President Obama repeated it in a speech three days later. “Four out of five schools will be labeled as failing,” Mr. Obama said at Kenmore Middle School in Arlington, Va., in March. “That’s an astonishing number.” Now a new study, scheduled for release on Thursday, says the administration’s numbers were wildly overstated. The study, by the Center on Education Policy, a Washington research group headed by a Democratic lawyer who endorses most of the administration’s education policies, says that 48 percent of the nation’s 100,000 public schools were labeled as failing under the law this year. The center is the only research group that has compiled an annual report of how many schools nationwide have run afoul of No Child Left Behind. The center calls its 48 percent figure an estimate, but it is based on a tally of schools that 49 states have reported as failing. (New York has not yet released its 2010-11 figures.) Final numbers are not expected until next year, but the center said the 48 percent estimate was unlikely to change by more than 1 percentage point. Forty-eight percent, up from 39 percent in 2010, is the highest proportion of schools labeled as failing since President George W. Bush signed the education law in 2001. Schools acquire the label when they fail to raise student reading and math scores enough to keep up with testing targets set by their states. Mr. Duncan, in a statement issued on Wednesday, brushed aside the discrepancy. “Whether it’s 50 percent, 80 percent or 100 percent of schools being incorrectly labeled as failing, one thing is clear: No Child Left Behind is broken,” Mr. Duncan said. Asked why the Education Department’s projection was so far off, Justin Hamilton, a spokesman for Mr. Duncan, said, “Our intention was to look thoughtfully at the data and show how the law would impact schools and students if left unchanged.” Almost everybody agrees that the law is broken. With Congress making little progress rewriting it, the administration announced this fall that it would issue waivers of its central requirements to states that outlined credible plans to hold schools accountable for student progress. Eleven states have applied for the waivers. An additional 28 states have said they intend to apply; applications for a second round in the waiver process are due in February. Back in March, Margaret Spellings, Mr. Duncan’s predecessor as education secretary under Mr. Bush, accused the administration of floating an exaggerated projection of failing schools to build support for a rewrite or reauthorization of the law in Congress. “They’re overstating the numbers to make a political point for reauthorization,” Ms. Spellings said. Jack Jennings, the president of the Center on Education Policy and a Democrat, said Wednesday: “I still don’t understand why their estimate was so far off. Obviously they didn’t use the right methodology.”