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On his blog, physicist Steve Hsu wrote the following:

Assuming a normal distribution, there are only about 10,000 people in the US who perform at +4SD and a similar number in Europe, so this is quite a select population (roughly, the top few hundred high school seniors each year in the US).

If you extrapolate the NE Asian numbers to the 1.3 billion population of China you get something like 300,000 individuals at this level, which is pretty overwhelming.

Can you explain Steve's statement in plain English – to non-statisticians using only common arithmetic operators like $+$, and $-$?

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    $\begingroup$ Are multiplication and division allowed? $\endgroup$ Commented Jun 15, 2016 at 1:18
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    $\begingroup$ To whom it may concern: Nothing about this question seems unclear to me. I don't see that this needs to be closed. $\endgroup$ Commented Jun 15, 2016 at 1:55
  • $\begingroup$ See @Dimitriy V. Masterov's comment. I thought we were looking for self contained questions and not ones that relied on external links. There's no way to answer this without reading the blog post. $\endgroup$
    – John
    Commented Jun 15, 2016 at 4:34
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    $\begingroup$ There are several problems with this reasoning: (1) distribution of IQ scores is not prefectly normal (espetially in tails), (2) there are cultural and social factors influencing scores, so they may not be comparable, (3) the tests are designed rather tu measure intelligence of "average" people, not geniuses (otherwise there would be too many unanswerable questions for non-geniuses) so they do not provide accurate estimates about the "tails" of distribution (i.e. the geniuses and the intellectually disenabled). I'd say that such estimation is a very rough approximation (in either direction). $\endgroup$
    – Tim
    Commented Jun 15, 2016 at 11:57

1 Answer 1

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Steve Hsu is using the augmented 68–95–99.7 rule to calculate what fraction of the population lies within 4 standard deviations of the mean, assuming IQ has a normal distribution.

Given how these tests are constructed, the mean IQ is around 100 with standard deviation of 15. Standard deviation is a standard measure of spread for data (denoted by the Greek letter $\sigma$). If it is small, everyone's score will be clustered tightly around $100$. If it is large, scores will be more dispersed.

Using the Wiki table linked above, we can see that about 0.999936657516334 of the population will have IQ between $100-4 \cdot 15=40$ and $100+4 \cdot 15=160$ (plus or minus 4 standard deviations from the mean). That leaves $$1-0.999936657516334=0.00006334$$ with scores below 40 and above 160. We only care about geniuses, so that gets cut in half to $0.00003167$ (since the distribution is assumed to be symmetric). If the US has a population of 322 million, that gives us $0.5 \cdot (1-0.999936657516334) \cdot 322,000,000 = 10,198$ geniuses.

To get the Chinese numbers, he's assuming that they have the same standard deviation, but a mean that is $0.5$ standard deviations higher (so $107.5$). This is grounded in the NE Asian PISA tests results, which are more of a scholastic achievement test rather than a test of IQ. The two assumptions are that achievement score distribution looks like the IQ distribution and that the Chinese resemble NE Asians.

Assuming this is the case, this means that to make it over 160, you only need (160-107.5)/15=3.5 standard deviations instead of 4. Using the 3.5 $\sigma$ row in the Wiki table, this gives $$0.5 \cdot (1-0.999534741841929)\cdot 1,300,000,000=302,418$$ geniuses, which is fairly close to SH's estimate.

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  • $\begingroup$ That doesn't get your 300,000 Chinese geniuses though. More information from the article should be included in the question. $\endgroup$
    – John
    Commented Jun 15, 2016 at 1:34
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    $\begingroup$ @John Based on the PISA results, he's assuming that they have the same standard deviation, but a mean that is .5SDs higher (so 107.5). This means that to make it over 160, you only need (160-107.5)/15=3.5 standard deviations instead of 4. This gives .5*(1-0.999534741841929)*1,300,000,000=302,418, which is close to SH's estimate. $\endgroup$
    – dimitriy
    Commented Jun 15, 2016 at 2:12
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    $\begingroup$ That should probably be in your answer since A) it isn't in the question; and B) it's very likely the questioner really wanted to know about the large discrepancy. $\endgroup$
    – John
    Commented Jun 15, 2016 at 4:33
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    $\begingroup$ Thank heaps. I'm stuck in the outback of Northern Thailand without access to statisticians. $\endgroup$ Commented Jun 16, 2016 at 4:49
  • $\begingroup$ @GodfreeRoberts Glad to help. If this answered your question, please select this as the answer. $\endgroup$
    – dimitriy
    Commented Jun 16, 2016 at 5:14

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