2 - THEORY OF STATISTICS.
It will be observed that if n, be very small compared with
ny €,; approaches, as it should, the standard error for a sample
of n, observations.
We omit, in this case, the allied problem whether, if the
difference between p, and p, indicated by the samples were
real, it might be wiped out in other samples of the same size
by fluctuations of simple sampling alone. The solution is a
little complex as we no longer have &=p.q,/(n; + ny).
Example v.—Taking the data of Example iii., suppose that
we compare the proportion of tall plants amongst the offspring
resulting from cross-fertilisations (viz. 50 per cent.) with the
proportion amongst all offspring (viz. 29/68, or 42:6 per cent.).
As, in this case, both the subsamples have the same number
of observations, n, =n,= 34, and
20 39 IN
n= 00x 1) ~ 0-060
or 6 per cent. Asin the working of Example iii., the observed dif-
ference is only 1°25 times the standard error of the difference, and
consequently it may have arisen as a mere fluctuation of sampling.
Example vi.—Taking now the figures of Example iv., suppose
that we had compared the proportion of girls of medium hair-
colour in Edinburgh with the proportion in Glasgow and
Edinburgh together. The former is 41'1 per cent. the latter
435 per cent., difference 24 per cent. The standard error of
the difference between the percentages observed in the sub-
sample of 9743 observations and the entire sample of 49,507
observations is therefore
Bin, nN 39,764 Y= ;
ep = (43'5 x 56:5) (ress — 0°45 per cent.
The actual difference is over five times this (the ratio must, of
course, be the same as in Example iv.), and could not have occurred
as a mere error of sampling.
REFERENCES:
The theory of sampling, for the cases dealt with in this chapter, is generally
treated by first determining the frequency-distribution of the number of
successes in a sample. This frequency-distribution is not considered till
Chapter XV., and the student will be unable to follow much of the literature
until he has read that chapter.
Experimental results of dice throwing, coin tossing, etc.
(1) QUETELET, A., Leltres . . . . sur la théorie des probabilités ; Bruxelles,
1846 (English translation by O. G. Downes; C. & E. Layton, London,
1849). See especially letter xiv. and the table on p. 374 of the
French, p. 255 of the English, edition.
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