During the recent struggle over collective-bargaining rights in Wisconsin, a number of left-of-center observers, including New York Times columnist Paul Krugman, pointed out that students in unionized Wisconsin do better on average than students in non-unionized Texas. The obvious conclusion, or so we were led to believe, is that teachers’ unions lead to better education.

There is, however, a problem with this argument. Drawing on data from the 2009 National Assessment of Educational Progress, the political commentator David Burge pointed out that white students in Texas outperform white students in Wisconsin, black students in Texas outperform black students in Wisconsin, and …