K. Pearson 
in 
of the modal frequency and to a reduction of the extreme frequencies which are 
inconsistent with the Gaussian curve. This is actually shown in my Plate 11 
and referred to in my text, and corresponds to a sensible deviation from the 
Gaussian law. It was open to Dr Ranke to attribute this exaggeration of the 
modal at the expense of the extreme variates to heterogeneity in the material. 
But he had no right when the material for a judgment was before him in my 
memoirs to conclude that, because the general distribution of frequency was not 
on the average incompatible with the Gaussian law, the deviation of a particular 
constant of the distribution from its Gaussian value might not be most significant. 
This case of the deviation of the Ranke's measurements from the Gaussian form 
is of special interest, for it is not one of asymmetry, but of a non-Gaussian type 
of symmetry. Dr Ranke suggests that because my test of goodness of fit shows 
that the Gaussian curve is " quite a good fit," my generalised method of deahng 
with frequency is idle. Assisted by a mathematician he ought to have recognised 
that the expression /S2 — 3 (which measures whether the frequency towards the 
mean is emphasised more or less than that required by the Gaussian law) had a 
sensible value, and that my method not only led to the discovery of this deviation 
but provided a method of allowing for it in the description of the frequency. Had 
Dr Ranke read my memoir on errors of observation (Phil. Trans., Vol. 198 A, 
pp. 274—286), he would have recognised that the two tests (a) whether special 
physical constants of the distribution satisfy the Gaussian law, and (b) whether 
the general distribution of frequency satisfies within reasonable limits the 
Gaussian law, are not necessarily identical. Finally had he concluded that 
(b) for the Altbaierisch crania was satisfied, but not (a), and that there was thus 
no necessary discrepancy between my memoir of 1894 and the statement in 
Palin Elderton's paper of 1900, he might indeed have fallen back on his customary 
assumption that when frequency is not Gaussian it is heterogeneous. But at any 
rate had he adopted this course he would have avoided the appearance of criticising 
his author without endeavouring to understand what the meaning of his investi- 
gations was, or striving to elucidate them by a study of his other memoirs on the 
same subject. 
I do not wish to say anything further on this point. I want merely to 
indicate by these two out of several cases that the reader must not look for a 
really impartial statement of my position from Drs Ranke and Gi'einer. I am 
quite unable to account for the peculiar tone they have at times given to their 
criticism. 
(2) With this preamble I should like to divide my reply under these headings : 
(A) The need for generalised frequencj^ curves, even in anthropological 
science. 
(B) The nature of the assumptions made in the Gaussian theory and their 
insufficiency. 
