W. R. Macdonell 
231 
and n'=25, and therefore from Elderton's Table P = '99; that is, in 99 out of 
every 100 trials we should get a more peaked polygon than the one representing 
the actual observations. The fit may therefore be considered perfect. 
These two illustrations show that we must be most cautious in using the 
peaked or multimodal appearance of a polygon representing the distribution of a 
character in a short series of crania, as an argument in favour of heterogeneity of 
race, seeing that the peaks may well be due entirely to random sampling*. 
In view of Pearson's article in Biometrika, Vol. Ii. Part ill. on " Professor 
Aurel von Torok's attack on the Arithmetical Mean," I have added to Table IX. a 
column showing the difference between the mean and the mode, and its probable 
error. Looking at the absolute values of the difference, we observe that in only 
three cases does it exceed 1 mm. In most of the remaining cases it is only a very 
small fraction of a millimetre or a point, and when compared with its probable 
error is quite insignificant. 
(11) On the Correlation of Cranial Characters. 
In addition to the 27 pairs of characters which C. D. Fawcett exhibited in her 
Table Xlll.f, I have selected seven other pairs of important characters, and in 
arriving at the 34 coefficients of correlation I have calculated the sum of the 
products of the pairs of measurements, instead of forming correlation tables; by 
this method, somewhat greater accuracy is attained, especially in dealing with 
circumferences and indices. Thus, let a, h be the measurements of two characters 
in an individual ; and il/^ the means, and and a.^ the standard deviations of 
the characters, N the total number of individuals ; then r, the coefficient of 
correlation, = S {xy)jNa-ia.., and 
^^^^^^S{M,-a){M,-h)IN 
= S{M,M^-aAL-hM, + ah)IN 
= {NM,M, - M,8 (a) - MS{h) 4- S {ah)} /N 
= ^)-^«. 
The following table gives the coefficients of correlation, and includes C. D. 
Fawcett's Table XIII. for the sake of comparison. 
* See C. D. Fawcett, loc. cit. p. 4.54, and Pearson in Biometrika, Vol. ii. pp. ,341 — 34.3, where 
he deals with greatest forehead breadth and greatest skull breadth in 2000 Hungarian skulls, 
t Loc. cit. p. 455. 
