III. Contributions to the mathematical theory of evolution

Abstract

(1.) If measurements be made of the same part or organ in several hundred or thousand specimens of the same type or family, and a curve be constructed of which the abscissa x represents the size of the organ and the ordinate y the number of specimens falling within a definite small range δx of organ, this curve may be termed a frequency-curve. The centre or origin for measurement of the organ may, if we please, be taken at the mean of all the specimens measured. In this case the frequency-curve may be looked upon as one in which the frequency—per thousand or per ten thousand, as the case may be—of a given small range of deviations from the mean, is plotted up to the mean of that range. Such frequency-curves play a large part in the mathematical theory of evolution, and have been dealt with by Mr. F. Galton, Professor Weldon, and others. In most cases, as in the case of errors of observation, they have a fairly definite symmetrical shape and one that approaches with a close degree of approximation to the well-known error or probability-curve. A frequency-curve, which, for practical purposes, can be represented by the error curve, will for the remainder of this paper be termed a normal curve. When a series of measurements gives rise to a normal curve, we may probably assume something approaching a stable condition; there is production and destruction impartially round the mean. In the case of certain biological, sociological, and economic measurements there is, however, a well-marked deviation from this normal shape, and it becomes important to determine the direction and amount of such deviation. The asymmetry may arise from the fact that the units grouped together in the measured material are not really homogeneous. It may happen that we have a mixture of 2, 3, . . . n homogeneous groups, each of which deviates about its own mean symmetrically and in a manner represented with sufficient accuracy by the normal curve. Thus an abnormal frequency-curve may be really built up of normal curves having parallel but not necessarily coincident axes and different parameters. Even where the material is really homogeneous, but gives an abnormal frequency-curve the amount and direction of the abnormality will be indicated if this frequency-curve can be split up into normal curves. The object of the present paper is to discuss the dissection of abnormal frequency-curves into normal curves. The equations for the dissection of a frequency-curve into n normal curves can be written down in the same manner as for the special case of n = 2 treated in this paper; they require us only to calculate higher moments. But the analytical difficulties, even for the case of n = 2, are so considerable, that it may be questioned whether the general theory could ever be applied in practice to any numerical case. There are reasons, indeed, why the resolution into two is of special importance. A family probably breaks up first into two species, rather than three or more, owing to the pressure at a given time of some particular form of natural selection; in attempting to procure an absolutely homogeneous material, we are less likely to have got a mixture of three or more heterogeneous groups than of two only. Lastly, even where the heterogeneity may be threefold or more, the dissection into two is likely to give us, at any rate, an approximation to the two chief groups. In the case of homogeneous material, with an abnormal frequency-curve, dissection into two normal curves will generally give us the amount and direction of the chief abnormality. So much, then, may be said of the value of the special case dealt with here.