Most of the nation’s elite colleges and universities fall short of a benchmark that Amherst College surpassed five years ago: More than one-fifth of its students come from families poor enough to qualify for federal Pell grants.

The share of Pell grant students at top-tier schools resonates now for at least two reasons.

First, selective colleges are in the midst of releasing admission decisions for the fall term. It will be interesting to see whether this recruiting cycle has widened access for needy students at elite schools or narrowed it. How many Pell-eligible students will the Ivy League admit when its schools release decisions at 5 p.m. Thursday?

Second, President Obama is developing a system to rate colleges. Pell grants are likely to factor into federal ratings because they are a widely accepted measure of commitment to access.

At Amherst, a private college in Massachusetts with about 1,900 students, 21 percent of undergraduates are Pell-eligible this school year. As recently as the 2005-06 school year, the share was 13 percent.

To raise the share that much requires not only admitting more needy students but also expanding financial aid. The maximum Pell grant, $5,645, covers a tenth or less of the full cost of attending an elite private college.

Some colleges do not have the resources to make up the difference for more than a small share of students.

Some, however, have endowments big enough to fund more financial aid. But, for whatever reason, they do not.

“You really have to have top-to-bottom support and enthusiasm for access and opportunity,” Amherst President Carolyn A. “Biddy” Martin told The Post in a recent visit to Washington. “It has to be faculty and students as well as administration, trustees and alumni — and obviously donors.”

Martin said Amherst’s drive to reshape its student body began well before she took the helm in 2011. The school, which has a $1.8 billion endowment, has plowed huge resources into financial aid — $46 million this school year. The aid budget, after adjusting for inflation, has more than tripled since 1992.

All that aid is essential because the full price of tuition, fees, room and board at Amherst tops $58,000 a year.

For Martin, college access is not an abstraction.

A former chancellor of the University of Wisconsin at Madison and provost of Cornell University, Martin grew up in rural Virginia, between Lynchburg and Roanoke. Her parents were skeptical of the value of college and did not think they could afford to send her. She went to the public College of William and Mary with help from scholarships.

“I think about this a lot,” Martin said.

Martin said that even with Amherst’s wealth, the open-access policy forces hard choices. “There are always trade-offs,” she said. Sometimes that means postponing a big remodeling project. The Amherst student center, she said, could use a makeover. It is, Martin said, “too small and not laid out in a way that draws students to it and gives them the space they need.”

Another issue is the amount of aid available for students from middle-income families. At Claremont McKenna College in California, federal data show about 14 percent of undergraduate students are Pell-eligible. Hiram E. Chodosh, president of Claremont McKenna since July 1, said the importance of the statistic should not be overstated.

“We have to be very careful not to create a ‘barbell effect,’” Chodosh told The Post in a recent visit to Washington, referring to a scenario in which a college might load up on full-paying students from affluent families as well as full-ride students from poor families. In that case, students in the middle — with some need but not high need — could get squeezed out.

Chodosh said he wants a “commitment to socioeconomic diversity across the spectrum.” He talked about recruiting students from families with low, moderate, middle and even “lower-upper-middle” income.

Measuring that kind of diversity could be tricky. Federal education statistics tend to focus on poverty.

A Washington Post analysis of 41 highly selective colleges and universities — drawn from U.S. News & World Report listings — found eight in which at least one out of five undergraduates is Pell-eligible. The University of California at Berkeley had the highest share, 33 percent.

Here is a ranking of the Pell-eligible share for Top 20 schools on two U.S. News lists, one of them for national universities and the other for national liberal arts colleges. The Pell share is drawn from federal data on the 2011-12 school year.

(On Tuesday afternoon, after this story was published, Harvard University pointed out that federal statistics include many students who are enrolled in its non-degree continuing education programs. If those students are excluded from the count, Harvard’s Pell-eligible share was 17 percent in 2011-12. The chart has been updated to reflect that information. Originally it reported the Harvard share as 11 percent.)

School Pell share in percent
UC-Berkeley 33
Smith College 24
Emory University 23
Grinnell College 23
Vassar College 22
Amherst College 21
Columbia University 21
Massachusetts Institute of Technology 20
Wellesley College 19
Williams College 19
Stanford University 18
Wesleyan University 18
Cornell University 17
Pomona College 17
Rice University 17
Harvard University 17 (excluding continuing education)
University of Chicago 16
Brown University 15
Dartmouth College 15
Swarthmore College 15
Vanderbilt University 15
Bowdoin College 14
Claremont McKenna College 14
Duke University 14
Georgetown University 14
Haverford College 14
Northwestern University 14
University of Pennsylvania 14
Yale University 14
Carleton College 13
Davidson College 13
Hamilton College 13
Johns Hopkins University 13
Harvey Mudd College 12
Princeton University 12
University of Notre Dame 12
California Institute of Technology 11
Colgate University 11
Middlebury College 10
Washington and Lee University 9
Washington University in St. Louis 7

Note: The U.S. Military Academy and U.S. Naval Academy, both highly ranked by U.S. News, are not listed here because they do not charge tuition.