258 " Goodness of Fit " in Statistics and Physics 
how a single value of A for each value of B cannot lead to any measure whatever 
of the "goodness of fit." I suggest the following method of determining a suitable 
value for cr^^. Let us assume that the arrays are homoscedastic, i.e. that the 
physicist's difficulty in measuring A is the same for all values of B. This will not 
always be true, but is a fair working hypothesis for many cases. Then we have 
where ct^ is the standard deviation of all the measurements made on A (without 
regard to the value of B), a much more rehable quantity than the standard 
deviation of any array values of A. Further tj^j.^j is the correlation ratio of A 
on B or 
_S {nj, (TOj, - mf] 
where «l is the mean value of all measurements of A and N is their total number. 
Thus 
^ (1 - 
can be readily determined as soon as Wi, and tj^^.^ have been found. 
Of course it is needful for a test of this kind that the number of measurements 
of A should considerably exceed the number of values of B tested. It would fail 
entirely if only one value of A were taken for each value of B, however numerous 
the latter might be. We must have some basis on which to determine the error 
made in single determinations of A. This is a point I think often overlooked by 
the physicist. A fairly good determination — I mean a quantitative determination — 
of the goodness of fit of theory to observation could be made from 10 series of 
8 observations of A corresponding to 10 values of B. But no measure of "goodness 
of fit" could be found from 80 observations of A corresponding to 80 values of B, 
and yet the latter system would probably make the greater appeal to most physicists. 
I do not see how quantitatively to obtain any measure of the goodness of fit of 
theory to observation in the latter method of procedure. It is not unusual to 
determine the mean square residual, i.e. 
but before we can really make use of this, i.e. find ^ and so P, the probability of 
as great or greater divergence between theory and observation, we must know 
a^^^^ and this can in no way be deduced from such a system of observations. In 
fact without a knowledge of a^^ip — unit in which — m^"^ is to be measured — 
the mean square residual is as delusive as the ocular comparison of a graph of 
the theoretical and observed results, where all turns on the arbitrary scale of the 
vertical ordinate. 
Illustrations. As illustrations I will take some of the data connecting length 
of arc with loss of carbon per coulomb provided in a recent memoir by Professor 
Duffield*. My only reason for taking this material is that it is recent work and 
* R. S. Proc. Vol. 92, A, p. 125. 
