230 PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



tion, 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 lias shown that the constants of 

 normal correlation are significant, if not completely descriptive, even in the case of 

 skew correlation.t It seems desirable to take, somewhat out of its natural order, the 

 subject of the present memoir, partly because the formulae 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 formulae seems to have been disregarded by various 

 authors in somewhat 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. Notwithstanding 

 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 compared with those of the primitive group,J 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 ri 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. 



* " Mathematical Contributions to the Theory of Evolution. III. Regression, Heredity, and Pan- 

 mixia," ' Phil. Trans.,' A, vol. 187, pp. 253-318. 



f " On the Significance of BRAVAIS' Formulae for Skew Correlation," ' Roy. Soc. Proc.,' vol. 60, 

 pp. 477-489. 



J This will be sufficiently indicated in the latter part of the present memoir, but has been more fully 

 dealt with in a paper on the " Influence of Selection on Correlation," written, but not yet published. 



