256 
" Goodness of Fit " in Statistics and Physics 
We are now in a position to appreciate the influence of the various factors in 
the solution for this case. I hold that as ffi,^ must be given the theoretical or 
calculated value, so also must the standard deviations of the arrays, i.e. aj^. But 
as we do not know the sampled population in this case we must obtain ct^,^ in 
precisely the same manner as we find m^, i.e. not by taking a value obtained from 
the individual array*, but by values graduated from the whole sample. Further 
it does not appear correct to take a J' = ay,Jnj,. The latter is only true when the 
number in the array is a priori fixed, that is to say is not itself provided by the 
random samphng. In the latter case we must take a^,^ = o\,Jn"p, where 
"""p = (l - I) + 2. 
to a second approximation and this modifies considerably the standard deviations 
of the small frequency arrays. If in the above example we use v^,, the theoretical 
frequency of the array, instead of the value n"p, we find, still using the theoretical 
^^n,,< X" = 8-6166 giving P= -986. Slutsky*, who has used both the observed 
frequencies and the observed standard deviations! of the individual arrays, 
finds = 9'!''^ ^■iid P = -980. It will be seen that the correction for n"j, is 
the more important factor. In this case no marked changes are produced 
by using observed quantities instead of graduated values, but I think that in 
short series very fallacious results might be reached by this process, and the present 
paper is written to suggest caution at this point, and o%^^ are as definitely 
at fHp values in the sampled population, not in the sample itself. 
Application of "Goodness of Fit'' Theory to testing Physical, Technical or 
A strono mical Measure m ents. 
In these cases there is no question in the ordinary sense of a frequency surface. 
The physicist makes a few measurements of a variate A for each of a series of values 
of a variate B. He plots the mean of his measurements for A to each value of his 
variate B, and he enquires whether the curve given by his series of mean values for 
A is closely approximate to some theoretical curve. It will be seen that his problem 
is very similar to that of the statistician. He has a number of means for the A 
variate, m^, ... nij, and he considers whether they are good fits to a theoretical 
curve — the statistician's regression curve. Obviously in this case these means are 
non-correlated as approximately in the statistical case. Further the variability of 
a mean will be given definitely by ofjjVrij, — not now as an approximation. Here 
?!p is the number of observations in the array on which m^, depends while a^^^ is 
* Loc. cit. p. 81. 
1 In the case of the 3-4 array of one with observed (r„ = 0 Slutsky says this is due to random 
sampling and extrapolates a standard deviation ; this is of course only a first slight step towards the 
proper graduation of the whole system of array variations. In the case of the array of two for 
23-24, the observed value is a„ = 1-9148, which is just as much an inconsistency due to random 
sampling, but this value although about ^ of the real value is retained and used by him. 
