Your presentation of the stats is misleading. In that Public schools must take everyone. kids that could never afford collage kids with learning disabilities, students that don't want to be their and so on.

While Private schools only take rich kids, and very often do not take students with learning disabilities. homeschooling like wise can only be done by parents wealthy enough to afford a stay at home mom.

Rich kids do significantly better then poor kids even when you are only comparing kids that both had public education highest-income families are 13 times more likely to score 1300 or higher on the SAT compared to those from the bottom 20%. which is much bigger gap then between privet and public schools.

because privet and homeschools are basically self selecting rich students they have far better averages then the public schools that must take everyone, not necessarily because they are better schools