I refer to this thanks to Felix Salmon, whom I shall simply quote bacause I could not say it better myself:
There aren’t a lot of studies of public vs private schools, but the ones which do exist generally show no difference at all in educational outcome, once you control for the socio-economic status of the kids being admitted. Essentially: middle-class kids who grow up with two well-educated parents and lots of books around the house will generally do very well in school no matter where they go.
Which is why I am always try to be a calming voice to well-educated middle-class parents in PPS who are dismayed at the erosion of support for public education. It is generally not their kids that suffer.