VII. Mathematical contributions to the theory of evolution.— IV. On the probable errors of frequency constants and on the influence of random selection on variation and correlation

Abstract

(1) In earlier memoirs by one of the present authors, methods have been discussed for the calculation of the constants (a) of variation, normal or skew, (b) of correla­tion, when normal. The subject of skew correlation would now naturally present itself, but although several important conclusions with regard to skew correlation have been worked out, there are still difficulties which impede the completion of the memoir on that topic. Meanwhile Mr. G. U. Yule has shown that the constants of normal correlation are significant, if not completely descriptive, even in the case of skew correlation. It seems desirable to take, some what out of its natural order, the subject of the present memoir, partly because the formulæ involved have been once or twice cited and several times used in memoirs by one of the present writers, and partly because the need of such formulæ seems to have been disregarded by various authors in some what too readily drawing conclusions from statistical data. Differences in the constants of variation or of correlation have been not infrequently asserted to be significant or non-significant of class or of type, or of race differences, without a due investigation of whether those differences are, from the standpoint of mathematical statistics, greater or less than the probable errors of the differences. Not withstanding that every artificial or even random selection of a group out of a community changes not only the amount of variation, but the amount of correlation of the organs of its members as com pared with those of the primitive group, it has been supposed that correlation might be a racial constant, and the approximate constancy of coefficients of correlation of the same organs in allied species has been used as a valid argument. In the like manner differences in variation have been used as an argument for the activity of natural selection without a discussion of the probable errors of those differences. In dealing with variation and correlation we find the distribution described by certain curves or surfaces fully determined when certain constants are known. These are the so-called constants of variation and correlation, the number of which may run up from two to a very considerable figure in the case of a complex of organs. If we deal with a complex of organs in two groups containing, say, n and n' individuals, we can only ascertain whether there is a significant or insignificant difference between those groups by measuring the extent to which the differences of corresponding constants exceed the probable errors of those differences. The probable error of a difference can at once be found by taking the square root of the sum of the squares of the probable errors of the quantities forming the difference. Hence the first step towards determining the significance of a group difference—i. e., towards ascertaining whether it is really a class, race, or type difference— is to calculate the probable errors of the constants of variation and correlation of the individual groups. This will be the object of our first general theorem.