IX. Mathematical contributions to the theory of evolution.—XIX. Second supplement to a memoir on skew variation

Abstract

In a memoir presented to the Royal Society in 1894, I dealt with skew variation in homogeneous material. The object of that memoir was to obtain a series of curves such that one or other of them would agree with any observational or theoretical frequency curve of positive ordinates to the following extent :—(i) The areas should be equal; (ii) the mean abscissa or centroid vertical should be the same for the two curves; (iii) the standard deviation (or, what amounts to the same thing, the second moment coefficient) about this centroid vertical should be the same, and (iv) to (v) the third and fourth moment coefficients should also be the same. If μ s be the s th moment coefficient about the mean vertical, N the area, x ¯ be the mean abscissa, σ = √ μ 2 the standard deviation, β 1 = μ 3 2 / μ 2 3 , β 4 = μ 4 / μ 2 2 , then the equality for the two curves of N, x ¯ , σ, β 1 and β 2 leads almost invariably in the case of frequency to excellency of fit. Indeed, badness of fit generally arises from either heterogeniety, or the difficulty in certain cases of accurately determining from the data provided the true values of the moment coefficients, e. g ., especially in J- and U-shaped frequency distributions, or distributions without high contact at the terminals ; here the usual method of correcting the raw moments for sub-ranges of record fails. Having found a curve which corresponded to the skew binomial in the same manner as the normal curve of errors to the symmetrical binomial with finite index, it occurred to me that a development of the process applied to the hypergeometrical series would achieve the result I was in search of, i. e ., a curve whose constants would be determined by the observational values of N, x ¯ , σ, β 1 and β 2 .