10 



VARIATION AND CORRELATION IN THE CRAYFISH. 



as following the normal law within the limit of errors due to random 

 sampling, neither of these constants should differ from zero by at most 

 more than twice the probable errors given. But the values of the con- 

 stants actually obtained, with a single exception, greatly exceed this 

 limit. Hence there can be no doubt that we are dealing here with sig- 

 nificantly skew variation. In order to make plain the character of these 

 distributions with which we are dealing, figs. 2, 3, and 4, have been pre- 

 pared. They show the frequency polygons and their fitted curves for the 

 variation in the following characters: Fig. 2, length of cephalothorax; 

 fig. 3, propodite of leg I; fig. 4, carpopodite of leg I. 



The curve for each of these three polygons is of Pearson's Type i, 

 having the range limited in both distributions. The equations to the 

 curves are as follows: 



,3.3037 A X \16.0392 



Length of cephalothorax, y 40.0539 (1 + J 



PropodHaof leg,, = 30.9939 



27.98927 



Carpopodite ol leg ,, = 43.3066 ( 



14. 3542 



.22.9167 



It is evident from the diagrams that the curves give excellent gradua- 

 tions, quite as good as could be expected when dealing with less than 300 

 individuals in the observational series. There are some points regarding 

 the curves which need especial mention. In the first place, we note from 

 the equations that the theoretical range is large in each case. The actual 

 values are as follows: 



Length of cephalothorax, theoretical range =33. 7542 mm. 

 Propodite of leg i, theoretical range = 42.5581 mm. 



Carpopodite of legi, theoretical range = 14.7730mm. 



These ranges evidently exceed considerably those observed, but this 

 excess is practically entirely due to the extreme "tailing out" of the 

 theoretical curve toward large values, i. e., on the + side of the mode. 

 The truth of this statement is shown by the values of the following table : 



1 The values in these columns are the centers of the extreme classes of the observed ranges. 



2 It should be remembered that the apparent discrepancy between these values and those of a t and 

 a, as given in the third equation supra arises from the fact that the constants of the equation are given 

 in units of grouping, while here they are in millimeters. 



