436 Variation and Correlation of Skull Measurements 
As is shown in the table, the coefficients of correlation are higher in 
the rat than in the averages from the human subject in every case, except 
the cranial capacity and length of skull where the female rat is slightly 
low. The correlation is decidedly greater in the rat in the case of the 
capacity as related to the width and height, while in the case of the length 
and the product of the three diameters the results for the rat are close 
to those for man. Here again we notice that in the female the coefficients 
of correlation are slightly greater than in male in the case of the human 
skulls (except Aino) while in the rat the reverse is true. We can not lay 
too much stress on this relation, however, since as is shown in the column 
marked “ difference ” the size of the probable errors shows the differences 
to be without significance. Thus although, the general tendency is to 
show that in the human skull, except Aino, the coefficients of correlation 
are slightly greater in female nevertheless any definite statement must be 
postponed until we have data sufficiently abundant to further diminish 
the value of the probable errors. 
(f) Comparison between the observed and predicted values of skull 
measurements.—In Table I we noticed that the mean values of the male 
characters are always greater than those of the female, the differences 
always being more than three times the probable errors. The greatest 
difference was found in the length of nasal bone, the least in the width of 
the cranium, while the remaining characters gave intermediate values. 
The question now arises: How the several characters will be related if 
the length of the entire skull of the male is reduced to that of the female ? 
In other words, is the male skull to be considered as an overgrown female 
skull, or the female skull an undersized male skull? ‘To answer this 
question a number of characteristic equations were prepared. ‘These equa- 
tions will enable us to determine the probable values of the characters in 
both sexes. The form of the characteristic equation is as follows: 
Yaytr7—® 
where X, Y are the two characters under consideration, 7, 7 are the two 
respective means of the arrays, ox, 6, are also the two respective standard 
deviations and y is the coefficient of correlation. The following table 
was made in order to show the values of the characters when the lengths 
of the entire skulls were equated. 
As is shown in Table X, when the length of the entire skull of the male 
is equated to the observed length of the female skull and vice versa, the 
sexual differences become very small. The closeness of agreement between 
observed and predicted values of the several characters varies with the 
