M. A. Lbwenz and K. Pearson 
371 
at once realise why Dr Lee considered a diametral product formula could be 
better extended from one local race to another than a multiple regression 
circumference formula. She did not attempt a product circumference formula. 
In view of the fact that our formulae vary so with race, it is venturesome to 
form a mean equation and treat it as an " inter-racial formula," but for com- 
parison with Dr Beddoe's results it is perhaps desirable to do so*. We have as 
mean formula for males : 
Probable C= 244-6 + -01977 P (/). 
We would only warn our readers to use such a formula with caution. For 
example, never to use it for females, and only in preference to one of the above 
four formulae when no evolutionary race relationship can be predicted between 
the above races and the race to be investigated. 
We now turn to the probable error of these results. We have : 
Probable Error of Individual Prediction. 
Formula 
(«) 
(0 
(d) 
(e) 
Mean 
Probable Error ... 
48 
41 
39 
35 
40-4 
To test the value of these results, 15 skulls (as nearly as possible every 
thirtieth skull) were taken at random from the Naqada measurements ; 15 skulls 
from the Theban records (as nearly as possible each tenth skull), and similarly 
15 from the Altbaierisch series. In the case of the Naqada crania 8 had an 
error greater and 7 less than the probable error ; in the case of the Theban the 
numbers were 6 greater and 9 less, and for the Altbaierisch 9 greater and 6 less ; 
or total 23 greater and 22 less than the probable error, which may be considered 
very satisfactory. The mean error of the 45 skulls as found from the mean 
formula is 47-5, corresponding to a probable error of 47-5 x '8454 = 40'2. The 
theoretical value of this probable error is 40-4. There can therefore be, we think, 
small doubt that our mean formula will predict the capacity of a single individual 
skull from circumferential measurements with a probable error of about 40 cubic 
centimetres. Dr Lee's diametral product formula f showed a probable error of 
43 for Aino (/s, 55 for German cfs and 52 for Naqada (/s, say a mean value of 
50. Actual application to 80 crania gave only a mean error of 49, or a probable 
error of about 42. I'hus we should expect the present formulae to give slightly 
better results than the diameter product formulae, but for reasons to be given 
later we hold them quite inferior to tho.se formulae for the living head. 
* A more correctly inter-racial result is given on p. 386 below. The above is really a mean intra- 
racial result. 
t See loc. cit. pp. 234-7. 
47—2 
