M. Greenwood 
75 
If we write p = , q 
n 
we find for (3) and (4) 
n 
p-q 
n+2 
and if m and ?t are both absolutely large, 
(q - pf 
l+2 m 
n 
m (p + e)(q- e) 
1 + 
m 
■(3 A), 
in 
1+6—1 + 
= 3 + 
m (jo + e) (J - e) 
.(4 A). 
1 + 
If now m be small relatively to n, 
(q-pT 
m(p + e) (q - e) 
If n be small relatively to m, 
and /3 2 = 3 + 
1 
m (p + e)(q- e) 
4(g-j>) a 
n(p + e){q-e) 
6 
Mill 
d /3 2 = 3+— — 
?i( j p + e)(^-e) 
After exhibiting these results, Pearson remarks* : " Both forms result- 
...(5). 
.(6). 
-for n or 
m large and the product of either with p and q not small — in /3 X = 0 and /3 2 = 3, 
i.e. in the symmetry and mesokurtosis, which are for practical purposes closely 
enough represented by the Gaussian curve. But if m and n be commensurable, 
and either p or q moderately small, this result by no means follows." 
It is accordingly plain that in all cases of m and n both small the use of 
a "normal" curve with s.D. = \lmpq is inappropriate. When p = q the condition 
of mesokurtosis is fulfilled and the divergence from "normality" reduces itself 
to the difference between the Gaussian and Pearson Type II curves. The 
accompanying table illustrates this in a particular example. 
A Second Sample of 10, after a first Sample of 100 ; p = q 
Comparison of Series with Curves (Totals = 100). 
50. 
Successes 
Hypergeometric 
Series 
Normal Curve 
S. D. \ l: nM 
Normal Curve 
S.D. sj[n + l)pq 
Curve of Type II 
/ x s \ 1022177 
0 
•146 
■221 
•333 
•188 
1 
1-243 
1 -122 
1-408 
1-331 
2 
4-931 
4-349 
4-843 
5-091 
3 
12-017 
1 1 -447 
1 1 -702 
12-107 
4 
19-922 
20-452 
19-728 
19-729 
5 
23-480 
24-817 
23-972 
23-105 
6 
19-922 
20-452 
19-728 
19-729 
7 
12-017 
11-447 
] 1 -702 
12-107 
8 
4-931 
4-349 
4-843 
5-091 
9 
1 -243 
1-122 
1 -408 
1-331 
10 
•140 
■221 
•333 
•188 
Op. cit. (1907), pp. 371—2. 
10—2 
