78 BULLETIN 16 9, UNITED STATES NATIONAL MUSEUM 



from which they were, in fact, drawn do differ in the mean number 

 of serrations. 



Similar differences associated with provenience occur in several 

 other characters and show that Piilodus mediaevus and P. montanus 

 are distinct. They could not be certainly separated if they occurred 

 together, although even in that case the significantly greater varia- 

 tion of the combined samples would lead one to suspect that the 

 population was heterogeneous. 



If we accept the specific groupings finally adopted as valid, some 

 interesting conclusions regarding variability and the value and signifi- 

 cance of various characters for taxonomy in these animals are possible. 



The length of P4, the most useful single dimension as this is far the 

 commonest tooth in multituberculate collections, has a coefficient of 

 variation of 9.3 ±1.6 in the sinclairi group. This is high, but com- 

 parably liigh coefficients have been recorded for linear dimensions of 

 teeth of mammalian species. ^^ In the montanus group this coefficient 

 is 5.3 ±0.6. This dimension is thus much less variable in the avail- 

 able sample of the latter species and to that extent seems a more 

 reliable taxonomic character for it than for the smaller sinclairi. ^^ 

 On the other hand, the length of Mi in the sample of sinclairi is very 

 constant, coefficient of variation only 4.4 ± 1.1, and in montanus some- 

 what more variable, coefficient 5.7 ±1.35. 



Thus appear the interesting facts that in sinclairi P4 is highly 

 variable and Mi little variable in length, v/hile in montanus both are 

 moderately, and about equally, variable. As a result of these facts, 

 the ratio length P4 : length Mi is very much more variable in sinclairi 

 (standard deviation 0.22 ±0.06) than in montanus (0.04 ±0.01). This 

 is also accentuated by the further fact that in montanus, but not in 

 sinclairi, these two dimensions are positively correlated, that is, that 

 in montanus the larger premolars tend to be associated with the larger 

 molars, while in sinclairi the available data show no such tendency. 

 Another expansion of this same unexpected and important fact, clearly 

 visible on the scatter diagram (fig. 7), is that in sinclairi the line of 

 regression of length Mi on length P4 is horizontal or even slightly 

 inclined downward to the right, coefficient nearly zero or a very small 



32 E. g., in Pal. Sin., ser. C, vol. 5, fasc. 5, Helga Pearson gives coeflBcients up to 11.4 for M^ of one side 

 in one sex of a single homogeneous human race and coefficients up to 8.1 for probably very homogeneous 

 groups of fossil suids. 



33 How misleading the best judgment may be when not aided by statistical treatment is shown by the 

 fact that although Gidley clearly relied on size of P4 chieily for specific separation (as shown by the nature 

 of his groupings and also by his unpublished specific names, all of which denote size), he placed the small 

 sinclairi specimens in one species but divided the large montanus into three species, although the variability 

 of the former Is nearly twice that of the latter. The misleading factor is that the absolute difference in the 

 extremes is less for the small than for the large species. Although this is the striking character to the eye, 

 it is not the essential factor either from a statistical or from a biological point of view. 



