I read one of those papers. Only between 10% and 25% of homeschool families completed the test. And that's of families that could be located because they had previously interacted with major homeschool testing companies; there isn't a national list of homeschooled kids and many parents homeschool precisely to avoid standardized testing. Plus in many cases the parents were the proctors. This is hardly a robust finding.

Site: National Homeschool Education Research Institute

First independent search result:

> The US-based NHERI describes itself as a leading research institute in the field of homeschooling. However, its neutrality is often questioned.

Asking NHERI to evaluate the effects of homeschooling is like asking Focus on the Family whether children do worse when raised by gay couples.

Aside from the horribly biased source, standardized testing scores are a terrible metric for judging education levels.