44 



VARIATION AND CORRELATION IN THE CRAYFISH. 



index values as one variable, and the length of the cephalothorax as the 

 other variable. Now, it is obvious that since cephalothorax does not enter 

 into the index fraction, there can be no spurious correlation here. The 

 resulting table is table 21. 



Table 21. — Correlation surface for the correlation between the index 

 Meripodite I/Propodite I, and the length of the cephalothorax. 



This table shows at once that there is a distinct correlation between 

 the two variables, and that this correlation is negative. Calculating from 

 the frequency distribution given by this table and using Sheppard's cor- 

 rection of the second moment, we find the following values for the mean 

 and variability of the index: 



Mean index - 49.428 ± 0.133 

 Standard deviation of index = 3.325 ± .094 



For the correlation between the index and absolute dimensions, length 

 of cephalothorax we find 



r = - 0.4621 ± 0.0397 



This is clearly a sensible value in comparison with its probable error, 

 and we must conclude, since there is no possibility here of any spurious or 

 "arithmetic" correlation, that there is a real organic correlation between 

 the index formed by dividing the length of the meripodite of the cheliped 

 by the length of the propodite of the same leg, and the absolute size of 

 the body as measured by the length of the cephalothorax. In other 

 words, the proportion existing between the length of the meripodite and 



