T HE PUBLIC high schools in Washington, DC , were once looked on with wonder. Overcoming deep-seated poverty (three in four pupils are classified as poor) and racial segregation, the district dramatically increased its graduation rate. In 2012 only 56% of high-school students graduated. By 2017 that rate had climbed to 73%. Arne Duncan, Barack Obama’s education secretary, touted the district’s results as an example of “what can happen when schools embrace innovative reforms.”

Then the truth emerged. It began with media reports on shenanigans at Ballou High School, an all-minority and entirely poor high school in the southeastern corner of the nation’s capital. Graduation rates had gone from 50% in 2012 to 64% in 2017. When auditors examined the district’s records, they found that 34% of all diplomas in 2017 year were improperly awarded. Many went to students who seldom showed up at school. Graduation rates at Ballou have since sunk back to Earth.

Nationally, high-school graduation rates have increased at a steady clip even while other measures of learning and achievement—international exams, state-mandated standardised tests, college-admissions test scores—have been flat or even slightly negative. That could be because children are doing better, or because schools are lowering standards.

There are some pockets of real success—high-performing charters in cities have helped many poor, minority students most at risk of dropping out if left in traditional public schools. But for the rest of the country the warning lights are starting to flash. The state of Alabama—which posted a remarkable 17 percentage-point increase in graduation rates between 2011 and 2015—has since admitted that its numbers were inflated. From Charlotte, North Carolina, and Atlanta, Georgia, to New York City and Los Angeles, credible accusations of graduation-rate inflation have emerged.