264 THEORY OF STATISTICS. 
The student should note that in both cases the standard-devia- 
tions given are standard-deviations of the proportion of male 
births per 1000 of all births, that is, 1000 times the values given 
by equation (2). These values are given by simply substituting 
the proportions per 1000 for p and ¢ in the formula. Thus for 
the first column of Table I. the proportion of males is 508 per 
1000 births, the mid-number of births 2000, and therefore— 
508 x 492\} 
w=("go00) =112 
11. In the above illustration the difficulty due to the wide 
variation in the number of births in different districts has been 
surmounted by grouping these districts in limited class intervals, 
and assuming that it would be sufficiently accurate for practical 
purposes to treat all the districts in one class as if the sex-ratios 
had been based on the mid-numbers of births. Given a sufficiently 
large number of observations, such a process does well enough, 
though it is not very good. But if the number of observations 
does not exceed, perhaps, 50 or 60 altogether, grouping is 
obviously out of the question, and some other procedure must be 
adopted. 
Suppose, then, that a series of samples have been taken from 
the same material, /; samples containing n, individuals or observa- 
tions each, f, containing n,, Js containing nm, and so on: What 
would be the standard-deviation of the observed proportions in 
these samples! Evidently the square of the standard-deviation 
in the first group would be pq/ny, in the second pg/n,, and so on: 
therefore, as the means tend to the same values in all the groups, 
we must have for the whole series— 
Fmpg(D4 Ler lay : 
7 n,n, 
But if H be the harmonic mean of ny, Ny By 
eh Bale 
Zn, nyt, 
and accordingly 
g=t0, (5) 
That is to say, where the number of observations varies from one 
sample to another, the harmonic mean number of observations in 
a sample must be substituted for n in equation (2). 
Thus the following percentages (taken to the nearest unit) of