Julia Ryan at The Atlantic observes that when correcting for demographics, public schools outperform private schools:

Sarah Theule Lubienski didn’t set out to compare public schools and private schools. A professor at the University of Illinois at Urbana-Champaign, she was studying math instructional techniques when she discovered something surprising: Private schools—long assumed to be educationally superior—were underperforming public schools.

She called her husband, Christopher A. Lubienski, also a professor at the university. “I said, ‘This is a really weird thing,’ and I checked it and double checked it,” she remembers. The couple decided to take on a project that would ultimately disprove decades of assumptions about private and public education.

Studying the National Assessment of Educational Progress and the Early Childhood Longitudinal Study, they have found that, when controlling for demographic factors, public schools are doing a better job academically than private schools. It seems that private school students have higher scores because they come from more affluent backgrounds, not because the schools they attend are better educational institutions. They write about these conclusions—and explain how they came to them—in their book, The Public School Advantage: Why Public Schools Outperform Private Schools. Here’s an interview with the Lubienskis about their work, edited and condensed for clarity and length.

This is not a surprise, but it does not matter for two reasons:

• The quality of an educational experience is largely determined by the abilities of the students.
• It creates social connections which serve to preserve a de facto aristocracy.

The biggest determinant of school quality has always been the “quality”, for lack of a better term, of the students.

To the degree that you put good students together in a self contained environment, their educational experience will likely improve, though this will be at the expense of those students, frequently poorer and more heavily pigmented, on the outside.

As to social connections, just look at Harvard, whose undergraduate program is widely considered to be one of the most overrated in academe,^* they remain very exclusive.
The reason for this is the quality of the student body, and the social connections that are created by going there. 

It seems like half of the Wall Street banksters went to Harvard, for example.

Nothing binds like those old school ties.

^*So says my Radcliffe (Harvard) graduate, former college president, step-mom.