I was listening to physicist and science writer Leonard Mlodinow on an All in the Mind podcast this morning. He was talking about his new book, Subliminal, which is about recent neuroscience results on neural processes that operate in the absence of conscious awareness. During the podcast, which was quite good, Mlodinow quoted a result that said 95% of all professors think they are above average, then he went on to say that we all know that only %50 can be. It’s unfortunate that Mlodninow, who wrote an excellent book on probability theory, would make such a statement. I am sure that he knows that 50% of all professors are better than the median but any number greater than one could be greater than the average (i.e. mean). He used average in the colloquial sense but knowing the difference between median and mean is crucial for the average or should I say median person to make informed decisions.
It could be that on a professor performance scale, 5% of all professors are phenomenally bad, while the rest are better but clumped together. In this case, it would be absolutely true that 95% of all professors are better than the mean. Also, if the professors actually obeyed such a distribution then comparing to the mean would be more informative than the median because what every student should do is to simply avoid the really bad professors. However, for something like income, which is broad with a fat tail, comparing yourself to the median is probably more informative because it will tell you where you place in society. The mean salary of a lecture hall filled with mathematicians would increase perhaps a hundred fold if James Simons (of Chern-Simons theory as well as the CEO of one of the most successful hedge funds, Renaissance Technologies) were to suddenly walk into the room. However, the median would hardly budge. (Almost) All the children in Lake Woebegon could really be above average.
Jun 18, 2013: I adjusted the last sentence to avoid confusion.