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Apr 5 '20 at 17:36 history bounty ended J. Mini
Mar 30 '20 at 18:02 comment added Bryan Krause I'd also say that it isn't that it must be this way. If there was some single dominant factor in IQ, the distribution might not be normal at all: it could be bimodal, for example. But we do observe that it is approximately normal, and we should not be surprised to find this.
Mar 30 '20 at 18:00 comment added Bryan Krause @J.Mini Not the raw test score, the underlying thing that is being measured. But yes, IQ distributions get fiddled with: that's the normalization procedure. When you design an IQ test, you don't yet know how the scores on the test will map to IQ scores. You design the test, give it to lots of people, and use the distribution you observe to decide what score is "100" (the median or mean score) and what score is "85" (one standard deviation below the mean, or an equivalent percentile).
Mar 30 '20 at 17:51 comment added J. Mini So we assume that the raw test score is a sum of independent random variables that we do not know the distributions of. But as it's a sum of independent random variables, it must have some normal distribution with unknown parameters. We then fiddle around with that a bit to get the nice IQ graphs that we all know and love. Is that right?
Mar 30 '20 at 17:37 comment added Bryan Krause @J.Mini The factors being summed/averaged are not the things on the test, it's the underlying construct that IQ is meant to measure. It's the sum of all the genes with a role in cognition, all the books read to you as a child, the diet you eat, the amount of lead you absorb, etc etc.
Mar 30 '20 at 17:27 comment added J. Mini I'm unconvinced. If the raw test score were the average of some independent factors, then we might have a shot at using the CLT, but sums don't behave as nicely. You could normalise, but I doubt that we know the population parameters.
Mar 30 '20 at 17:17 history edited Bryan Krause CC BY-SA 4.0
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Mar 30 '20 at 17:09 comment added Bryan Krause @J.Mini Quoting the first sentence of the linked Wikipedia article: "In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a bell curve) even if the original variables themselves are not normally distributed." See also aidanlyon.com/normal_distributions.pdf
Mar 30 '20 at 17:05 comment added J. Mini "Complex traits are predicted to have a roughly normal distribution based on the central limit theorem" - which version? The classical one that I know and love seems insufficient. Have I missed a lemma?
Mar 30 '20 at 16:08 history answered Bryan Krause CC BY-SA 4.0