XIV.—REMOVING LIMITATIONS OF SIMPLE SAMPLING. 289 
standard-deviation in some case of sampling exceeds the standard- 
deviation of simple sampling, two interpretations are possible : 
either that p and ¢ are different in the various universes from 
which samples have been drawn (i.e. that the variations are 
more or less definitely significant in the sense of § 13, Chap. XIII), 
or that the results of the events are positively correlated inter 
se. If the actual standard-deviation fall short of the standard- 
deviation of simple sampling two interpretations are again 
possible, esther that the chances p and ¢ vary for different 
individuals or sub-classes in each universe, while approximately 
constant from one universe to another, or that the results of 
the events are negatively correlated inter se. Even if the 
actual standard-deviation approaches closely to the standard- 
deviation of simple sampling, it is only a conjectural and not 
a necessary inference that all the conditions of * simple sampling ” 
as defined in § 8 of the last chapter are fulfilled. Possibly, for 
example, there may be a positive correlation » between the 
results of the different events, masked by a variation of the 
chances p and ¢ in sub-classes of each universe. 
Sampling which fulfils the conditions laid down in § 8 of 
Chap. XIII., simple sampling as we have called it, is generally 
spoken of as random sampling. We have thought it better to 
avoid this term, as the condition that the sampling shall be 
random—haphazard—is not the only condition tacitly assumed. 
REFERENCES. 
go generally the references to Chap. XIIL, to which may be 
aadaea— 
(1) PEARSON, KARL, ‘“ On certain Properties of the Hypergeometrical Series, 
and on the fitting of such Series to Observation Polygons in the Theory of 
Chance,” Philosophical Magazine, 5th Series, vol. xlvii., 1899, p. 236. 
(An expansion of one section of ref. 10 of Chap. XIII., dealing with the 
first problem of our § 14, i.e. drawing samples from a bag containing 
a limited number of white and black balls, from the standpoint of the 
frequency-distribution of the number of white or black balls in the 
samples, ) 
(2) GREENWOOD, M., “On Errors of Random Sampling in certain Cases not 
suitable for the Application of a ‘ Normal Curve of Frequency,’ Bio- 
metrika, vol. ix., 19183, pp. 69-90. (If an event has succeeded p times in 
n trials,what are the chances of 0, 1, . . . m successes in m subsequent 
trials! Tables for small samples.) 
EXERCISES. 
1. Referring to Question 7 of Chap. XIII, work out the values of the 
significant standard-deviation o, (as in § 10) for each row or group of rows 
there given, but taking row 5 with rows 6 and 7, 
19