Taking the Calculus BC test already puts you in the upper echelon of high school math students though. According to a quick google only 15% of high schoolers will take the BC exam, so getting a 5 is something only 5% of students achieve.
That's the student providing 2.7 bits of information ("I want to take this exam") and getting an additional 1.4 bits from the result. By contrast, being valedictorian is about 10 bits of information, and being in the 99th percentile on a test (1510+ on the SAT) is 6.6 bits. Scoring 1600 on the SAT is 12 bits.
So the SAT can easily give you 7 times as much info as the AP BC, and it's more about your aptitude than your achievement.
You don't need information theory to tell me that 1% is less than 5%. Obviously scoring in the top 1% of a test everyone takes provides more information than getting a 5 on an AP test. But that 5 still provides, as you put it, 4ish bits of information, which is a lot more than the 0 you have without it or other standardized tests.
How many bits do you really need to know that you're a good candidate for continued education? Is being in the top 5% really not enough?
Zero bits, because everyone is a good candidate for continued education.
From the point of view of MIT, admitting you has a significant downside: they have to turn down someone else, someone who might have been able to do better research than you do and donate more as an alumnus. But from the point of view of an individual person, more education is always better, because your alternative to education is not admitting someone better; your alternative to education is ignorance.
(Schooling may not be the best way to get that education, however... far too many people opt for ignorance after their schooling.)
I didn't say the 5 gives you 4ish bits of information. I said it gives you 1.4 bits of information. That's only 35% of 4, not very 4ish at all. It's barely in the order of magnitude centered on 4 (1.26 to 12.6).
> I didn't say the 5 gives you 4ish bits of information. I said it gives you 1.4 bits of information.
Which is obviously wrong. The self-selection does provide a lot of information. Students who believe themselves unlikely to score well are unlikely to take the test, which is why so many students get a 5.
No, I was considering them as happening to potentially different people at potentially different times. It's definitely true that if you somehow know you have scored 1510+ then finding out that you scored 1600 only gives you an extra 5.6 bits of information, and if you know that you scored 1600 then the fact that you scored 1510+ gives you 0 additional bits of information. And the correlation with being valedictorian is also pretty strong.
Scoring 1600 intersection being valedictorian definitely does not give you 22 bits of information. Maybe 13.
Also being valedictorian gives you less information than 10 bits about your relative countrywide or worldwide standing.
