§ 21. Perhaps one of the best illustrations of the legitimate application of such principles is to be found in Mr Galton's work on Hereditary Genius. Indeed the full force and purport of some of his reasonings there can hardly be appreciated except by those who are familiar with the conceptions which we have been discussing in this chapter. We can only afford space to notice one or two points, but the student will find in the perusal, of at any rate the more argumentive parts, of that volume[18] an interesting illustration of the doctrines now under discussion. For one thing it may be safely asserted, that no one unfamiliar with the Law of Error would ever in the least appreciate the excessive rapidity with which the superior degrees of excellence tend to become scarce. Every one, of course, can see at once, in a numerical way at least, what is involved in being ‘one of a million’; but they would not at all understand, how very little extra superiority is to be looked for in the man who is ‘one of two million’. They would confound the mere numerical distinction, which seems in some way to imply double excellence, with the intrinsic superiority, which would mostly be represented by a very small fractional advantage. To be ‘one of ten million’ sounds very grand, but if the qualities under consideration could be estimated in themselves without the knowledge of the vastly wider area from which the selection had been made, and in freedom therefore from any consequent numerical bias, people would be surprised to find what a very slight comparative superiority was, as a rule, thus obtained.
§ 22. The point just mentioned is an important one in arguments from statistics. If, for instance, we find a small group of persons, connected together by blood-relationship, and all possessing some mental characteristic in marked superiority, much depends upon the comparative rarity of such excellence when we are endeavouring to decide whether or not the common possession of these qualities was accidental. Such a decision can never be more than a rough one, but if it is to be made at all this consideration must enter as a factor. Again, when we are comparing one nation with another,[19] say the Athenian with any modern European people, does the popular mind at all appreciate what sort of evidence of general superiority is implied by the production, out of one nation, of such a group as can be composed of Socrates, Plato, and a few of their contemporaries? In this latter case we are also, it should be remarked, employing the ‘Law of Error’ in a second way; for we are assuming that where the extremes are great so will also the means be, in other words we are assuming that every amount of departure from the mean occurs with a (roughly) calculable degree of relative frequency. However generally this truth may be accepted in a vague way, its evidence can only be appreciated by those who know the reasons which can be given in its favour.
But the same principles will also supply a caution in the case of the last example. They remind us that, for the mere purpose of comparison, the average man of any group or class is a much better object for selection than the eminent one. There may be greater difficulties in the way of detecting him, but when we have done so we have got possession of a securer and more stable basis of comparison. He is selected, by the nature of the case, from the most numerous stratum of his society; the eminent man from a thinly occupied stratum. In accordance therefore with the now familiar laws of averages and of large numbers the fluctuations amongst the former will generally be very few and small in comparison with those amongst the latter.
[1] Essai de Physique Sociale, 1869. Anthropométrie, 1870.
[2] As regards later statistics on the same subject the reader can refer to the Reports of the Anthropometrical Committee of the British Association (1879, 1880, 1881, 1883;—especially this last). These reports seem to me to represent a great advance on the results obtained by Quetelet, and fully to justify the claim of the Secretary (Mr C. Roberts) that their statistics are “unique in range and numbers”. They embrace not merely military recruits—like most of the previous tables—but almost every class and age, and both sexes. Moreover they refer not only to stature but to a number of other physical characteristics.
[3] As every mathematician knows, the relative numbers of each of these possible throws are given by the successive terms of the expansion of (1 + 1)10, viz.
1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1.
[4] That is they will be more densely aggregated. If a space the size of the bull's-eye be examined in each successive circle, the number of shot marks which it contains will be successively less. The actual number of shots which strike the bull's-eye will not be the greatest, since it covers so much less surface than any of the other circles.
[5] Commonly called the exponential law; its equation being of the form y = Ae−hx2. The curve corresponding to it cuts the axis of y at right angles (expressing the fact that near the mean there are a large number of values approximately equal); after a time it begins to slope away rapidly towards the axis of x (expressing the fact that the results soon begin to grow less common as we recede from the mean); and the axis of x is an asymptote in both directions (expressing the fact that no magnitude, however remote from the mean, is strictly impossible; that is, every deviation, however excessive, will have to be encountered at length within the range of a sufficiently long experience). The curve is obviously symmetrical, expressing the fact that equal deviations from the mean, in excess and in defect, tend to occur equally often in the long run.