Thursday, May 14, 2015

"Super High" IQ Scores - Why They Are Impractical & Meaningless

Parade  magazine contributor Marilyn vos Savant was once claimed by The Guinness Book of Records to have the highest IQ ever recorded. By one measure- a test first taken by her in 1956 - it was claimed to be 228 or a mental age of 22 years and 10 months at the age of 10.  A recent article ('The 21 Smartest People In The World')  at also claimed to have a list of the "world's highest IQs" with a number even higher than vos Savant's and many with IQs higher that Albert Einstein's (160). But is this really possible? Do such high IQ scores even have any credible meaning?

 In the case of vos Savant, according to Wikipedia:

" Alan S. Kaufman ,a psychology professor and author of IQ tests, writes in IQ Testing 101 that "Miss Savant was given an old version of the Stanford-Binet (Terman & Merrill 1937), which did, indeed, use the antiquated formula of MA/CA × 100. But in the test manual's norms, the Binet does not permit IQs to rise above 170 at any age, child or adult. And the authors of the old Binet stated: 'Beyond fifteen the mental ages are entirely artificial and are to be thought of as simply numerical scores.' (Terman & Merrill 1937). ...the psychologist who came up with an IQ of 228 committed an extrapolation of a misconception, thereby violating almost every rule imaginable concerning the meaning of IQs."

 The list of the "world's smartest people" featuring over a dozen with higher IQs than 160 (for Einstein) is also daft and a classic example of what author Charles Seife would call "proofiness", discussed in his excellent book, : 'Proofiness: How You're Being Fooled By the Numbers'.  In page after page Seife decries the use of numbers not merely to lie but to baffle with bullshit. No better example of such proofy twaddle can be seen than in the website below:

But alas, super high IQ scores are also rife with proofiness. Further understanding of what I am about can be grasped by reference to the standard (normal) Gaussian distribution or "Bell curve"  on which the distribution of IQ scores is based, e.g.

Note the percentages of the population decreasing at both the high and the low ends, corresponding to the number of standard deviations  (stds) from the zero point or mean.

We see, for example, that by a 2.2 std dispersion we are effectively at a population fraction of 2 percent - which is the cutoff threshold for qualifying in Mensa. Generally, this is taken as an IQ of 133-35. (Depending on the test used). Note also, how the corresponding population applicable is decreasing as the number of stds gets larger from the central point. At about 2.6 std we are near the threshold (or just slightly beyond it) for the top one percent or Intertel members. At 3 std (about 145 IQ) one qualifies for the Poetic Genius Society.

Using a base of 300 million as the U.S. population, the Bell curve proportions would translate into 6 million who'd qualify for Mensa entrance, and 3 million who'd qualify for Intertel.  By about 3.3 stds we reach the 0.1 percent cutoff and the threshold for the Triple Nine Society. The population qualifying is one-tenth that for Intertel or 3 million/ 10 = 300,000. Divide 300,000 by 300,000,000 and you get 1,000 - hence the Triple Nines are also known as the "One in one thousand society".  

Note how already we are near the effective upper limit of the Bell curve in terms of being able to discriminate between the IQ groups. At near 4.5 stds, however, there is the Mega Society which will have a threshold at the 0.0001 % level. Or doing the same math as I showed in the previous paragraph, only 1 in a million qualifying. This would imply 300 in the U.S. population of 300 million - and with an IQ of 171. But what does that even mean, and is it useful?

As the relevant article in Wikipedia points out:

"No professionally designed and validated IQ test claims to distinguish test-takers at a one-in-a-million level of rarity of score. The standard score range of the Stanford-Binet IQ test is 40 to 160.The standard scores on most other currently normed IQ tests fall in the same range. A score of 160 corresponds to a rarity of about 1 person in 30,000 (leaving aside the issue of error of measurement common to all IQ tests), which falls short of the Mega Society's 1 in a million requirement."

In other words, there aren't even any IQ tests that currently exist which achieve the level of discrimination to identify a would-be Mega Society member! The highest score achievable (and hence measurable) within the constellation of intelligence allows no higher than 160 - which was Einstein's IQ.

Consider the mean or 0 mark which sets the "average IQ" at 100. This means half score above this level and half below. At about 4.5 std, the threshold for the Mega Society we are already in uncharted territory. This is why it is likely impossible to design a test adequate to ferret out those 300 potential members in the U.S. The Mega Society insists it has "unsupervised IQ tests that the test author claims have been normalized using standard statistical methods" but this is very doubtful.  Most psychometricians concur that by 3 std (145) to 4 std (160 IQ) one is already at the limit of the measurable validity for testing IQ in any useful way. Beyond 4 std (again, Einstein's level) the difference between such scores blurs.

Again, in considering such stratospheric IQs one ought to look not only at the std dispersion from the mean but the area under the curve. At 4.7 std the area is essentially nil. What about claims of a 300 IQ? This is even more preposterous. We are now talking about 13.33 std from the mean. As one contributor put it on an IQ- statistics site:

"An IQ of 300 has no useful meaning because there aren't enough people in the world (or intelligent beings in the history of the cosmos) to make it useful".

Or, again, no test that could possibly discriminate that single potential member from 6 billion people. (Other estimates put the proportion in even more rarefied terms, i.e. 1 in 50 billion - or nearly 1 out of every other person who's ever walked the Earth).  In effect, it means that any article bragging on one or more people with supposed IQ scores exceeding Einstein's is preposterous. Even the Mega Society entry threshold is likely a Macguffin, given that in reality there is no living human with an IQ that high that can be satisfactorily verified at the necessary level of statistical confidence.

Even if such a "Mega" person really existed (irrespective of how many claim to be Mega Society members) would one really see a qualitative difference, say from an "Einstein-level" IQ person? I doubt it. I can't see a Mega Society member being able to do any more than Einstein did, including developing the tensor calculus to use in his General Theory of Relativity.

Thus as one of the web psychometricians put it, even a 150 IQ person (near the top of most IQ test standards) would not see a significant difference between himself and a 160 or 170 IQ person. Where the real differences would be seen is where the areas registered for standard deviations are the most.  For example, between a normal IQ person and a Mensa (upper 2 percent) IQ person. Their interests, and even vocabulary would likely be so different as to approach the analogy of an alien trying to communicate with a typical earthling.

But this is precisely why Mensa as an organization was created in the first place, to establish a communal civic space where gifted individuals could meet and converse without being put down by pejoratives (Dweebs, or "geeks") from "normals".  Intertel was launched for similar reasons, though it is interesting to note that many Intertel members also belong to Mensa - simply because it is vastly larger and there are more opportunities for social exchanges.

As an interesting side note, at Monsignor Edward Pace High in N. Miami one of the first  segregations transpired after we all took an IQ test (Stanford -Binet) in 10th grade. In the immediate aftermath I noticed a loss in the diversity of the student body with many of those perceived "slower" (but often more interesting, joking and friendly) leaving Pace for good. I was also encouraged to leave, but for a different reason:  to attend Nova High where I could advance at my own pace and not be held back by a less challenging curriculum. I decided not to take the advice because it would mean 100 extra miles a day total transporting by my dad.

The takeaway from this blog post? Don't trust any claims of people with IQs over 160!

Sample questions from the "Sigma Test" similar to the one to qualify for the Mega Society

1) Several faucets were used to fill up six tanks. For one hour, all the faucets discharged water in a reservoir, which distributed it between four of these tanks: A, B, C and D. After that, for one hour, the faucets discharged water in a double funnel which directed half of the water to tanks E and F and the other half to the reservoir which, in turn, continued to distribute its water between tanks A, B, C and D. With this, tanks A, B, C and D were full. To fill tanks E and F up, it was necessary to use one faucet, which, for two hours, distributed its water between tanks E and F. After this all the six tanks were full. What was the number of faucets initially used? (Note: all the faucets had the same water flow rate and all the tanks had the same volume).
2)  Several rectangles are drawn on a plane surface in such a way that their intersecting lines form 18,769 areas not further subdivided. What is the minimum number of rectangles that must be drawn to form the described pattern?
3)  Several straight line segments are drawn on a plane surface in such a way that their intersecting lines form 1,597 areas that are not further subdivided. What is the minimum number of line segments that must be drawn to form the described pattern?
4)  1 + 10^1,234,567,890 triangles are drawn on a plane surface. What is the maximum number of areas, not further subdivided, that can be formed as these triangles intersect each other? (Contributed by Rodrigo de Almeida Rodrigues)
5)  According to Fermat’s Last Theorem, a^n + b^n = c^n has no solutions for n > 2 (a, b, c and n must be positive integers). In 1992, I proved this in a simple, yet incorrect manner. This was my reasoning: Fermat’s Theorem is a generalization of Pythagoras’ Theorem, which asserts that the sum of areas of the squares drawn on the legs (short sides) of a right triangle equals the area of a square drawn on the hypotenuse of the same right triangle (a^2 + b^2 = c^2). If we try to generalize that theorem, going from 2 to 3 dimensions (a^3 + b^3 = c^3), we have a triangular prism formed by displacement of a right triangle along an axis perpendicular to its face, as illustrated by the figure below. 
We can construct a cube on one of the three quadrangular faces of that prism. Two of those faces correspond to the legs of the right triangle (ADFB, BFEC) while the larger face corresponds to the hypotenuse (ADEC). It is possible to construct a cube on one of the faces, implying that the 4 sides of that face have the same length. This affects the whole prism, causing the cube constructed on the other face to have the same size than that constructed on the first, for if AB=BF and BF=BC, then AB=BC. In that way, no cube can be constructed on the third face, for if AC represents the hypotenuse, then AC cannot be equal to to AB. Therefore, a^n + b^n = c^n has no solution for n=3. Following the same line of reasoning, we can show that it has no solution for any number of dimensions larger than 2. What is the error in this proof?
6) A certain gear system consists of 5 concentric, superposed discs: A, B, C, D and E, which are mounted on a solid platform, taken as a stationary reference. The discs have different sizes and spin at different speeds. All the discs spin at constant rates, some clockwise, some anticlockwise. Each disc has a red dot on its surface, and initially all these red dots are not lined up. At a given moment, all the discs start to spin simultaneously, each at its own speed, without any contact between them. It takes 7 minutes for disc A, 13 minutes for disc B, 17 minutes for disc C, 19 minutes for disc D and 23 minutes for disc E to complete a full 360-degree spin. After a certain time, all the red dots were aligned, disc A being in the same position that it was 2 minutes after the discs started to spin, disc B being in the same position that it was 3 minutes after the discs started to spin, disc C being in the same position that it was 4 minutes after the discs started to spin, disc D being in the same position that it was 7 minutes after the discs started to spin, and disc E being the same position that it was 9 minutes after the discs started to spin. How much time elapsed from the moment the discs started to spin until the discs reached that configuration for the first time?
7) In 1993, in an essay about Science and Religion, I described a project regarding the possibility to build an “invisibility machine”. On describing the details, I realized that some problems were insolvable, not only because of technological limitations but also for physical reasons imposing theoretical and possibly insurmountable limits. The project starts from the central idea that in order to make an object invisible, it is necessary for an external observer looking in its direction to visually stop noticing its presence. This can be done in the following way: A sphere is constructed, and its whole external surface is covered with minute, high-resolution TV cameras and monitors. Millions or even billions of cameras and monitors are to cover the whole sphere in such a way that each monitor transmits the image captured by a camera located in the point diametrically opposite to that monitor. The result will be as shown in the figure below. 
The image of the object (blue square) is captured by a camera located in point A, which transmits the image to a monitor in point M. As a result, an observer in point O will see the blue square as if there were nothing in front of him. In that way, everything inside the sphere will be invisible to the external observer. But this scheme presents two problems. One of them can be solved in theory while the other one is insoluble. Indicate those two problems and explain why one of them can be solved but the other one cannot..
8) The porous and gray “lead” inside a pencil consists of a mixture of graphite and clay. The ratio of graphite to clay is not known. On writing on a sheet of paper, a fine layer of “lead” remains on the surface of the sheet. Describe a method for calculating the mass of “lead” in the dot of the letter “i”. You may use only US$10 to buy the material needed for the experiment..
9)  We have a cylinder with a radius of 50 cm and a tape measure 0.01 cm thick. The height of the cylinder equals the width of the tape measure. The thickness of the tape measure is invariable and one of its wider sides is inextensible. What is the minimum length of tape necessary to wind it around the cylinder 9 times, all rounds overlapping, as in a roll of scotch tape. The top and base of the cylinder may not be covered with tape. The solution must be given with 14 significant digits and it is not allowed to cut the tape or cut or deform de cylinder.
10)  Describe a practical and fast method that can be used with good precision to determine the number of words in a person’s vocabulary..     


Thomas Chatterton said...
This comment has been removed by the author.
Copernicus said...

Sorry, I still dispute a PRACTICAL size sample can be found. For IQ 300, as I noted you are looking at half of all those who ever lived. It is preposterous then to think one could find even one such person in a relatively contemporary time frame. Even the Mega society has failed to define a proper statistical basis by which to vet its test.

If you have actual sources to validate your claims, i.e. "there may be enough data in the higher standard deviations to be of significance" I'd like to see them. Please give journal, volume, authors, titles.

Copernicus said...

Here's the Wikipedia take on William Sidis:

"He entered Harvard at age eleven and, as an adult, was claimed to be conversant in over forty languages and dialects. It was later acknowledged, however, that some of the claims made were exaggerations, with a researcher stating :

'I have been researching the veracity of primary sources of various subjects for about twenty-eight years, and never before have I found a topic so satiated with lies, myths, half-truths, exaggerations, and other forms of misinformation as is in the history behind William Sidis'"

Thomas Chatterton said...
This comment has been removed by the author.
Copernicus said...

"however, your quest for a PRACTICAL size sample for IQ's at circa 250-300 cannot possibly be satisfied, because by very definition their numbers are so very low and always will be unless the world's human population increases well beyond the human species' ability to feed it! "

Actually, it's not my "quest" at all, but to me a delimiting parameter as already noted in the blog article that it is implausible that it even exists. I mean half of all the people that ever lived? And that's the conservative lower bound Granted that it is true that IQ tests display wide variability but the work of Dr. Abby Salny of Mensa (Mensa Research Journal and others shows that there is a reasonable measuring template - with "causal deterministic" parameters- that can be assessed but within limited bounds.

Going to the SIGMA society, Poetic Genius Society or other IQ groups doesn't help as one is not getting any independent assessments or research analyses. They will basically tell you what you wish to hear, or expect to hear - in order to validate their own selection strategies.

Anyway, on that note this subject is closed. Let us simply agree that many IQ claims, especially for celebs etc, ought to be treated with caution - especially when they approach the 160 limit.