The Paretian Distribution of Intelligence

ONE OF THE CENTRAL problems of the ability theory of personal income distribution has been that income has been found empirically to be distributed according to Pareto’s law, N = Ax-B where N is the number of persons receiving x income or more. A and B are constants, the latter being about 1.5 and a normal distribution being assumed for ability (A). The purpose of this paper is to analyze this latter assumption.

Historically, the justification for the assumption of the normal distribution of intellect was based upon analogy with physical traits such as height and weight. Originally, even income was a priori assumed to be distributed normally until Pareto’s empirical evidence destroyed this belief. His findings indicated a lognormal distribution with a skewness to the right having a flat tail.

The error of the historical assumption of normal distribution of intelligence was aggravated by the development of intelligence tests designed to fit the normal curve. This was done by greatly limiting the range of item difficulty on tests of intelligence and loading the tests heavily with very simple items, in comparison with the universe of knowledge. If, for example, test items are selected by randomly choosing entries from a comprehensive encyclopedia, the number of correct answers indicates that most people know only a very small amount of the world’s collected knowledge, while a very few know from ten to a thousand times the norm. Such extreme inequalities demonstrate that knowledge is diffused more in line with Pareto’s law than with a normal distribution.

Why has this escaped notice for the more than sixty years that intelligence testing data have been statistically analyzed? Quite obviously because the assumption of the normal curve biased the selection of questions. “Who wrote Hamlet?” is the middle question on the information subtest of the Wechsler Adult Intelligence Scale and is missed by half the testees. Out of the entire universe of possible information items, this is a very simple one. Yet it is the item of median difficulty on this test.

Assuming that ability, as measured by achievement, has a Paretian rather than a Gaussian distribution, the question must still be answered why intelligence is not analogous to normally distributed physical traits. If one thinks about this question, it seems that all behavioral traits such as knowledge of physics, tennis-playing ability, bridge-playing ability, etc., are distributed according to Pareto’s law, i.e., there is a very uneven distribution of ability with a few people having an ability of many magnitudes the ability of the average person.

This evidence — which seems to contradict the normal distribution of physical traits — can be reconciled by taking the well-known fact that normally distributed factors interacting multiplicatively will yield a lognormal distribution (one consonant with Pareto’s law). One can assume that the physical aspects of the brain, as with other physical traits, are distributed normally, but that within us these factors interact multiplicatively to yield a lognormal result. Normal differences in physical traits yield lognormal differences in behavioral results.

This is true in every field from physics to football. The implication for the ability theory of personal income distribution is that there is no longer a contradiction between the distribution of ability and the distribution of income.

The implications of the above for the IQ scale are profound. Dashed are the hopes of those who argue circularly that the IQ scale is an interval scale. On the Paretian assumption, the intervals are quite different from those based on the Gaussian assumption. From Pareto’s equation, assuming B = +1.5 and A = 100, then N = 100 when x = 1. We have here what is probably the closest approach to an absolute intelligence scale which is also a ratio scale where a score of 1 is the absolute lowest score. From the equation we find x = 2 is the 64th percentile; x 3 (that is, 3 times the ability or knowledge) is the 80th percentile; x = 5 is the 95th percentile; x 10, the 96th percentile; x = 50, the 99.7 percentile; x = 100, the 99.9 percentile.

We can say on the assumption of the Paretian distribution that a person in the 99.9th percentile is more than 50 times as intelligent as a person in the 50th percentile. This is a measure which was statistically impossible using the old Gaussian scale.

The above, of course, may be vitiated by the invalidity of any particular test, but it does show that the old divisions of intelligence levels — dull, dull-normal, bright-normal etc. — create larger distinctions among people than is justified, when compared with the huge distinctions among individuals at the higher levels, who have, heretofore, all been lumped into the homogeneous category “gifted.”

The Paretian distribution of intellect and other behavioral traits imply that the great achievements of mankind depend upon a very few men who are many factors superior to the average man, superior to a degree that could not be conceived under the misconceptions of the Gaussian distribution.

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Source: Instauration magazine, March 1979