The NCES released results of the National Assessment of Educational Progress (NAEP) for private schools. Here is what they found:

Students at grades 4, 8, and 12 in all categories of private schools had higher average scores in reading, mathematics, science, and writing than their counterparts in public schools. In addition, higher percentages of students in private schools performed at or above Proficient compared to those in public schools.

The report also looks at Catholic, Lutheran, and Conservative Christian schools individually. They concluded:

Where differences existed at grades 4 and 8, students in Lutheran schools generally outperformed those in Conservative Christian schools. In some grade/subject combinations, Lutheran school students outperformed Catholic school students, and Catholic school students outperformed Conservative Christian school students.