Karl Pearson 
247 
Thus we see that the condition for normality of the array mean, i.e. ^B.^ = 3, is 
by no means so nearly satisfied as is the condition for symmetry of the array mean, 
j,B^ = 0. For, while the latter might be fairly closely approximated to by an array 
of 15, even if the array distribution were not norma], yet the former even in the 
case of normality and of a big sample would give ^JSg = 4, a wide deviation from 
the normal. Thus the array must be fairly considerable for the distribution of 
its means to become reasonably Gaussian. As a rule the distribution will be 
reasonably symmetrical but is leptokurtic, i.e. ^fi.^ > 3, and therefore is to be 
described by a curve of my Type VII or one of the form 
rather than by a curve of the normal type|. Practically multiple frequency of 
this form is at present undiscussed and we are thrown back on treating the arrays 
as giving normal distributions of their means. Thus the assumption made by 
Slutsky in the paper cited p. 248 can only be considered as approximative, and 
that assumption was not legitimate until its degree of approximation had been 
investigated. It is singular that the goodness of fit theory can actually be applied 
with greater accuracy to test physical laws than to test regression lines. 
It is clearly only in the large arrays that the kurtosis {^B.^ — 3) approaches the 
normal value zero. What must be understood by "large" can easily be estimated 
roughly. For 
- 3 = nearly = Ib/n^, roughly. 
Hence if = 75, ^S,, would equal 3-2, which is certainly a limit to what mav 
be roughly treated as a normal distribution. Accordingly when we assume a 
normal distribution for the means of an array, we must remember that this is 
really very rough in the case of the smaller arrays, and that we only do it in default 
of a better theory. At the same time it must be noted that the small arrays will 
have less weight than the larger, and the error made in assuming their distribution 
normal will be of far less significance for the same deviation J. Thus far we have 
shown that (i) the means of different arrays are uncorrelated, (ii) that the standard 
deviations of these means are given by ay^jVu"^, where n" nj,[l — 2, 
* This agrees with the result given by Isserlis (loc. cit. p. 31) only when h,, = N, i.e. the "array" 
is as in his case a marginal total. 
f For the curve to be of normal type 5 must be large. If the array were normal in the sampled 
population and the sample large, then s = (2/?^, + 25)/10, and this would be only 8-5, if iip were 30. 
Thus the appUcation of Gaussian theory to samples with even minimum arrays of 30 can only be 
approximative. 
J The deviations in the means of the small arrays are likely, however, to be much more irregular 
and greater than in the case of the large arrays. 
