Destruction of the Normal Paradign

[Quentin Burrell]

Stephen

Thanks for this. 

Throughout my teaching career, I always referred to the Normal distribution - the capital letter signifying that it is a technical term rather than its common usage interpretation. I just wish that this was more widely adopted.

I recall my first interview for an academic post and was asked "Mathematicians believe in the Normal distribution because it is an established physical fact; Physicists believe in the Normal distribution because it is an established mathematical theorem. What do you think?"

(I think that the interviewer did give an attribution for the source of the question but I was too nervous to remember it. Maybe someone can enlighten me after all these years?)

BW

Quentin


On 27 Sep 2010, at 17:14, Stephen J Bensman wrote:

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> End of Chapter 5 of my book.  I am really having fun with this things
> 
> Stephen J. Bensman
> 
> LSU Libraries
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> Louisiana State University
> 
> With his memoirs on skew variation in homogeneous materials Pearson accomplished the destruction of the normal paradigm.   In his obituary of Pearson, Yule (1936) declared, “I should count it one of Pearson’s greatest contributions in this field…that he enforced attention to the extraordinary variety of distributions met with in practice, illustrating the thesis with example on example and creating in this way little less than a revolution in the outlook of the ordinary statistician” (p. 81).   Although Pearson’s system of curves is seldom used today, Eisenhart (1974, p. 451) noted that these curves played an important role in the development of statistical theory and practice with the discovery that the sampling distributions of many statistical test functions appropriate to analyses of small samples from normal, binomial, and Poisson distributions—such as chi-squared and t—are represented by particular families of Pearson curves either directly or through simple transformation.  Moreover, according to Eisenhart, the fitting of Pearson curves to observational data was extensively practiced by biologists and social scientists in the decades that followed these memoirs, and he observed, “The results did much to dispel the almost religious acceptance of the normal distribution as the mathematical model of variation of biological, physical, and social phenomena” (p. 451).
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> From the perspective of this book’s topic, one of the most the most interesting, if controversial, examples of this is the work of Cyril Burt on the distribution of human intelligence.  Burt (Mazumdar, 2004; Mcloughlin, 2000; Vernon, 2001) was the preeminent British professional psychologist from 1930 to 1950, being made Knight of the Royal Garter in 1946.  He worked for the London County Council as Britain’s first educational psychologist.  Burt stemmed from the same intellectual tradition as Galton and Pearson, being a member of the Eugenics Society, and in 1932 he succeeded Spearman as professor of psychology at University College London, continuing that institution’s Galton-Pearson statistical tradition and pioneering the integration of biometric techniques into psychology.  Like Galton, Burt believed that IQ was a function of nature, not nurture.  In a paper on the distribution of intelligence Burt (1957) defined intelligence “in the technical sense given to it, explicitly or implicitly, in the work of Spencer, Galton, Binet, and their followers, namely, ‘the innate general factor underlying all cognitive activities’” (p. 173), and he hypothesized that it should follow a moderately asymmetrical distribution.  The reason for this was that he postulated this type of distribution as a function of two possible genetic modes of inheritance: 1) in certain cases, the deviation studied may act as a recognizable trait dependent on a single, major gene; and 2) in other cases, it is apparently determined by the joint action of a large number of genes.   He summarized the effect of these two interacting genetic modes in the following manner:
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>    …The ‘major genes’ seem comparatively rare, but each will produce
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>                effects that are large and…for the most part detrimental; the ‘polygenes’
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>                must be much more numerous, but their effects will be too small to be identified
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>                individually.  With this double assumption, the resulting distribution would
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>                take the form, not of the normal curve, but of an asymmetrical, bell-shaped
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>                curve of unlimited range in either direction.  p. 166.    
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> Burt identified this curve with the Pearson Type IV, citing Pearson’s first memoir on skew variation that this was the prevailing in zoological and anthropological material.  In this memoir Pearson (1895) described the Type IV as having “Unlimited range in both directions and skewness” (p. 360) and speculated that the reason for its prevalence in zoological measures was due to the “inter-dependence of the ‘contributory’ causes” (p. 412).  Burt (1957) found that empirical evidence derived from intelligence tests produced statistical constants implying curves—slightly leptokurtic and negatively asymmetric—consistent with his twofold genetic hypothesis.  In a follow-up paper Burt (1963) tested frequency distributions obtained from applying IQ tests to large samples of the school population and found that the distributions actually observed were more asymmetrical with longer tails than predicted by the normal curve.  The best fit to the data was the Pearson Type IV.  According to him, the assumption of normality led to a gross underestimate of the number highly gifted individuals in England and Wales—31.7 persons with IQs above 160 predicted by the normal curve as against the 342.3 such persons predicted by the Type IV. 
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>             After his death in 1971 Burt came under assault for shoddy research methods, falsification of data, and supposedly fictitious research assistants.  As a result of these attacks, the British Psychological Society Council found that Burt was a “scientific fraud” in 1979.  The assault produced a reaction, whereby the original assaulters themselves came under assault.  Mazumdar (2004, p. 6) points out that it is difficult to separate the question of Burt’s science from politics.  Burt was formed in an era when hereditarianism and eugenics were the norm, and in the egalitarian atmosphere of post-war Britain such views were considered antiquated and unjust.   The assault on Burt was led largely by psychologists, who were passionate environmentalists.  As a result of the battle, Burt was partially rehabilitated.  In 1992 the British Psychological Society Council (1992) resolved that no universally accepted agreement was possible on this matter, declaring, “The British Psychological Society no longer has a corporate view on the truth of the allegations concerning Burt” (p. 147).  In a book of readings on the measurement of intelligence Eysenck (1973)—himself a highly influential but controversial British psychologist of German descent—included Burt’s 1963 article proving that the Pearson Type IV best fitted the distribution of human intelligence, stating that Burt’s view on the applicability of the Pearson Type IV to the distribution of IQ is “probably correct” (p. 37).  Evaluating Bert’s findings, he stated that the normal and Type IV curves are not very dissimilar, but, as Burt pointed out, there are marked differences at the extremes.  Referring to these marked differences at the extremes, Eysenck, stated that “from the social point of view these may be very important indeed” (p. 37).  For example, at upper IQ extreme they do increase the probability of persons capable of doing high level science in the population.  Eysenck’s book has the following dedication:  “To the Memory of Cyril Burt, who taught me.”    
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> Pearson did not entirely dethrone the normal distribution, which still plays a central role in statistical theory.  Snedecor and Cochran (1989, p. 40) list four reasons for this.  First, the distributions of many variables such as heights of people, the lengths of ears of corn, and many linear dimensions of manufactured articles are approximately normal.  These authors state that in fact any variable whose expression results from the additive contributions of many small effects will tend to be normally distributed.  The second reason listed by Snedecor and Cochran is that for measurements whose distributions are not normal, a simple transformation of the scale of measurement may induce approximate normality.  Two such transformations—the square root and the logarithmic—are indicated by them as being often employed.  According to Elliott (1977, p. 33), the Poisson is made to approximate normality by the square root transformation, whereas most distributions in scientometrics and information science require some form of the logarithmic transformation, converting them into the lognormal distribution.  The third reason listed by Snedecor and Cochran is the normal distribution is relatively easy work with mathematically, and their fourth reason is that even if the distribution in the original population is far from normal, the distribution of sample means tends to become normal under random sampling as the size of the sample increases.  This contradiction between the importance of the normal distribution in statistical theory and its relative infrequency in reality creates a tension, which caused Geary (1947) to emphasize the importance of testing for normality and to recommend that the following warning be printed in bold type in every statistical textbook: “Normality is a myth; there never was, and never will be, a normal distribution” (p. 241).  The tension between importance and infrequency caused George Box (1976), R.A. Fisher’s son-in-law, to compare the role of the normal distribution in statistics to the general role of the mathematical model in science as a whole thus:
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>  
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> In applying mathematics to subjects such as physics or statistics we
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> make tentative assumptions about the real world which we know are
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> false but which we believe may be useful nonetheless.  The physicist
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> knows that particles have mass and yet certain results, approximating
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> what really happens, may be derived from the assumption that they
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> do not.  Equally, the statistician knows, for example, that in nature
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> there never was a normal distribution, there never was a straight line,
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> yet with normal and linear assumptions, known to be false, he can
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> often derive results which match, to a useful approximation, those
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> found in the real world.  p. 792. 
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>  
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> What Pearson accomplished can be easily deduced from that above.  By proving that most reality is not random and additive but causal and multiplicative, he converted the normal distribution from a universal descriptor of reality into a mathematical, mental construct for the distribution of error, against which to test reality.  Given that much of reality is multiplicative, whereas error is additive—and the logarithmic transformation converts data from multiplicative to additive—the Galton-McAlister law of the geometric mean, which Pearson rejected as a descriptor of reality due to still being based upon Gaussian axioms, now has an important role as a law of error in statistical tests of significance.
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