Editorial 
2G7 
This principle, which is ahnosfc self-obvious, was, however, overlooked by Pearson 
in his paper " On the Application of ' Goodness of Fit ' Tables to test Regression 
Curves and Theoretical Curves used to describe observational or experimental 
Data," in Bioineirika, Vol. xi, pp. 239 — 261. 
One of the objects of that paper was to investigate the probable errors and 
frequency distributions of errors in the mean and standard-deviation of an array. 
If we have an array of a first variate corresponding to a small subrange of a second 
variate in a sample of N, the law of distribution of the means and standard- 
deviations of such arrays when many samples of N are taken had not been 
investigated at the time Pearson wrote. If there be individuals in such 
a sample, then the problem differs froui the ordinary problem of the distribution 
of means and standard-deviations in a sample of size iip, in the fact that rip in the 
case of the array varies from sample to sample. Hence we cannot straight away 
assume that if be the mean number in the array then afij'^np and a,7j,/v'2?i^^ 
will be the standard-deviations of the distributions of means and of standard-devia- 
tions of the arrays ; still less do we know how far it is legitimate to suppose these 
distributions approximate to the Gaussian or normal type. As the problem is an 
exceedingly important one the writer asked Miss Eleanor Pairnian to revise his 
work of 1916 by introducing where needful the fourth-order products. This she 
has done with certain additions and expansions. 
(a) From the equation on p. 239 we have : 
mean {Snip) = mean S (— )" S 
where 2 is a summation for every value of a from 1 to oo 
But . mean 8n,jp8np = n.^p (j 
and the regression relation is accordingly 
8h,„, = Blip . 
Tip 
(IT 
} 
Substituting this we see that every term vanishes and accordingly Snij, = 0, 
not merely to a high order of approximation, but absolutely. In other words the 
mean of the means of any array — notwithstanding that the number'in that array 
will vary — ^is equal to the mean of that array in the sampled population. 
(6) We have for the ^^th array : 
'^h^ + ^"^''- -\ + 8np ' 
but Slip = S {8n,jp) and S {n,jpiCg) = TipMp, 
18—2 
