I’ve designed a mental calculation test to study it with Rasch analysis. It consists of 65 items of widely varying difficulty, to be solved in 20 minutes. If you’re interested in taking it, e-mail me at the e-mail address on the Contact page, and I should respond within ~12 hours.

# Tag Archives: math

# 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:

- An IQ test which yields a score at the interval or ratio level of measurement.
- 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:

- 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.
- Perform Rasch analysis on the test.
- 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.

# In His own image

If you describe reality in mathematical or logical terms, as for instance in the Cognitive-Theoretic Model of the Universe, it’s natural to think of the cosmos as the output of a function in some extended metaphysical sense, which brings a whole new meaning to Genesis 1:27: “So God created mankind in his own image, in the image of God he created them; male and female he created them.”

Mind you, I don’t attribute any particular theological or philological significance to this textual association. It’s a striking coincidence which may incite further thought, nothing more.

# Pure math and correspondence to reality

I’ve never been able to get interested in pure math because it fundamentally fails the test of *adaequatio rei et intellectus*. You expect me to buy that you can rearrange 1 sphere to make 2 or that 0.999… == 1? It’s bad enough that these abstractions obviously can’t be realized in the physical world, but what’s worse is that they’re provably correct statements within the context of the relevant mathematical systems. What that shows is nothing more or less than that those logical systems don’t correspond to reality, which to me is the fundamental drive for all philosophy in the extended sense: theorizing and speculating for the sake of it, pondering about the mysteries of the universe until they are mysteries no longer.

Just to be clear, **I am not claiming that 0.999… =/= 1 within the standard formulation of the real number system**. What I **am** claiming is that I have zero interest in working with abstractions according to which 0.999… == 1, as that requires a completed infinity and a physical infinity can never be completed.

I’m also well aware that pure mathematics does sometimes give rise to practical applications, sometimes ones that couldn’t have been foreseen when the underlying concept was originally formulated. But what I’m talking about here is pure math for the sake of pure math. Maybe some people enjoy that. That’s fine.