324 THEORY OF STATISTICS. 
The square of the standard-deviation is given by the sum of 
the terms in col. (4) less the square of the mean, that is, 
rr=np { gr-1+20n- )gn=rp + 80g spn Bde |- np, 
But the series in the bracket is the binomial series (q+ p)"! 
with the successive terms multiplied by 1, 2, 3, . . . It therefore 
gives the difference of the mean of the said binomial from -1, 
and its sum is therefore (n — 1)p +1. Therefore 
oZ=np{(n-1)p +1} — n%p? 
= np — np? =npq. 
7. The terms of the binomial series thus afford a means of 
completely describing a certain class of frequency-distributions— 
v.e. of giving not merely the mean and standard-deviation in 
each case, but of describing the whole form of the distribution. 
If &V samples of n cards each be drawn from an indefinitely large 
record of cards marked with 4 or a, the proportion of A-cards 
in the record being p, then the successive terms of the series 
N(q +p)" give the frequencies to be expected in the long run of 
0, 1, 2, . . . 4-cards in the sample, the actual frequencies only 
deviating from these by errors which are themselves fluctuations 
of sampling. The three constants XN, p, n, therefore, determine 
the average or smoothed form of the distribution to which actual 
distributions will more or less closely approximate. 
Considered, however, as a formula which may be generally 
useful for describing frequency-distributions, the binomial series 
suffers from a serious limitation, viz. that it only applies to a 
strictly discontinuous distribution like that of the number of 
A-cards drawn from a record containing 4’s and a’s, or the number 
of heads thrown in tossing a coin. The question arises whether 
we can pass from this discontinuous formula to an equation 
suitable for representing a continuous distribution of frequency. 
8. Such an equation becomes, indeed, almost a necessity for 
certain cases with which we have already dealt. Consider, for 
example, the frequency-distribution of the number of male births 
in batches of 10,000 births, the mean number being, say, 5100. 
The distribution will be given by the terms of the series 
(0-49 40-51)1900 and the standard-deviation is, in round numbers, 
50 births. The distribution will therefore extend to some 150 
births or more on either side of the mean number, and in order 
to obtain it we should have to calculate some 300 terms of a 
binomial series with an exponent of 10,000! This would not 
only be practically impossible without the use of certain methods 
of approximation, but it would give the distribution in quite 
200