That would be an interesting study to read. We know private and home schooled kids do far better than public schools.
To be honest many if not most private schools cherry pick the better students...and they can decline a student for various reasons so their stat's are higher.
The public schools have to take anyone..good or bad. Smart or dumb.
I'm not familiar with home schoolers system but I expect there is some areas where you cannot compare them to public schools as well..kids who the parents have a problem with go back to the public school.