Miscellanea 
187 
sample values of the constants for those of the sampled population, we do not m any way alter 
our original supposition that we are considering the distriVjution of random samples of size 11. 
We have still p—\ degrees of freedom, if we have p categories of frequency. 
The process of substituting sample constants for sampled population constants does not mean 
that we select out of possible samples of size w, those which have precisely the same values of 
the constants as the individual sample under discussion. Cleai'ly the given sample has definite 
moment-coefficients, and if there be p frequency categories the first p—\ moment-coefficients 
together with the size n of the sample would suffice to fix all the frequencies of the. p categories*. 
Hence no deviations from the " probable constitution " would be possible if we confined our 
attention to samples of ii tied to the constants of the given sample ! In using the constants of 
the given sample to replace the constants of the sampled population, we in no wise restrict the 
original hypothesis of free random samples tied down only by their definite size. We certainly do 
not by using sample constants reduce in any way the random sampling degrees of freedom. 
What we actually, do is to replace the accurate value of which is luiknown to us, and 
cannot be found, by an approximate value, and we do this with precisely the same justification as 
the astronomer claims, when he calculates his probable error on his observations, and not on the 
mean square error of an infinite pojDulation of errors which is unknown to him. The whole of this 
matter was very fully discussed (pp. 164—7) in my original paper dealing with the x^., P test. 
The above re -description of what seem to me very elementary considerations would be 
unnecessary had not a recent writer in the Journal of the Royal Statistical Societi/\ appeared to 
have wholly ignored them. He considers that I have made serious blunders in not limiting my 
degrees of freedom by the number of moments I have taken ; for example he asserts (p. 93) 
that if a frequency curve be fitted by the use of four moments then the n' of the tables of 
goodness of fit should be reduced by 4. I hold that such a view is entirely erroneous, and that 
the writer has done no service to the science of statistics by giving it broad-cast circulation in 
the pages of the Journal of the Royal Statistical Society. 
What he would obtain if he placed this restriction on his samples is not the for the distri- 
bution of samples of size k, but of samples which give definite moments. The absurdity of this 
manner of approach is at once obvious, if as I have suggested, we consider the p first-moments, 
as there is no reason why we should not do, — for these are just as much "fixed" as the first four — 
and the conclusion must be that we can learn nothing at all about variation from our sample ; 
for we have p frequency groups and ^-tying conditions. 
When we wish to find the probable error of a mean or a standard deviation, we do not start 
by fixing down these characters to their values in the individual sample; we suppose them 
to take all the possible values they could take by sampling, and after we have reached our 
measure of variation we then put into our formula the sampled values, to give an approximate 
value to the functions reached, because we are in ignorance of the I'eal values in the sampled 
population. 
The writer in the Journal of the Royal Statistical Society speaks as if I applied x^ to a con- 
tingency table starting by fixing the marginal totals. As far as I am aware I am not gviilty of 
this. My conception of contingency is very different from my conception of x^. I started my 
conception of contingency with the idea not of a random sample, but with the idea that some 
function of frequencies alone without regard to their relation to the measured characters would 
lead to the value of the correlation. Naturally I started from the deviation of the individual cell 
contents from the same cell contents on the basis of independent probability, as determined by 
the marginal totals. There was no questif)n of sampling in the matter. In now fairly usual 
notation I termed 
* This is Thiele's method of representing frequency distributions, 
t Vol. Lxxxv. p. 87, 1922. 
