38 
On the Polsson Law of Small Numbers 
Now cr is the stanciard deviation of variations in /Xo and therefore 
Similarly a > is the standard deviation of variations in the mean and therefore 
a'-^^' = fi^/N. Lastly the product cr^ ai^^''>'fi„fj,^' measures the correlation between 
deviations in fx^ and /ti/ and is known to be ix-^jN*. 
Thus we have : 
q'<^»' {m4 - + {p - gY 1^1 - 2 (jj - q) 
»-ry V,/ = -^^ - iM.? + p-,x, - 2pfi,]. 
Butt ,x, = vpq{l+S{n-2)p<j], j ' 
I (iv). 
ljL., = npq{ p- q\ fJ^, = npq j 
Whence after some purely algebraical reductions we deduce: 
\/Nq 
Formulae (v) and (vi) are very important; they enable us to obtain the 
probable errors for n and p when a binomial limited in the present manner is 
fitted to a frequency distribution|. 
We see at once, that as n grows large and q grows small 
(Ty= (Tfj approaches the limit '^'2jN, 
or the probable error, •67449 \^2/N, of j) and q is finite. But being finite cr,i 
becomes infinitely great, or the probable error of n indefinitely large. Thus when 
the n of the binomial is very large, q being very small, the probable error of its 
determination is so great that its actual value is not capable of being found 
accurately. Again, suppose N embraced 200 observations, the probable error of q 
would be of the order '07 ; if N corresponded to only eighteen observations, then 
the probable error of q would be of the order •22. It is clearly wholly impossible 
* Biometriha, Vol. ii. "On the Probable errors of Frequency Constants," see p. 275 (iv), p. 276 (vii), 
and p. 279 (xii). 
t Phil. Trans. Vol. 186, A, p. 347, 1895. 
X There is no difficulty in obtaining the probable errors of n and p from the more general values 
in (ii). In this case 
<r., = (r,. 
.PI 
The values of , <t^_^ and r^^^^ for different values of /3i and /Sg have been tabled by Khind, Biometriha, 
Vol. VII. pp. 136—141. 
