In the latest in a series of embarrassing snafus, the testing company Pearson miscalculated thousands of city students’ scores on tests to screen for giftedness.

According to the Department of Education, 2,698 children who were told they did not hit the city’s threshold for gifted and talented program eligibility actually made the cut. Another 2,037 children had scores high enough to apply to citywide gifted programs, but were told they could apply only to less elite district programs.

Together, Pearson miscalculated the scores of 13.2 percent of test takers — and undercounted the number of test takers to hit the city’s threshold by 30 percent.

Originally, the department had said that just 9,020 students hit the eligibility threshold. The updated numbers mean that far more students are competing for seats in the city’s gifted programs than ever before, even though the city changed the admission process this year to make it harder to game.

Pearson and the department disclosed the massive error — to which they said parents had alerted them — just before 4:30 p.m. today, just hours before the deadline for eligible families to finalize their applications with the city Department of Education. But officials at the testing company and the department have known for a week about the mistakes, according to people with knowledge of the situation.

On Thursday afternoon, Chancellor Dennis Walcott told GothamSchools, “I don’t have any information on that.” Today, he said in a statement that a top Pearson testing official had “personally apologized to me in our conversations this week.”

The president of Pearson’s Learning Assessment division, Scott Smith, said in a statement that the company — which also has a contract with the state to produce its annual exams — would bring in outsiders to figure out what went wrong. The company did not weigh test-takers’ ages correctly; inaccurately converted students’ raw scores to rankings; and used the wrong formula to come up with the overall score that the city would use to rank students, he said.

“The fact that these errors occurred is simply unacceptable to Pearson as we fully understand the importance of accurate scoring,” said Smith, who also sent a letter to parents apologizing for the errors.

The mea culpa represents the latest in a series of local embarrassments for Pearson, one of the world’s publishing companies and for-profit testmakers. Last year, the company drew fire for producing state exams that included nonsensical reading passages and some errors.

“What does drive my anxiety is [Pearson’s] ability to deliver on the contract,” Regents Chancellor Merryl Tisch said at the time. “The mistakes that have been revealed are really disturbing.”

“We were given the same assurances of quality control and heightened standards after last year’s problems,” said Public Advocate Bill de Blasio, a mayoral candidate. “It’s time for the Department of Education to reassess its relationship with the company.”

But Walcott signaled that the city was prepared to continue to work with Pearson. “I have told the company’s officials in no uncertain terms that I expect this will never happen again,” he said.

For now, parents are left to sort out whether their children are newly eligible for gifted programs and, if so, where to apply. The department plans to tell families whether they were affected in the next 24 hours and will issue new score reports in the next 10 days. The department is also extending the application deadline from today until May 10.

The timeline is likely to have far-reaching consequences, according to Robin Aronow, a consultant who helps families with their school searches. “If this new setback postpones notifications of placements even further, it has a domino effect on not only G&T placements, but all other wait listed public school students, as well as private schools which are waiting to learn if certain families will withdraw their contracts to go to G&T programs,” Aronow said.

She said the news of Pearson’s errors came after a fraught application process that included a new test, a later timeline, and an aborted attempt by the city to change admissions rules.

“To now be told that Pearson miscalculated the scoring is another blow, just as parents are completing their ranking of schools,” she said. “While more children have now scored higher, the odds of acceptance may go down.”

Parents have been reporting oddities in the test results since they came out earlier this month. “I think people who have two 99s and don’t have a 99 [overall percentile score] certainly would like an explanation. If it’s wrong it’s a big mistake that affects their whole schooling,” said Gary Rubinstein, a mathematician and Stuyvesant High School teacher whose daughter took the G&T screening exam this year.

A parent whose child was screened for gifted programs this year said she was glad the department identified the errors but hoped it would communicate with families quickly and carefully. “This is already such a stressful time for many parents and this only creates more anxiety,” she said.

Even as he questioned the city’s recent endorsement of a Pearson-produced curriculum, UFT President Michael Mulgrew said in a statement that the city was wrong to deflect blame onto Pearson for the errors in gifted test scoring.

“Thousands of children and parents get the wrong results on a very important test. Only after parents urge an investigation does the DOE act; it then blames the testing company and tries to bury the announcement on a Friday afternoon,” Mulgrew said. “Mayor Bloomberg may wonder why parents have so little faith in his management of the schools.”

This story has been updated to show that while the city plans to inform families about whether they were affected by the errors within 24 hours, it will not be sending new score reports immediately. New score reports will be sent by April 29, officials said.