XIII.—SIMPLE SAMPLING OF ATTRIBUTES. 9
and the standard error of the difference €5 the samples being
independent, will be given by
f cepa] Suid 5
€12 pa, +.) . . (9)
If the observed difference is less than some three times €, it
may have arisen as a fluctuation of simple sampling only.
(6) If, on the other hand, the proportions of 4’s are not the same
in the material from which the two samples are drawn, but », and
py are the true values of the proportions, the standard errors of
sampling in the two cases are
6 =pq/m € = Poe]
and consequently
=~ +P . : (6)
Lo Le
If the difference between p, and p, does not exceed some three
times this value of ¢,, it may be obliterated by an error of simple
sampling on taking fresh samples in the same way from the same
material.
Further, the student should note that the value of €, given by
equation (6) is frequently employed, in lieu of that given by
equation (5), for testing the significance of an observed difference,
The justification of this usage we indicate briefly later (Chap.
XIV, § 3). Here it is sufficient to state that, if » be large,
equation (6) gives approximately the standard-deviation of the
true values of the difference for a given observed value, and hence,
if the observed difference is greater than some three times
the value of ¢, given by (6), it is hardly possible that the true
value of the difference can be zero. The difference between the
values of , given by (5) and (6) is indeed, as a rule, of more
theoretical than practical importance, for they do not differ largely
unless p, and p, differ largely, and in that case either formula will
place the difference outside the range of fluctuations of sampling.
Example iii.—The following data were given in Qu. 3 of Chap.
ITI. for plants of Lobelia Julgens obtained by cross- and self-fertilisa-
tion respectively :—
Parentage Cross-fertilised., Parentage Self-fertilised,
Height— Height—
Above Average. Below Average, Above Average. Below Average.
17 1% 12 22
The figures indicate an association between tallness and cross-
fertilisation of parentage. Is this association significant of some
real difference, or may it have arisen solely as an “error of
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