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As another poster has already said, the problem here is that high-performing students are often high-performing because of the enrichment given at home.

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Nearly all academically successful students are homeschooled. It does not matter if they attend school or not. I say "nearly" because there are always a few exceptions.

The only thing a government school does is send home a curriculum for the parents to follow.

My husband worked for du Pont for many years. The scientists and engineers he worked with went home every night and made certain that their children knew their math facts by rote. Then they corrected the new-new math misunderstandings and made certain their children were well grounded the math algorithms that they learned in their mud hut, third-world, elementary schools.

These scientists and engineers bought Hooked on Phonics for their pre-schoolers, and at the first sign of academic difficulty hired tutors.

Their homework was checked for neatness, spelling and grammatical errors, and completeness.

Vacations and trips to museums were planned to reinforce topics taught in school.

These children were read to every night.

TV and video games were tightly regulated.

Books and magazines filled their homes.

Their children attended concerts, ballet, theater productions, art exhibits, and recitals of all kinds.

These children participated in theater, dance, music, and art lessons.

These children attended summer camps that specialized in their children's talents.

In essence these parents did everything I did as a homeschool parent. Who takes the credit when these well-educated children raise the standardized scores of their government school? YES! Of course! The government school!

There are two very significant differences between the children of the scientists and engineers who attended school and true homeschoolers:

First, the schooled children did not have as much time to express their creativity or pursue their own academic interests.
Second, they didn't have as much time for play to burn off calories and build strong muscles and bones, had worse school diets, and were fatter.