TABLES FOK TESTING THE GOODNESS OF FIT 
OF THEORY TO OBSERVATION. 
By W. PALIN ELDERTON, Actuary. 
[Received October 18, 1901.] 
On the Test for Random Sampling. 
Any theoretical description by means of curve or series is ceteris paribus 
admissible as a graduation of a given set of frequency observations, provided the 
observed values do not differ from the values provided by this theory by more 
than the reasonable deviations due to random sampling. There may be utilitarian 
reasons (e.g. relative fewness of descriptive constants, or their easy calculation) or 
philosophical reasons (e.g. general theories as to the nature and distribution of 
causes producing frequency phenomena) why we should adopt one theoretical 
description rather than another, but apart from such reasons that theoretical 
description is best, which describes the observed frequencies with the "greatest 
probability." By " describing the observed frequencies with the greatest proba- 
bility" we understand a good although conventional test of fitness. Suppose the 
theoretical description of the frequencies to be the actual distribution of the 
whole population ; we ask in how many cases per 100 in a series of random 
samplings should we differ from the theoretical distribution by as wide a system of 
deviations as that observed, or by a still wider system ? In other words we want 
to find out the probability P that in random sampling deviation-systems as great 
as or greater than that actually observed will arise. This point has been dealt 
with in a paper by Professor K. Pearson published in the Philosophical Magazine*, 
and it is there shown that if there be n' = w -f 1 frequency groups in the series, 
and nir and m^' be the theoretical and observed frequencies in any group, it is 
necessary to find 
( (squares of differences of theoretical) \ 
1 and observed frequencies j | 
theoretical irequency / 
* On the Criterion that a given System of Deviations from the Probable in the case of a Correlated 
System of Variables is such that it can be reasonably supposed to have arisen from Eandom Sampling, 
Vol. L. pp. 157—175, July, 1900. 
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