IQ scores are used to interpret "the intelligence" of individuals and to predict their function in an academic or workaday situation. The primary argument for doing so is that IQ scores "fit the curve", the "bell curve" which mathematicians call the "Gaussian probabiilty distribution curve" or "normal probability distribution function". But legitimate objections have been ignored.First, what statisticians call "a Gaussian curve" is conventionally built from "quantitative" measures (linear or ratio measures, such as temperature or length, whereas IQ scores -- if more than mathobabble -- are "qualitative" (ordinal, a matter of "degree", as in the Moh-Brinell measure of hardness of minerals). Actually, the jargon, "quality-quantity" is so much quibble-quack.
In 1940 Harvard University psychophysicist S. S. Stevens initiated the theory of measurement scales (our best metrical foundation), defining a measurement scale by the transformation group which leaves the scale invariant and also by the scale's "allowable statistics" ("statistics which make sense").
Transformation groups are part of group theory (the mathematics of symmetry), developed by a 16-year-old Frenchman, Évariste Galois (1811-1832). ("Modular groups" now encode satellite signals for our news, sports, and entertainment programs from "across the world"). Groups explain arithmetic and its many number systems. Behind each group used in physics is a conservation law, which echoes religious faith ("Oh, Thou Unchanging!").
I'll explain the "transformation" of "transformation group" by familiar sentence-forms.
I show you a drawing, labeled "John walks to the store", with a figure labeled "John". Curved marks in the drawing symbolize John's walking. An arrow identifies "The Store". Each word of this gramatically active sentence-form can be cartooned.
But consider the passive transformation of this sentence: "John is going to the store". Now all words can be cartooned -- except the auxiliary verb "is". But, here, "is" functions as a language-device which vanishes in the transformation from passive to active sentence-form.
FORM INVARIANCE FAILS!
Physics students observe a similar failure in "Coriolis force", which "explains" high and low pressure circulation. Tracked from earth-perspective, a wind current from the north seems to curve westward (because of the earth's rotation). Curving and Coriolis force vanish when the wind current is tracked relative to a space-plane.
Similar now-you-see-it-now-you-don't effects occur in factor analysis, the tool of the psychometrician. Subtest scores may correlate positively and plot as dimensions or coordinates. Charles Spearman found a common dimension among these correlations and labeled it "g" for "general intelligence".
Later, Lewis Thurstone showed that g vanishes in shifting the axes! (Stephen Jay Gould Gould devotes several pages of his unique book, The Mismeasure of Man, to g.)
What does this mean? AN ORDINAL MEASURE, such as mineral hardness and such as IQ measure, might claim -- but nothing more! -- that it can fit the ISOTONIC TRANSFORMATION GROUP WHICH PRESERVES ORDERING. And these purely ordinal measures are ALLOWED THE "AVERAGE" KNOWN AS MEDIAN ("the middle of the ordering" as representative). But we can show that mineral hardness and IQ scores cannot claim "the linear group" and the "average" of arithmetic mean because of MEASURE-CLOSURE-FAILURE when one attempts to ADD such purely ordinal measures.
Let's see where this condition succeeds. MEASURE-CLOSURE for LENGTH occurs because A LENGTH CONCATENATED WITH A LENGTH CAN BE MATCHED IN PARALLEL BY A SINGLE EXTENSION OF THE SAME LENGTH. (The double-extension is equivalent to the single extension, and vice versa.) So 2 meters + 3 meters = 5 meters. In any case wherein we need 5 meters of length (wood, steel, whatever), the 2 meter + 3 meter combination will do as well as a single extension of 5 meters. That examples MEASURE-CLOSURE!
Similarly with weight, in killograms or pounds or any accepted unit of weight. And other successes can be demonstrated.
But consider the Moh-Brinell scale of mineral hardness. Chalk is assigned the scale value of ONE, while diamond is assigned the scale value of NINE. TEST: Attempt to scratch mineral A by mineral B; if a scratch-mark remains on A, B is HARDER than A, otherwise not so, and the converse REST is performed. But how many pieces of chalk add up to the hardness of a diamond? Dig?
So apply a similar argument to IQ scores. If I combine a 50 IQ with a 50 IQ with a 50 IQ, is this equivalent to a 150 IQ? Would you hire three persons with IQ of 50 to do the work of a 150 IQ person? Of course, not! We have MEASURE-CLOSURE-FAILURE!
As another failure, we cannot PREDICT from IQ scores. There is much evidence of PREDICTION-FAILURE in the literature, despite the noise of psychometrists. But we need only ask the following. Given three combined meters of wood (or whatver), we can predict that this will fill out the space and the function of a single 3-meter piece. Can we predict that three persons of IQ 50 will fulfill the function of a person with 150 IQ? Of course not. So IQ scoring has PREDICTION-FAILURE.
And we saw above that IQ scoring has CLOSURE-FAILURE.
And elsewhere, we show that IQ scoring fails the Eudoxian-Archimedean Condition necessary for ADDING MEASURES. So we have Eudoxian-Archimedean FAILURE for IQ scoring. (Three strikes [failures] and you're out!)
In consequence:
- You CAN'T ADD IQ SCORES!
- AND, IF YOU CAN'T ADD IQ SCORES, YOU CAN'T FIND THE ARITHMETIC MEAN ("AVERAGE") OF A SET OF IQ SCORES.
- AND IF YOU CAN'T FIND THE ARITHMETIC MEAN, YOU CAN'T USE IT TO DERIVE THE VARIANCE OF THE IQ SCORES.
- AND, IF YOU CAN'T FIND THE VARIANCE OF IQ SCORES, YOU CAN'T DERIVE THE STANDARD DEVIATION FROM THE VARIANCE.
- AND, IF YOU CAN'T FIND THE ARITHMETIC MEAN AND STANDARD DEVIATION OF IQ SCORES, YOU CAN'T USE THEM TO DERIVE THE NORMAL DISTRIBUTION ("BELL") CURVE OF IEQ SCORES!
So the whole business -- the primary business of "psychometrics" is NONSENSE! As much nonsense as adding the numerals on uniforms of athletes and trying to form such a curve!