AFAIK (and wikipedia tells), this is not how IQ works. For measuring intelligence, we get an “ordinal scale”, i.e. a ranking between test-subjects. An honest reporting would be “you are in the top such-and-so percent”. For example, testing someone as “one-in-a-billion performant” is not even wrong; it is meaningless, since we have not administered one billion IQ tests over the course of human history, and have no idea what one-in-a-billion performance on an IQ test would look like.

Because the IQ is designed by people who would try to parse HTML by regex (I cannot think of a worse insult here), it is normalized to a normal distribution.
This means that one applies the inverse error-function with SD of 15 points to the percentile data. Hence, IQ is Gaussian-by-definition. In order to compare, use e.g. python as a handy pocket calculator:

from math import *

iqtopercentile = lambda x: erfc((x-100)/15)/2

iqtopercentile(165)

4.442300208692339e-10

So we see that claims of any human being having an IQ of 165+ is statistically meaningless. If you extrapolated to all of human history, an IQ of 180+ is meaningless:

iqtopercentile(180)

2.3057198811629745e-14

Yep, by current definition you would need to test 10^14 humans to get one that manages an IQ of 180. If you test 10^12 humans and one god-like super-intelligence, then the super-intelligence gets an IQ of maybe 175 -- because you should not apply the inverse error-function to an ordinal scale, because ordinal scales cannot capture bimodals. Trying to do so invites eldritch horrors on our plane who will parse HTML with a regex.

The 15 should be (15.*sqrt(2)) actually, resulting in iqtopercentile(115) = 0.16 as it should be rather than 0.079 as your expression gives, iqtopercentile(165) = 7.3e-6 (i.e. 7 such people in a city with 1 million inhabitants in average), and iqtopercentile(180) = 4.8e-8 (i.e. several hundred such people in the world).

(Note also that in python (x-100)/15 returns an integer whenever x is an integer.)

Yeah, I agree with everything you wrote here. For extra irony, I also have Mensa-certified IQ of 176. (Which would put me 1 IQ point above the godlike superintelligence. Which is why I am waiting for Yudkowsky to build his artificial intelligence, which will become my apprentice, and together we will rule the galaxy.)

Ignoring the numbers, my point, which I probably didn’t explain well, was this:

There is an upper limit to biological human intelligence (ignoring new future mutations), i.e. getting all the intelligence genes right.

It is possible that people with this maximum biological intelligence are actually less impressive than what we would expect. Maybe they are at an “average PhD” level.

And what we perceive as geniuses, e.g. Einstein or von Neumann, that’s actually a combination of high biological intelligence and many other traits.

Therefore, a genetic engineering program creating thousand new max-intelligence humans could actually fail to produce a new Einstein.

Congrats! This means that you are a Mensa-certified very one-in-a-thousand-billion-special snowflake! If you believe in the doomsday argument then this ensures either the continued survival of bio-humans for another thousand years or widespread colonization of the solar system!

On the other hand, this puts quite the upper limit on the (institutional) numeracy of Mensa… wide guessing suggests that at least one in 10^3 people have sufficient numeracy to be incapable of testifying an IQ of 176 with a straight face, which would give us an upper bound on the NQ (numeracy quotient) of Mensa at 135.

(sorry for the snark; it is not directed at you but at the clowns at Mensa, and I am not judging anyone for having taken these guys seriously at a younger age)

Regarding your serious points: Obviously you are right, and equally obviously luck (living at the right time and encountering the right problem that you can solve) also plays a pretty important role. It is just that we do not have sensible definitions for “intelligence”.

IQ is by design incapable of describing outliers, and IMHO mostly nonsense even in the bulk of the distribution (but reasonable people may disagree here). Also, even if you somehow construct a meaningful linear scale for “intelligence”, then I very strongly suppose that the distribution will be very far from Gaussian at the tails (trivially so at the lower end, nontrivially so at the upper end). Also, applying the inverse error-function to ordinal scales… why?

On the other hand, any regular reader of LW will (1) be aware that LW folks as a population are extremely smart and (2) notice that Viliam is demonstrably one of the smartest here, so the Mensa test got something right.

Of course any serious claim to be identifying people five standard deviations above average in a truly normally-distributed property is bullshit, but if you take the implicit claim behind that figure of 176 to be only “there’s a number that kinda-sorta measures brainpower, the average is about 100, about 2% are above 130, higher numbers are dramatically rarer, and Viliam scored 176 which means he’s very unusually bright” then I don’t think it particularly needs laughing at.

Well, Mensa sucks at numbers since its very beginning. The original plan was to select 1% of the most intelligent people, but by mistake they made it 2%, and when they later found out, they decided to just keep it as it is.

“More than two sigma, that means approximately 2%, right?” “Yeah, approximately.” Later: “You meant, 2% at both ends of the curve, so 1% at each, right?” “No, I meant 2% at each.” “Oh, shit.”

If you divide a standard Gaussian at the +2 sigma boundary, the probability mass to the left will be 97.5% and to the right (“the tail”) -- 2.5%.

So two sigmas don’t mean 2.5% at each end, they mean 2.5% at one end.

On the other hand, if you use a 4-sigma interval from −2 sigmas to +2 sigmas, the probability mass inside that interval will be 95% and both tails together will make 5% or 2.5% each.

Apparently, Mensa didn’t get any better at math since then. As far as I know, they still use “2 sigma” and “top 2%” as synonyms. Well, at least those of them who know what “sigma” means.

Therefore, a genetic engineering program creating thousand new max-intelligence humans could actually fail to produce a new Einstein.

Only if what makes von Neumanns and Einsteins is not heritable. Once you have a genetic engineering program going, you are not limited to adjusting just IQ genes.

AFAIK (and wikipedia tells), this is not how IQ works. For measuring intelligence, we get an “ordinal scale”, i.e. a ranking between test-subjects. An honest reporting would be “you are in the top such-and-so percent”. For example, testing someone as “one-in-a-billion performant” is not even wrong; it is meaningless, since we have not administered one billion IQ tests over the course of human history, and have no idea what one-in-a-billion performance on an IQ test would look like.

Because the IQ is designed by people who would try to parse HTML by regex (I cannot think of a worse insult here), it is normalized to a normal distribution. This means that one applies the inverse error-function with SD of 15 points to the percentile data. Hence, IQ is Gaussian-by-definition. In order to compare, use e.g. python as a handy pocket calculator:

4.442300208692339e-10

So we see that claims of any human being having an IQ of 165+ is statistically meaningless. If you extrapolated to all of human history, an IQ of 180+ is meaningless:

2.3057198811629745e-14

Yep, by current definition you would need to test 10^14 humans to get one that manages an IQ of 180. If you test 10^12 humans and one god-like super-intelligence, then the super-intelligence gets an IQ of maybe 175 -- because you should not apply the inverse error-function to an ordinal scale, because ordinal scales cannot capture bimodals. Trying to do so invites eldritch horrors on our plane who will parse HTML with a regex.

The 15 should be (15.*sqrt(2)) actually, resulting in iqtopercentile(115) = 0.16 as it should be rather than 0.079 as your expression gives, iqtopercentile(165) = 7.3e-6 (i.e. 7 such people in a city with 1 million inhabitants in average), and iqtopercentile(180) = 4.8e-8 (i.e. several hundred such people in the world).

(Note also that in python (x-100)/15 returns an integer whenever x is an integer.)

Yeah, I agree with everything you wrote here. For extra irony, I also have Mensa-certified IQ of 176. (Which would put me 1 IQ point above the godlike superintelligence. Which is why I am waiting for Yudkowsky to build his artificial intelligence, which will become my apprentice, and together we will rule the galaxy.)

Ignoring the numbers, my point, which I probably didn’t explain well, was this:

There is an

upper limitto biological human intelligence (ignoring new future mutations), i.e. getting all the intelligence genes right.It is possible that people with this maximum biological intelligence are actually

less impressivethan what we would expect. Maybe they are at an “average PhD” level.And what we perceive as geniuses, e.g. Einstein or von Neumann, that’s actually a

combinationof high biological intelligence and many other traits.Therefore, a genetic engineering program creating thousand new max-intelligence humans could actually

failto produce a new Einstein.Congrats! This means that you are a Mensa-certified very one-in-a-thousand-billion-special snowflake! If you believe in the doomsday argument then this ensures either the continued survival of bio-humans for another thousand years or widespread colonization of the solar system!

On the other hand, this puts quite the upper limit on the (institutional) numeracy of Mensa… wide guessing suggests that at least one in 10^3 people have sufficient numeracy to be incapable of testifying an IQ of 176 with a straight face, which would give us an upper bound on the NQ (numeracy quotient) of Mensa at 135.

(sorry for the snark; it is not directed at you but at the clowns at Mensa, and I am not judging anyone for having taken these guys seriously at a younger age)

Regarding your serious points: Obviously you are right, and equally obviously luck (living at the right time and encountering the right problem that you can solve) also plays a pretty important role. It is just that we do not have sensible definitions for “intelligence”.

IQ is by design incapable of describing outliers, and IMHO mostly nonsense even in the bulk of the distribution (but reasonable people may disagree here). Also, even if you somehow construct a meaningful linear scale for “intelligence”, then I very strongly suppose that the distribution will be very far from Gaussian at the tails (trivially so at the lower end, nontrivially so at the upper end). Also, applying the inverse error-function to ordinal scales… why?

On the other hand, any regular reader of LW will (1) be aware that LW folks as a population are extremely smart and (2) notice that Viliam is demonstrably one of the smartest here, so the Mensa test got

somethingright.Of course any serious claim to be identifying people five standard deviations above average in a truly normally-distributed property is bullshit, but if you take the implicit claim behind that figure of 176 to be only “there’s a number that kinda-sorta measures brainpower, the average is about 100, about 2% are above 130, higher numbers are dramatically rarer, and Viliam scored 176 which means he’s very unusually bright” then I don’t think it particularly needs laughing at.

It was not my intention to make fun of Viliam; I apologize if my comment gave this impression.

I did want to make fun of the institution of Mensa, and stand by them deserving some good-natured ridicule.

I agree with your charitable interpretation about what an IQ of 176 might actually mean; thanks for stating this in such a clear form.

Well, Mensa sucks at numbers since its very beginning. The original plan was to select 1% of the most intelligent people, but by mistake they made it 2%, and when they later found out, they decided to just keep it as it is.

“More than two sigma, that means approximately 2%, right?” “Yeah, approximately.” Later: “You meant, 2% at both ends of the curve, so 1% at each, right?” “No, I meant 2% at each.” “Oh, shit.”

What? 2 sigma means 2.5% at each end.

That sentence is imprecise.

If you divide a standard Gaussian at the +2 sigma boundary, the probability mass to the left will be 97.5% and to the right (“the tail”) -- 2.5%.

So two sigmas don’t mean 2.5% at

eachend, they mean 2.5% atoneend.On the other hand, if you use a 4-sigma interval from −2 sigmas to +2 sigmas, the probability mass inside that interval will be 95% and both tails together will make 5% or 2.5% each.

Apparently, Mensa didn’t get any better at math since then. As far as I know, they still use “2 sigma” and “top 2%” as synonyms. Well, at least those of them who know what “sigma” means.

Only if what makes von Neumanns and Einsteins is not heritable. Once you have a genetic engineering program going, you are not limited to adjusting just IQ genes.