M. Greenwood 
71 
and the "probable error" of mp is + '67449 \/mpq. This is the classical text-book 
case. Its limitations are obvious. If either p or q be very small unless m is very 
large indeed, and for all values of p and q wheti m is very small, the normal curve 
does not approximate closely to the binomial. 
Consider this problem. A certain bacillus is stated to occur in the mouths of 
2 per cent, of all normal persons. Twenty persons have been examined and the 
bacillus was isolated from two of them. Is this observation consistent with the 
truth of the hypothesis ? 
Let us find the chance that in 20 trials two or more successes would be met 
with if p = -02, q = -98. 
By direct calculation we find this chance to be about 1 in 17. If we use a 
"normal" curve with Standard Deviation ^20 ("02 x '98), the chance proves to 
be rather less than 1 in 25, or the probability determined in this way is only 
68 per cent, of the real value. Of course when the number of trials is so 
small we could not expect a continuous function effectively to represent the 
binomial expansion, but even for m large the inadequacy of the " normal " curve, 
in the case of p ~ q not small, must be insisted upon. I think the best way of 
making this clear arithmetically is from a consideration of the moment coefficients 
of the binomial (p + q) m . 
With the ordinary notation we have : — 
yu, 2 = c'mpq, 
fjL 3 = &mpq (p -q), 
/a 4 = c i mpq (1 + 3 (m — 2)pq\, 
and /? 1= *L, & = % 
In cases like the present, c may be taken as unity. 
For a " normal " curve to be a good fit to the binomial, /3, should be very small 
and /3 2 nearly equal to 3. 
Take as an illustration the values of fix and /3. 2 for different values of m where 
p = m, q = -9S. 
We obtain : 
m 
ft 
100 
■4702 
3-4502 
WO 
•2351 
3 2251 
300 
•1567 
3-1501 
400 
•1176 
3-1126 
500 
•0940 
3-0900 
600 
•0784 
3-0750 
700 
•0672 
3-0643 
800 
•0588 
3-0563 
900 
•0522 
3-0500 
1000 
•0470 
3-0450 
