22 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



It is apparent from inspection of fig. 3, or of the frequency distri- 

 butions in table 1, that the variation in Ceratophyllum in respect to leaf- 

 number is very skew. In order to bring out more precisely the amount 

 of this skewness, as well as other important facts regarding the nature 

 of the distributions, I have analyzed a portion of the data by the methods 

 given by Pearson, in his fundamental memoir on "Skew Variation" 

 (Pearson, '95) and its supplement (:01). I have dealt in this way with 

 two of the distributions made up of all the whorls on each plant. The 

 first of these distributions (No. 177, table 1) is that resulting if Series 

 I, II, and III are combined, that is, it is what is obtained by adding 

 together distributions 44, 61, and 62. It represents the variation of 

 the character under consideration in the Carp Lake population of Cera- 

 tophyllum, taken as a whole. As has been pointed out in the preceding 

 section, all the Carp Lake plants came from the same spot, and only 

 differed in the time of collection. It is clearly evident from the diagrams 

 in fig. 3 and the constants in table 2 that these three collections do not 

 differ from each other significantly. The differences between the con- 

 stants for the totals of these three series, as given in table 2, are small 

 in comparison with their probable errors, which 'are themselves small, 

 as we are dealing with relatively large numbers. Hence the series may 

 be considered as three random samples of a homogeneous population and 

 may be combined if it is desirable in order to bring out a particular 

 point. The other distribution, which I have graduated for comparison 

 with this, is that of plant 2, Series IV (No. 74, table 1). This plant was 

 the largest single plant in our collections and for that reason was chosen 

 for analysis. Others would have given essentially the same results, 

 but with larger probable errors. 



The problem of how best to treat analytically distributions of the 

 sort given by Ceratophyllum is, as has been emphasized by Pearson 

 (:01, p. 456), a very difficult one. We have, in the first place, in these 

 distributions discrete variation, the observations proceeding by unit 

 steps, while the mathematical function which we take to represent them 

 is continuous; further, the distributions are markedly skew; the range 

 is very small, giving less than ten observations to determine the moments 

 from; and finally there is no approach to high contact at one end of the 

 range. Hence, the determination of the true moments is by no means 

 a simple matter. Proper corrective terms for use in such cases have 

 not yet been worked out. Sheppard's corrections are theoretically not 

 applicable. In the present instance by actual trial I obtained much 

 worse results with Sheppard's corrections than with the raw moments, 

 and consequently for lack of anything better the latter have been used 

 in graduating these two distributions. 



