I don't know about the US but private schools here in the UK are rather more than that (we pay a total of about $15K a year).
According to the U.S. Census, U.S. public schools spent $9,138 per student in 2006. This varies from $14,884 in New York to $5,437 in Utah:
http://www.census.gov/Press-Release/www/releases/archives/ed...
According to the council for private education, the average private school tuition in the U.S. for 2007-2008 was $8,549 at all levels, and $6,733 at elementary level. Non-religious schools were significantly more expensive than religiously-affiliated schools:
http://www.capenet.org/facts.html
It would be hard to directly compare spending because one school might offer more than another. For example, a private school in Orange County, California, may provide laptops to all students, expensive science labs, and fancy football fields (and would have to pay a lot for land), while a public school in rural Nevada may not have any of that stuff. That wouldn't prove that private education was more expensive or less efficient, just that the people in one region wanted different things and were willing to pay more for those things.
The cost for our children at a Catholic elementary school here in the U.S. is about half that.