56 Oil the Poissoti Laiv of Small Nimibers 
The actual results as giveu by tlie binomial and the Poisson-Exponential are : 
TABLE XI. 
Number of Deaths . . . 
0 
1 
2 
o 
J^ and over 
Observed 
109 
65 
22 
3 
1 
Biiioniial 
108-6 
66-4 
20-2 
4-1 
0-7 
Exponential ... 
108-7 
66-3 
20-2 
4-1 
0-7 
Actually if we woi-k to two decimal places in the frequeDcies we have j^- = 'Q\ 
for both binomial and exponential, or the goodness of fit is practically identical. 
In this case it seemed worth discussing the binomial fit more at length. 
Taking the moment coefficients about the mean we have : 
(i) Mean = ?i(/ = -0100. 
(ii) = H^xy = -0079. 
(iii) yu.;; = np<i {p — q) = \-)dO,oi5'2. 
(iv) /Aj = iipq (1 + Snjjq — Opq) = 1-64-3,373. 
We have already discussed the binomial from (i) and (ii), giving "x; for goodness 
of fit = "6096. Using (ii) and (iii) we have for the binomial 
200 (-985,739 + -014,261 )«-=^-", 
giving ;)^'- = -665. 
Using (iii) and (iv) we have : 
200 (-979,524 + •020,057)^'''^°^ 
giving x' = ■''07. 
Putting : = and /8j = fx ^jfi.?, 
we have : — 3 = (1 — 6pq)jnpq, /8i = (1 — 4<pq)/'npq, 
and working from /3i and /S.^ we find : 
200 (-969,150 + ■0S0,S50y''"»', 
and in this case x'— 1'1286. 
This of course does not give a bad fit, but it is clear that working from the 
lowest fiwment coefficients, as we might anticipate, gives the best results. 
But if q be the chance of death from the kick of a horse, and n the number of 
men in an army corps, tlien the binomial should be 
200 (p + qy\ 
Now it is obvious that none of the binomials give, by their value of n any 
approach to the real number of men in an army corps. If we start with the 
