The state uses the city’s annual reports of violent episodes to designate certain schools as “persistently dangerous.” Those schools are required to take steps to reduce violence and to notify parents that they are entitled to enroll their children at a less violent place.

Mr. DiNapoli’s office found that one of the schools examined, the Choir Academy of Harlem, possibly should have been designated “persistently dangerous” in 2012-13 but was not because of the department’s underreporting. The school now has the designation. In 2013, the department decided to begin phasing out that school, citing concerns about safety expressed by parents, students and teachers. It will close in 2016.

The city’s Education Department said the incidents that the comptroller said had to be reported to the state did not have to be. In response, the comptroller’s office said that it had confirmed its interpretation of the requirements with the State Education Department.

The schools included two in each borough and were selected to represent a range of degrees of violence. Mr. DiNapoli’s office examined the incidents that were reported by the individual schools to the city department through the city’s own reporting system and determined whether they should have been included in the city’s annual report to the state and whether they were actually included.

Officials at Public School 83 Luis Munoz Rivera, in East Harlem, told the comptroller’s office that they did not even report all violent episodes to the city’s Education Department, keeping a separate paper system for their own purposes and reporting only cases in which the offending student had a pattern of disruptive behavior.