However, when one corrects for class variables (such as region of residence, handicappedness, college attendance expectations) this bias disappears. 1 2 3 This articles were selected as the first articles appearing in a google scholar search on "private schools socioeconomic correlation" and the only 3 on the first page which are deal with the issue of education "quality." That these are the first three might compensate for my confirmation bias in not leaving the first page.
While it can be said that enrolling your children in a private school is a good idea because of "possibly better retention rates, increased security and discipline, and greater opportunities for a variety of specialized school-day and extracurricular activities", I claim that it cannot be said that the education is better: a good student is a good student, regardless of where you put them.
I will be logging off as it is 2:18 am here. My claim is that any study which regresses for socioeconomic indicators will find no significant correlation between private vs. public status and educational quality.