273 THEORY OF STATISTICS. 
form a correlation-table between the true proportion p in a given 
universe and the observed proportion = in a sample of n observa- 
tions drawn therefrom. What we have found from the work of 
the last chapter is that the standard-deviation of an array of =’s 
associated with a certain true value p, in this table, is (pg/n)t; 
but the question may be asked —What is the standard-deviation 
of the array at right angles to this, <.e. the array of p’s associated 
with a certain observed proportion =? In other words, given an 
observed proportion w, what is the standard-deviation of the true 
proportions? This is the inverse of the problem with which we 
have been dealing, and it is a much more difficult problem. 
On general principles, however, we can see that if n be large, 
the two standard-deviations will tend, on the average of all 
values of p, to be nearly the same, while if » be small the standard- 
deviation of the array of =’s will tend to be appreciably the 
greater of the two. For if #=p +38, 8 is uncorrelated with p, 
and therefore if o, be the standard-deviation of p in all the 
universes from which samples are drawn, o, the standard- 
deviation of observed proportions in the samples, and os the 
standard-deviation of the differences, 
ol =o} +0}. 
But o} varies inversely as ». Hence if » become very large, os 
becomes very small, o, becomes sensibly equal to a, and therefore 
the standard-deviations of the arrays, on an average, are also 
sensibly equal. If n be large, therefore, [m(l—=)/n]} may be 
taken as giving, with sufficient exactness, the standard-deviation 
of the true proportion p for a given observed proportion 7. But 
if » be small, os cannot be neglected in comparison with a, oy, i8 
therefore appreciably greater than a, and the standard-deviation 
of the array of #’s is, on an average of all arrays, correspondingly 
greater than the standard deviation of the array of p’s—the state- 
ment is not true for every pair of corresponding arrays, especially 
for extreme values of p near 0 and 1. Further, it should be 
noticed that, while the regression of = on p is unity—a.e. the 
mean of the array of ’s is identical with p, the type of the 
array—the regression of p on = is less than unity. If we as- 
sume, therefore, that a tabulation of all possible chances, observed 
for every conceivable subject, would give a distribution of p 
ranging uniformly between 0 and 1, or indeed grouped symmetri- 
cally in any way round 05, any observed value 7 greater than 
0-5 will probably correspond to a true value of p slightly lower 
than , and conversely. We have already referred to the use of 
the inverse standard error in § 13 of Chap. XIII. (Case IL, p. 269). 
If we determine, for example, the standard error of the difference 
Tw