At the flagship public universities of Maryland and Virginia, students with economic advantages graduate at a higher rate than those who are less fortunate, a gap between haves and have-nots that troubles many schools around the country.

But at George Mason, the largest public university in Virginia, the opposite is true.

Sixty-eight percent of Mason students with enough financial need to qualify for federal Pell grants graduate within six years, according to a new national study. That’s four percentage points higher than the graduation rate for students at the Northern Virginia school who do not qualify for the grants.

The study from the nonprofit Education Trust, released Thursday, provides what is perhaps the clearest picture to date of how individual colleges and universities nationwide perform in helping Pell-eligible students reach their goal: earning a diploma.

The analysis of data from about 1,150 public and private schools found that 51 percent of Pell-eligible students who started college full time in 2007 earned a bachelor’s degree by 2013. The rate for non-Pell students was 65 percent. But the average gap for individual schools was 5.7 percentage points.

“We want institutions to take this seriously and understand they need to be accountable for helping the students they enroll,” said Andrew H. Nichols, an analyst at Education Trust who authored the report. The nonprofit organization, based in Washington, seeks to raise achievement for disadvantaged students.

For decades, Pell grants have helped students of modest means attend college. The grants are now worth up to $5,775 per student per year, with a total cost of $31 billion a year. But national data on the success of those students has been scarce.

“When we’re spending this much money on a program, good or bad, we need to know what the results are,” said Sarah Butrymowicz, an editor for the education news Web site the Hechinger Report, which this year analyzed Pell grant graduation rates at dozens of major schools.

The government does not typically publish completion rates for Pell-eligible students by school. This month it released information on that topic, but federal officials acknowledge the numbers were flawed because they came from a student aid database that was not designed to generate graduation rates. Officials said they hope to improve their reports in coming years.

Education Trust gathered much of its data from schools and state agencies. It also paid U.S. News and World Report for information the magazine obtains in annual surveys for ranking colleges.

There are big blind spots in any system that measures only the success of first-time, full-time students, because so many study part time or transfer from one school to another. But for now the Education Trust study offers one of the best ways to compare schools on a crucial metric of social mobility.

In the District, the graduation rate for Pell-eligible students was 92 percent at Georgetown University and 81 percent at George Washington University. There were virtually no gaps in degree completion at either school related to Pell eligibility. Both are private schools.

At the University of Maryland at College Park, the Pell-eligible grad rate was 75 percent, 8 percentage points lower than for non-Pell students.

“We’re disappointed,” Barbara Gill, associate vice president for enrollment management at U-Md., said of the gap. “We would clearly like every student at Maryland to be successful, regardless of income.” Gill said the university in the past year has sought to intensify and broaden its support for students from low-income families.

At the University of Maryland Baltimore County, the Pell-eligible grad rate was 61 percent, 1 point below the rate for non-Pell students. Diane Lee, vice provost and dean of undergraduate education at UMBC, said the school is aiming to raise completion rate for all students. “It’s still nowhere where we need it to be,” Lee said.

Among UMBC’s key strategies: widely available tutoring and an early warning system for struggling students. Those who are getting a C or lower in a given class, Lee said, are flagged for special attention from faculty and advisers.

At the University of Virginia, the Pell-eligible grad rate was 84 percent, 9 points lower than that for non-Pell students. However, U-Va. officials said their latest figures show sharp gains for Pell-eligible students, with the graduation rate climbing to 90 percent.

“The consistently high achievement of Pell grant recipients at U-Va. is a testament to the quality of the students who choose to come to the university and to our strong focus on helping students succeed through effective advising and mentoring,” U-Va. President Teresa A. Sullivan said.

U-Va., a highly selective school, has one of the lowest shares of Pell-eligible students among public universities in Virginia. About 13 percent of its undergraduates qualify.

At George Mason, the Pell-eligible share is more than double.

“Finding talent wherever it may lie is part of what we do and have done at Mason for some time,” David Burge, vice president for enrollment management, said. “The Pell statistics are examples of how we can serve a broad spectrum of students.”

Burge said one key to raising graduation rates is to keep the price as low as possible, to minimize the pressure students feel to find jobs to pay their bills. At Mason, tuition and fees for Virginians total about $11,000 a year, not counting financial aid. Another key, he said, is to track and support those who must pause their studies, forcing them to take longer than the traditional four years to a degree.

“You have to look at each individual path to graduation,” he said. “When you talk about serving a broader audience, you have to take into account there are multiple ways to get to the end goal.”