T h e prind- a closer inspection shows that the great discrepancy between the Frequency and Probability
weadopt hi Curves is due in this particular case to an unusual concentration of the specimens in the
Curvesof16 nei&kbourhood of the arithmetical mean with a consequent deficiency towards the extremes,
Probability so that in spite of the high arithmetical figure obtained for the misfit the cephalic index
Meardngand m Fourth and Fifth Dynasty males is really much less variable than in some other
importance periods where the misfit of the curves is not so of great. This fact is betrayed, apart from 'misnt . . i t , . j > r
and * prob- ocular observation which would have detected it in any case, by the small arithmetical
a eerror. value 0f r, the ‘ probable error/ It is evident, therefore, that to appreciate the meaning of
any misfit it is necessary to take into account the place in the scale at which it occurs and
to check its mere arithmetical figure by a reference to r the ' probable error.’
Next, with regard to the employment of r itself. A comparison of the various series
shows that the smallness or greatness of r does not necessarily depend on the size of the
series. For instance, in the cephalic index of the first five series of males, for which r has
been calculated (Chart I, Fig. i, series a, b, c, e, f), the following are its values:
In Early Predynastic r is . . . 2-30 with a misfit of 30 per cent, in a series of 53.
In Late Predynastic r is . . . . 1*83 „ 16 „ „ 125.
In First Dynasty r i s .................... 1-92 „ 34 „ ,, 41.
-In Fourth and Fifth Dynasties r is 1-73 „ 46 „ „ 54.
In Sixth to Twelfth Dynasties r is 2-03 „ 12 „ „ 172.
From which we see that the largest series of all the five has the second highest
figure for r, while a series only a third as large has the lowest figure. There is no reason
therefore to fear that the value of r is delusive and merely dependent on the number of
specimens.
On the other hand, this comparison suggests that the amount of misfit is dependent
to a considerable degree on the size of the series. Except in the case of the Fourth and
Fifth Dynasties, which is somewhat exceptional, there is an evident ratio, the misfit
decreasing regularly as the number of specimens increases. But.ifthe other indices, the
vertical, upper facial, ortho-prognathism, and nasal (Chart I, Figs. 2, 3, 4, 5), be reviewed
it wffl appear that there is no constant law. While it is true in general that in the
smallest series the misfit is much greater than in the large series, yet the contrast is only
observable when the series are extremely disproportionate in length. When the number
of specimens is at all comparable there is evidence that the misfit is not always dependent
on the size of the series. In the vertical index, for instance, it is considerably less for the
Late Predynastic series of 105 examples than for the Sixth to Twelfth Dynasties, where
there are 171 examples; and again, various series which are of equal length differ greatly
in the degree of misfit
It seems, then, that it would be justifiable to argue about homogeneity and heterogeneity
of series from the evidence of the misfit taken in connexion with r, provided only it
be remembered that when two series are extremely disproportionate in regard to the
number of specimens of which they are composed, the misfit is likely to be much more
considerable in the smaller than in the larger series.
An application of such a criterion to the first series of males (Chart I, Fig. 1, series a ,
b, c, e) shows that there is a far less degree of homogeneity in the Early Predynastic than
in any of the three others. The misfit and the values of r are on the whole greater in all
the tables, excepting only those of cranial capacity, than they are in any of the other periods,
viz. B, c, E. It might be supposed therefore that in the Early Predynastic, if in any period,
the irregularity of the curves and the high values of the probable error demonstrated the
presence of more than one component race. Can such an inference be safely drawn ?
On strict statistical principles, as laid down by the biometricians, an inference of the kind
would be illegitimate. The chances against the occurrence in a pure race of such a curve
as that displayed by the vertical index of the Early Predynastic males (Chart I A, Fig. 2)
have been calculated by the elaborate mathematical method suggested in Biometrika,
vol. i. pp. 155-63. The result showed that the chance was only thejsame as the chance
against many events which are recognized in daily life as quite probable. That is to say
there was, on biometrical principles, no evidence of heterogeneity in the series, but there
was equally little evidence of homogeneity. And we need hardly point out that had the
form of the question been not ‘ what are the chances of such a curve occurring in a pure
race,’ but ‘ what are the chances of such a curve occurring in a mixed race ’ the odds
against the latter hypothesis would have been infinitesimal.
It may be true that the paucity of examples may be an excuse for the biometrician
arriving at no conclusions in this case, but we cannot refrain from pointing out that the
material has been sufficient to enable us to definitely recognize the composite nature of
the races in this and in other periods represented by the diagrams in Charts I and II of the
supplement.
Since, however, other methods of analysis have enabled us to decide the question
of heterogeneity already we need not labour the interpretation of the Probability Curves.
Accordingly we shall content ourselves with comparing the several periods together in
respect merely of their relative characteristics, and shall note subsequently how far these
are in harmony with our independent deductions (see pp. 132, 133).
We may leave out of consideration the Fourth and Fifth Dynasty series from Medum
and Deshasheh, as they are the only specimens which come from Middle Egypt, that is to
say, from a very different locality.
M a l e s .
Cephalic Index (Chart I, Fig. 1). The least consistent period is evidently the
Roman, which has a very large value for r and a great misfit even within these wide limits ;
and next to this the most inconsistent is the Early Predynastic series. Curiously enough^
however, the most consistent series is the Late Predynastic, which is almost matched
by the Sixth to Twelfth Dynasties with a slightly larger figure for r, but a less degree
of misfit within that larger area. The Eighteenth Dynasty from Abydos shows a high
degree of consistency, in marked contrast to the series of the same date from Shekh Ali.
The Twelfth to Fifteenth Dynasties from Hou are peculiar in exhibiting a considerable
misfit in the neighbourhood of the mean (cf. the high figure of r ). Of the others the
Ptolemaic and the First Dynasty diagrams give the impression of a comparative consistency
which is disturbed by some factors of small importance.
Vertical Index (Chart I, Fig. 2). One series only is strikingly inconsistent and that
is the Early Predynastic with a very high figure for r and a great misfit. It is again
the Roman senes which stands next to the Early Predynastic in degree of inconsistency,
it has the same high value for r, but the misfit is less in amount and occurs in a -less
significant place. The most consistent periods are the Late Predynastic and the Twelfth
to Fifteenth Dynasties from Hou, in both of which the value of r is small and the curve
is very symmetrical. The Sixth to Twelfth Dynasties form a very regular curve which
» remarkable for its great width at the apex. The Eighteenth Dynasty from
Shekh Ali and the First Dynasty look like nearly consistent series which have perhaps
T h e principles
which
we adopt in
treating the
Curves of
Probability.
Meaning and
importance
of ‘ misfit ’
and ‘ probable
error.’
Statement
of the
limited conclusions
which we are
willing to
draw from
the Curves.