One interesting thing is how widespread homeschooling is. Classified by household income, the percentage of children home schooled is essentially constant for the first three categories (under $25,000, $25,000-$50,000, $50,000-$75,000), a bit lower for the top ($75,000+) category--presumably because higher income parents have easier access to the private school alternative. By race, the rate is higher for non-hispanic whites than for blacks, but only by about a factor of two; interestingly, hispanics have about half the rate of blacks.
By parental education, home schooling percentages increase with increasing education through a bachelor's degree but are slightly lower for families where the highest parental education is a graduate degree than for those where it is a bachelor's--again, the differences are not enormous.
The one big effect is that families with two parents only one of whom work are much more likely to home school than other families--5.6% of their children are home schooled, compared to an overall average of 2.2%. And families with three or more children are somewhat more likely to home school than smaller families. Neither is surprising.
Another table on the site has data on reasons parents gave for home schooling. The most common "most important" reason was concern with the environment at other schools. The second most common, given by just under a third of parents, was "to provide religious or moral instruction." [In a comment on an earlier post, I reported those as the figures for "one reason" rather than "most important reason," which was a mistake; about 2/3 of parents gave it as one of their reasons].
All of which suggests that the common negative stereotype of home schoolers as poorly educated religious fundamentalists trying to isolate their children from the polluting effect of wicked ideas such as evolution is seriously inaccurate--no doubt such people exist, but the data suggest that they are a minority of all homeschoolers.
None of which I find terribly surprising. I am an atheist with a PhD, my wife is a mainline Christian with a masters degree. The first homeschooling family I knew, some forty years ago, contained two boys. At the time I knew them, one was the under 21 chess champion of the U.S., the other the under 14 champion.
Which is not to suggest that those cases are typical either.