Are Private Schools More Cost Effective Than Public Ones?

Mark J. Perry at Carpe Diem (which is quickly becoming one of my favourite economics blogs) thinks so. He states:

Average private school tuition ($6,600) was about 1/3 less than the spending per pupil in public schools ($9,620) in 2003-2004 (the most recent year available), and average Catholic school tuition ($4,254) was less than half of public school spending per student...

Private schools can educate students at a lower cost, with more teachers per 1000 students, than the public schools. Reason: Private schools must have significantly fewer non-instructional administrative employees, and therefore significantly lower administrative expenses than their public counterparts.

Dr. Perry might be right, but I don't see how he can infer that from the data he presents (in another post Dr. Perry presents data about administration levels, but it's not clear those levels are causing this cost difference). The underlying (and unstated) assumption made is that the students at the public schools are similar to the ones at private schools. But I suspect they're not - I suspect that there are a higher proportion of students with behavioural, physical and emotional problems in public schools than in private. If that's the case, then some, if not all, of the cost difference might come from cream skimming, not from differences in administration.