C. D. Fawcett 
441 
Generally it will be realised that the cranial characters are not highly variable 
like the parts of plants, and that there is considerable agreement between results 
from very different races. 
Turning now to the graphical and analytical representation of cranial variability, 
it was impossible to give diagrams of the 47 characters dealt with for the two sexes. 
Accordingly a choice was made of 12 characters, and the curves calculated and 
plotted in the 24 cases corresponding to these characters for the two sexes. 
The laborious calculations required are largely due to Dr Alice Lee, and for the 
draughtsmanship involved in converting her numbers into diagrams we have to 
thank Mr Karl Tressler, formerly one of the demonstrators in the Department 
of Applied Mathematics at University College, London. The general fit and 
smoothness of Mr Tressler's diagrams is the best verification of the accuracy and 
completeness of Dr Lee's work. 
Now these diagrams are given with a view to bringing out a number of points. 
First and foremost to emphasise that no stress ought to be laid on the " peaks " of 
such frequency distributions as occur in most craniological investigations. These 
" peaks " are chiefly due to the fact that we are dealing with very small random 
samples. Few crauiologists can work with more than 30 to 100 measurements of 
any character in one sex. In the next place .sex-determination is only a, probable 
determination ; further, diversity of age, occasional foreign skulls, postmortem de- 
formation and other causes produce heterogeneity. Lastly where a cemetery or 
local burial place has been plundered, we are certain to get family groups of skulls. 
All these causes tend to emphasise the irregularity of the distribution even l)eyond 
the limits of random sampling. We can only ask, are these small samples more 
variable — variability being measured by the standaid deviation — than the largest 
and most homogeneous series known to us, such a sample for example as the 
" Altbayerisch " ? If they are not, and the Naqadas are certainly not, then it is 
very unwise — -nay, quite unjustifiable — to argue from a series of peaks as to racial 
mixtures. Even if like peaks occur in the two sexes for several chai'acters the 
sources of heterogeneity given above are often ample explanation. 
What we do see from these diagrams is that the curves which have already 
proved themselves sufficient for many frequency distributions in other living forms 
will suffice to graduate and smooth the frequency polygons obtained for short 
series of craniological measurements. They will serve for comparative purposes as 
soon as further series of craniological measurements have been reduced in the 
statistical manner advocated in this memoir. When once the craniologist has 
learnt to look upon his many peaked sample as the disturbed form of such smooth 
distributions, he will hesitate to make statistically unwarranted racial differentia- 
tions on the basis of such peaks. The question will then become : Is any system 
of peaks, allowing not only for random sampling, but for the above-mentioned 
sources of heterogeneity, really significant ? In the state of our present ignorance, 
it is safer to be sceptical as to significance, than to build up on a statistically 
unsound foundation elaborate theories of racial mixture and racial relationship. 
Biometrika i 48 
