I wanted to add- in general if you make a simple calculation or model of some real phenomena, such as taking the mean, median, or mode of a real set of data- and the conclusions you then draw from each are fundamentally incompatible, then you made a mistake somewhere in drawing those conclusions.

The replies to my original comment are basically mostly people making this same category of mistake and arguing which is "correct" - the average or the mode of the same distribution. Unfortunately, I chose a really bad example where people already hold strong almost religious or political opinions, rather than something generic that just illustrates the math.