Distribution Matters

Kevin Drum gives a little statistical lesson about how thresholds have very different effects when they apply to the center or the tail of a statistical distribution.  In the case of violent criminals, a slight shift in the distribution has a dramatic effect on the tails, but for education, a slight shift has only a slight effect on the average.  Climate change could have a dramatic effect on the number of extreme weather events despite having only a small effect on the average temperature for the same reason.  Read his full post for more, but here is an excerpt that gets at the most salient concept.

…when you expose huge numbers of children to lead, as we did with leaded gasoline after World War II, what you’re essentially doing is moving the bell curve to the right. For most people, that makes very little difference. But for a few who were already on the edge, it pushes them over into a life of violent crime. And when you move a bell curve, the area under the rightward tail increases a lot. The diagram below illustrates this:

What this means is that a small effect from lead can have a very big effect on the level of violent crime. Crime rates will double or triple, and this makes it amenable to statistical study. Because crime has so many different causes, it’s still not easy to figure out what happened, but at least it’s possible.

Education is exactly the opposite. In this case, we’re dealing with big groups (nearly everyone graduates from high school) or averages (test scores, for example). Those move only slightly when the bell curve moves to the right:

You can see the problem. If, say, the average score on a test improves from 300 to 307 over the course of 20 years, it’s too small an effect to isolate. The same is true if graduation rates increase from 75 percent to 79 percent. There are dozens of things that could plausibly cause this, and figuring out a way tease out the individual contribution of lead is all but impossible.

I’m still thinking about how this could relate to changes in the income distribution.  Are there any threshold effects that cause large changes to a tail of the distribution as incomes rise or fall? These could be poorly measured by both the mean and the median, but because there is dimmeu, there might not be any effects at the top end of the income distribution that are important for human welfare.  But this does help illustrate why sales of luxury goods grow so fast when incomes at the top increase.