Karl Pearson and David Heron 
311 
where m r is the theoretical frequency and m,.' the observed. Now obviously in 
any distribution where even one m r is zero, the value of ^- must be infinite 
whatever may be the values of the other m r 's or m r "s. That is, if the theoretically 
expected frequency on any base element is numerically zero, the probability against 
the whole curve becomes infinite. Thus, for example, suppose a system of 
frequencies like the following, a type which is continually arising in Mendelian 
work : 
Class 1 2 3 4 5 
Theoretically expected frequency ... 595 827 68 0 96 
Actually observed frequency ... 594 828 67 1 96 
" Now it does not need a mathematical measure of any kind to tell one that in 
this case the theoretical and actual distributions are in very close agreement. 
Yet because the theoretical frequency on class 4 is zero, the probability by 
Pearson's test is literally infinite against the observed distribution being regarded 
as a random sample of a population distributed in accordance with the theoretical 
frequencies. Pearson has indeed himself noted what is essentially this same 
difficulty in using the test on ordinary frequency distributions*." 
Now what does this paragraph exactly signify ? Interpret it in coloured balls ; 
white, red, black and yellow balls are placed in a bag in large numbers in the 
proportions of 595 : 827 : 68 : 96. There are no green balls in the bag. One 
such green ball is said to have been drawn. Theory says it is an impossibility, 
and the criterion of goodness of fit says its improbability of occurrence is infinite. 
We can conceive no logical theory doing anything else. Dr Pearl does in fact tell 
us that cases in which a ball, classified as green, comes out of the theoretical 
white, red, black and yellow ball bag are of " a type which is continually arising in 
Mendelian work." This at any rate is a frank admission. As a matter of fact 
with his arbitrary division between "over 30-egg" and "under 30-egg" hens, 
we are not surprised that "green" balls appeared not only in ones, but in twos 
and even in fours, and in a few cases to even more, although this is attributed to 
" physiologically extremely favourable " matings (loc. cit. p. 248) as apart from 
gametic theory f. What Dr Pearl is seeking is a plastic theory or an elastic 
record, not a real criterion of goodness of fit, which must give no finite proba- 
bility when green balls come from a bag which contains no green balls ! Dr Pearl 
continues : 
" The point noted obviously limits greatly the applicability of Pearson's test, 
and in a most unfortunate direction. Tests of goodness of fit are much needed in 
Mendelian work [we cordially agree !]. But it is just here that the classes where 
* This is a complete misunderstanding. Pearson says that you must not in the case of continuous 
variation make use of classes which theoretically have each less than unit frequency, where the record 
goes by individual units only. 
t Until Dr Pearl publishes the actual record of each bird, and not merely its class-index, and the 
same for its ancestors, it is impossible to estimate his degree of justification for the theoretical 
treatment of his results. 
