612 BELL SYSTEM TECHNICAL JOURNAL 



observed distribution and the corresponding Poisson distribution. In 

 the first place, the sample considered will necessarily consist of a 

 finite number of trials instead of an infinite number as assumed in 

 the mathematical theory, and the trials may not be completely inde- 

 pendent or entirely uniform. Secondly, even if the individual sample 

 possessed the ideal characteristics assumed in the mathematical 

 formulation, the actual series of samples must be finite and the samples 

 may be interdependent and far from uniform. The size of the samples 

 relating to the economic, geographic, and time divisions ordinarily 

 used in statistical work generally varies considerably. The effect of 

 modifying the original mathematical assumptions to correspond with 

 some of these actual conditions is illustrated by Figs. 6-8, which show 

 various theoretical frequency-distributions plotted on Fig. 1 or Fig. 2 

 for comparison with the corresponding Poisson distributions. 



The finiteness of the number of trials w not only makes impossible 

 the occurrence of values of c greater than the value of n, but also tends 

 to produce a general trend away from the Poisson distribution. This 

 is illustrated by the four typical finite binomial distributions shown in 

 Fig. 6, which have a definite curve and slope toward the left which 

 becomes more pronounced as n is decreased.^ Interdependence of the 

 trials constituting a sample will also tend to give the resulting dis- 

 tribution a slant, to the right if the correlation is positive, to the left 

 if the correlation is negative.** Thirdly, even though the trials are 

 independent, if they are not uniform, there will be a tendency for the 

 distribution to slant to the left. 



The requirement that A^, the number of samples in the actual 

 series, be finite introduces a somewhat different kind of deviation 

 from the theoretical Poisson distribution. The observed relative 

 frequency F, which is compared with the theoretical probability P, 

 is an integral multiple of 1/iV, so that, since N is finite, the points 

 representing the observed distribution (except those at P = and 

 P = \, for which the ordinates are plus and minus infinity, and which, 

 therefore, never appear on the graph) are all in the finite range between 

 the two horizontal lines P = l/N and P = l—1/N. Not only is the 

 occurrence of points outside this range impossible, but the points 

 near its extremes, being determined by a comparatively small number 

 of samples, are of less significance than those near the center. 



To call attention to these facts all observed distributions shown 

 here have been represented, as in Fig. 4, with the vertical line rep- 



' A more detailed discussion of the effect of finite sampling will be found in the 

 paper by G. A. Campbell previously referred to. 



* See "Explanation of Deviations from Poisson's Law in Practice," by "Student," 

 Biometrika, Vol. 12, pp. 211-215, 1919. 



