426 Mr Udny Yule, Fluctuations of sampling 
. . . * 
samples of » individuals 0, 1, 2, ete. of whom shew the first! 
character, are given by the terms of the binomial series (¢— p)” or) 
# me ea) eo 2) 
C0 
( 
| 
| 
| 
The mean of the series is np: its standard deviation is (npq)?. If, 
for example, data are available for a number of litters of moderate | 
size, litters of the same size n may be grouped together and for 
each such group the numbers of litters with 0, 1, 2, etc. recessives | 
may be compared with the frequencies to be expected as given by 
the binomial series. 
Where the sub-groups are of considerable size but not very | 
numerous, as in counts of peas on different plants or of maize 
grains on different cobs, this complete comparison becomes | 
impossible and we have to fall back on a simple comparison of the | 
standard deviation of the observed proportions with expectation. 
So far as I know, not even this has been done in many instances. 
The work of R. H. Lock on maize, referred to further below, is a. 
notable exception. For a number of samples, all of the same size | 
n, the standard deviation of the proportion is ( pq/n)?: if the sizes | 
of the samples vary, for n should be substituted the harmonic | 
1 41.) 
mean H of the numbers in the samples where H-7y™ & il 
N be the number of samples. With the use of Barlow’s Tables ~ 
(Spon), for giving the reciprocals, 1/H is readily evaluated: the 
values of 1/n are written down straight from the tables, their 
arithmetic mean r=1/H is formed, and the expected standard | 
deviation is ( qr)’. 
The following illustrations of these theorems were for the most | 
part obtained some years since and have frequently been used as | 
examples in lectures, but with one exception they have not hitherto 
been published. I have put together these notes in the hope that 
they may stimulate those who are carrying out actual experiments | 
to make more extensive and detailed tests on the same general — 
lines. - The work seems to me well worth doing. 
I know of no data sufficiently extensive to give a thoroughly | 
satisfactory test against the binomial distributions. From — 
A. D. Darbishire’s data respecting his crosses of Japanese waltzing — 
mice with albino mice (Biometrika, 111., 1904, p. 1) tables can be | 
compiled in the required form, but there are only 121 litters for — 
the case in which the expectation is 25 per cent., and 132 for the 
expectation 50 per cent. These numbers are too small for any 
close agreement with theory to be expected in the case of the 
* Cf. Yule, Introduction to the Theory of Statistics, x11. § 11. 
