Raymond Pearl 
39 
Considering first the skewness, it is seen to be in all the series positive, or the 
mean is greater than the mode. Further we note that in all cases the value is low. 
Whether the values can be considered significant however can only be determined 
by an examination of the probable errors. The formula for the probable error of 
the skewness is ■67449 ^ ~- *. The limiting values for this probable error for 
values of n ranging from 197 to 529 as in the present case are respectively "0589 
and "0359. Having regard to the number of cases on which the calculations are 
based it appears that in six out of the nine cases tabulated the skewness can be 
regarded as certainly or pi'obably insignificant. In all of these six cases the 
skewness is less than thrice its probable error, in two cases it about equals its 
probable error, and in one is less. The remaining three cases out of the total 
(skewness, "1635, •2161 and '1769) are very probably or certainly significant. In 
general we may safely conclude, I think, that, in the case of tlie tueight of the brain, 
the distance from the mean to the mode will he very small. If the mean and mode 
do not coincide the mean will he greater than the mode. This agrees with 
Miss Fawcett'sf conclusion for the most important skull characters in the Naqada 
race. MacdonellJ finds, however, that in the case of the English, considering the 
same skull characters, " if we were to draw the curves, the mean would be found in 
half the number of the curves to be less, and in the other half to be greater than 
the mode." In neither brain- weight nor skull series does there appear to be any 
definite preponderance in the value of the skewness of one sex over the other. 
We may turn now to the other constants, which are of most significance in 
determining whether the distribution may be considered normal within the limits of 
error ; viz., V/Si, /Ss, and the criterion. The probable error of ^= '67449 ^Z^) 
ranges in value for our series between "0718 (w = 529), and '1177 (?« = 197); 
that of {= -67449 y^^) between 1437 (n = 529) and -2354 {n = 197) ; and that 
of the criterion "67449 between -2873 C« = 529) and "4708 (?i = 197). 
Considering the probable errors of it is seen at once that of the eight 
"total" series three give certainly insignificant values ('0958, '1139, '1113) 
for V/^ij two others give values which are probably insignificant ('1694, "2254). 
One ("2298) is probably significant ; and the two remaining values (•3623 and 
•4029) are certainly significant. The Bavarian " young " % series gives a certainly 
significant value for VA- Taking next the deviation of from 3 in comparison 
with the probable error of /Sa we see that in four cases (3 — /Sj = — '1031, •1628, 
—'1396, "1329) /S2 differs from 3 by an insignificant amount. In one case ('2036) 
the difference is less than twice the probable error and hence may be considered 
* The formulae for the probable errors of the analytical constants are given on p. 278 of Pearson's 
memoir on "The Mathematical Theory of Errors of Judgment," loc. cit. supra. 
+ Loc. cit. p. 443. 
% Biometrika, Vol. in. p. 227. 
