186 
Slew VaridMon, a Rejoinder 
populations cases of 24 (Trinidad), 26 (Cuba), 27 (Nicaragua) and 34 (Colombia), 
the women beginning to bear at 13 and continuing to 50, have been recorded ! 
It will, I think, be realised by the impartial reader that Ranke's statements : 
Fiir den Anthropologen ist also mir das Gaussclie Gesetz von Wichtigkeit. Fiir seine 
Probleme beansprucht es aber auch vollige Giiltigkeit (S. 327) 
and 
So haben anderseits auch die Werte von Variationsumfiingen die bislang aufgefunden sind, 
keinerlei weitere Erkenntniss gebracht (S. 324) 
fall wide of the mark. Ranke has either very much circumscribed the field of the 
anthropologist, or he has not in this, as in other cases noted in his paper, studied 
the literature of the subject, or finally he has disregarded the results reached. 
It is quite true that the range cannot always be determined and being 
determined does not always give a very good result. The reasons for this are not 
far to seek. For example, If a discrete quantity has for its minimum 0 units the 
start of the curve must naturally fall on the negative side of the origin — since its 
area measures frequency between — "5 and + '5. Ranke would probably find 
something mysterious in this "reichen oft ins Negative." Actually it is to be 
expected, especially if due allowance be given for the probable error of the range. 
In most biometric statistics, we cannot as in the case of births deal with 20,000 
to 30,000 cases and get small errors for our constants. We have only perhaps 500 
to 2000 cases and even less than this in craniology. This may denote an error of 
14 to 17 p.c. in the calculated range, and it is quite possible that the range may 
"reichen ins Negative." Take the case worked out by me* of the number of 
Mullerian glands in the forelegs of % swine. The range theoretically calculated 
is 18 glands with a probable error of + 2"54 ; the start of this range ought not to 
have exceeded — "5. It is actually — "82 with a probable error of '16. The actual 
skewness of this distribution is '31 with a probable error of '02. The distribution 
is accordingly significantly asymmetrical. 
I have cited these cases as sufficient for our present purpose, but there are 
many other cases in which the discovery of the range has been of biological or 
special anthropological interest, e.g. the earliest appearance of certain diseases in 
childhood, the range of cancer attacks, the first occurrence of signs of puberty, etc. 
It has been applied also effectively to a number of zoological and botanical data. 
A more striking case, perhaps, of usefulness is the limit to high barometric pressure 
obtained by dealing with the frequency statistics of barometric height at series of 
stationsf. Throughout the whole of the stations of the British Isles dealt in, 
there is sensible skewness of distribution, and with one Irish exception, which is 
sensibly mesokurtic, the whole series of curves are platykurtic, and this deviation 
from noi'mality cannot be chance, but is a significant character of the frequency 
distributions. In these cases the limit to high pressure has been found, and appears 
to be a constant of considerable physical importance for the local climate. 
* Pearson and Filon : Phil. Tram. Vol. 191 A, p. 289. 
t Pearson and Lee : Phil. Trans. Vol. 190 A, p. 423 et seq. 
