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<p>I’m considering the perceptive observation by JHS (post #11) that the number of AIME qualifiers is comparable to the number of SAT+ACT perfect math scores. </p>
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<p>That’s consistent with the calibration of AIME qualification as having roughly 2-3 times the raw selectivity (number of attainers) and about the same psychometric selectivity (considering that SAT is taken by non-mathematicians) as a perfect math SAT.</p>
<p>Perfect AMC-12 score is probably comparable to taking a large number, such as 20 or more, of AP exams with perfect scores. Both achievements are beyond the ordinary use of the credential, such as passing to the second stage of the olympiad or getting college credit. They require a specific “fit” of skills to the task (speed for AMC12, endurance on AP, accuracy) and often a deliberately pursued goal of attaining a maximum result. Both can involve sandbagging, such as people who doubt they can reach the IMO gunning for a perfect score on the first-round exam, or people who could have graduated high school early, staying and continuing to accumulate test credits like high scores on an arcade game. </p>
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<p>This is consistent with, and follows from, a calibration of AIME qualification as similar to SAT math perfect score. </p>
<p>The questions raised here (thread title and OP question, post #1, post #11) were about the selectivity of AIME qualification, especially first-time qualification in grade 12, and how these compare to a perfect math SAT. </p>
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<p>Some evidence was posted, as you might have noticed. First, insofar as these are effectively IQ tests, the similarity in raw number of qualifiers is one indicator. Second, the higher number of SAT/ACT perfect math scores dictates that most are coming from people who did not qualify for AIME. Only half of the AIME qualifications per year are from new individuals (the mean number of qualifications is about 2 per individual, maybe a little higher). The rate of repeat performance for SAT math 800s is much lower than that of AIME requalification. </p>
<p>Taking this into account, at most 1 in 4, and it could be lower, of SAT/ACT perfect scorers are coming from AIME qualifiers. If every single AIME qualifier gets an 800, and for each AIME qualifier, two additional AMC contestants also reach the SAT 800, it would still leave a sizable horde of non-math-specialists getting the 800s. A more realistic assumption is to extend the SAT math scale to a normal distribution extending past 800, assume that AIME qualification lies somewhat on this scale, and model the SAT outcome distribution of AIME qualifiers and near-qualifiers to assess the likely number of 800s from this population. In particular, the imperfect correlation between SAT percentiles and AMC percentiles would tend to break any assumption that the AMC contestants are monopolizing the high SAT math scores. There is an independent source of those, namely the smart non-mathematicians. </p>
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<p>They get 800 after spending years of preparation in mathematics, sometimes through competitions (“math team”) and sometimes other things. That includes: math classes, math clubs, math books, math puzzles, math websites, math camps, math games; academic summer/weekend/afterschool activities and enrichment programs; and equivalent activities in computer programming, puzzle solving, science, engineering or hacking. The ability to solve an entire math SAT develops early, but the ability to do this reliably enough to have a good chance at 800 no matter when taken, generally comes with practice at other skills such as test-taking, more advanced math, or competitions. The age at which this occurs will vary with ability, interest and specialization. </p>
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<p>There’s maybe a one- or 1.5-year difference in the time needed to learn the additional mathematics, so if first-time AIME qualification is roughly an SAT 800, an 800 in 7th grade should be similar to qualifying in the 8th or 9th grade. The data do support this.</p>
<p>The number of 800s in 7th grade (for talent testing programs) is about 1 in 1000 or less, out of 100-200000 taking the test per year over the past decade, so in the low hundreds at the very most. It could be much lower, such as a few dozen per year; College Board only published data by 50-point score intervals and there was a rapid decrease in the upper tail of the distribution. The number of AIME test takers in 8th grade and below last year was 195, with 459 separate “qualification events” in that population, and 59 of the AIME takers in 7th grade. Accounting for the probably large fraction of non-US qualifiers in the AMC data could alter the figures by a modest factor such as 1.5 or 2 but it cannot change the take-home message that 7th grade perfect math SAT is comparable in rarity to an 8th (and maybe 7th!) grade AIME qualification. The selectivity does seem to be similar between the two exam metrics, though I would never have suspected that before looking at the data.</p>
<p>This is from the 2010 AMC statistics. Qualification for AIME has gotten easier faster than the SAT math section has been weakened. In earlier years reaching AIME would have been relatively more difficult.</p>