THEORY OF STATISTICS. 
Working from the observed proportion of green seeds, viz. 0:2532 
instead of the theoretical 0:25, we have 
s= 7/025 x 0°75/7125 = 0:0051, 
and similarly the divergence from theory is only some 3/5 of the 
standard error, as before. 
It should be noted that this method must not be used as a test 
of association by comparing the difference of (4B) from (4)(B)/N 
with a standard error calculated from the latter value as a 
“theoretical number,” for it is not a theoretical number given 
a prior: as in the above illustrations, and (4) and (B) are themselves 
liable to errors of sampling. If we formed an association-table 
between the results of tossing two coins XV times, o= ,/&.}. 3 
would be the standard error for the divergence of (4.8) from the 
a preore value n/4, not the standard error for differences of (4.5) 
from (4)(B)/N, (4) and (B) being the numbers of heads thrown 
in the case of the first and the second coin respectively. 
Case II.—Two samples from distinct materials or different 
universes give proportions of A’s p, and p, the numbers of 
observations in the samples being n, and =, respectively. (a) Can 
the difference between the two proportions have arisen merely as a 
fluctuation of simple sampling, the two universes being really 
similar as regards the proportion of A’s therein? (8) If the 
difference indicated were a real one, might it vanish, owing to 
fluctuations of sampling, in other samples taken in precisely the 
same way? This case corresponds to the testing of an association 
which is indicated by a comparison of the proportion of 4’s amongst 
B’s and fs. 
(¢) We have no theoretical expectation in this case as to the 
proportion of 4’s in the universe from which either sample has 
been taken. 
Let us find, however, whether the observed difference between p, 
and p, may not have arisen solely as a fluctuation of simple 
sampling, the proportion of 4’s being really the same in both cases, 
and given, let us say, by the (weighted) mean proportion in our 
two samples together, z.e. by 
py ELT Poly 
0" nm +m, 
(the best guide that we have). 
Let ¢, €, be the standard errors in the two samples, then 
& = PoQo/ My & = PQo/ Ma 
If the samples are simple samples in the sense of the previous 
work, then the mean difference between p, and p, will be zero, 
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