: THEORY OF STATISTICS. 
if p is, say, 1/3, and not 1/2 owing to the black balls, for some 
os tending to slip through our fingers. (Cf. Chap. XIV. 
S 4. 
9. It is evident that these conditions very much limit the 
field of practical cases of an economic or sociological character 
to which formule (1) and (2) can apply without considerable 
modification. The formule appear, however, to hold to a high 
degree of approximation in certain biological cases, notably in 
the proportions of offspring of different types obtained on crossing 
hybrids, and, with some limitations, to the proportions of the 
two sexes at birth. It is possible, accordingly, that in these cases 
all the necessary conditions are fulfilled, but this is not a necessary 
inference from the mere applicability of the formule (cf. Chap. 
XIV. § 15). In the case of the sex-ratio at birth, it seems 
doubtful whether the rule applies to the frequency of the sexes in 
individual families of given numbers (ref. 9), but it does apply 
fairly closely to the sex-ratios of births in different localities, 
and still more closely to the ratios in one locality during 
successive periods. That is to say, if we note the number of 
males in a series of groups of » births each, the standard-deviation 
of that number is approximately a/mpg, where p is the chance 
of a male birth; or, otherwise, a/pg/n is the standard-deviation 
of the proportion of male births. We are not able to assign an 
a priors value to the chance p as in the case of dice-throwing, 
but it is quite sufficiently accurate for practical purposes to use 
the proportion of male births actually observed if that proportion 
be based on a moderately large number of observations. 
10. In Table VI. of Chap. IX. (p. 163) was given a correlation- 
table between the total numbers of births in the registrationdistricts 
of England and Wales during the decade 1881-90 and the pro- 
portion of male births. The table below gives some similar figures, 
based on the same data, for a few isolated groups of districts con- 
taining not less than 30 to 40 districts each. In both tables the 
drop in dispersion as we pass from the small to the large districts 
is extremely striking. The actual standard-deviations, and the 
standard-deviations of simple sampling corresponding to the mid- 
numbers of births, are given at the foot of the table, and it will 
be seen that the two agree, on the whole, with surprising closeness, 
considering the small numbers of observations. The actual 
standard-deviation is, however, the larger of the two in every case 
but one. The corresponding standard-deviations for Table VI. of 
Chap. IX. are given in Qu. 7 at the end of this chapter, and show 
the same general agreement with the standard-deviations of simple 
sampling ; the actual standard-deviations are, however, again, as 
a rule, slightly in excess of the theoretical values. 
269