CHAPTER XIV 
SIMPLE SAMPLING CONTINUED: EFFECT OF 
REMOVING THE LIMITATIONS OF SIMPLE SAMPLING. 
1. Warning as to the assumption that three times the standard error gives the 
range for the majority of fluctuations of simple sampling of either sign 
—2. Warning as to the use of the observed for the true value of p in 
the formula for the standard error—3. The inverse standard error, or 
standard error of the true proportion for a given observed proportion : 
equivalence of the direct and inverse standard errors when = is large— 
4-8. The importance of errors other than fluctuations of ‘‘simple 
sampling ” in practice: unrepresentative or biassed samples—9-10. 
Effect of divergences from the conditions of simple sampling: (a) 
effect of variation in p and ¢ for the several universes from which the 
samples are drawn—11-12. (5) Effect of variation in p and ¢ from one 
sub-class to another within each universe—13-14. (¢) Effect of a 
correlation between the results of the several events—15. Summary. 
1. THERE are two warnings as regards the methods adopted in 
the examples in the concluding section of the last chapter 
which the student should note, as they may become of importance 
when the number of observations is small. In the first place, he 
should remember that, while we have taken three times the 
standard error as giving the limits within which the great 
majority of errors of sampling of either sign are contained, 
the limits are not, as a rule, strictly the same for positive and 
for negative errors. As is evident from the examples of actual 
distributions in § 7, Chap. XIII, the distribution of errors is not 
strictly symmetrical unless p=¢=05. No theoretical rule as 
to the limits can be given, but it appears from the examples 
referred to and from the calculated distributions in Chap. XV. 
§ 3, that a range of three times the standard error includes 
the great majority of the deviations in the direction of the 
longer “tail” of the distribution, while the same range on the 
shorter side may extend beyond the limits of the distribution 
altogether. If, therefore, p be less than 0°5, our assumed range 
may be greater than is possible for negative errors, or if p be 
a 
)Y 
x