XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 301 
unnecessary detail: as a matter of practice, we would not have 
comp ‘ed a frequency-distribution by single male births, but 
would certainly have grouped our observations, taking probably 
10 births as the class-interval. We want, therefore, to replace the 
binomial series by some continuous curve, having approximately 
the same ordinates, the curve being such that the area between 
any two ordinates 7, and y, will give the frequency of observations 
between the corresponding values of the variable x, and z,. 
9. It is possible to find such a continuous limit to the binomial 
series for any values of p and ¢, but in the present work we will 
confine ourselves to the simplest case in which p = q¢=05, and the 
binomial is symmetrical. The terms of the series are 
od n(n-1) n(n-1)(n-2) 
NE) 1 1b 5 Ag + —t i +....% 
The frequency of m successes is 
£2 
N(3) [m|n—m 
and the frequency of m+ 1 successes is derived from this by 
multiplying it by (n-m)/(m+1). The latter frequency is 
therefore greater than the former so long as 
n—m>m+1 
n-1 
mM THT 
Suppose, for simplicity, that = is even, say equal to 2%; then the 
frequency of % successes is the greatest, and its value is 
12 2 
%=N@) kk (1) 
The polygon tails off symmetrically on either side of this greatest 
ordinate. Consider the frequency of + x successes ; the value is 
=NG 2k. | 24 9 
= eh 123) 
and therefore 
veo BDE-DE=2) .... (i=2+1) 
Yo (k + 1)(% + 2)(k + 3) EN CE) 
1 2 3 xz-1 
Joi. 300). ufr-%0 . 
= +3) 2) 3 ry EY ! 
(143 1+72\1+7) . (1+ (143) 
or 
od