272 



Method of Presenting Eesults. Results of statistical inquiries into vari- 

 ation can best be presented by frequency of error curves, and tliese will be used 

 wherever possible. The abscissa will in all cases be made to represent the size of 

 the organ, the ordinate the percentum of individuals having the particular size. 



To convert variations in one organ into the terms of another organ the scheme 

 of distribution will be used with the formula given by Galton for comparing one 

 such curve with another. The process of comparing any curve "a" with any 



of "a" 

 curve "b," multiply each of "a's" height by .^ f "v," 



The Q of any scheme of distribution is one-half the difference between any two 

 grades. The same grades in the two carves to be compared being used to determine 

 their Q for this purpose, 25 per cent, and 75 per cent, are suggested as most con- 

 venient by Galton. 



Ideally the variations occurring in a single organ expressed by a frequency 

 of error curve, when a large number of individuals have been examined, will 

 form a symmetrical curve which is called a "normal." Such a curve may always 

 be expected when the material under consideration is of a single origin and has 

 developed under the same environment. Unfortunately for non-mathematical 

 evolutionists, the converse does not seem to be the case, for a symmetric curve 

 may be made up of two symmetric curves with axes not far apart, a fact that can 

 only be determined mathematically. Says Pearson, ''There will always be the 

 problem: Is the material homogeneous and a true evolution going on, or is the 

 material a mixture? To throw the solution on the eye in examining the graph- 

 ical results is, I feel certain, quite futile." 



It is not hoped that the data can be treated with the mathematical refine- 

 ment suggested l)y Pearson, nor is it probable that such treatment of our material 

 will become absolutely necessary, since there can be but little question of the 

 unity of origin of the material in any given small lake. 



While usually, as stated above, the curve resulting from the study of a large 

 number of specimens will be symmetric, it will frequently be asymmetric. Sam- 

 ples of the difTerent sort of curves actually observed are given. 



Asymmetric curves may be the result, 



1. Of the selective influence working on one side of a symmetric curve and 

 be then found in more or less mature specimens. 



2. Of the reaction to a change in the environment and indicative of a muta- 

 tion or change in the mean specific form. 



3. Of the double origin of the material under consideration, and may then 

 have a great variety of forms, from slightly asymmetric curve to one with a broad 

 top or with many peaks. 



