Among high performing countries, what is the relationship between the percentage of private (and government-aided private) school students and performance on international assessments?

Below, I include the 2006 PISA mathematics scale score for each country, as well as the lower secondary private school enrollment (first percentage) and upper secondary private school enrollment (second percentage). The PISA mathematics test is administered to a stratified sample of 15-year old students from nations large and small.

Finland (548) ? 4 percent and 14 percent
Korea, Republic of (547) ? 19 percent and 49 percent
Netherlands (531) ? 83 percent*
Switzerland (530) ? 7 percent and 7 percent
Canada (527) ? 6 percent*
Japan (523) ? 7 percent and 31 percent
New Zealand (522) ? 16 percent and 27 percent
Belgium (520) ? 57 percent and 58 percent
Australia (520) ? 33 percent and 21 percent
Denmark (513) ? 24 percent and 2 percent

United States (474) ? 9 percent and 9 percent

*Based on World Bank statistics that did not differentiate between lower and upper secondary schools.

Most high performing countries had a healthy percentage of students enrolled in a private or government-aided private school sometime in their secondary school career. Switzerland and Canada are oddballs; both countries enrolled fewer private school students than the United States.

Removing the Netherlands and Canada from the cohort, an average of 21 percent of students attended a private lower secondary school and an average of 26 percent of students attended a private upper secondary school. Both exceeded the OECD, EU19, and US averages by ten or so percentage points.