On 26 February 2015, Mike Pesca said that most bears cannot be smarter than the average bear, “that’s a tautology.” It is not. It is a property of the mean and symmetric distributions. If a distribution is asymmetric (also known as skewed), most of the data points in that distribution can be greater or less than the mean.

Just a few days ago Pesca pointed out that 50% + 1 is not the same as a majority if there are just a few data points. Although my claim does not rest on having a few data points, it can be illustrated with three data points. If two people have IQs of 110 and one has an IQ of 90, the average IQ is 103.33 and most people in this group are smarter than average. It works with large groups as well. For example, almost everyone has more legs than average since as long as one person has less than two legs then the average number of legs will be less than two.

My claim rests on the notion that although the term “average” is ambiguous, most people take “average” to refer to the arithmatic mean. The only “average” for which Mike Pesca’s statement is always true is the median. If he wants to claim that when he says “average,” he means “median,” I cannot argue about his intent, but that seems unlikely.

Also, I agree with the notion that it relatively common that half of the data points are above and half are below the mean of the distribution. It is just that it is not tautological.

Here is a example in which most data are above the mean.

skew <- scale(rnorm(1000) - rexp(1000, .1))
hist(skew, col="gray")

mean(skew)
## [1] -2.277963e-18
sum(skew > mean(skew))
## [1] 624
(pct.abv.avg <- round(100*sum(skew > mean(skew))/length(skew),2))
## [1] 62.4

This means that for these data 62.4% of the scores are greater than average as long as, by “average” you mean “mean.”

To be clear, as we measure it intelligence is normally distributed, and normal distributions are symmetric. So, it is true empirically that half of the bears are smarter than average, but it is not tautological.