Is it certain
that the Law
of Errors is
applicable to
anthropometry?
Assuming
that the Law
of Errors is
applicable to
these series,
how should
they be
interpreted ?
cannot become appreciably more long-headed or appreciably more round-headed. And to
superpose a Binomial Curve over the Frequency Curve in these cases would necessitate the
assumption that the tendency of such series was to vary equally to either side of their
respective arithmetical means. Though this may be theoretically possible it is difficult
to conceive it possible in practice. The variation of the long heads would on that
hypothesis exhibit specimens with little more breadth than the geometrician’s straight
line, and that of the broad heads would result in oblongs of which the width was greater
than the length. Rather than believe in the possible occurrence of such phenomena, we
should suppose that when a series is of such a character that its arithmetical mean is near
the extremity of known variation the curve derived from it would be not binomially
symmetrical but sensibly askew in outline, that is to say, it would not conform to the
normal curve of the Law of Errors.
We cannot then regard it as yet demonstrated that the Law of Errors expresses the
distribution of most characters in mankind. We are, however, willing to accept the hypothesis
for the moment, pending further inquiry, and to see to what conclusions it may lead.
Quetelet, to recall the details of the last chapter, showed that in a pure series
measurements would yield a curve closely approximating to the Binomial Curve, and this
brings us at once to the question which underlies the whole statistical treatment, viz. what
is a pure series ? That is a matter which Quetelet did not fully consider. For him any
series is pure which conforms to the principle of the Binomial Curve1, but he never treated
of those which do not so conform. It has been usual to draw the corollary that when
a series fails to conform to the Gauss-Laplacian Law it must, in the absence of any
peculiar circumstances to explain it, be regarded as impure. But Quetelet did not deal
with this aspect of the subject, and therefore did not furnish the necessary criteria for the
anthropologist who is desirous of ascertaining whether a curve of measurements represents
one race or more than one. Consequently, although the Probability Curve has frequently
been used by anthropologists since his day, it has always been handled with considerable
uncertainty and misgiving. Of late the subject has been taken up by various highly-
skilled mathematicians, who have devised formulae by means of which they claim to
interpret the significance of any given curve. Such formulae, however, which are difficult
for the layman to master and to handle, give very little assistance in determining the
nature and meaning of a variation from the normal distribution unless they are illustrated
by the actual graphic representations. On the other hand, mere ocular inspection of
a curve will often suggest a clear interpretation, of which the abstruse formulae by themselves
give no idea. Thus, in the present instance, a professed biometrician would declare
that, judging by his canons, our curves were perfectly consistent with unity of race, but not
inconsistent with any one of a dozen other possibilities ; yet judged by eye they seem to
conform in some cases at least to precisely those conclusions which we have reached by
other and independent means. And we would point out that the statisticians themselves
do not invariably dispense with empirical treatment. It is unnecessary to go further for an
illustration than Professor Karl Pearson’s Huxley Lecture for 1903. Amongst the problems
with which he was dealing in connexion with the inheritance of mental and moral
characteristics, as illustrated by statistics collected from schools, was that of the ‘ athletic
power’ of the school-children. But instead of accepting unquestioningly the data in the
schedules returned by the teachers as he had done in other cases, he thought well to
1 T h ou gh it is now known that an apparently regular curve ma y be really heterogeneous and composed o f
several irregular curves, see above, p. 74.
exercise his judgment upon this particular point1. We do not suggest that he may not
have been perfectly justified in doing so, but it is evident that such a procedure, legitimate
as it may be, vitiates the claim to exactitude which is based solely on the infallibility and
necessity of mathematics. It is an admission, in short, of the validity of empirical methods
when employed by a properly trained and disciplined intelligence.
In practice we greatly doubt whether the system, which according to its authors is
the only safe one to adopt for the elucidation of complex craniological problems, provides
so useful an instrument as they represent. In the very simple form explained in the last
chapter a mathematical treatment of series may give evidence corroborative of the results
obtained by other means, but by itself it has proved inadequate to suggest these results.
And the employment of the more complicated methods advocated by the biometricians
leads to no new developments, but merely confuses the issue. In support of our view
we may quote the verdict of Mr. Porter, who has made a complete study of the subject on
our behalf for the purposes of the present work. He informs us that ‘ the erudite and
complex development of the subject presents many interesting points judged by the
standard of mathematics, and that it may quite conceivably be of use in some branches of
biology, but that, in his opinion, the more advanced portions of the subject are absolutely
useless for the series which we are considering and the work on which we are at present
engaged.’
It may possibly be objected that deficiency of material is a sufficient explanation
of the failure of the biometrical methods to produce any conclusive results. But it
must be urged that the series from Egypt are as large as any that the craniologist
can hope to obtain so long as he is careful rigidly to distinguish between disparate
elements2. I f they fail in this instance it is not likely that they will prove of much
service in ordinary craniological inquiries, where the amount of material is usually much
less in quantity.
That we may not be accused of bias in regard to this matter we have published
a complete set of tabulations in the supplement (Charts I and II), where the Curves of
Probability are accompanied by such mathematical calculations as will facilitate the work
of those who wish to experiment with statistical methods.
We may now proceed to state how far any general conclusions can fairly in our opinion
be drawn as to the character of the several series published in the supplement. The
reader should have the charts of the supplement before him at this point. There are
only four properties of the curves which are of apy real importance to the non-mathematical
reader. These are the misfit and r which should indicate the homogeneity or heterogeneity
of the series, and m and r which represent the prevalent type in the series and
the stability of that type.
The first question to be approached is that of the homogeneity or heterogeneity of
series. A review of the first few periods will enable us to decide how it should be treated.
An important point is brought out by the diagram of the cephalic index for the males of
the Fourth and Fifth Dynasties (Chart I e , Fig. 1 ) , viz. that misfit alone is an insufficient
criterion of homogeneity. In that diagram the misfit amounts to as much as 46 per cent.,
which prima facie might seem to be strong evidence of the impurity of the series. But
1 J o um . A n ih rop . In s t., vol. xxxiii, p. 199. * A n examination o f the schedules led me at once to the conclusion
that much o f this resemblance w as wholly spurious. Certain schools . , . prided themselves on an athletic reputation,
hence two brothers or two sisters at such schools are usually returned as an athletic pair.’
2 See Appendix, p. 130, note on the N agada Series.-
Assuming
that the Law
of Errors is
applicable to
these series,
how should
they be
interpreted ?
At best the
mathematical
treatment
cannot
elicit new
results, but
can only
corroborate
those
obtained
by other
methods.
The principles
which
we adopt in
treating the
Curves of
Probability
and Frequency.