Sorry I have no links handy but i will pass on a perosnal observation.
I recently completed this summers Vacation Bible School sessions at my Church.
I had 10-25 kids in the groups I dealt with over the course of 10 weeks.
Keep in miond we are talking about Bible vacation School and not camp or a general social gathering.
Now I thought all the kids I dealt with were good and smart kids. But ithin the group I had 2 home schoolers and without a doubt they were more mature, more settled, quicker to answer and be right then the rest of the group which would be Public school kids.
Not a scientific study I understand but it is not the first peroanl experience bewtween public schoolers and home schoolers that I have had and always the home schoolers seen to ME to have the edge.......
Our private school test scores are much higher than neighboring public schools (even the best public schools.) However, I do think it’s possible that more educated people who care more are more likely to send their kids to private school and therefore their grades are higher.
I do firmly believe that my daughter benefits from her (Catholic) education, but you would need to have a study that compared apples to apples.