We take the total number of Advanced Placement, International Baccalaureate and Cambridge tests given at a school each year and divide by the number of seniors who graduated in May or June. I call this formula the Challenge Index. With a few exceptions, public schools that achieved a ratio of at least 1.00, meaning they had as many tests in 2015 as they had graduates, were put on the national list. We rank the schools in order of that ratio, with the highest (27.32) this year achieved by BASIS Oro Valley, in Oro Valley, Ariz., which takes the top spot for the second straight year.

I think 1.00 is a modest standard. A school can reach that level if only half of its students take one AP, IB or Cambridge test in their junior year and one in their senior year. But this year, just more than 10 percent of the approximately 22,000 U.S. public high schools managed to reach that standard and earn placement on our list. On our list, the top 220 schools are in the top 1 percent nationally, the top 440 in the top 2 percent, and so on.

2. Why do you count only the number of tests given, and not how well the students do on the tests?
Some schools brag about their high passing rates on AP or IB, meaning the percentage of test-takers who scored 3, 4 or 5 on the 5-point AP exam or 4, 5, 6 or 7 on the 7-point IB exam. Passing scores make students eligible for credit at many colleges and universities.

I decided not to count passing rates in this way because I found that many high schools kept those rates artificially high by allowing only top students to take the courses. AP, IB and Cambridge are important because they give average students a chance to experience the trauma of heavy college reading lists and long, analytical college examinations. Research has found that even low-performing students who got a 2 on an AP test did significantly better in college than similar students who did not take AP.