Not because of some magic formula that only the private schools have. Because they have to pay, you have a better class of parent and a better class of student. Sounds elitest, but its the truth. Programs and facilities for special needs children, programs for gifted children, programs such as head start, athletics, the band, busing, etc. Of course there would.
Schools are better because of the parents, teachers and students.
If you take a bunch of bums and give them each homes on a street in Beverly Hills, pretty soon that street would look like skid row. You take lousy students with their lazy parents and send them to a private school, pretty soon the private will be just like the public school those hoodlums were kicked out of. In fact, the reverse was true. They are required to use the Authorized King James Version. They are required to memorize scripture.