6 VARIATION AND CORRELATION IN THE CRAYFISH. 



GENERAL DISCUSSION. 



The measurements upon which this work is based are given in the 

 form of correlation tables in the Appendix (tables 22 to 32). The first 

 thing which impresses one in studying these tables is the fact that we are 

 dealing here with very high correlations, and almost perfect linearity of 

 the regression. High correlation denotes, of course, a relatively great 

 degree of constancy in the proportionality of the correlated parts. Our 

 tables show that in the crayfish, so far as the range of the characters we 

 have considered is concerned, there is very little variation in the array of 

 one variable associated with a given type of the other. This means that 

 there is a very strong tendency for definite and particular conditions of 

 the different characters to be associated together. The fact that the 

 characters which were chosen for this study are generally so highly cor- 

 related makes them especially favorable for the discussion of our prob- 

 lems, since high values for the coefficients of correlation give us low 

 values for the probable errors, and hence we shall be able to estimate the 

 probable significance of small differences with considerable accuracy. 



It seems desirable to collect together in one place the constants 

 measuring the degree of variation and correlation exhibited by the 

 characters studied. These constants are given in tables 1 and 3. These 

 form the fundamental reference tables; the data from them will be used 

 in succeeding portions of the paper, to answer special problems. Table 

 1 gives the following variation constants for the distribution of each of 

 the eleven characters: Mean, mode, standard deviation, coefficient of 

 variation, moment coefficients, /8 n $ 2 , kurtosis (/3 2 3), skewness, and 

 modal divergence. The unit for the "physical" constants is 1 mm., while 

 for the algebraical constants the unit is the unit of grouping, the value 

 of which in millimeters is given in the second column of the table. 



From table 1 we note at once the following general facts regarding 

 variation in the characters under consideration: 



(a) The relative variability in proportion to the size of the thing 

 varying is of roughly the same order of magnitude in practically all the 

 characters studied. With the single exception of the great chela (propo- 

 dite of leg i), the coefficients of variation all fall between 11 and 14.3 

 per cent. The great chela is very significantly more variable than any 

 of the other characters. It is of some interest to compare our results on 

 this point with some of the data which have been given by other workers 

 for variation in other Crustacea. Table 2 makes possible such com- 

 parison, and from the data there given one would conclude that in the 

 dimensions of the body the crayfish is proportionately a much less variable 

 form than is Eupagurus, and slightly more variable than Gelasimus. 



