388 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
curves, the integral of the frequency in all suggested forms of the frequency curve 
being not expressible in terms of undetermined constants. Valuable as is the 
method of percentiles for representing popularly the numerical facts of anthro- 
pometry, it is to be regretted that percentile statistics are replacing the raw material 
in so many publications. The raw material of Professor WELDOoN’s crab-measure- 
ments and BowpircH and PortsEr’s child-measurements ought to be preserved and 
circulated in print, as a means of developing and testing statistical theory. 
(25.) Example V. Length-Breadth Index of 900 Bavarian Skulls.—The following 
statistics are taken from Tables .—-VI., VIII.-X., inclusive, of J. Ranke’s ‘ Beitriige 
zur physischen Anthropologie der Baiern, Miinchen, 1883.’ They include all the 
material, which may be treated as typically ‘‘ Alt-Baierisch,” both male and female 
skulls. 


| 
Index. Frequency. Index. | Frequency. | Index. Frequency. 
70 1 80 71:5 | 90 10 
71 1 81 82 91 8 
72 0 82 116 92 3 
73 2°5* 83 98 | 93 15 
74 15 84. | 107 94 2 
79 3°) 85 82 95 15 
76 125 86 74 | 96 0 
77 17 87 58 | 97 0 
78 37 88 345 I 98 1 
79 55 89 | 19 | 99 0 
} I 






We find, as before, 
Position of centroid-vertical, 83°07111, 
a=, 3°468, Yo = 103°532 (for normal curve), 
Wy = 12°027166, B,= ‘0078995, 
us, =  3°707179, Bo =  3°649553, 
ju, = 527°91696, r= 1242734, 
Ob "111388, Skewness = 0321186, 
m = 7'213867, vy = 853,771, a = 11°69583, Yo = 107°4706. 
Thus we see that the curve is again of Type lV. This result seems of considerable 
significance, but it requires, of course, wider examination of cases than I have yet 
been able to make. But, so far as I have gone, in both anthropometric and 
biological statistics, whether relative or absolute measurements of organs, the 
frequency curves all deviate from the normal curve—however slight the deviation— 
in the direction of Type IV. hat is to say, the distribution of chances upon which 
the frequency of variation of an organ depends, appears to resemble the drawing of a 
* Indices such as 73°5 have been divided between 73 and 74 groups. 
