It is well known that school results are mostly explained by socioeconomic status (I read somewhere up to 75% of the variance can be explained that way). So if you don't take into account other factors that graph doesn't prove anything other than people who can afford private schools and homescholing are on average richer. My guess is that the rest of the data can be explained by smaller class sizes and better resources.
If you want more research this [0] article has a good bibliography on the effects of socioeconomic diversity in schools
Socioeconomic status doesn’t explain why parents with only high school degrees who homeschool in the US will still have kids performing at the top of public school kids’ scores, while strong parental involvement does. Over and over the only truly meaningful external factor in student performance is parental involvement, of which homeschooling is highly indicative.
If you want more research this [0] article has a good bibliography on the effects of socioeconomic diversity in schools
[0] https://repository.law.umich.edu/cgi/viewcontent.cgi?article...