8 Tlie Fundamental Problem of Practical Statistics 
(ii) Examining the odd terms we see that within the range 
3 3n 
"1+2/3 ~ n + 2 m 
positive deviations are less frequent than negative deviations, but beyond this 
range the positive are more frequent than the negative. In other words large 
excesses are more probable than large defects, but small excesses are less probable 
than small defects from the result of our first sample. 
(iii) It is not legitimate to keep the term in in — , for this is really the 
order of the lowest term in }i*la\ Without including that term our degree of 
approximation will only be to Ij'^m. 
(iv) For the very narrow case of Q — P, or equal chance of success and failure, 
of course the odd terms vanish. 
If we include in this case the lowest term in /(''/(j'* we have 
1 h- 1 + 2/) \ 1 l + 3p + 3p2 
^1 ^ Qg 2 (7- \ m{l + p)J 12(7^ rn{l + p) 
The case oi P = Q is, however, of such secondary importance that this second 
approximation on Laplace's lines is of little significance*. 
(v) If our second sample is so large that terms in -j= are negligible, then 
V //i 
our frequency becomes 
_ 1 ^ 
This i-esult was reached by Laplace before Gauss. Neither stated the very narrow 
limitations of the formula. 
Besides the condition that m is to be very large, we must also note that neither 
Q nor P can be very small. We obtain quite different results, if we suppose 
m large and P small so that mP remains finite. 
(vi) Thus the Gauss-Laplacian distribution fails: 
(a) for small samples. Its whole method of deduction is then wrong for 
Stirling's Theorem is invalid ; 
(6) when the sample is large, but the probability of occurrence is small, so 
that inP is finite and small. 
The whole of this subject has now-a-days been reconsidered under the topics 
of " Small Samples " and the " Law of Small Numbers." 
* A method of investigating tbe frequency constants of sucli a distribution has recently been given 
by Forsyth, Messenger of Mfitliematic^, Vol. xlviii. pp. 131-44. 
