358 
MR. R. A. FISHER ON THE MATHEMATICAL 
we have, when x is in all cases small compared to m, 
as a first approximation. In those cases, therefore, when x 2 is a valid measure of the 
departure of the sample from expectation, it is equal to 2L ; in other cases the approxi¬ 
mation fails and L itself must be used. 
The failure of equation (7) in the general problem of finding the best values for the 
parameters may also be seen by considering cases of fine grouping, in which the majority 
of observations are separated into units. For the formula in equation (6) is equivalent to 
where the summation is taken over all the observations, while the formula of 
equation (7), since it involves n 2 , changes its value discontinuously, when one 
observation is gradually increased, at the point where it happens to coincide with a 
second observation. 
Logically it would seem to be a necessity that that population which is chosen in 
fitting a hypothetical population to data should also appear the best when tested for 
its goodness of fit. The method of the optimum secures this agreement, and at the 
same time provides an extension of the process of testing goodness of fit, to those cases 
for which the x 2 test is invalid. 
The practical value of x 2 Fes in the fact that when the conditions are satisfied in 
order that it shall closely approximate to 2L, it is possible to give a general formula 
for its distribution, so that it is possible to calculate the probability, P, that in a random 
sample from the population considered, a worse fit should be obtained ; in such cases 
X 2 is distributed in a curve of the Pearsonian Type III., 
d f (|) ^ 
or 
df cc L 2 e -L dh, 
where n f is one more than the number of degrees of freedom in which the sample may 
differ from expectation (17). 
In other cases we are at present faced with the difficulty that the distribution L 
requires a special investigation. This distribution will in general be discontinuous (as 
is that of x 2 ), but it is not impossible that mathematical research will reveal the existence 
of effective graduations for the most important groups of cases to which x 2 cannot 
be applied. 
