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Michael W. Ferguson writes in The Inappropriately Excluded:

The probability of entering and remaining in an intellectually elite profession such as Physician, Judge, Professor, Scientist, Corporate Executive, etc. increases with IQ to about 133. It then falls about 1/3 by 140. By 150 IQ the probability has fallen by 97%!

Is that claim backed by trustworthy data?

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    @Ryan : The fact that someone cites sources doesn't mean that the accurate reflect the research on the topic. – Christian Sep 20 '16 at 17:36
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    One comment. Moderate autism is associated with high IQ. Source So the correlation in your study may be due to the relation between autism and IQ, and have no relation to intelligence. – user35897 Sep 20 '16 at 17:46
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    @Ryan That's a straw man argument. Saying that they are not necessarily good is not the same as saying they are bad. Why would he ask to begin with? Because he doesn't consider himself qualified enough to evaluate the sources himself. – called2voyage Sep 20 '16 at 18:22
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    @called2voyage I could have sworn there was things like Noteable claims and Expressing doubt in the rules of Skeptics when asking questions. I apologize if trying to see if a mostly unknown claim made by seemingly one person held either of these traits in any way by having Op provide more than a copy paste of the claim itself and a, "Is it true" sentence. – Ryan Sep 20 '16 at 18:44
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    @Ryan : I ask the question because I consider this to be a significant claim where I don't know whether it's true or isn't. That's what expressing doubt is about. I don't make up my mind based on what the article says but I ask here for an answer. Peer review of claims is a valuable concept. – Christian Sep 20 '16 at 19:45
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Executive summary of what follows:

  • No, Ferguson does not support the claim in the question title, nor the more quantitative one in the question body, with trustworthy data.
  • The claim is the result of a mathematical calculation based on assuming something (IQ among "elite professions") to be normally distributed that we have no reason to think is normally distributed.
  • Ferguson's references do not appear to give any evidence that it is normally distributed.
    • One reference does show that one particular subpopulation of the "elite professions" has an IQ distribution not obviously inconsistent with a normal distribution.
  • Incidentally: Even if the claim were correct, Ferguson's article also doesn't do a great job of justifying the "inappropriately" and "excluded" parts of the title.

What's the actual question?

In order to figure out what would count as trustworthy data to support the article's assertion, let's begin by looking at its argument. It goes like this:

  • In the population at large, IQ is normally distributed with mean 100 and standard deviation 15. (This isn't stated explicitly but it is assumed; see below.)
  • In the population of people in "elite professions", IQ is normally distributed with mean ~125 and standard deviation ~6.5.
  • If you look at the ratio of these two normal distributions -- telling you the relative likelihood that someone with a given IQ is in an "elite profession", you will see that it increases up to IQ 133 and then decreases, becoming very small by (say) an IQ of 150.
  • This indicates that while being quite intelligent helps you get into the "elite professions", being extraordinarily intelligent actually makes it less likely that you do.
  • This is a bad thing because we want smarter people in the "elite professions", for their sake and for society's.

The first claim is (at least approximately) true by definition; IQ tests are designed to give normally distributed results with that mean and standard deviation.

The second claim is the main thing that might or might not be backed by good evidence.

The third claim is simply a matter of calculation, and I believe the calculation is correct. (This is the claim in the body of the question.)

The fourth claim is a reasonable deduction from the previous ones, but we should distinguish between "makes it less likely" and "makes it more difficult" and be aware that correlation is not causation. (This is the claim that might justify "excluded" in Ferguson's title.)

The fifth claim is reasonable if the previous ones are correct, but again there are some caveats. (This is the claim that might justify "inappropriately" in Ferguson's title.)


The distribution of IQ in "elite professions"

The claim I'm going to focus on here is the second: that "elite profession" IQs are normally distributed with mean about 125 and s.d. about 6.5.

More specifically, the most contentious part of this claim -- which Ferguson assumes but never actually states -- is that IQs within the "elite professions" are something like normally distributed. This is the assumption behind the graph labelled "Excluded Hi IQ people", and the assumption behind the claim that within the "elite professions" "99.98% have IQs between 99 and 151".

I do not see any evidence that any of the references cited by Ferguson support this claim, nor does it seem likely on its face to be true.

If this claim is not true, then everything quoted in the question is unsupported: the peak at 133, the 3x falloff to 140, the 30x falloff to 150. All of these figures come from looking at the ratio of two normal distributions.

It's not, on the face of it, a claim we should expect to be correct. IQ in the population at large is approximately normally distributed, but that says nothing about the distribution in a highly-selected population like "elite professions" (whatever exactly that is taken to mean). For instance, consider the toy model where membership in those professions is selected at random from people with IQ at least 120; the resulting distribution will be very far from normal. (It will also, as it happens, have roughly the mean and standard deviation cited by Ferguson, though I don't think it's plausible that the actual elite-profession IQ distribution looks much like this.)

(Ferguson's conclusions are quite sensitive to the exact shape of the distribution at the tails. For instance, people with IQ>150 are about 0.04% of the general population; if Ferguson's claimed normal distribution is right then they are about 0.006% of the "elite professions"; if the actual relationship between the general population and the "elite professions" followed Ferguson's curve up to its peak at and IQ of 133 and then stayed there (meaning no falloff at all, no "exclusion", but the same advantage for someone with IQ 160 as for someone with IQ 133), the figure would be about 0.5% -- which is much bigger than 0.006%, but still very small in absolute terms.)

So, this claim is key to the argument, Ferguson never states it explicitly, and on the face of it it doesn't seem likely to be true. It could be true, even so. What's the evidence?


Evidence from Ferguson's references

Let's go through the references looking to see what they have to say about that claim. In the body of the article, Ferguson cites two studies.

  • Gibson & Light, Intelligence among university students, Nature 213. The first thing to say is that the title is actually Intelligence among university scientists. Ferguson summarizes their finding thus: "Gibson and Light found that 148 members of the Cambridge University faculty had a mean IQ of 126 with a standard deviation of 6.3." This is definitely misleading in that the study was specifically of scientists. Unfortunately, I don't have a copy of that article and can't find it available online, but nothing I have seen suggests that Gibson & Light look at the shape of the distribution of IQs and find it normal. [EDITED to add:] See below for more about this; another answerer found a copy of this paper and there is actually some information about the distribution in here.

  • Matarazzo & Goldstein, The intellectual caliber of medical students, Journal of Medical Education v47i2. PDF on Gwern's website. Note that this is about students rather than actual professionals. This article says nothing at all about the shape of the distribution, unsurprisingly since it's mostly aggregating the findings of other studies. It does remark that its first author has found no correlation between IQ and placement in medical examinations, which weakly suggests that if people with truly exceptional IQs are unable to become doctors this may not do much harm to the medical profession.

Other references in Ferguson's article, in order of appearance:

  • A nation deceived: how schools hold back America's brightest students, ed. Colangelo, Assouline & Gross. This is all about schools, and so far as I can tell it doesn't say anything about the IQ distribution in elite professions.
  • Understanding and being understood: the impact of intelligence and dispositional valuations on social relationships, Denissen. (Not explicitly cited in the body of Ferguson's article.) This PhD dissertation is about what its title says it's about. It doesn't say anything about the IQ distribution in elite professions.
  • Gibson & Light: discussed above.
  • Exceptionally gifted children, Gross. This looks at a small number (15 selected from a group of 40) of exceptionally-smart children, what was done with them in school, and (in the second edition) how they were doing in their twenties. It doesn't say anything about the IQ distribution in elite professions. It does purport to find (I don't have the book so can't judge how well this claim is supported) that such children have far better outcomes if they are substantially "accelerated" at school rather than staying with their equal-age peers.
  • Exceptionally and profoundly gifted students: an underserved population, Gross. Online. Same themes as Gross's book discussed above. Doesn't say anything about the IQ distribution in elite professions.
  • Exceptionally gifted children: long-term outcomes of academic acceleration and nonacceleration, Gross. Online. Same themes as the two previous references. Doesn't say anything about the IQ distribution in elite professions.
  • Matarazzo & Goldstein: discussed above.
  • Meritocracy, cognitive ability, and the sources of occupational success, Hauser. Link given unfortunately no longer works. Copy on Gwern's website. This one does talk about IQ distributions a bit, and but has little to say about "elite professions". (There are a few graphs from which we can read off approximate 10th centile, lower quartile, mean, upper quartile and 90th centile IQ for poorly-specified groups of "college professors" and "medical occupations -- MD or equiv.". We can't tell very much from that about the shape of the distribution.
  • Children above 180 IQ Staford-Binet, Hollingsworth. Online. As the title suggests, this is all about children and tells us nothing about the distribution of IQ in elite professions.
  • The making of a scientist, Roe. Copy on Gwern's website. Specifically about scientists rather than "elite professions" generally. Does have a chapter about intelligence of scientists, indicating unsurprisingly that scientists are very smart. Doesn't say anything about the shape of the distribution. If we trust Roe's rough conversions between her test results and IQs, the figures are high enough to be hard to square with Ferguson's claim that exceptionally smart people are being "excluded".
  • Intelligence and personal influence in groups: four nonlinear models, Simonton, Psychological Review 92. First page online. I don't have this paper and haven't found a copy online, but it seems to be purely theoretical: Simonton made up some models and described their implications. (On the other hand, Ferguson's article seems to cite an empirical result from it. Maybe I'm wrong about its being purely theoretical, or maybe Ferguson is misusing it.)
  • The outsiders, Towers. Provided link no longer works. Working link. Claims that very intelligent people often have difficulty in society. Doesn't say anything about the distribution of IQ in elite professions.
  • The empty promise, Towers. Described as "currently not available" in Ferguson's bibliography. I think this is it. Does give some information about distributions of IQ in various groups (though it's not clear that anything there quite corresponds to "elite professions"). They don't look very normally-distributed to me.

A very general difficulty in this sort of study

Investigations of the sort done by, say, Hauser are very likely to miss very-high-IQ people altogether, unless such people are an explicit focus of the investigation. So, for instance, Hauser looks at the population at large and breaks them down into maybe 10-30 different professional groups. In any of those groups, the extraordinarily-high-IQ people are going to be a small minority; there just aren't many of them in the population to begin with. People with IQ > 150 are about 0.04% of the population. Hauser plots the 90th percentile of large groups like "college professor", and we shouldn't expect that to tell us anything about whether people with an IQ of 150 or more can get jobs as college professors. In the United States there are about 120k people with an IQ of 150+ and about 2M college professors, so even if all the high-IQ people were college professors you still wouldn't see them at the 90th percentile of college professors. If, more realistically, 10% of the high-IQ people were college professors, then they would be about 0.6% of the college professors, and their presence would make rather little difference to the 90th percentile. Or, of course, to the mean and standard deviation.

Finer-grained data from Gibson & Light

A reference I didn't find a copy of, the Nature paper by Gibson & Light, turns out to have finer-grained information about IQ distribution than the others discussed above. See the answer by Taw -- the paper apparently has only a chart, but Taw has estimated the actual numbers by measuring the chart. For a specific group of people (scientists on the faculty at the University of Cambridge), there are figures at a resolution of 5 IQ points; they do in fact look fairly normally distributed, with roughly the mean and standard deviation used by Ferguson.

As Taw says, Gibson & Light's numbers fit a normal distribution well enough that they wouldn't entitle us to reject the hypothesis of normal distribution at the 5% (or even the 15%) level. There is, of course, quite a gap between that and citing them as evidence that the distribution actually is normal, especially at the tails (Gibson & Light have a sample size of 148, and they found no IQs above 145 or below 110). For Gibson & Light's numbers to be strong evidence of normal distribution, we would need that distribution to fit much better than any distribution differing substantially from normal, and we just don't have enough data for that to be the case.

For instance, we get an almost exactly equally good fit (measured by log-likelihood of the observations estimated by Taw from the chart in Gibson & Light) for a pdf that's parabolic but clipped where it crosses 0, with peak at 126.6 and dropping to 0 at a distance of +-15. For the avoidance of doubt, I don't think it's at all likely that the real distribution looks much like that, and if it did then it would indicate an "inappropriate exclusion" of very high IQs; the point is just that it's quite different from a normal distribution but the fit is just as good.

More to the point, we can get a pretty decent fit from a probability distribution designed to describe selection for higher IQ applied to the general population. Consider the "logistic function" a(x) = 1/2 (1 + tanh (x-t)/s). This is 0 for very small x and increases smoothly to 1 for very large x; most of the increase takes place near x=t, on a scale defined by the parameter s; for instance, we have a(t-s)~=0.12 and a(t+s)~=0.88. Suppose that your chance of getting an academic post at Cambridge doing science is a(your IQ), with parameters t=124.7 and s=6.1; so e.g. with an IQ of 100 your chance of success would be about 0.03% and with an IQ of 140 it would be about 99.3%. And suppose that we take the population at large -- IQ distributed normally, mean 100, standard deviation 15 -- and give everyone a probability of ending up a Cambridge scientist that's proportional to a(x). Note that there is definitely no "exclusion" of very high IQs here; higher is always better. How well does this fit the numbers in Gibson & Light? Worse than the best-fit normal distribution but still well enough to pass Taw's chi-squared test at the 5% level. (The p-value is somewhere around 0.07, versus 0.17 for the best-fit normal distribution.)

Evidence from the references: summary

That's all the references. So far as I can tell, Gibson & Light is the only one that offers anything like evidence that "elite professions"' IQ distribution is close to normal. The evidence it offers (1) is for one specific subset of one specific "elite profession" only and (2) from a sample too small to give strong evidence about what happens at the tails of the distribution. The numbers Gibson & Light report do look fairly consistent with normal distribution (with roughly the parameters claimed by Ferguson), but they are not strong evidence for that particular distribution over others that would lead to very different conclusions from Ferguson's. The other references offer no further support for the normal-distribution hypothesis. No plots (other than one very crude one in the last reference, which looks extremely not-normal to me and in any case is based on a very small sample), no Kolmogorov-Smirnov or similar statistical tests. We have some means and standard deviations, some 10/25/50/75/90 percentiles, but nothing more detailed and in particular nothing that looks at the shape of the tails (which Ferguson would need to justify his claim about what happens at an IQ of 150, for instance).

So, I say that Ferguson's article does not offer any "trustworthy data" to speak of for the key claim that IQs of people in "elite professions" are close to normally distributed with mean ~125 and s.d. ~6.5. And that normal distribution is essential for his conclusion.


Some other dubious things in Ferguson's argument

This isn't directly relevant to the specific claim in the question here, but since "inappropriately excluded" is in Ferguson's title I think it's worth a bit of a look at whether that language would be justified if the claim were true. Readers interested only in the specific claim under discussion can safely stop reading here.

Let's suppose arguendo that in fact Pr(in elite profession | IQ) does drop off as one looks at extremely high IQs.

  • Ferguson calls the very high-IQ people inappropriately excluded. For this to be reasonable, there would need to be actual exclusion and it would need to be inappropriate.

    • Is there exclusion? (Some other possible hypotheses: very-high-IQ people may find the "elite professions" boring, or very cleverly analyse the costs and benefits of getting into them and decide to do something that saddles them with less student debt, or something of the sort.) Ferguson offers neither evidence nor argument for this. He tells a handwavy just-so story about how "it is an artifact of a culture that fails to provide them with audience or followers ... the leaders are not persuaded and often won't even understand the advice", but telling stories is easy and giving good evidence that the stories match reality is hard, and Ferguson has done only the former.

    • Is it inappropriate? (Some other possible hypotheses: very high-IQ people don't outperform in most "elite professions" but expect better treatment or higher pay; very high-IQ people tend to have personality quirks that make them not work so effectively with others; very high-IQ people tend to have outright intellectual deficits in particular areas that make them ineffective.) Ferguson mentions that Towers and Sternberg (none of whose work is in Ferguson's bibliography) propose alternative hypotheses of this kind. Ferguson does offer a little bit of evidence that they're wrong: he quotes Roe's book mentioned above as showing that top scientists have very high intelligence. But "top scientists" is a very different group of people from "people in elite professions". It could well be (and anecdotally it seems likely to be true) that science is exceptional among professions in how valuable intelligence (in the what-IQ-tests-measure sense) is for its practice.

The nearest thing to an explanation for Ferguson's "exclusion", if it should happen to be real, that I find in his references is the one suggested by Gross's work on exceptionally intelligent children: if such a child is expected to stay with their equal-age peers up to age 18, then there's a danger that they find the educational process boring and demoralizing, which may mean that they drop out of education, or resent being with the children they have to spend time with and never learn to relate to them in healthy ways, either of which will be bad for their prospects of entering "elite professions". Ferguson does discuss this and what he says about it seems plausible; but none of it seems like a good fit for the term "exclusion".

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  • I have access to Gibson, Light (1967) and analyzed it in my answer; it actually does support Ferguson's claim. – Taw Jul 26 at 5:36
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The claim directly follows if we assume "elite professions" have a normally distributed IQ with mean = 126, standard deviation = 6.7. I will show that this a plausible hypothesis for the distribution of the IQ of University of Cambridge scientists. Namely, based on Gibson, Light's 1965 data for Cambridge scientists, we fail to reject the null hypothesis that the scientists' IQ has this distribution.

Failing to reject the null hypothesis does not mean the null hypothesis is True. Moreover, this conclusion is based on a single study which only looks at 150 Cambridge scientists in a single year. Based on these limitations, you can decide for yourself whether or not it is reasonable to assume that the IQ of "elite professions" is normally distributed with the claimed parameters.

In [Gibson and Light, Nature, Intelligence among University Scientists (1967), 441-443] they give a histogram which contains the following data (Note this is incorrectly cited as "Intelligence among university students" in the blog), enter image description here Note that I derived these numbers from the histogram using pixel measurements, and from my measurements the number of total scientists sums to 150. In the article they state there were 148 scientists spread across 33 departments in the study, so my numbers are probably off by +/-1 in various places. I report my unrounded numbers from the pixel measurements for this reason.

Based on this data, which is already binned, I used a chi squared test. The null hypothesis is that the data was generated by a normal distribution with mean = 126.5, sigma = 6.3. This is the sample mean/standard deviation, as reported in Gibson, Light. The p-value is .1674, so we fail to reject the null hypothesis (assume the standard .05 significance level). Ferguson uses mean = 126, sigma = 6.7 in his article, and I'm not sure why he uses those numbers. The difference is not large enough to make a meaningful difference. enter image description here If you want to see this calculation/plots for yourself, you can use my colab notebook. From both Pearson's chi squared test and graphical inspection, the normality assumption seems reasonable.

As for the claim in the question: It's not possible to calculate the probability of becoming elite at a given IQ, because we don't know how many "elites" there are. We can calculate the probability up to a constant though, so we can compare the probability at different IQs. Namely,

probability becoming elite at IQ = number elite at IQ / number general population at IQ
= (number elite * % elite at IQ)/ (number general * % general at IQ) 
==> % elite at IQ / % general at IQ = probability becoming elite at IQ * K

where K is size of the general population divided by the size of the "elite" professions. I'll call this probability * K the "relative score".

The general population's IQ is Normal(100, 15^2) by definition, and Ferguson's blog uses Normal(126, 6.7) for the IQ of the "elite". Based on this, we see that the relative score for IQ = 133 is 14.58, the relative score for IQ = 140 is 8.77, and the relative score for IQ = 150 is .932. Ferguson again seems to have taken some liberties with rounding, as this gives a 40% and 93.6% reduction respectively, but this is close enough to the claimed 33% and 97% to be reasonable.

For context, the score for an IQ of 100 is .0012. So someone with 150 IQ is over 750 times more likely to become "elite" compared to the average joe with 100 IQ. Someone with 133 IQ is 11780 times more likely to become "elite".

Overall, there are some inconsistencies (i.e. rounding issues, citation issue) with the blog post, but they aren't in the favor of the author, which makes me conclude that these are due to sloppiness/general confusion rather than bad intent. As stated before, you can decide for yourself whether a) 148 cambridge scientists across 33 departments are representative of "elite professions" and b) whether or not the data is convincing enough to conclude that the IQ of the scientists is normally distributed.

More generally I should point out that I don't in general agree with the blogpost and find some parts to be logically unsound. However, Gareth's answer is unfair to Ferguson, and to directly answer the question: yes, there is truthworthy data to back up the claim. I think that the biggest flaw by far in the blogpost is that it implies that high IQ individuals are somehow rejected or discouraged from the "elite professions". This is not clear and the blog seems to not give any evidence for this.

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    Testing for normality on binned frequencies with sample sizes as low as this is really weak, and the chi square test is far from the most powerful. This paper compares power, and for a beta(2,2) distribution the best tests have no more than 50% power. – Dave Jul 27 at 7:39
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    I'm not a statistician. If I knew my answer was going to get flagged I would have just posted a picture of the histogram from gibson, light. You can see just by looking at the plot of the histogram in gibson, light, that assuming the IQ distribution is normal is a plausible assumption (i.e. look at the blue lines of histogram). The top voted answer makes it seem like ferguson pulls this assumption out of nowhere. – Taw Jul 27 at 8:48
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    It seems to me that for this sort of question it's hard to avoid "original analysis" of the sort found in this answer. Ferguson makes a claim; he cites Gibson & Light who have some data relevant to the claim; one can't really answer the question without saying something about what evidence (if any) the data provide for that claim. That means doing at least a little bit of mathematics. – Gareth McCaughan Jul 27 at 13:42
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    We're looking for confirmation that the assumption of elite-IQ normality is correct, so it seems to me that it would be more appropriate to set non-normality as the null and normality as the alternative hypothesis. With the way you've set up the hypothesis test, you could simply collect very little data and fail find a significant deviation from normality. At the very least, you need a power calculation - failing to reject a null hypothesis is utterly unsurprising if the sample size isn't big enough, and really tells you very little about the validity of the null. – Nuclear Hoagie Jul 27 at 13:48
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    @GarethMcCaughan: There is actually a way to answer questions like this without original research – finding published research articles that addressed the claim. If these don'tt exist, then we may be out of options for skeptics.SE (and the claim might not be that notable after all if nobody cares to investigate it). – Schmuddi Jul 27 at 17:51

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