190 
Miscellanea 
But the is 
(a' + 6')(«' + c- )y (a' + 6')( 6' + c^') y / (ft- + c')(c' + (^')y ic+d')ih'+d') y 
+ 
{a' + b'){a' + e') {a + b') (b' +d') {^a' + c') {c' + d') (c +d') (b' + d') 
M M M JI 
( «2 fe2 g2 ^2 
l(a' + 6')(a' + cO («' + 6')(&' + c^') (a' + c') {c' + d') (c' + d') (b' + d') 
there being i^ree degrees of freedom or we must take n'= i in calculating the probability P, this 
may be written 
M[p.ipi. p.iPu P.2P2. J 
where p'.2, p'l. and are the four percentage numbers of the marginal categories in the 
sampled population. Now we do not know these percentages in that population and we do what 
every physicist, every astronomer, and — till I saw the jjaper by my critic in the Journal of the 
Statistical Society I should have said — every statistician does, supply the unknown constants 
from the sample, which leads us to 
2= — M{ab-cdY ^ 
^ {a + b){a + c){b + d){c + d) 
as used in my memoir of 1912*. 
The problem I had and still have in view is the variability in samples of definite size — with 
no other restriction than sample size. The solution of that problem is absolutely comparable 
with that of any discussion of the probability of an observed result in the theory of probable 
errors. We have in the bulk of such cases constants involved which concern the distribution in 
an unknown population, and we supply those constants from the sample itself. 
As I have already noted the probable error of a mean is 
•67449 J^^P 
By this we understand that the means of samples restricted solely by their size M from an 
indefinitely large population of moment-coefficients fx^' about a fixed origin will have a 
variability determined by the above formula. But when we proceed to give both /xj' and (12 the 
values determined from the sample we know, we do not add in the manner of my Royal Statistical 
Society critic, " but in doing so the type of samples is reduced to those having the mean and 
standard deviation of the sample." If we did, this selection of samjjles would clearly have no 
variation of mean or standard deviation at all ! In fact probable errors would be meaningless, 
unless we drew our samples from a population already fully known to us, in which case we should 
not in 99 of cases want to sample it at all. 
In the same way when we use the marginal totals of the sample in formulae like (S) we do not 
thereby reduce our samples to those having constant marginal totals, we m«rely take the best 
approximation available to the proper value of x^, and the fact that x,\ as found from the sample, 
is only an approximation to the true was fully r'ecognised and discussed in my original memoir 
in the Philosophical Magazine. 
It only remains to say that the following sentence of my critic's paper seems to me based 
upon a fallacious principle and apparently flows from a disregard of the nature of probable 
errors in general. 
" It should be pointed out that certain of Pearson's Tables for Statisticians and Biometricians, 
namely Tables XVII, XIX and XX, together with XXII (Abac to determine rp) are all calculated 
* On a novel method of regarding the association of two variates classed solely in alternative 
categories. Drapers' Company Research Memoirs, Cambridge University Press. 
