FORT UNION OF CRAZY MOUNTAIN FIELD, MONT. 



75 



from the sinclairi group, but in the grouping on external cusps of Mi 

 it is clearly distinct. Or, as another example, the gidleyi and douglassi 

 groups are not well distinguished in either of these diagrams, but are 

 far apart in the scatter diagram of length Mi and length P4 (as they 

 are also in others not reproduced). By continuing this process, eight 

 groups were eventually distinguished. A smaller number than eight 

 was inconsistent with the distribution of two or more characters, 

 and a larger number not necessary to interpret the distributions. 



Figure fi.— Histogram and corresponding roughly fitted normal curve of length of P4 in Ptilodus montanus 

 from the Qidley and Silberling Quarries. The distribution is the same as that for this species included 

 in flg. 5, 6. 



59 



H 

 I 



b 



, '''MONTANUS 



'2.0 



5.0 50 



LENQXH 



8.0 



9.0 



P4 



Figure 7.— Scatter diagram of length of M: and length of Pi for all Gidley and Silberling Quarry specimens 

 of multituberculates that show both of these measurements, with separation into species based on this 

 and all other characters. 



The two most abundant groups {sinclairi and montanus in the classi- 

 fication as finally worked out) were then tested for homogeneity, by 

 statistical methods which showed that their deviations from normal 

 form in any case are not of probable significance, and then for numer- 

 ous continuous variates of each of these groups the mean, standard 

 deviation, coefficient of variation, and the standard errors ^° of each 



"> I have given standard errors throughout. 

 0.6745 times this figure. 



The probable error, which has nothing to recommend it, is 



