## 21 September 2009

### Perfection "squared" on standardized tests

I came across an article about a student who got a perfect score on both the ACT and the SAT. (These are the two standardized tests used for university admissions in the US; generally schools on the coasts use the SAT and schools in the interior of the country use the ACT, although this is a vast generalization. The geographical separation seems to be a function of where the tests originated, in Iowa and New Jersey respectively.

This article (which I'm not linking to because I found it by googling a student, and the student is probably already not happy that this is all over the Internet) points out that less than 1 percent of students get a perfect score on each of these tests. (As you'll see below, this is quite an understatement.) I think we're supposed to come to the conclusion that less than 1 in 10000 students would get a perfect score on both.

But of course scores on these tests are positively correlated! So the probability of getting a perfect score on both tests is much higher than the product of the probability of getting a perfect score on each. (I don't think knowing that would help you on the SAT. But it's been a while. In my day they were out of 1600.)

This article indicates that 294 of the high school seniors graduating in 2008 got a perfect score on the SAT, and 514 out of 1.4 million got a perfect score on the ACT. Wikipedia puts the number of SAT takers at 1.5 million per year; let's knock this down to 1 million since some people take the test more than once and we're talking about the total number of students. So the probability that a random student who takes both tests gets a perfect score on both is something like (294/1000000) (514/1400000), which is about one in 1.3 million. The number of students taking both tests is less than this (many people only take one of the two), so assuming independence there should be less than one student per year who gets a perfect score on both tests.

But a quick glance at the Google results will convince you that there are a few students per year who pull this off.