170 
Skew Variation, a Rejoinder 
xuorden according to iny critics, it was actually written in 1895 and its substance 
given in academic lectures of the same or the following year; it was not published 
till some years afterwards owing to that want of leisure for preparing matter for 
press which every teacher who has to lecture four hours a day will appreciate. 
Dr Ranke says that had the paper been written, it would not have influenced his 
judgment. That is quite possible, and I only cite the matter here to indicate 
the tone adopted by my critics. 
A very similar instance occurs in Dr K. E. Ranke's treatment of my fitting 
of Professor J. Ranke's data for 900 Altbaierisch crania. I used this example 
purposely because it had already been used by Stieda. I should not myself have 
mixed, even for the cephalic index, (/" and $ data as Stieda did, but I wished to 
compare the results reached by the generalised curve with those reached by the 
Gaussian curve. I actually spoke of the resulting curves in my memoir, whether 
the generalised curve or the Gaussian, as being " quite good for this type of 
statistics*." My object of course was to show that the generalised method did 
not fail where the Gaussian succeeded, but surpassed it. Now how does Dr Ranke 
treat this instance ? He cites an example (actually inserted-f- by me !) in a memoir 
by Palin Elderton giving the Ranke'sche Messungen as an illustration of my 
method of testing goodness of fit in the case of the normal curve. Undoubtedly 
as I said in 1894 the Gaussian curve is quite good for J. Ranke's data, but it does 
not follow that the Type IV. frequency curve does not give a better fit, and is not 
significant for constants which the Gaussian process cannot deal with. Now it 
was open to Dr Ranke to test the values given for the distance from mode to 
mean, the skewness and the other constants in the case of the Altbaierisch crania. 
I have given the probable errors of these constants in my memoir : On the Mathe- 
matical Theory of Errors of Judgment, etc.. Philosophical Transactions,\o\. 198 A, 
see p. 278. Had Dr Ranke fairly tested my results he would have found that 
the asymmetry was not significant, and that the mode sensibly coincided with the 
mean, but that the constant /3o, which should equal 3 for the normal curve, has a 
value 3'(i5 with a probable error of only about "11. Now this constant and its 
probable error have no relation at all to any particular theory of variation. 
They follow quite easily from the general Gaussian theory. It is accordingly 
extremely improbable that Ranke's measurements are truly given by a Gaussian 
distribution in all their features. The fact that /3.2 is > 3 points to an emphasis 
* p. 389. 
t The same remark applies to the illustrations of goodness of fit given by Fawcett, cited in footnote 
Ranke u. Greiner, p. 326. The reference to Powys is inexact ; his paper shows that in at least three 
cases the Gaussian curve is quite impossible. Dr Macdonell's work on the English skull shows that at 
least in 4 out of 13 cases the asymmetry is significant. " Die englische Schule," by which Dr Ranke 
refers to workers in my Biometric Laboratory has not discovered a truth which had escaped me ; they 
have shown that the Gaussian curve is of wide applicability, but not of universal truth in anthropometric 
measurements. This result was reached with a view to testing whether the theory of inheritance, so far 
as it is based on the Gaussian theory, might be safely applied to human characters. In testing this 
validity of the Gaussian theory, it was of course needful to have a more general theory from which to 
determine the chief physical constants involved in non-Gaussian distributions. 
