12 VARIATION AND CORRELATION IN THE CRAYFISH. 



We see at once that in the case of the distributions for breadth of head 

 and for the length of the carpopodite of leg in, there is a certainly signifi- 

 cant deviation from the mesokurtic condition of the normal curve. The 

 distributions for the propodites of legs I and n are probably significantly 

 different from the condition of mesokurtosis. In the other cases the values 

 of /3 2 3 are less than twice the probable error and hence taken singly 

 would have to be considered as probably not significant. Due weight 

 must, however, be given to the fact that all the distributions show devia- 

 tions from mesokurtosis in the same direction. 



((/) Putting all the data together, we may safely conclude that the con- 

 tinuous variation in the characters of the crayfish which we have studied 

 can not be adequately described by the normal or Gaussian curve. Both 

 in respect to the symmetry of the variation about the mean or modal con- 

 dition and in respect to the kurtosis the crayfish distributions deviate in 

 a uniform manner from the normal curve. They demand skew curves for 

 graduation. 



We have in table 3 the coefficients of correlation, together with their 

 probable errors, for every possible pair of the eleven characters studied. 

 It is at once evident that we are dealing here with very high correla- 

 tions. The lowest coefficient in the table is 0.84, and from this the 

 values run up to as high as 0.973. These high correlations are in accord 

 with what has been found by other workers who have studied correlation 

 in other Crustacea. (Of. Yerkes, 1901, and Schuster, 1903, for example.) 

 The correlations between the different characters of the crayfish here 

 studied are of the same order of magnitude as has generally been found 

 for the correlation between bilaterally homologous organs or characters.* 



In order to show graphically the closeness to linearity of the regres- 

 sions between these different characters of the crayfish we have prepared 

 the diagrams shown in figs. 5 and 6. Fig. 5 shows the regression of the 

 length of the propodite of leg I on the length of the cephalothorax, and 

 fig. 6 the regression of the length of the meripodite of leg in on the 

 length of the meripodite of leg I. These two cases were taken at random 

 merely as samples of what holds generally for all the other characters 

 studied. 



* Cf ., for example, a table of the coefficients for such bilateral characters given as 

 table in in Davenport (1903). 



