274 
PEOFESSOR K. PEAESON ON THE MATHEMATICAL THEORY 
(a.) General physical characters of the nature of the distribution without regard to 
the special frequency of errors of particular sizes. 
(6.) Agreement between theory and observation in the general distribution of 
errors of each j^articular size. 
I propose to investigate these classes of considerations separately. 
(10.) (a.) General Physical Characters of a Normal Distribution. 
While the analytical processes by means of which the normal curve is deduced are 
extremely varied—sometimes very simple (Hagex), sometimes very complex 
(Poisson), there is confessedly or tacitly involved an axiom of the following kind :— 
(a.) Positive and negative errors of the same size are equally frequent. Sometimes 
this result is disguised by assuming that the actual error is the sum of an indefinitely 
great number of small elementary errors which are equally likely to be positive or 
negative. Whatever process of proof be followed the result is the same—the normal 
distribution gives a symmetrical distribution of errors, and this is its first general 
physical character. Now in an immense number of cases of deviations from the 
mean, such as occur in organic nature, this symmetry is quite unknovm ; such distri¬ 
butions I have spoken of as skew frequency distributions,* and their characteristic 
feature is that the mode or position of the maximum frequency diverges from the 
mean. The ratio of the distance of the mode from the mean to the standard 
deviation I have treated as a measure of the “ skewness ” of the distribution. It will 
vanish when the curve is symmetrical or when the sums of all odd powers of the 
errors are zero. Thus if n be the number of observations, nyup the sum of the 
pth powers of the errors, [Xj,= 0 for a normal distribution if p be odd. For most 
practical purposes the labour of investigation compels us to confine our attention to 
the question of whether /xg is sensibly zero. 
But the normal distribution not only involves a condition as to the odd moments, 
hut also one which must sensibly hold in the case of each pair of even moments.! 
The simplest of such relations is expressed by 
= Syf 
Now this relation and its extensions to higher moments have nothing whatever to 
do with the symmetry of the normal distribution—with the equal frequency of errors 
of the same size, whether positive or negative. They depend really upon two 
additional axioms, which are again confessedly or tacitly assumed in the course of the 
proof, namely :— 
(^.) That there are an indefinitely great number of cause-groups associated in 
producing each individual error. 
* ‘ Phil. Trans.,’ A, vol. 186, p. 343 et seq. 
t ‘ Phil. Trans.,’ A, vol. 185, p. 108. 
