Jonathan Ichikawa looks at a a study regarding private and public schools. Apparently the overwhelming support for the thesis that private schools are better disappears when you adjust for socioeconomic status. When you compare people of one class level, they do slightly better on standardized tests (well, really one standardized test) in public schools than in private. I don't think this shows that public schools are on the whole better, though.
This study doesn't seem to differentiate between two very different kinds of private school. There are the elite private schools like the one I went to for high school, and then there are the smaller, usually religious schools, which have a much lower tuition, often hire teachers with much less training, and don't attract students who are anywhere near as good. I suspect that many such schools are worse in most academic ways than the public schools in their area. Schools like the one I went to are just so clearly superior to the local public schools that I can't accept Jonathan's conclusion. That's not going to be representative of private schools on the whole, though. I'd like to see a study that compares students who went to a school like the one I went to with students in public schools alongside a separate category of those in smaller and less-funded private schools.
This may also be affected by what kind of students of each class level would end up in a private school. I suspect there are other factors. Parents might think a very smart person would do well enough in public school, but someone less smart would need private education to do better. This could be true at each economic level. Poorer families will have a harder time getting their kids into private schools unless their students are good enough to get their tuition covered, but there might be even smarter students at the public schools whose parents don't think they need as much of a boost as the parents of those who end up in private school. I suspect this is more true with rich kids and less true with poor kids, but it's plausible enough that it might have an effect.
Jonathan didn't mention one conclusion of the article he linked to, and this is one that I think is correct. The article suggests that one of the reasons private schools tend to do better is because the parents of students there are more involved with their kids. This is especially so among the bread and butter of private school kids, the ones from upper middle class families. Someone who grows up with parents who don't value or model reading or learning for its own sake isn't as likely to seek private education for their children. So the ones with better educational environments for their children are sending their kids to private schools. This is inflating the scores of private school students, and adjusting for class removes some of that inflation, because poorer homes tend to have less of this valuing of learning for its own sake. So on the whole I think the issue is just more complicated than it seems at first, both to those who think the private schools are all better and those trying to use this study to show that public schools tend to be better.