W. R. Macdonell 
227 
for which the frequency is greatest, and the " skewness," or ratio of distance 
between mode and mean to standard deviation, with its probable error*. 
Dealing first with the skewness, we observe that in 13 cases it is positive, or 
the mean is greater than the mode, and in 13 cases it is negative, or the mean is 
less than the mode. Further, in only two of the 13 cases of positive skewness, 
viz., NB and HjL $, and in only two of the 13 cases of negative skewness, viz., 
H and G'H ^, can the skewness be considered as certainly significant when 
compared with the probable error; of the remaining cases two of the negative 
skewnesses, L $ and U %, and one of the positive, NH %, are perhaps significant; 
the rest are certainly insignificant. We cannot, therefore, conclude, as C. D. Fawcett 
was able to do with her Naqada skullsf, that if the mean and the mode do not 
coincide, the mean will be almost invariably greater than the mode. In our series 
if we were to draw the curves, the mean w^ould be found in half the number of the 
curves to be less, and in the other half to be greater than the mode. 
We will next examine the constants /Sj and l3o,. If A = 0 and /3o = S, the 
curve representing the distribution is the normal curve, and, therefore, in order to 
see how far the skew curves diverge from the normal we must first observe how 
much /3i and /S^ differ from zero and 3 respectively, and then estimate the 
significance of the difference by comparing it with the respective probable errors. 
Columns 8, 9 and 10 enable us to make the comparison. Taking the V/^i column 
first, we note that in only four cases out of the 26 is V/^i certainly significant when 
compared with its probable error, and these four are precisely those which we have 
had occasion to notice as exceptional in regard to skewness, viz., H ^, O'H 
NB (/■ and HjL $ . In six cases \/ fi^ is greater, but not more than 1^ times greater 
than the probable error, and in the remaining 16 cases V/^i is less, sometimes very 
much less than the probable error. 
A comparison of the 3 — /Sg column with the probable error of /Sg leads to 
similar results ; in only three cases is the difference certainly significant ; in the 
most unfavourable of the remaining cases the difference is less than twice the 
probable error. Similar remarks apply to a comparison of the criterion with its 
probable error. 
Our examination of the constants of the curves thus confirms C. D. Fawcett's 
conclusion J : Witlt series of skull measurements such as the liresent, which are long 
for the craniologist, if short for the statistician, we shall reach for most practical 
purposes adequate graphical representations of the frequency by using the normal 
curve of deviations, y ~ yt^e ~ 
This being the case, I do not propose to calculate and plot any of the skew 
curves, but will content myself with tracing two normal curves, the one repre- 
* For the formulae for calcnlatiug these probable errors, see Pearson, Fhil. Trans. Vol. 198, A, 
p. 278. 
t C. D. Fawcett ou p. 443 says only 5 out of 24 cases have negative skewness, her table shows G, but 
this does not affect her argument. 
% Loc. cit. p. 443. 
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