K. Pearson 
187 
I again am forced to consider that Ranke has not been aware of what has been 
published, still less what has been done in this matter. He appears to base his 
conclusions chiefly on my first paper on skew-variation, and he has not noticed the 
fact that being the first paper much has to be coi'rected in the light of more recent 
work in the past ten years. Ranke speaks of the writer's : 
Andwendimg von allerlei grosseren raid kleineren Aiidci'ungen in seiner Methode ad hoc in 
eine fiir den vorliegenden Zweck nicht zu unwahrscheinliche Form bringcn (S. 324). 
Now I contend that this gives a grossly unjust description of the paper in 
question. Had Ranke read recent literature, he would have been aware that the 
great difficulty with frequency distributions is to obtain the true values of the 
" moments " fi'om records which merely give data for arbitrary " Spielraume," often 
far too large and usually selected by the observer without any regard to the needs 
of the computator. My method is one based on the method of uioments, but to 
deduce the moments from given data is the i-eal difficulty which Ranke never for an 
instant seems to grasp or at any rate refer to. The standard deviation (which he 
appears to consider sufficient for anthropologists) will vary, and often very sensibly, 
with the nature of the grouping of the data. This difficulty was very present in 
my mind in 1894, and is constantly referred to in my memoir, the "allerlei grossere 
und kleinere Anderungen in seiner Methode " are no changes in method at all 
but attempts to obtain some approximation to the true moments of the data. It 
was not till 1898 that Sheppard showed the correct manner of calculating the 
moments from the raw data in his important memoir on frequency constants*, for 
one type and one type only of frequency distribution. The curves calculated by 
Sheppard's method, now in general use, would give Vjetter results undoubtedly 
than are to be found in my memoir of 1894. Further, however, Sheppard's 
method applies only to curves with high contact with the horizontal axis 
at both ends. It leaves us still in doubt as to how to find the moments of 
curves, which cut the axis at the end of the range or are asymptotic at one or 
both ends to the vertical axis. At such ends of the range, the real solution lies 
in recording the frequency for very small elements, but this was not provided in 
any of the statistics which were then before me. It is just these cases of limited 
range at one or both ends which present difficulty in the determination of the 
moments. The difficulty will be familiar to all statisticians, if it has escaped 
Ranke. To some extent it is met in my memoir on the systematic fitting of curves 
issued in April, 1902f. Yet granting all these difficulties what do we find in my 
memoir of 1894? An analysis of the cases in which range is dealt with seems 
justified by the charges made : 
Example I. Range determined of Cambridge Barometric Heights. There is 
nothing physically improbable in the result. 
Example VI. Range found for enteric fever runs from —I 'Ab years to about 
385 years. The probable error of the range is not given, but the whole difficulty 
* Proc. London Math. Society, Vol. xxix. p. 353 et scq. t Biometrika, Vol. i. p. 265. 
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