Sorry my point was expressed so convolutedly.
I think you understood my point obliquely. The survey in the article said that when economic and community factors are taken into account, private and public school students performed similarly on the tests. I think that yes, what you said in your first paragraph is correct, those families who send their kids to private schools probably have more money and different community factors than the general public-school families in the area. This is not the case in just tough neighborhoods, though.
In a nice neighborhood, the local public school might be great, with lots of educational opportunities and money for extracurricular activities, but the private school might offer equestrian classes or something, along with a smaller class size and more challenging classes. I have the feeling that for most people the local private schools generally offer better education in reading and math than the public schools do.
I do not at all support vouchers, though, and I agree with you that a voucher program would not be successful. I’m not sure what you meant here:
Unless other reasons are posited for why private schools are better, I don’t see a logical justification for it.
By “it”, did you mean vouchers, or a reason why private schools are better than public schools?
If you meant vouchers, then I agree with you completely. If you mean the latter, we can get into it in future posts. I am, however, biased. 
I wonder how private school kids would compare to public school kids in science. I know that in many conservative Christian schools, students get the religious story of the age of the Earth or where stuff comes from, along with a “scientific” explanation of the Flood burying all the dinosaurs and creating the fossils, etc. It’s too bad they didn’t compare the schools in that subject.
Debbie