It's not cherry-picking the data to attempt an apples-to-apples comparison. In aggregate, students educated at home or in a private school will score higher on the ACT than students educated in a public school. Relative to the population educated in a public school, however, the population educated at home or in a private school has, in aggregate, parents more highly educated and more heavily invested in their children's education. To what extent, then, the difference in achievement can be attributed to superior instruction (rather than to superior parenting) will remain in doubt unless some effort is made to control for the differences between the respective populations of students.
It should also be emphasized, perhaps, that the comparison that should concern parents is not public school vs. private school vs. homeschool, but the family's particular public school options vs. its particular private school options vs. its particular homeschool options. Not all public schools are created equal, nor are all private schools, nor or all homeschools.
For what it's worth, my wife and I have elected to use a combination of homeschooling and private schooling. The options we've had, however, are not available to all families, nor do our values necessarily correspond to those of others.