C. D. Fawcett 
443 
These results were in fact first obtained from the measurements, but at a later 
date further skulls were found available for measin-ement. The third column gives 
the unit in terms of which the corresponding moment constants in the next three 
columns are expressed. Then follow the values of the numerical constants ySj, ^2 
and the criterion 6 + 3/8i — 2/3„. Finally the mean and mode in the usual units 
and the skewness of the distribution. The actual equations to the 24 curves 
deduced from these constants with the origin in the same unit as the mean are 
given on the diagrams below. In these curves the unit of y is one individual per 
unit of X, and the unit of x is the unit given in the third column of Table XII. 
As a matter of practical use, it is sufficient in these cases to treat y as given 
by a scale of absolute frequency, which is indicated on the vertical to the left of 
the diagram. 
Several points may be drawn from these analytical results. 
(a) The skewness is negative in five cases only, or, looking at the diagrams, 
the mode is gi'eater or falls to the right of the mean in only 5 out of 24 cases. 
But supposing the skewness to be merely a result of random sampling and not due 
to any bias in the organs in question, the probable error of skewness would be 
•67449 V3/2'rt where n = number of individuals dealt with*. We therefore con- 
clude that for 11 = 81 to 144 observations, the probable error of skewness would be 
between "09 and '07. Thus in four out of the five cases, the skewness is only about 
half its probable error ; in the fifth, tlie auricular height of the female, the skewness 
= — '1450 + '0624, and this may possibly be considered as significant. Of the 
positive skewnesses 8 are insignificant, and 11 certainly or probably significant. 
We may therefore conclude that : 
In measurements on the skull, if the mode and mean do not coincide, the mean 
will almost invariably he greater than the mode. 
(b) Next let us examine the probable errors of V/3i, /So find the criterion -f. 
The probable error of V/:?! runs in our case from "14 to '18 roughly, according 
to the number in the series, that of /Sg from about '28 to '37 and that of the 
criterion from about '56 to "74. We notice at once that fi^ differs from 3 in 
only very few cases by an amount which is significant having regard to its 
probable error ; the same is again true of the criterion, which differs from zero by 
quantities of the order of the probable error. V /3i has deviations from zero, which 
are upwards of double its probable error in two or three cases, but on the whole we 
may conclude that : 
With series of skull measurements such as the present, ivliich are long fur the 
craniologist, if short for the statistician, we shall reacli for most practical purposes 
adequate graphical representations of the frequency by using the normal curve of 
deviations: 3/ = 2/oe~^"-°'^'. 
* Pearson : " On the Mathematical Theory of Errors of Judgment," PliU. Trans. Vol. 198, A, p. 278. 
t Loc. cit. p. 278. 
