As usual, causation is not correlation, and consciously deciding to take your children out of public schools and either homeschool them or put them in an alternative school show's a
significant amount more involvement in your child's life than is the norm.
Also, that study was produced by an organization, (http://www.nheri.org/) that is specifically dedicated to producing studies about homeschooling. The titles of their reports all allude to defending the idea of homeschooling, and they put out guides to putting it into practice. The children for the study were "almost 12,000 home-school students from all 50 states who took three well-known standardized achievements tests", which aren't required for children to take, unless possibly if they live in Iowa or California. It doesn't seem unreasonable to think that poorly performing homeschooled children would not take these tests, since they have nothing to gain from them.
For what it's worth, all the anecdotal evidence I have from growing up and knowing a number of home-schooled children, it's certainly not guaranteed to work better than public schools, and I've never met an intelligent, thoughtful person to come out of it. Not that amazingly smart people can't result, but I've never met them, and I feel like that's the attitude you're up against. I hope there are more and better stats, because presenting ones like yours above will only steel the negative opinion of most people towards homeschooling.