The Two-Sigma Advantage Hypothesis

Here I present my hypothesis that an activity will be trivially, insultingly easy if you’re 2 standard deviations above the mean of people who engage in it. This means 2 standard deviations of the primary trait which confers success in that activity. Also, for convenience, I’m using normalized distributions against the general population, rather than real and potentially non-normal distributions or standard deviations of the group itself.

I formulated this in the context of IQ and academic success, but it extends to other traits and fields. For example, military service:

The largest impediment for today’s young people is health problems — specifically, obesity. Twenty-seven percent of young people in that age group aren’t eligible to join the military because of obesity, the report states, with another 37 percent ineligible due to other health problems such as asthma or joint problems.

If 64% of young adults are physically unfit for military service, then the average military-fit young adult is at the 82nd percentile relative to young adults. That will almost certainly be even higher among the general population because young adults are at the time of their life where they have the highest (potential, and often actual) physical fitness. Let’s round up to a clean 85th percentile. Then the average military-fit young adult has an F.Q. (“Fitness Quotient”) of 115. Therefore, we estimate that with an F.Q. of 145 (~99.7th %ile), you’ll be able to pass basic training without feeling particularly strained. Since almost everyone with an F.Q. of 145 or more will be male, we’ll double that and say this is the 99.4th %ile for American males.

Can anyone think of other examples which would lend support for or against this hypothesis?


A hypothetical method for estimating IQ rarity on the right tail

The distribution of IQ, like that of many other quantitative traits in nature, appears to approximate a normal distribution around the mean, but deviates from normality beyond roughly plus or minus 2 standard deviations from the mean. For example, a score four or more standard deviations above the mean would occur a rate of about 1 in 30,000 if IQ were normally distributed, but such scores are far more common than that in practice. Furthermore, we can only estimate how many standard deviations above the mean that score truly is, since we have no way to directly measure “absolute” intelligence levels, only relative ones. This poses a serious obstacle to meaningfully estimating the rarity, relative to the general population, of an IQ score far above the average (and also below the average, but that’s not what I’m concerned with here).

This barrier could be overcome by combining two possible psychometric tools:

  1. An IQ test which yields a score at the interval or ratio level of measurement.
  2. A theoretical distribution which closely approximates the actual distribution of IQ in the general population, even at extreme heights.

As far as I know, neither of these things truly exist, but they both have potential substitutes which might be accurate enough to be scientifically meaningful.

Tool 1 manifests as Rasch scoring. This has already been extensively studied and even implemented in freely available psychometric test analysis software, so I won’t discuss it further here.

Tool 2 is a hitherto a subject of arcane speculation, and I’m not aware of any studies into the true distribution of intelligence at the high end. However, by combining statistical theory with knowledge about the biological underpinnings of IQ, it may be possible to determine a distribution which corresponds reasonably well to reality. Cyril Burt, studying this question in 1963, concluded that the IQs of schoolchildren matched well with a Pearson Type VII curve. Because the IQs of adults and children undoubtedly have the same biological underpinnings, like number of neurons, glucose processing efficiency, etc., I suspect that this curve would, if its parameters were correctly adjusted, also closely approximate the true distribution of adult intelligence.

Because finding the correct parameters, even roughly, for such a curve is currently beyond my research capacity, I instead propose using the log-normal distribution, for which a conversion table from the normal distribution can be found at the bottom of this webpage.

Therefore, my proposed approach for estimating the true rarity of an extremely high IQ score:

  1. Design an IQ test with generous ceiling for exceptional scorers and administer it to a sample of people known to have high IQs, possibly along with an “anchoring” sample of people with less than high IQs.
  2. Perform Rasch analysis on the test.
  3. Convert the person’s Rasch score on that test, in estimated standard deviations above the mean of the general population, to a log-normal rarity score.

Example: Person A takes this test and scores 5.33 standard deviations above the mean of the general population in “absolute” intelligence, as estimated by their Rasch score. This corresponds to a “normalized” IQ of 180. According to our normal to log-normal conversion table, this score is as rare as an IQ of 162.6 would be if intelligence were perfectly normally distributed, giving us a final rarity estimate of about 1 in 60,000.

Obviously, this is all preliminary at best. The details certainly need to be ironed out, and even the fundamental ideas may turn out to be flawed. I’m currently thinking of this as the “Bob Semple tank” of right-tail psychometrics: it doesn’t fully work, but I don’t see anyone else coming up with better ideas.

Edit (April 8, 2021): This article was inspired by a letter from Grady Towers to Darryl Miyaguchi.


Shiny IQ

The probability of someone having an IQ of 155 of higher, assuming a mean of 100 and standard deviation of 15, is 1 in 8,137.

The probability per encounter of a wild Pokémon in a main series Pokémon game from Generations II to V being shiny is, with a few exceptions, 1/8,192.

Hopefully a comparison like that will help readers, especially those in a certain age bracket, to understand how exceptional such high IQs are by statistical necessity, and how skeptical you should be whenever figures like that are casually tossed around. You can easily play through all of Pokémon Emerald, Diamond, etc., without encountering a single shiny Pokémon.

I probably played those games for a total of hundreds if not thousands of hours, but that was before I was a confident Google user, so I didn’t know what shiny Pokémon were. I wonder how many I missed without knowing…


Typing speed and IQ

I wonder what the correlation between typing speed and IQ is. I doubt it would be very high in the general population owing to the confounding variable of computer proficiency, although that probably correlates somewhat with IQ as well. But as for plateau speed for experienced typists, I bet the correlation is at least moderate, considering how much it resembles a simple “speed” task like on the Processing Speed Index of the Wechsler tests.

I’m one of the fastest typists I know, way faster than even most professional transcriptionists, but still nowhere near the world’s elite. Here are some of my TypeRacer statistics:

Avg. speed (last 10 races): 124 WPM

Best race: 157 WPM

Rank (WPM percentile): 99.8% [remember that this is relative to people who play a competitive typing game, so the percentile in the general population would almost certainly be far higher than even this]

But all of this still pales in comparison, at least from a simple numerical perspective, to my 200 WPM run on the “captcha” task, which allows for some uncorrected typos:

200 WPM at 97% accuracy on the TypeRacer captcha test