Mr. Richman still maintains that some small percentage of nnoncitizens vote in American elections. But the debate over this study has moved on. It’s no longer about whether millions of illegal votes were cast, but whether there’s any evidence for noncitizen voting at all.

The study’s bold claims fell apart because of something called response error: the possibility that people taking a survey don’t answer a question correctly — in this case, a question about being American citizens.

There is always a tiny amount of response error in surveys. Respondents might not understand the question. Or they might understand it, but mark the wrong answer by mistake, if the survey is self-administered. An interviewer, if there is one, could accidentally record the wrong answer. Such errors usually aren’t a problem large enough to change the results of a survey.

But both the survey and the question posed by researchers were unusual. The survey — the Cooperative Congressional Election Study — was huge, with 38,000 respondents in 2008. And the group in question — noncitizens — was very small, just 339 of those respondents.

The problem is that even a tiny amount of response error among the 38,000 respondents could have been enough to contaminate the results of the tiny group of noncitizens.