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. 
7