Intelligence Filters

By Fred Vaughan

The objective of reaching high IQ ceilings with standardized tests for intelligence to qualify members into societies such as Prometheus is complicated by severe statistical problems. I will ignore here the difficulties associated with the actual construction of the tests that do not saturate with high scores for testees with moderate intelligence and assume that we are dealing exclusively with such properly constructed tests and testing procedures. We are left with the problem that selecting testees at random from the general population would require millions of individual testees to achieve the desired confidence of percentile rankings for norming at the 4-sigma level and higher. This difficulty is largely circumvented, however, by implementing some form of advanced screening of the testees to augment the random selection process. Of course, to be helpful the characteristics of the screening process must be accurately known. This is in essence what was done, whether purposely or inadvertently, to facilitate the norming process of the LAIT and Mega tests.

If an IQ test were administered to individuals selected at random off the street, one would expect approximately one in thirty thousand of these people to score at the 4-sigma level. Therefore, in order to get meaningful norming data (a rule of thumb in statistics being that a sample size must be on the order of 100 in order to make statistically valid statements concerning the sample) to calibrate the test at the 4-sigma level, one would need to administer the test to three million people. An extremely difficult requirement. However, if the test results that are used in norming the test are the scores of individuals who have been pre-selected for high intelligence, the number of required testees can be dramatically reduced.

How Intelligence Filters Work

In the previous article Paul Maxim has indicated that the average IQ’s of individuals who have taken the SAT is 1.33 times that of the population in general. (Actually, what he said was, "…SAT testees are slightly more intelligent than the population as a whole, by a factor of about 4/3, …" which I have interpreted as stated above.) Whether the actual number is 1.33 (which common sense cautions me to doubt!) or not I will not debate; it is certainly greater than unity because extreme pressures are definitely brought to bear on the brighter high school students in our civilization to take the test as a prerequisite for matriculation at a reputable college or university. Now Paul has raised (by having supposed an incorrect answer to) the unformulated question, "How much larger is the percentage of SAT testees whose IQ’s are at the 4-sigma level than for the population at large?" He concluded it would also be larger but only by the same factor of 1.33, but that is in error; it must, in fact, be larger than this by orders of magnitude if he is correct about the mean.

Selection as exemplified by having taken the SAT imposes a conditional probability (See the note on Bayes’ theorem above.) on the selected individual’s likelihood of possessing the characteristic to a given degree. For example, if it is learned that an individual taken at random from the general population has previously submitted to taking the SAT, his likelihood of having an IQ at the 2-sigma level (roughly 133) or higher suddenly jumps from 0.02 to 0.5 (very like the collapse of a wave function when an observation is made in quantum mechanics). The two probabilities are what the terms "2-sigma" on the standard IQ distribution and "mean" of any distribution tell us. The conditioned percentage is roughly twenty five times higher than for individuals about whom we know nothing.

The IQ values of individuals satisfying the condition of having taken the SAT are not necessarily (and would typically not be) normally distributed with regard to intelligence. We will assume with Paul that whatever the form of the distribution, it has a mean of 133! The condition of having taken the test must by any valid heuristic argument be a filter for high intelligence such that the more intelligent one is, the more likely he/she will be to have taken the SAT. One could not sample at random and include individuals based on a single condition and get a distribution whose mean is so considerably different than the parent distribution otherwise; intensive selection is necessarily involved. The fact that the mean is so different from the parent population virtually assures that the resulting distribution, in addition to having a higher mean, has progressively higher percentages of individuals achieving the upper 1-, 2-, 3- and 4-sigma IQ levels. As we have seen, the percentage at the 2-sigma level is already 25 times higher. So that Paul’s reaction to the percentage of such conditioned testees outnumbering the percentage of unconditioned testees by a factor of 1.33 is certainly in error. How much greater the percentage is in comparison to that of the parent population depends on the actual degree of selection for IQ manifest by this condition. The percentage of SAT participants that actually have IQ’s above the 4-sigma level is not to be obtained by deductive reasoning based on the conditioned mean as Paul attempts; it is an empirical fact that may or may not be available to Paul and/or Ron but it will never be deduced by pure reason from the mean! (The fact that we were able to make such an assessment at the 2-sigma level was the result of the happy coincidence of the mean of the SAT distribution --according to Paul-- corresponding to the 2-sigma level of the parent distribution.) However, as an exercise I have tried to determine a reasonable heuristic model of possible SAT selection filters for didactic purposes. I have defined the conditional probability of an individual taken at random who has an IQ of x having been "selected" to take the SAT as the function, h(x,sigma,xo). The function has been parameterized such that the effects of both the level (xo) and the degree of selection (sigma) about this level can be investigated. The representation is as follows:

h(x,sigma,xo) = [Integral from 0 to x of f(x,sigma,xo)]

where f(x,sigma,xo) is a "Normal" distribution given by the equation:

f(x,sigma,xo) = [1/(2 pi)1/2 ]exp[- (x - xo)2/2 sigma2 ]

So that h(x) is being represented by an "error" function. (Later I will investigate a somewhat different form and show that for realistic assumptions there are certain inevitable results which negate Paul’s hypothesis.) This functional family of curves is plotted in figure 1 for a range of sigma and xo values with each curve supporting the purported SAT distribution mean. The forms which slope more gradually to the right seem intuitively more like what the selection pressure must actually be.

[The MathCad plot could not be brought into the on-line version so please refer to the hard copy Gift of Fire.]

Figure 1: Family of Intelligence Filters Resulting in a Mean of 133

The resulting conditioned distribution of intelligence of those individuals who have been selected from the general population to take the SAT, for example, would be obtained from the following equation:

DSAT(x,sigma,xo) = h(x,sigma,xo) f(x,16,100)

in accordance with Bayes theorem where f(x,16,100) is just the standard distribution of intelligence for the general population. To normalize the conditioned distribution it must be divided by the integral of the distribution over the entire range of intelligence as follows:

NDSAT(x,xt,sigma) = DSAT(x,sigma,xo) / I(sigma,xo)

where

I(sigma,xo) = [ integral from 0 to infinity of DSAT(x,sigma,xo) ]

and for example, I(10,139) := 0.01937 and I(5,120) := 0.105, etc..

The mean of the normalized conditioned distribution can be determined as:

m(sigma,xo) = [ integral from 0 to infinity of x times DSAT(x,sigma,xo) ]

where for example the means, m(15,155), m(12.5,146), m(10,139), m(7.5,134), etc. all equal 133, as shown in figure 2. Since 133 is the purported mean of the SAT distribution, (s,xo) pairs such as (15,155), (12.5,146), (10,139), (7.5,134), etc. have been chosen in the examples here as satisfying all we know of the distribution. The first few of these being at least reasonable conjectures concerning the information we do not know, in particular the percentages of individuals at various IQ levels who have taken the SAT as shown in figure 3. This figure shows normalized conditioned distributions resulting from the various related filters. It is easily seen that if the selection were merely a high pass filter (for which case we would have sigma going to 0 such that everyone with an IQ over the threshold would certainly have taken the test and those below it certainly would not have, then the distribution would just be the renormalized portion of the normal distribution of individuals taken at random above the threshold. In this case, to maintain the average IQ of 133, the level (threshold) would have to be set at about 126 as can be seen in figure 1. This is unrealistic on several accounts.

Now, the probability that a member of NDSAT(x,xt,sigma) is n-sigma can be determined by the following integral:

P(x > xn-sigma) = 1 - [ integral from xn-sigmato infinity of NDSAT(x,xt,sigma) ]

Then to calculate the ratio of the percentage of individuals with n-sigma intelligence in the normalized conditioned distribution, NDSAT(x,sigma,xt), and parent distribution, f(x,16,100), the following equation must be used:

R(sigma,xo) = [ integral from 164 to infinity of NDSAT(x,xt,sigma)] / [ integral from 164 to infinity of f(x,16,100)]

[The MathCad plot could not be brought into the on-line version so please refer to the hard copy Gift of Fire.]

Figure 2: Determination of Xo for Intelligence Filters Resulting in Distributions with Mean 133

[The MathCad plot could not be brought into the on-line version so please refer to the hard copy Gift of Fire.]

Figure 3: Family of Normalized Distributions Representative of Persons Having Taken the SAT

This ratio is over 100 times the value Paul has assumed if we address only realistic values of sigma.

Other functions could be used to estimate h(x). While the mean must be maintained at 133, the conditioned probability of exceeding the 4-sigma level can be considerably increased for realistic values of selection intensity.

Other Useful Filters in High IQ Testing

Criteria for having taken the GRE, MCAT and other tests also constitute filters for intelligence. Perhaps there are others of the tests that are only given to selected individuals as well. Their evaluations would all be similar to that for the SAT and all would ultimately be determined empirically and not by deduction.

Concerning the condition of having had access to an article in OMNI magazine, one must say that it is an unknown filter, but whereas having taken the SAT acts somewhat as a high pass filter, this condition probably is more of a band pass filter, i. e., there is a range of IQ’s between which the subject matter of OMNI may appeal. My intuition tells me that this range begins somewhere well above the mean of the general population and shuts off well below the 4-sigma level. One can not say very much more about it than that. To make assumptions and attempt to confirm or impugn hypotheses based on them would be foolish without actual data.

Concerning the condition of having taken the "Worlds Toughest IQ Test," one can suppose much more! The percentage of merely normal or subnormal people to actually take the test and pay out money to get it graded must certainly be minuscule. On the other hand, the percentage of people in the 4-sigma range who would be told about the test independent of where it happened to appear would be large. I at least was not an OMNI reader when I was presented with a blank copy of the LAIT by someone who ended up scoring considerably lower on it than I. I find it easy to believe that this may have been typical. This scenario is particularly likely when the test appears in a magazine targeted at people just below that level who are the main contacts of the 4-sigma population to the rest of the world. I would readily believe, based on what I know of people, in particular and in general, that this condition would constitute an extremely effective high pass filter.

When one considers that the testees that took Ron Hoeflin’s and Kevin Langdon’s tests (besides those that were actually used in norming the tests) typically satisfied several qualitative conditions constituting filters for high intelligence one can readily understand why the LAIT and Mega testees would have a much higher dosage of 4-sigma individuals than one without an appreciation for such processes might consider justified.

I reiterate my hope that the membership committee will perform analyses as suggested above (but with concrete data as a basis) to justify membership criteria that we could unabashedly present to skeptics.