(I'm a nooby in probability)

So why are IQ test results normally distributed? Or more precisely what are the hypothesizes and theorems that imply this distribution?

Has it to do with the central limit theorem? (But this theorem is about the arithmetic mean of iid variables. I dont see iid variables here: I suppose it's not one person repeating the test. Is it the skills given at a person that is considered as a random variable?)

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    IQ is adjusted by fiat so that the distribution of scores is normal with a mean of 100 and a S.D. of 15. You can always map the scores so that any distribution, whatever it is, becomes normal. – Ron Maimon Jun 3 '15 at 21:25
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    @ Ron Maimon If you can adjust the mean and S.D., you can adjust the shape of the distribution too?? – LLuu Jun 3 '15 at 21:38
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    The (over a century old) history of IQ tests includes a number of dubious statistical practices, such as "removal" of inconvenient data and "re-designing" the questions to produce the "expected" results. It cannot really be said to measure "human intelligence" as we have since come to understand that term. The "defined" normal distribution is more imposed than observed. But it is so useful for some parties to "sieve" people by abusing the number that it has been difficult to get its application dropped entirely... – colormegone Jun 4 '15 at 3:07
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    @LLuuUsErI132UOutOfMemory: You adjust the distribution by defining the original score I(n) as some increasing function of the number n of correct answers, then find the empirical distribution f(I) of the unnormalized I scores in the population, and then you reparametrize I by defining a new score function g so that f(I(g)) dg/dI is a Gaussian with mean 100 and s.d. 15. You can always do it by relabling the x axis, every 1-d distribution can be reparametrized to any other. They also have baskets of gender biased questions, and they also adjust the test to make sure female and male IQ is equal. – Ron Maimon Jun 9 '15 at 2:29
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    @TonyK: IQ is normal by definition, you parametrize the raw score by brute force so that the distribution ends up normal. If you use any natural mathematical metric for difficulty of questions, like "size of search space" in a chess problem, or "number of steps of deduction" in a mathematical problem, basically anything, in the natural metric, the distribution of humans would be a power-law like distribution with a heavy tail and different individuals would perform astronomically better at some tasks than others. – Ron Maimon Jun 9 '15 at 2:34

It has been an empirically observed fact that many "naturally" observed traits, like height or IQ, are NOT empirically normally distributed. At the very least they can't be truly normally distributed because they are always non-negative. But even more than that, before non-negativity is violated, it has been observed that the "tails" (values enough standard deviations away from the mean) tend to have higher probability than predicted by a normal distribution for the population, at least for certain traits. The only thing you can say is that if you take many samples and compute the mean, then the empirical mean for the sample should be approximately normally distributed under mild assumptions if you have enough samples (this is the central limit theorem).

As an aside, if you'd like a speculative theory for why many traits appear "somewhat normal", just consider the possibility that many factors affect the trait, e.g. many genetic factors and many environmental factors. If you have many factors and their effects are additive and you don't have too crazy distributions for each factor's effect, and the factors are independent enough, then the accumulated effect should be somewhat normal basically by the central limit theorem.

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    True, but if you look at the general population trends, the bell curve approximates a normal distribution. We all know that IQ cannot be negative, but the mean is enough standard deviations away from 0 that the curve is essentially a normal distribution curve. To your point about the tails being too common, is that because sampled traits such as IQ must occur in integral sample counts $x\in \mathbb{N}$, thereby reducing the sample probability resolution to being no better than $\frac{1}{N_{samples}}$? – FundThmCalculus Jun 3 '15 at 21:25
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    @FundThmCalculus No I'm saying that if you fit a normal to a large sample of say height, then when you evaluate the likelihood of the data under that normal you will tend to get significantly lower likelihood than an equally large sample under that normal distribution, and the reason is that the outliers are either too common and/or too far away from the mean. – user2566092 Jun 3 '15 at 21:29
  • So this is not a consequence of a theorem under reasonable hypothesizes. The shape of the distribution could be asymetrical for exemple, but the results of the tests have been compiled and give an aproximate normal distribution? But then perhaps smart people avoid IQ tests, does that change something? (again: nooby) – LLuu Jun 3 '15 at 21:56
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    What you are missing is that if you don't have a definition for the x axis, you can always parametrize the x axis so whatever distribution is normal. I said it already 4 times here, this answer is incorrect. The IQ test is defined so that the population results are normal with mean 100 and s.d. 15, this replaced the earlier definition of 100 times the ratio "mental age"/"chronological age" which gave age-dependent results, but at least had an objectively defined x axis, and identified prodigies. The modern IQ "g" is just defined to be normally distributed, it's true by definition. – Ron Maimon Jun 9 '15 at 2:36
  • @RonMaimon Ok, thanks for your answer. What should I do if the answer is a comment? – LLuu Jun 10 '15 at 17:28

I suggest the distribution of IQ's may be log-linear (not log-normal). Such distributions often called a Gibbs distribution (who first applied it to the distribution of energy and built a strong foundation for thermodynamics (1878), eg Boltzmann) can be applied to like positive-definite variables that have an 'energy' connotation.

t works for me with natural remotely sensed imagery (hurricanes, sea ice, rough ocean surface, cold front occurrences), even heart beat variation BUT only above some threshold. On occasion the down (below average) side can be also log-normal (not necessarily of the same slope).

I'm looking for some data with a sample size large enough to resolve the large deviations from most probable. If anyone has a good suggestion in that regard, please pass it along.

If it turned out to be the case, and IQ's had a log-linear distribution, I would suggest that the IQ variable is acting as an 'energy'. If so I would ascribe the 'energy' to a person's ability to concentrate/focus (as in the colloquial phrase 'brain energy' which would involve the reduction in confusion/entropy associated with tasks.

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