“This information is useful but has to be used with caution,” he said. “It’s that middle ground. It’s not useless, but it’s not perfect.”

Yet a promising correlation for groups of teachers on the average may be of little help to the individual teacher, who faces, at least for the near future, a notable chance of being misjudged by the ranking system, particularly when it is based on only a few years of scores. One national study published in July by Mathematica Policy Research, conducted for the Department of Education, found that with one year of data, a teacher was likely to be misclassified 35 percent of the time. With three years of data, the error rate was 25 percent. With 10 years of data, the error rate dropped to 12 percent. The city has four years of data.

The most extensive independent study of New York’s teacher rankings found similar variability. In math, about a quarter of the lowest-ranking teachers in 2007 ended up among the highest-ranking teachers in 2008. In English, most low performers in 2007 did not remain low performers the next year, said Sean P. Corcoran, the author of the study for the Annenberg Institute for School Reform, who is an assistant professor of educational economics at New York University.

The high margin of error for most scores, something the city refers to as the confidence interval, is another source of uncertainty, Dr. Corcoran said. In math, judging a teacher over three years, the average confidence interval was 34 points, meaning a city teacher who was ranked in the 63rd percentile actually had a score anywhere between the 46th and 80th percentiles, with the 63rd percentile as the most likely score. Even then, the ranking is only 95 percent certain. The result is that half of the city’s ranked teachers were statistically indistinguishable.

“The issue is when you try to take this down to the level of the individual teacher, you get very little information,” Dr. Corcoran said. The only rankings that people can put any stock in, he said, are those that are “consistently high or low,” but even those are imperfect.

“So if you have a teacher consistently in the top 10 percent,” he said, “the chances are she is doing something right, and a teacher in the bottom 10 percent needs some attention. Everything in between, you really know nothing.”

In New York, the rankings face an additional set of issues. The state tests on which they were based became, over time, too predictable and easy to pass, and this summer the state began to toughen standards. Daniel Koretz, a Harvard professor whose research helped persuade the state to toughen standards, said that as a result it was impossible to know whether rising scores in a classroom were due to inappropriate test preparation or gains in real learning. Rankings that include the tougher standards will not be available until the next academic year.