190 
Skew Variation, a Rejoinder 
Using Stirling's theorem they showed that if the mth be the largest term in the 
binomial, then the sum p,. of all the terms from ni — r to m + r is very nearly 
given by : 
2 /"'■ _if 1 _il 
P> = ^-^ e ■i<r-da;+ --.=-e'i<T'^ (viii), 
v27r<7Jo v27ro- 
where a = ^1 Npq, 
Here we have the first appearance of the probability integral as representing 
a series of discontiruioas binomial terms. In fact when N' is fairly large Laplace 
and Poisson show that sums of terms of the binomial are closely given by the areas 
of the probability curve. It is an approximate result based upon Stirling's theorem, 
and it does not for a moment involve making N infinitely large, or the spacing 
apart of the binomial terms very small. This representation of a number of finite 
terms by the probability integral seems to be unfamiliar to Ranke and Greiner, 
but no practical statistician would calculate the sum of r terms in the binomial 
{p + q)^' for even moderate values of N. He would simply calculate the standard 
deviation a — ^Npq of the binomial and turn up tables of the probability integral. 
This fundamental property of the normal curve, i.e. that it closely represents a 
discontinuous series, is passed over in silence by my critics. It is the very purpose 
for which the probability integral was originally introduced by Laplace. In other 
words it arises without any consideration of (i) continuity of variation, or (ii) equal 
probability of negative and positive deviations. 
It will be observed that the above approximation to the binomial, i.e. to 
{p+q)'^ is symmetrical, but we can easily allow for some degree of asymmetry. 
Still writing a — \/Npq, and for the binomial 
A = (1 - ^pq)KNpq), ^ = /3, - 8 = (1 - i}pq)l{Npq). 
I have shown*, i/^ being the maximum term in the binomial, that the rth term 
from the maximum is given by : 
- £ (1 - ft + 4'/) - Wft '+ Jv'ft S (1 - fft + - etc. 
>/r=y,e (ix). 
The term in r/a was, I believe, first added by Poisson, and expresses his attempt 
to allow for asymmetrical variation. Edgeworth expanding the exponential has 
adopted for his asymmetrical curve, a form easily deduced from (ix), 
,,=,..-,^.|,-iv'ft(r-i^)| 
It will thus be seen that the normal function and the probability integral arise 
naturally from the expression for a single term or a series of terms of the binomial 
polygon. This is their historical origin and the historical origin of the conception 
of asymmetrical variation. Instead of the complex form given above resulting 
from Stirling's theorem, I approached the subject by looking at the relation of the 
* Phil. Trans. Vol. 186 A, p. 348, footnote. 
