372 University of California Publications in Geological Sciences 



also whether it could belong to group B; and (2) if it could belong to either, then 

 determining the one in which it should be placed, i.e., finding a discriminant func- 

 tion by which the specimen may be categorized. Utilization of many measurements 

 and adjustment for correlation are desirable here also. 



The technique employed in Mahalanobis's D* and discriminant analyses can be 

 visualized roughly by extension from bivariate analysis. If two measurements are 

 taken on each tooth of a series, the results can be plotted as a series of points on an 

 ordinary (Cartesian) graph, with the breadth, say, being plotted on the ordinate, 

 the length on the abscissa. Thus each tooth is represented by a point on the graph. 

 An additional dimension — for example, crown height — would require a three-di- 

 mensional, "solid" graph, but each tooth could still be represented by a single point. 

 If a total of n measurements were taken, an n-dimensional space would be re- 

 quired to represent each tooth as a point. It would be desirable to have the points 

 in the cluster that represents a series of teeth so arranged that equal densities of 

 points would form concentric spheres around the centroid. If measurements unad- 

 justed for correlation and disparity in gross size were used, the cluster would be 

 ellipsoid. But by removing mutual correlation and by putting the measurements in 

 standard form (making the mean-differences comparable) the spheroid shape 

 would be attained. If similar operations were performed on measurements of 

 other series of teeth and these all placed in the same n-dimensional space, the 

 amount of separation between the clusters of points representing teeth would be 

 directly related to the distance between their centroids, and further, measurements 

 of any questionable specimen, placed in this n-dimensional space, would assume a 

 position indicating its affinities with the various clusters. 



The multivariate technique used in this study demands that the data meet three 

 requirements: (1) that the frequency distribution of the measurements of each 

 variable approximates a normal curve, (2) that the intercharacter correlation is 

 linear, and (3) that the joint distribution of the variables approximates a multi- 

 variate normal distribution (Mahalanobis, Majumdar, and Rao, 1949: 120-121, 

 137-138). If the first two requirements are fulfilled by the data, the probability 

 that the data also fulfill the third is likely but not proved, since the first two require- 

 ments are necessary but not the only determinants for the third. No single test is 

 available to demonstrate that the third requirement has been met, but the previous 

 experience of many biologists using morphological data supports the assumption 

 of a normal distribution for the separate variables. 



Mixing the sexes could upset the normal distribution expected for measurements 

 taken on material of one sex. A bimodal distribution might be predicted, but it has 

 not been found for the canid measurements. Hildebrand's (1951) data, and data 

 gathered for this study, indicate that canid males average only 3 to 5 per cent 

 larger than comparable females of the same species. The potential bimodality 

 seems to be subsumed in general morphological variability. 



The length of the maxillary tooth row in Canis latrans ochropus was tested for 

 a normal distribution by the cfti-square test. The low value obtained indicated a 

 reasonably good fit to a normal curve. In addition to this test, several other varia- 

 bles were plotted on normal-curve graph paper; all described fairly straight lines, 

 as normally distributed data will. 



