Liberty

APPENDIX II

PATTERNS OF VARIATION

VARIATION SEEMS to pervade the universe. Even where once it was believed not to exist, further study and refinement of measurement reveals its presence.

When one views the members of another race, with which he is unfamiliar, they all seem to be alike until on further acquaintance their differences come to be more and more evident to him; eventually he finds them to be as great as the differences among members of his own race. It is the same with other species of life, and with the unlive formations of nature.

Once it was believed that the physical unit, the atom, lacked variation but now physicists are said to believe that even atoms vary. Everywhere variation seems to exist, in everything.

The complexity of compounded variation surpasses our comprehension, A person’s fingerprint will distinguish him from every other person; or a toe print; or the hairs on his head; and so on through a long list of features, each of which exhibits differences. When these features are considered in their seemingly endless number of combinations, the differences between any two persons is found to be so great that one wonders how any similarity between any two persons is to be noticed. It should be clear that a knowledge of variation causes the “average man” to dissolve into an abstraction, not found modeled anywhere in actual life.

Variation, in its rough and crude expression of random occurences, seems to be disorderly and chaotic. When observed in this form, it seems to be the result of pure “chance,” and to deny any purpose; it seems to reveal nothing but the “carelessness of nature.” But we shall see that this interpretation is highly doubtful.

A little over a century ago the foundations were laid for a science of the phenomenon of variation by the French mathematician Laplace (1749-1827). He began the work of dealing with variations so as to reveal the similarities of their patterns.

The newly developing science of variation was applied to astronomy by Carl Friedrich Gauss (1777-1855), the German mathematician and astronomer.

Adolphe Quetelet (1796-1855), the Belgian, deserves credit for the general principles of variation, reported in works published in 1835 and later. Though an astronomer, he extended his studies of variation to many other types of data such as temperature, the price of grain, and the heights and chest-measurements of men. Quetelet’s findings led to what later became known as the “normal curve” of variation.

The normal curve exhibits a symmetrical mathematical series. When graphed, it becomes a smoothly bell-shaped curve. About two-thirds of the total area of this curve lies within vertical lines placed at a distance of one standard deviation on each side of the vertical center, or the average of the series of data.

Quetelet found that all the data he studied fitted this style of curve fairly well.

The super-salesman of this new concept of order within the seeming chaos of variation, and of the predictable nature of variation, was Francis Galton. In 1899, in his classic book on Natural Inheritance, he had this to say:

It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed one, starts potentially into existence.

And:

It [the “normal curve” of variation] reigns with serenity and in complete self-effacement amidst the wildest confusion.… Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitudes, an unsuspected and most beautiful form of regularity proves to have been latent all along.

Galton asserted that this knowledge of the nature of variation, had it been known in ancient Greece, would most certainly have been personified and deified. And so it might. Galton himself spoke of it as being of a cosmic order.

Galton, as one of the pioneers in the discovery and interpretation of variation, may be excused for what now appears to be over-simplification. It now appears that the “normal curve” type of pattern in variation is not the only one. More accurately, as we now believe, Galton might well have waxed eloquent about the laws of variation in possible designs where more than a single pattern is allowed to unfold itself. These variations, in more than the single pattern of which Galton spoke, all have within them an orderliness which is concealed by their usually shuffled arrangement. Variation in each instance seems to fit into one or another mathematical function of the variable.

One formation of variables, more pertinent than the “normal curve” to many of the matters with which the social scientists deal, is that found in income variations and wealth variations. It is the “harmonic series” of magnitudes, wherein, if we represent the largest as 1, the second largest will be found to be ½, the third largest , the fourth largest ¼.…

The presence of this curve is revealed in its pattern from the largest end of the array, where large size and low frequency is found, to the other end where small size and high frequency is found. A tolerance must be allowed for the latter end of the array, in essentially all instances of data that has been derived from observed events. This is because wherever the law of limitation applies — which is universal except in things like distance or time, where observations are of unlimited magnitude — the harmonic series runs out and the frequency falls off unavoidably; the result is, under these conditions, the appearance of a “skewed curve.”

The pattern of variability of the harmonic type appears over a wide range of phenomenon. Vilfredo Pareto (1848-1923) found it to prevail among income data. In addition to income and wealth, it appears in the demand for any product or service, in the size of cities and towns in any settled national unit, in the frequency in the use of various words by one person, to mention a few that have been studied. We may even suspect to find that the harmonic series describes the variation of human abilities, as will be revealed only when an over-all measurement of ability has been developed.

One further point should be mentioned about variation. The fact that variation seems to fit into certain definite patterns as to type (the “normal curve,” the harmonic series, etc.) does not mean that the intensity of variation is the same wherever the pattern is the same. On the contrary, the intensity of variation differs widely. The size of one species of animal varies more intensely than another; the weight of one species of life varies more intensely than another; the color of one species of flower varies more intensely than another. … It is found, for instance, that the seedlings of the apple are highly variable in their commonly-observed characteristics, whereas the seeds of some other plants yield much more similar offspring.