44 
On the Poissou Lmv of Small Numbers 
Thus we see that if the true value of the frequency be 5 for the average sample, 
it will only lie outside the range 1 to 10 in '674 + 1-370 = 2-044 cases per cent., or 
the odds are 49 to 1 that the value found will be from 1 to 10. 
On the other hand it will lie outside the range 2 to 8 in 4-043 + 6-809 = 10-852 % 
of cases, or once in about 9 trials the frequency will lie outside this range. Or, 
again, once in about every four trials (25-8 °/^) the result will fall outside the 
range 3 to 7. 
On the other hand if we write a = \/b (1 - -005) = 2-23047, we have - 4-5 
and + 5 5 as the deviations from a mean 5 of all beyond 0-5 and above lO'S, 
giving a;/a- = — 2-0175 and + 2-4658 respectively. These cut off tail areas of 
•02181 and -00684, respectively. Thus in 2-865 — uot 2-044 — per cent, of cases 
we should assert that the fi'equency would lie outside the range 1 to 10, or the 
odds that it would lie inside this range are now only about 34 to 1, not 49 to 1. 
Calculated from the Gaussian the frequencies outside ranges 2 to 8 and 3 to 7 
correspond to 10-1% and 26-2 7o of the trials instead of 10-9 7o and 25-8 7o- If 
we take for the standard-deviation of our Gaussian V«pg — = 2-21171, we find 
that the odds in the first case are still only 35 to 1, but the percentages in the 
other two cases are ll'S and 25-8. 
It will be clear that near the centre of the curve — especially when we equalise 
the excess and defect of the Gaussian by taking equal ranges on both sides — ^it 
does not give bad percentages of frequency, but that it does not lend itself to 
the accurate determination of the range for reasonable working odds such as 
50 to 1. 
It will be noted that the total area in excess and defect of 2 and more 
= 23-782 + 26-503 = 50-285, or corresponds very nearly to the "probable error." 
Actually the Gaussians with standard deviations of 2-23047 and 2-21171 give 
probable errors of 1-504 and 1-492 respectively, so that the Gaussian with 1-5 as 
the probable error is very nearly accurate. 
Table II gives the Poisson-Exponential ; it will enable the reader to appreciate 
the range of probable variation in small frequencies. Thus we realise that in 
37 °/„ of cases in which the true frequency is 1, the cell will be found empty ; 
in 13-5 per cent, of cases it will be empty when the actual frequency is 2, and in 
5 7o of cases when the frequency is 3 and in 1-8 7o when the frequency is 4. These 
results indicate how rash it is to assume that a sample 4-fc)ld table with one zero 
quadrant signifies perfect dependence or association in the attributes of the 
material sampled. The second line below gives the percentages of cases that 0 
would appear in a cell when the actual number to be expected is that in the first 
line calculated from Table II on the usual theory of a priori probabilities : 
Actual 
0 
1 
'3 
0 
Jf 
5 
0 
7 
8 
9 & over 
Percentage . . . 
63-21 
23-25 
8-55 
3-15 
1-16 
0-43 
0-16 
0-06 
0-02 
0-01 
