If the city’s Department of Education received a report card, it would get an “F” in math.

The schools chancellor, Joel Klein, said yesterday that his staff had drastically miscalculated the city’s high school graduation rate – a disclosure that, had it been made months earlier, could have saved Mayor Bloomberg serious embarrassment.

Rather than decreasing slightly, as the city previously reported, the percentage of students graduating on time actually soared to 58% – the highest graduation rate since the department started tracking the number 20 years ago, Mr. Klein said.

In February, the mayor took heat when he reported that despite a high profile attempt to boost student achievement, the city’s graduation rate had dipped a percentage point, to 53%.

During his re-election campaign, Mr. Bloomberg asked voters to judge him on the success of his school reforms.

“Obviously, I’m not pleased that there was an error,” Mr. Klein told reporters gathered yesterday at the education department’s headquarters at Tweed Courthouse. The disclosure put Mr. Klein in the awkward position of lamenting the miscalculation and celebrating the rosier outcome.

The mayor’s office referred calls to the education department.

The problem came to light in March, when several schools stepped forward to complain that the department had gotten their graduation numbers wrong. Mr. Klein said the department had made two different errors in calculating the figures.

At Hillcrest High School in Queens, for example, 61% of students actually graduated; the department originally said the number was 32%.

After recalculating the rates, the department hired the accounting firm Ernst & Young to independently verify the numbers. Mr. Klein said he would retain the company each year to assess the city’s graduation rates, but that he does not plan to have it audit the department’s other data.

Even before this blunder, there was great confusion over the city’s numbers.

During the election, Mr. Bloomberg’s Democratic challenger, Fernando Ferrer, continually questioned the mayor on the city’s graduation and dropout rates.

A New York University education professor, Robert Tobias, the former director of assessment and accountability for the Department of Education, said he was alarmed by the mistake.

“When you see a revision like this one it shakes your confidence in the accuracy of the data,” Mr. Tobias said. While 58% is an impressive number, he said, “the problem is the credibility of all of these accountability statistics is really being called into question.”

Mr. Tobias was in charge of calculating the graduation rate between 1986 and 2001. He suggested that the city should hire an independent entity to conduct all assessments.

“While the higher graduation rates are encouraging, we need to be able to have confidence in the assessment and reporting of the numbers,” the president of the United Federation of Teachers, Randi Weingarten, said. “Let’s have one transparent standard with an independent source verifying the numbers each year.”

Even the city and state can’t agree.

According to state figures, only 44% of city students graduate in four years, although the state does not include students who passed the high school equivalency test or who graduated in August as opposed to June. The state also includes a group of severely disabled students that the city excludes from its calculation.

Mr. Klein said that in the future he wants to harmonize the formulas used by the state and the city.

The chancellor also seized on yesterday’s announcement to tout higher graduation rates at the city’s new small schools.

Since gaining control of the city’s school system, the Bloomberg administration has dismantled dozens of large, failing schools and opened 149 new small schools in their place. It plans to open 35 more in the fall.

About 73% of students attending small schools graduated on time this year – a rate based on just 15 schools with graduating classes.

Education analysts said rates at those schools could be higher because small schools serve fewer English-language learners and special education students.

As for his own graduation, Mr. Klein said yesterday that while he has just finished four years in the system, he has no plans to leave anytime soon.