188 
Skew Variation, a Rejoinder 
turns upon the great changes introduced into the range by different methods of 
calculating the moments. More recent investigations, in which the sexes are 
separated, the moments more accurately determined, and larger numbers dealt with, 
give far better results for zymotic diseases. I presume that one character being 
age, however, Ranke and Greiner would dismiss these data from consideration 
under any circumstances. 
Example VII. Guesses at 9 tints. Possible range 1 to 9, i.e. curve to run 
from "5 to 9'5. Observed guesses run from 1 to 8. Theoretical range of 11 instead 
of 9. The paucity of the observations gives a probable error of at least 20 to 30 
per cent, in the determination of the range,. and the result is rather better than 
might have been anticipated. 
Example VIII. Ratio of forehead to body length in Garcinus moenas, observed 
range 30, calculated range 51. This range is probably not very close but it is 
not in any way that I can see impossible. The material is probably dimorphic. 
E.cample XI. H. de Vries's data for Ranuncidus hulbosus. Actually observed 
range 5 to 10 petals. Calculated range 5 to 11 petals. 
Example XII. H. de Vries's data for a race of Trifolium repens. Actually 
observed range 0 to 10 high blossoms. Theoretical range in complete agreement. 
Example XIV. Pauperism percentages for 632 cases. Observed range 18 for 
the year 1891 dealt with. Calculated range 31. This range gives 2 units of 
negative pauperism. Its probable error is, perhaps, 14 per cent. 
It will be seen that out of the seven examples in which range is calculated only 
three reichen ins Negative, and that this reichen is well within the limits of the 
errors arising on the one hand from random sampling and on the other from the 
defective methods of determining the moments, which were alone available in 1894. 
While quite appreciating the honour done me when other workers use my methods, 
I must decline to be responsible in any way for their application of my fonuulae. 
I have so often found that their failure to fit my curves is due to a misapprehension 
of my luethods or to actual errors in arithmetic, that I have long given up any 
attempt to set such matters right. The frequent assumption made that statistical 
methods can be applied without adequate mathematical training is the source of 
most of the slips in this matter*. 
So far then I think we may conclude that Ranke is completely unjustified both 
in his statement that the Gaussian curve fully describes all the frequency that is 
of importance to the biologist, and in his attempt to discredit any result of 
scientific value which flows from endeavouring to measure such differences from 
the Gaussian law as we find in the distance between mode and mean, the skewness, 
the kurtosis and range of man}' actual frequency distributions. 
A good illustration, by no means unique, of this is F. Eeinohl : Die Variation im Andrikeum der 
SteUaria Media, 1903. He finds it impossible to fit certain distributions with my curves, owing to 
ignorance of the full literature and to faulty determination of the moments. He then argues from this 
want of fit to biological conclusions. 
