## Monday, November 03, 2003

### (for stats fans only) a complete aside about genetics and iq

My recreational-occupational reading this past week has been on intelligence testing. Far more could be said about the issue than anything I would ever bother to write up for the weblog. But, anyway, one issue that's been gnawing at me is the suspicion that statements to the effect that "50% of IQ is genetic" is often read as implying that IQ variation is much more constrained than it actually is. This led me to do some math to figure out what actually is implied by the statement.

Imagine that IQ is "real," that it is normally distributed, that we have some way of measuring it exactly, and that we can also exactly measure the contribution of genes to IQ. We can focus on people who have genes that are exactly average for the part of IQ that is genetic, and we can ask what kind of range the middle 95% of IQs these people with exactly-average-genes would have relative to the entire population. We can do this by looking at the expected "typical variation"--that is, how well an "overachiever" does who is at the very top end of this middle 95% compared to an "underachiever" with the same genes who is at the bottom end of this middle 95%. If IQ was 100% genetically determined, then the concept of "over" and "underachiever" wouldn't even have any meaning, because everybody with exactly-average genes would have exactly-average-IQs, meaning that our overachiever and underachiever would both score in the 50.0th percentile. On the other hand, if IQ was 0% genetically determined, then the distribution of IQ among people with exactly-average-genes would be the same as the distribution of the population as a whole, meaning that our overachiever would score in the 97.5th percentile and our underachiever would score in the 2.5% percentile.

So, if IQ was not-at-all genetic, there would be 95 percentage points of difference between our over- and underachiever with exactly-average-genes that we would need to explain by nongenetic factors, while if it was all genetic there would be 0 points of difference left to explain. The question is what the difference between the over- and under-achiever would be if IQ was 50% genetic, given what is statistically meant when people say "50% genetic." The answer, after a drumroll, is that the overachiever would score around the 91.3rd percentile, and the underachiever would score around the 8.7th percentile. So where there were 95 percentage points of variation difference under the not-at-all genetic scenario, there are still more than 72 percentage points out there to be understood in the fifty-percent-genetic scenario.

Saying that a trait is 25% genetic may seem like a lot, but for this same logic the overachiever-with-exactly-average-genes would score in the 95.5th percentile for the overall population and the underachiever would score in the 4.5th percentile--91 percentage points of difference compared to the 95 points if the trait was not-at-all genetic. Even when a trait is 75% genetic, which is about as high a claim about heritability as one sees, there would still be a difference of more than 66 percentage points between our overachiever (who would score in the 83.3rd percentile overall) and our underachiever (who would score in the 16.6th percentile overall).