430 
Miscellanea 
bability of any observed result we have only to calculate the terms of this series. The mean 
of this frequency distribution is well known to be mp, and its standard deviation V mpq. When 
neither p nor q is very small, then for practical purposes (when m of course is of a fair size, not 
a few units) it is quite adequate to calculate the probabihties of occurrence in such a single class 
frequency by the tables of the probability integral. The Mendelian worker who wishes to be more 
exact has merely to evaluate the terms of the binomial, or if m be too large, to approximate to 
those terms by the aid of my Type III curve, which falls for the cases excepted above much 
closer to the binomial than the normal curve does*. 
Two observations may here be made: First, if according to Mendelian theory there must be 
zero frequency in any particular class, then the improbability of any occurrence in that class is 
infinite. Secondly, the ^- test can be applied to any number whatever of Mendelian classes, 
to s classes individually and all the remainder, or in a particular case to one class and all the 
remainder. When applied to one class and all the remainder, it reduces to exactly the process 
given above, i.e. to the application of the probability table on the basis of the standard deviation 
Vmpq; or, referring to the genesis of the )^ test, which started by replacing the binomial by a 
normal curve, to using if we please the binomial. There is thus no contradiction between the two 
methods. This absence of contradiction was indicated by me in Biometrika, Vol. ix. p. 312, 
and was perfectly familiar to Weldon, who applied the binomial test to cases where it is proper 
to apply it. Now let us examine Dr Pearl's remarks in the Ught of these observations. He 
writes of Weldon's and Johannsen's use of the standard deviation of the binomial for entering 
the probability tables f : "In the first place it assumes the Gaussian distribution of errors, an 
assumption not often strictly warranted, as Pearson has clearly shown, and in many cases 
grossly in error." Now this statement requires considerable modification, (i) it does not assume 
the Gaussian law of errors, but the fact that the Gaussian or better Laplacian integral 
for practical purposes gives adequately the frequencies of the binomial when extreme values 
of m, p and q are not included. Dr Pearl makes no attempt to show that Weldon (or Johannsen 
for the matter of that) was dealing with such extreme cases, but repeats the suggestion of his 
previous paperj that Weldon somehow blundered, (ii) Further Dr Pearl cites for my view on 
the subject my paper "On the Influence of Past Experience on Future Expectation §." In 
that paper, which deals as we shall show below with an entirely different matter, the standard 
deviation is not V mpq, but the standard deviation characteristic of a second sample, so that 
criticism on this ground is out of place. Incidentally, however, I do recite in that paper the 
warning as to the use of the Laplacian integral when m, p, or q are extreme, and it was open 
to Dr Pearl to investigate whether Weldon had used the test in such cases. 
Let us look again at Dr Pearl. He writes: "The test gives a measure of the goodness 
of fit of the M;We. distribution, and only that. Now besides being interested in that point the 
Mendelian worker quite as often wants to know, in addition, something about the probabiHty 
that particular classes observed are significantly different from the expected. To that sort of 
knowledge the x^ test helps him not at all. It is an 'all or none' sort of method||." 
Now Dr Pearl's italics absolutely give him away, for they demonstrate that he does not in 
the least understand how to handle the x^ test or what it means ! Any test applied to s out 
of n classes must of course include the remaining n - s classes as a single group, for thek total 
frequency is fixed, when the frequencies of the s classes are given. The x^ test can be applied 
to a single class or to any number of classes up to w - 1, and it can only be a total misunder- 
standing of the test which can lead Dr Pearl to say that it is an "all or none" sort of method. 
Every test is an "all or none" test, if Dr Pearl means that it involves the total number observed. 
* The area of Type III curve up to any deviation from the mean is given by the incomplete 
r-function. Tables of that function have been long in hand and are approaching completion and 
publication. Delay has arisen only from urgent war work. 
•f Loc. ciL, p. 144. } Journal of Experimental Zoology, Vol. xiii. p. 203 et seq. 
§ Phil Mag., Mareli, 1907, pp. 365-378. || Loc. cit., p. 145. 
