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Vassar has taken steps to hold down spending on faculty and staff. Amherst and the University of Florida have raised new money specifically to spend on financial aid for low-income students. American University reallocated scholarships from well-off students to needy ones. Grinnell set a floor on the share of every freshman class — 15 percent — whose parents didn’t go to college.

Over the last decade, dozens of colleges have proclaimed that recruiting a more economically diverse student body was a top priority. Many of those colleges have not matched their words with actions. But some have.

These colleges have changed policies and made compromises elsewhere to recruit the kind of talented poor students who have traditionally excelled in high school but not gone to top colleges. A surprising number of such students never graduate from any college.

This education gap is a problem not only for the teenagers on the wrong end of it. It’s a problem for the American economy. The economic differences between college graduates and everyone else have reached record levels. Yet for many low-income children — even many who get A’s in high school and do well on the SAT — college remains out of reach. No wonder that upward mobility is less common in the United States than in many other rich countries.

To see which selective colleges are doing the most, and the least, to change the situation, The Upshot has analyzed data for every college with four-year graduation rate of at least 75 percent. We combined data on enrollment and tuition costs to measure how hard each college is trying to attract and graduate poor and middle-class students. The result is our College Access Index.

Vassar, the once all-female college in the Hudson River Valley, tops our index, with Grinnell placing second. About 23 percent of Vassar’s freshmen in recent years have received federal Pell grants (which mean they come from roughly the bottom 40 percent of the income distribution), up from 12 percent in 2007. After taking scholarships into account, the average annual cost of attending Vassar for lower-income students is about $6,000. Students cover much of that cost through campus jobs and loans.

The biggest theme to emerge from our analysis is that otherwise similar colleges often have very different levels of commitment to economic diversity. In this area, endowment is not destiny, and prestige is not destiny.

After Vassar, the top of the list includes some of the wealthiest colleges in the country, measured by endowment per student: Grinnell; Amherst College, in Massachusetts; Harvard; and Pomona, in Southern California. But other resource-rich colleges, including Swarthmore and the California Institute of Technology, have done substantially less.

In the middle of last decade, Lawrence Summers, then the president of Harvard, called attention to the issue and made Harvard’s financial-aid policies much more generous. Given Harvard’s prominence, other colleges began to look at the situation as well.

If the number of college graduates from low-income backgrounds is going to rise sharply, it’s true that most of them won’t come from selective colleges like Grinnell. But those colleges still have an outsize role.

For one thing, the low-income students who enroll there tend to graduate. For another, research has shown that the individual college attended by upper-middle-class students has little effect on their eventual earnings, after controlling for their SAT scores. But it does seem to matter for poor students. They get something extra from a top college.

That is why it’s worrisome that so many students are falling through the cracks — and that the student body at many elite campuses remains largely affluent. But it may not always be so.