K. Pearson 
1U3 
probability integral, when ho is discussing the probable errors of death rates, class 
indices and a multitude of other problems, and this when the binomial is skew and 
N relatively small ! It is the whole theory of current statistics which Ranke and 
Greiner are tilting at when they object to the use of what is equivalent to the 
Euler-Maclaurin theorem, i.e. the mathematical representation of a finite sum of 
terms by a definite integral*. 
There is another point also which may be noted here before we leave the 
binomial. The quantity /3i is a measure of the asymmetry. Now consider the 
ratio ^Jr), or the ratio of this measure of asymmetry to the kui'tosis. For the 
asymmetrical binomial we have /Sijrj = (1 — 4<pq)/(l — Opq). Since p and q if positive 
must give a product lying between 0 and |, this ratio cannot take any value 
between 0 and + 1. Hence any curve which gives for /Sj/t? a value less than unity 
cannot possibly diverge from the normal curve in the direction of a binomial 
series. We shall see the application of this later. 
C. (ii) The Galton-McAlister Carve. 
I have already referred to the attempt of Poisson to give the skew binomial by 
an extra term applied to the Gauss-Laplacian probability integral. Quetelet 
endeavoured to meet the asymmetry of frequency distributions by placing 
graphically skew binomials on top of the frequency polygon — a very rough and 
somewhat deceptive process. The next step in the advance was taken by Francis 
Galton, who in 1879 suggested that the geometrical mean and not the arithmetical 
mean is likely to give the most probable result in many vital phenomena -f-. Galton 
refers to Gauss's assumption that errors in excess or in defect are equally probable, 
and says "this assumption cannot be justified in vital phenomena." He cites 
especially the cases of errors in human judgment, guessing at temperatures, tints, 
pitch, etc. He appeals to Fechner's law in its simplest form as evidence to the 
contrary, and placing the matter in the hands of D. McAlisterj, the law of 
frequency 
V ITX 
was deduced, and methods for fitting this curve were discussed. The curve is 
well-known in England and also on the continent §. It is therefore curious to find 
Ranke and Greiner attributing this curve to Fechner's work which was not 
published till 18 years later. It was not till I had made a fairly complete set 
of experimental determinations of the kind supposed to give this curve, that 
I finally discarded it. Thus I asked audiences of 100 to 300 persons to match 
tints in several ways, I asked them to guess heights, to determine mid-lengths, 
to state which figiu'es in randomly distributed series were most closely circles, 
* See Lacroix : Traite du Calcul d[ff('rentid et integral, Tom. iii. p. 13(j. 
t R. S. Proc. Vol. 29, p. 365 et seq. " The Geometric Mean in Vital and Social Statistics." 
+ R. S. Proc. Vol. 29, p. 367 et seq. "The Law of the Geometric Mean." 
§ It is cited by Kapteyn, for example. 
Biomctrika iv 25 
