledged.