Destruction of the Normal Paradigm
Stephen J Bensman
notsjb at LSU.EDU
Wed Sep 29 11:35:05 EDT 2010
Thanks for the thanks. It is further validation that I have an accurate
grasp of the topic. In return I want to pose for you a conundrum that
bothers me.
It is well known that it is really impossible to distinguish the effects
of nature from nurture. These two interact, and, from the beginning ,
the main criticism of the hereditarian position has always been that it
is impossible to distinguish the effects of one from the other.
Unfortunately this leads to the uncomfortable fact that certain persons
and groups (ethnic, religious, social, national, etc.) do far better in
science than others, leading to highly skewed scientometric
distributions. For example, the historical contribution of Britain to
the advancement of science has been "disproportionate." I will let you
deduce the egalitarian implications from my use of this modern
diplomatic term.
In his 1869 book, Hereditary Genius, where he posited the normal
distribution of human intelligence, Galton gave the scores of students
on the test for mathematical honors at Cambridge for two years in the
1860s. I have created the graph below from his figures:
The position of the top wranglers such as Pearson, Keynes, etc., is
quite clear. I call this the first scientometric distribution, for it
is typical of the distributions found in article productivity,
citations, library use, etc.
My question to you is how do we go from an intelligence distribution
either of the normal or Pearson Type IV type to this type of
distribution. Burt was dealing with problem that the distribution of IQ
scores is not representative but an artifact of the way these tests are
structured. Or is the difference of the distributions a function of the
fact that in scientometrics we are drawing our samples only from the
extreme right on the intelligence scale?
This has always been a problem that has puzzled me.
Stephen J. Bensman
LSU Libraries
Louisiana State University
Baton Rouge, LA 70803
USA
notsjb at lsu.edu
From: ASIS&T Special Interest Group on Metrics
[mailto:SIGMETRICS at LISTSERV.UTK.EDU] On Behalf Of James Hartley
Sent: Wednesday, September 29, 2010 5:15 AM
To: SIGMETRICS at LISTSERV.UTK.EDU
Subject: Re: [SIGMETRICS] Destruction of the Normal Paradign
Thanks for this. In a slightly related matter colleagues may be
interested in a paper criticizing Burt's work on typographical design.
See:
Hartley J. & Rooum, D. (1983). Sir Cyril Burt and typography: A
re-evaluation. British Journal of Psychology, 74, 203-212.
Copes available on request (by post!).
James Hartley
School of Psychology
Keele University
Staffordshire
ST5 5BG
UK
j.hartley at psy.keele.ac.uk
http://www.keele.ac.uk/depts/ps/people/JHartley/index.htm
----- Original Message -----
From: Quentin Burrell <mailto:quentinburrell at MANX.NET>
To: SIGMETRICS at LISTSERV.UTK.EDU
Sent: Monday, September 27, 2010 7:10 PM
Subject: Re: [SIGMETRICS] Destruction of the Normal Paradign
Adminstrative info for SIGMETRICS (for example unsubscribe):
http://web.utk.edu/~gwhitney/sigmetrics.html 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:
Adminstrative info for SIGMETRICS (for example unsubscribe):
http://web.utk.edu/~gwhitney/sigmetrics.html
End of Chapter 5 of my book. I am really having fun with this
things
Stephen J. Bensman
LSU Libraries
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).
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:
...The 'major genes' seem comparatively rare, but each will
produce
effects that are large and...for the most part
detrimental; the 'polygenes'
must be much more numerous, but their effects
will be too small to be identified
individually. With this double assumption, the
resulting distribution would
take the form, not of the normal curve, but of an
asymmetrical, bell-shaped
curve of unlimited range in either direction. p.
166.
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.
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."
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:
In applying mathematics to subjects such as physics or
statistics we
make tentative assumptions about the real world which we know
are
false but which we believe may be useful nonetheless. The
physicist
knows that particles have mass and yet certain results,
approximating
what really happens, may be derived from the assumption that
they
do not. Equally, the statistician knows, for example, that in
nature
there never was a normal distribution, there never was a
straight line,
yet with normal and linear assumptions, known to be false, he
can
often derive results which match, to a useful approximation,
those
found in the real world. p. 792.
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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