I normally use this blog to talk about mathematical questions that interest me. However, I saw a math error in the New Yorker yesterday that I think is worth commenting on.
James Surowiecki has an article (The Mobility Myth) arguing that income mobility (the ability of groups to change their economic status) in the United States isn’t very high. At one point he makes the following statement: “The middle class isn’t all that mobile, either: only twenty per cent of people born into the middle quintile ever make it into the top one.”
This sounds pretty bad at first glance: Twenty percent isn’t that large of a percentage. However, the top quintile is, by definition, the top one-fifth; i.e., the top twenty percent. If the United States were to have perfect income mobility (in the sense that a parent’s economic status has absolutely no effect on that of their children) then we would see exactly twenty percent of children born into the middle class end up in the top quintile (as well as twenty percent in the bottom quintile and twenty percent in each of the other three quintiles). In other words, the argument that “twenty percent of people born into the middle quintile make it into the top one” is actually an argument in favor of high income mobility in the United States, not against it!
Surowiecki’s misunderstanding about what quintiles actually mean causes some of his other claims not to be as drastic as they sound. For example, “fewer than ten per cent [of the bottom quintile] get into the top quintile,” while still not that great, isn’t quite so bad once you realize that twenty percent is the goal. Similarly with “In San Jose, just thirteen per cent of people in the bottom quintile make it to the top.” (However, I should point out that, given the number of people we’re talking about here, the difference between twenty percent and thirteen percent is quite large in absolute terms.)
I’m not going to comment on the merits of the rest of Surowiecki’s column, as to do so would be to step too far from my area of expertise. I will add, though, that his data appear to come from the study “Where is the Land of Opportunity?: Intergenerational Mobility in the United States,” by Chetty, Hendren, Kline, and Saez. This document contains a wealth of information to interest data guys like me. See especially Table III and Online Appendix Table IV, which appear to be the sources of some of the numbers Surowiecki cites.