The city’s Department of Education is investigating a charter school housed in its own headquarters building following an allegation that student scores on a state test were doctored.

The person accused of test-tampering, Stephanie Clagnaz, left abruptly as Ross Global Academy’s principal in the middle of May. She is at least the fifth head of school to leave Ross Global since it was founded two years ago.

The brainchild of the multimillionaire philanthropist Courtney Ross, the charter school is publicly funded but privately run. It is modeled on an eponymous private school Mrs. Ross runs in the Hamptons.

In opening the school, Mrs. Ross received support from New York University, of which she is a trustee, as well as from the Department of Education, which authorized the school and even donated space in its own headquarters to house it.

For the past two years, uniformed children wearing Ross Global insignia on their sweaters have shared Tweed Courthouse’s ground floor with suited bureaucrats.

In that time period, relations between department officials and Ross Global administrators have soured.

Department officials last week described their concerns regarding Ross as “serious” and left open the possibility that the school could be placed on “probation” — a danger signal to parents and administrators — or even shut down.

“All the options are on the table,” the department’s executive director of charter schools, Michael Duffy, said. “Substantial issues have been raised with regard to performance at this school, and we take those very seriously.”

The concerns follow two years of trouble.

Saying working conditions were chaotic, many teachers quit and many were fired in the middle of the last school year, and nearly all staff members chose not to return to the school this year. Before Ms. Clagnaz’s abrupt departure in May, eight staff members had left by the middle of this school year, a school official said.

Many parents also pulled their children out last year, complaining of violence, disorder, and a disappointing curriculum.

Staff members and parents shared their letters with The New York Sun on condition of anonymity; several said that after leaving they had signed documents promising not to speak to the press about the school.

“Please, please help the other children that still attend Ross,” one parent wrote in a letter to the department. “I am frightened about them not having a chance to succeed.”

One group of parents and teachers sent their complaints in bulk, walking the winding staircase that separates the school from department officials to hand-deliver their letters.

Members of Ross Global’s board of trustees insist that their school is not in crisis. They say the school has turned an important corner in its second year, pointing to improvement in measures such as the percentage of parents who plan to re-enroll their children next year — up to 92% from 72% last year, they say — and that the school now has a balanced budget.

The officials said Ms. Clagnaz departed as a result of a mutual agreement with the board made after it conducted a review of her performance.

Ms. Clagnaz remains the official principal, though she left permanently in the middle of May.

The assistant principal, Julie Johnson, is handling her duties, but there is no acting principal in the building for the duration of the school year.

At least four school heads served and left before Ms. Clagnaz. One resigned before the school even opened. Another who was hired after an extensive search served less than 60 days but was paid nearly $54,000, a department investigation found.

Ms. Clagnaz had taken the reins when the allegation of test-tampering was made last year, by a former staff member.

The teacher wrote to the state and city education departments to say he had witnessed Ms. Clagnaz take two sets of state math tests home with her before turning them in to be graded.

State rules mandate that principals keep tests in “secure storage” immediately after students take them, and principals sign certificates vowing to have done so.

According to the teacher’s letter, before Ms. Clagnaz took the letters home, she asked another teacher to write an answer key for her.

That teacher, Stephanie Gantz, who has since left Ross Global, confirmed the letter’s account.

She said she did not know what the principal had done with the tests after receiving her makeshift answer key. Under pressure to show good results, the school had mounted a so-called math camp three weeks before the exam to prepare students, four former teachers said in interviews. Administrators hired a group of private tutors to run the camp.

Ross Global officials said they were cooperating with the investigation and knew nothing more about it than what had been reported by the teacher.

Reached by telephone last week, Ms. Clagnaz confirmed that she has left the school but did not say why she left.

Asked about the test-tampering allegation, she said, “I’m not going to address that. I don’t know anything about that.”

Ms. Johnson said there was no “math camp” this year.

Teacher turnover is another concern.

Eight staff members departed at the middle of this school year, including two main teachers; four associate teachers, and two language teachers, a member of the board of directors, Nicolas Combemale, said.

Several teachers said they left because they felt the school’s chaotic nature was hurting students.

“I never quit anything in my life,” Ms. Gantz said. “It was just a disservice for me to stay at the school and work for her, for those children. I wasn’t doing them any good.”

Mr. Combemale also pointed out as a positive that only 10 of 37 teachers do not plan to return to Ross next year. He said parents are also pleased, and that the school’s waiting list is growing and now includes almost 300 names.

To parents of prospective students, the image Ross Global Academy projects is gleaming. Every student gets a free Apple laptop, and lessons in Chinese begin in kindergarten. To enhance the school’s focus on globalization and cultural history, classrooms are lined with Roman sculptures.

In interviews, parents of children at the school said Ross Global has not lived up to its promises. They described a chaotic environment and said the curriculum is still more test-focused and rudimentary than the elaborate themed idea described in brochures.

Several said they were unhappy that summer school was abruptly canceled for this summer, and that another promise, Saturday school, has dissolved to an optional day that many children no longer attend from a mandatory staple.

The Roman sculptures seem symbolic of the damage done: Their smooth, alabaster-white torsos have been chipped, exposing areas of brown base.

A former teacher, Heather Dallas, said the children had broken the sculptures. “They would throw things on them, they would hang on them, they would take crayons and actually color on them,” she said. “They broke off the finger, they broke off the penis — whatever would cross a fifth- or sixth-graders’ mind to do with a naked statue in their room.”

Some parents have given up hope that the school will pull out of its slump; others already see signs of progress.

“The potential of what we are to become outweighs any of the current problems that we’re going through,” one father, Ian Pearce, said.

Another parent, Reada Edelstein, said she plans to keep her daughter at Ross for the sake of stability.

Ms. Edelstein said she does not fault the Department of Education but the school’s board of trustees for permitting what she described as constant chaos.

Besides Mrs. Ross, perhaps the most prominent member of the board is the dean of the Steinhardt School of Education at New York University, Mary Brabeck.

In an interview last week, Ms. Brabeck said the school, one of 25 partner schools to NYU, has shown “tremendous progress” this year.

Mrs. Ross is the widow of a former Time Warner chairman, Steve Ross. She has appeared on Forbes magazine’s list of the 400 richest Americans twice, and she has donated some of her own money to the charter school.

Her publicist at Rubenstein Public Relations did not respond to a request to interview Mrs. Ross.

Ross Global Academy was founded two years after Mrs. Ross’s daughter, Nicole, made a gift to NYU in honor of her mother.

The gift endowed a professorship: the Courtney Sale Ross University Professor of Education and Globalization.