GENERAL DISCUSSION. 11 



No great stress is to be laid on the great overestimation by the theo- 

 retical curves of the ranges on the plus side, for two reasons. On the one 

 hand, the constant a^, which gives the range on the plus side of the mode, 

 is subject to a large probable error. On the other hand, the total frequency 

 beyond the observed upper limit of the range as given by the theoretical 

 curve is so small as to be entirely negligible for all practical purposes. 

 This will be evident from mere inspection of the ends of the curves, with- 

 out going into the figures to prove it. 



Inspection of the curves leaves no doubt as to the substantial deviation 

 of the distributions from the symmetrical condition of the normal curve. 

 That the skewness is positive for all the characters studied is directly con- 

 nected with the fact that there is a very high degree of correlation between 

 these characters. Given a positive skewness for one character, say length 

 of cephalothorax, and we should expect that all other characters of the 

 same organism which are closely correlated in size with the cephalothorax 

 would also exhibit positive skewness. This is evident if we consider that 

 positive skewness in the character mentioned merely means that there are 

 more individuals with the cephalothorax larger than the modal condition 

 than there are individuals with it smaller than the mode. But if deviations 

 from the mode in cephalothorax length have associated with them corre- 

 sponding deviations in the same direction of the other size characters, 

 clearly we should expect a similar sort of skewness in the frequency dis- 

 tributions of these other characters. In the present case we actually do 

 find a very high degree of correlation or association existing between the 

 different pairs, and hence that all should show skewness in the same 

 direction is no more than is to be expected. 



The explanation of the fact that the direction of the skewness is positive 

 is possibly in part that individuals in different molts were included in the 

 sample. This, however, can at most account for only a minor part of the 

 observed condition. It seems to us idle to speculate as to the fundamental 

 causes concerned in the production of skew variation in cases of this kind. 

 We can only hope to determine them by experimental investigations 

 conducted ad hoc. 



(c) The kurtosis is in every case positive, or, in other words, /32>3. 

 In order to get an idea of the relative significance of this constant it is 

 necessary to compare the value of fi^ — 3 with the probable error of fB^ for 

 the normal curve. Working from the well-known formula we have, when 



Probable error of y8, = it 0.1564 



