2vn THEORY OF STATISTICS.
sampling ”% The proportion of plants above average height in the
two classes (cross- and selffertilised) together is 29/68. The
standard-deviation of the differences due to simple sampling
between the proportions of * tall” plants in two samples of 34
observations each is therefore
29139 1 aN
c= 53 X 88 X 2) =0120,
or 12:0 per cent. The actual proportions observed are 50 per
cent. and 35 per cent.—difference 15 per cent. As this difference
is only slightly in excess of the standard error of the difference,
for samples of 34 observations drawn from identical material, no
definite significance could be attached to it—if it stood alone.
The student will notice, however, that all the other cases cited
from Darwin in the question referred to show an association of
the same sign, but rather more marked. Hence the difference
observed may be a real one, or perhaps the real difference may be
greater and may be partially masked by a fluctuation of sampling.
If 50 per cent. and 35 per cent. were the true proportions in the
two classes, the standard error of the percentage difference would
be, by equation (6),
3
ar XO) 11 pr on,
and consequently the actual difference might not infrequently be
completely masked by fluctuations of sampling, so long as experi-
ments were only conducted on the same small scale.
Example iv.—(Data from J. Gray, Memoir on the Pigmentation
Survey of Scotland, Jour. of the Royal Anthropological Institute,
vol. xxxvii., 1907.) The following are extracted from the tables
relating to hair-colour of girls at Edinburgh and Glasgow :—
Of Medium Total Per cent.
Hair-colour. observed. Medium.
Edinburgh . ; 4,008 9,743 41'1
Glasgow : : 17,529 39,764 44-1
Can the difference observed in the percentage of girls of medium
hair-colour have arisen solely through fluctuations of sampling *
In the two towns together the percentage of girls with medium
hair-colour is 43-5 per cent. If this were the true percentage,
the standard error of sampling for the difference between per-
centages observed in samples of the above sizes would be-—
1 ry
cia = (435 x 560) x (ge + 57)
=(0'56 per cent.
ed
