Simple mathematics shows that commonly cited statistics on income inequality are often highly misleading. Phil Birnbaum is a “sabermetrician,” or baseball statistician (a la Moneyball), with a nice post on this topic (hat tip: Cafe Hayek). He picks on a Federal Reserve report on the U.S. distribution of income, pointing out a flawed interpretation of these statistics that is all too common. In particular, the Fed economists discuss shares of income flowing to various quantiles of the population as if the individuals within each quantile never change. There is a tendency to think of changes in other widely-used statistics on income distribution in the same way (e.g., Gini coefficients).

Birnbaum also mentions an interpretation of the Fed report by a New York Times writer that was even more egregious. For emphasis, Birnbaum changes the context from income quantiles to lottery winnings:

“‘Lottery winners picked up 10 percent higher jackpots in 2013 than 2010, keeping winnings disproportionately in the hands of those who already won so much.’ That would be an absurd thing to say for someone who realizes that the jackpot winners of 2013 are not necessarily the same people as the jackpot winners of 2010.”

As Birnbaum notes, the use of income shares by fixed quantile is largely a matter of data availability, as our ability to track individual households as they move through the income distribution is limited. However, tracking individual households would often lead to different conclusions about changing income shares over time. For example, the traditional approach (used by the Fed) is to act as if the top 1% by income is the same group before and after the period of analysis. In fact, if the income share flowing to the identical set of households were measured after the fact, it would look far different; it would be lower than the share quoted by the Fed report because people can migrate from the top 1% in only one direction: down. Similarly, the lowest quantile (1%, or whatever percent) either stay put or migrate upward in the distribution, so the ex-post income share of the same households would be higher than reported by the Fed.

Birnbaum provides simple numerical examples demonstrating the different implications of changes in income shares of fixed quantiles versus fixed sets of households. (His last example is flawed, as discussed in the comments.) He also demonstrates that the simple process of economic growth with broadly distributed benefits almost always leads to greater inequality based on fixed quantiles, but not based on fixed household groups. Here is a wonderful quote:

“It makes no sense at all to regret a sequence of events on the grounds that, in retrospect, it helped the people with better outcomes more than it helped the people with worse outcomes. Because, that’s EVERY sequence of events!”

As noted by John Cochrane in a recent address, inequality is often not so much a problem as it is a symptom of other issues. One of those issues can be economic growth. Inequality, in and of itself, is fairly meaningless as a guide to economic policy. If only the pundits treated it as such.