272 
Feccavimus ! 
(ii) Another method is to assume a Pearson Type III curve: y = y^x°-e~'''^, 
which is known to give a better approximation than the Gaussian to the binomial. 
We assume it to start at the beginning of the first subrange of the binomial and 
to have the same mean and standard-deviation. These conditions involve 
a + 1 
= n. 
« + 1 
Accordingly 8 = \\ y.w'^-' e-^^dx = |5 -}_-^^ F (a - ,s + 1 ), 
\. lip / A I (, A. y 
where A, = total frequency = —^^ 
Hence 
r(a + l). 
1\ 71 
S 
S 
1 
2 
N 
a(a-l)(a- 2) 
which are exact. Or, approximating 
•(H) 
Both methods agree to the terms in 
1\- 
1 , but the Gaussian appears to exagge- 
rate in the terms in 
n, N. 
(iii) We will now proceed to approximate on the basis of the moment- 
coefficients of the binomial. We have 
8 
S (Snp) s(s + l)S (8,1./) s(s + l)(s + 2)S (H') 
+ 
1.2 
1.2.3 
+ ... 
Here S (Bnp) = 0, and we will keep terms up to the order _ which involves 
lip 
proceeding to the fourth moment-coefficient. We find 
