42 VARIATION AND CORRELATION IN THE CRAYFISH. 



indicated by the high spurious coefficients has been overcome and there 

 is in every case an additional balance on the positive side. This shifting 

 of the correlation from the negative direction of the spurious through zero 

 to the positive side, as observed in the gross coefficients, can only be due 

 to the influence of a positive organic correlation between index and abso- 

 lute size. It is immaterial for the essential problem what the exact amount 

 of this correlation is, or how it shall be measured.* The figures show 

 beyond any doubt (a) that an organic correlation between the indices 

 and the absolute dimension exists ; (6) that this correlation is in the posi- 

 tive direction; and (c) that while it varies for the different characters it 

 is generally high, and certainly to be regarded as significant. 



The relation of gross and spurious coefficients in these index correla- 

 tions may be illustrated by fig. 8. 



FIG. 8. To illustrate the relation of gross and spurious coefficients 

 in index correlation. (Explanation in text.) 



In this diagram the semicircular arc is taken to represent the whole 

 possible range of values of a correlation coefficient from 1 through 

 to +1. The positive correlations are on the right of and the negative 

 on the left, the directions being indicated by the outside arrows. The 

 heavy line at A we may consider to represent the actual value of the 

 correlation between an index, say, carpopodite of leg n/cephalothorax, 

 and the length of the cephalothorax which one would find if he were 

 to calculate the index for each individual from a correlation table, and 



* Pearson has suggested the expression p p as a measure of the intensity of the 

 "net" organic correlation, in the case of indices. It is obviously defective, in that it is 

 not necessarily limited in the values it may take to the and 1 of a true coefficient 

 of correlation. There can be no doubt, though, that in a general way it measures the 

 "shift" of the correlation. This point has been more fully discussed and illustrated 

 in the papers of the present writer referred to above (p. 40). 



