Estimating IQ

ACT

Based on my own statistical analysis of NLSY NLSY97 and ACT ACT High School Profile: HS Graduating Class National Report data, we can compute the correlation between ACT scores and IQ (ASVAB scores) from 2002. In doing so, we find that for people with ACT scores between 25 and 36, the line of best fit between their highest composite ACT score and their IQ is

Another study did a similar analysis on the same dataset Coyle, finding a correlation of 0.76. It looks like their measure of IQ was better than the first study, which probably explains their higher correlation. The second study suggests (but doesn't outright state) that the model is something like

This is, in my opinion, the best model.

For the sake of completeness, I should mention an earlier study that was actually peer-reviewed examined data from the early 1980s Koenig and estimated the relationship between g and the sum of the ACT's English and Math scores. They found

Still, that was literally 40 years ago, so I'm skeptical its of much value now.

SAT

Coyle also used the same NLSY dataset to examine the association between SAT score and IQ. They find r~0.73. Again, reading between the lines, the implied model is something like

This is my preferred model.

Again, another study looked at the correlation between g and the math and verbal SAT test scores Scholastic assessment or g? The relationship between the scholastic assessment test and general cognitive ability in the early 1980s. They found

GRE & LSAT

I couldn't find any academic analysis linking LSAT to IQ, but Mensa accepted members with a score above 167 Mensa, suggesting a score of 167 corresponds to the 98th percentile in IQ (IQ~131). Likewise, the Triple Nine Society accepted a score of 173 Triple Nine Society, suggesting a score of 173 corresponds to the 99.9th percentile in iQ (IQ~146). These two data points suggest a linear conversion from LSAT score to IQ:

We can follow similar logic with the GRE after converting between scoring systems Staffaroni to achieve

However, I recommend not taking either model too literally. Mensa and the Triple Nine Society don't accept these test scores from modern versions of these tests. Moreover, these models imply some weird things for non-outliers. For instance, the LSAT model implies that someone who scores 154.6 on the LSAT has an expected IQ of 100. The average LSAT score is 152, and the set of students who take the LSAT almost certainly have a higher average IQ than 100.

Measurement Error

Most of the tests above correlate with IQ test scores at about r~0.8. However, IQ tests are, themselves, imperfect measures of intelligence and generally correlate with each other only at r~0.8. In other words, it appears the standardized tests above are about as g-loaded as normal IQ tests.

All the above makes sense if we broadly assume that all these standardized tests and IQ tests all correlate with g at about r~0.9. This would predict a correlation of r~0.81 between two different tests, which is basically what we see above.

For this reason, my general advice when trying to estimate IQ is to use the above models to compute a naive IQ estimate and then increase the z-score by about 11%. For instance, a 36 on the ACT naively correlates with an IQ of about 133, but after making this adjustment, it suggests an IQ of 137.

More General Strategy

Often, you don't actually have any strong evidence linking a standardized test to IQ. In this case, pretty much all you can do is

1. Assume the standard deviation (SD) of test-takers' IQs is 12 (rather than 15) - based on the fact 12 is the standard deviation generally seen within an institution Kaufman
2. Assume the correlation between test score and IQ is around r~0.8
3. Re-center the score based some kind of outside source - such as the results discussed later in the sequence

This may sound extremely simplistic, but I find the results generally line up pretty well with the sections above.

To estimate actual intelligence rather than IQ, you can assume a correlation of r~0.9 instead.

Aggregating Estimates

The math here is left as an exercise to the reader.

Suppose there is a single common intelligence factor, that each test correlates with that factor with correlation $r$, and that all the noise is independent. Suppose you take $n$ tests and record their z-scores.

The correlation ($r$) between the sum of those z-scores and your true intelligence z-score is given by

The variance in the sum of these scores is given by

and, so, slope between the sum and intelligence is

and, of course, we can simply multiply by $n$ to convert from the average score to the estimate of intelligence.

For instance, if $r=0.8$ or $r=0.9$, then the slope between the average test z-score and your intelligence z-score is given by

Given that (a) the above tests (ACT, SAT, LSAT) all correlate so strongly with IQ tests and (b) IQ tests themselves generally only correlate at about $r \approx 0.8$ with each other, it is reasonable to conclude that the above tests all generally correlate with intelligence at about $r \approx 0.9$.

So, for instance, if you take such an IQ test that correlates $r \approx 0.9$ with intelligence and score 130, the best estimate of your true intelligence is actually 127. If you take two such tests and average 130, the best estimate is 129. If you take three, it is 134. And so on.

If the correlation between each test and true intelligence are different, the math gets more complicated. However, as a special case where you have two tests with correlations $r_x$ and $r_y$, your true intelligence is best estimated by