R. Crewdson Benington 
129 
In comparing the cranial type contours of two races, it is desirable to have 
some measure of whether they diverge more than the limits of the probable error 
of random sampling. This point is not a very easy problem to solve because 
of course the measurements made are highly correlated*. We considered that 
some appreciation of significance of divergence might be obtained in the following 
manner : 
In the transverse vertical section we accepted the auricular line as a funda- 
mental line which in position, not in length, might be considered as a fixed base. 
From the mean value of the vertical axis we then added and subtracted twice its 
probable error. On the lesser vertical axis at the tenth points found in the usual 
way were then plotted up the mean right and left ordinates less twice their 
probable errors, and on the larger vertical axis divided in the same manner were 
plotted up the mean right and left points plus twice their probable errors. We 
thus obtained a diagram with a type transverse vertical section and on either side 
of it a second diagram, the outer one rather larger than the type, the inner rather 
less than the type. Within the space between these two contours practically fell 
the great bulk of the individual contours. We will speak of this space as the 
type zone. If two type zones for two races do not fall wholly or at definite points 
well outside each other, we suggest that the races represented by these types are 
either undifferentiated or closely related. As it was difficult to appreciate clearly 
the type zones on the scale of the actual contours, the type zone diagrams were 
drawn to double the scale on relatively thin paper. The two corresponding type 
zone contours of two races were then placed on a large glass plate below which 
was diffused electric lighting, and the correspondence or divergence of the zones, 
when the auricular lines and the vertical axes were superimposed were noted. 
The horizontal section was treated in exactly the same manner. Twice the 
probable error was added and subtracted at each end from the mean fronto- occipital 
axis, and these two greater and smaller fronto-occipital axes used as the bases for 
plotting the ordinates increased or decreased as the case might be by twice their 
probable errors. Thus a type zone was obtained in the case of the horizontal 
section. 
The same method was adopted in the case of the sagittal section, but it seems 
in this case somewhat less conclusive, because the nasion-gamma line, although 
close to the nasio-occipital line, is not necessarily coincident with it, and may 
therefore be itself variable and less a standard for comparison. While the line 
from nasion to lambda might form a more natural base, it could not be even 
approximately determined in the living head, and the nasion-inion line is also 
nearly as difficult of true determination. One of the points of Dr Benington's 
investigation was a comparison of crania and living head contours, and therefore 
the nasion-gamma line was preserved as base for the type zone of the sagittal 
contour. 
* And, further of course, the same number of skulls may not be used in every case. 
Biometrika vin 17 
