620 BELL SYSTEM TECHNICAL JOURNAL 



of particles visible siinullaneously if that number is more than three 

 or four. 



In several cases special measures have been taken to reduce the 

 variation of a and the resulting trend away from the corresponding 

 Poisson distribution. In general, a is made as nearly constant as 

 possible by making n and p constant throughout. In examples 

 (b6)-(b9) and (c7)-(clO), for instance, each sample consists of ap- 

 proximately the same number of calls, and in example (cl) four corps 

 were rejected because they were considerably larger than the others. 

 In these examples it is assumed that p is practically constant and 

 that by making n constant a constant average will be obtained. A 

 somewhat different adjustment to keep a constant is illustrated by 

 examples (al) and (a2), where, as the decay of the radioactive sub- 

 stance decreases the average number of a particles emitted in a given 

 solid angle per unit of time, the screen on which the particles strike is 

 moved so that it intercepts a greater angle. In some cases n may be 

 controlled much more easily than p, or vice versa, and a may be kept 

 constant by letting one factor vary and adjusting the other to com- 

 pensate, rather than by keeping both constant. 



Summary 



These examples of distributions which can be described by the 

 Poisson exponential are of a dozen quite different kinds. They include 

 eleven distributions found in published work on biometrics or statistics 

 and twenty-one which are new. The agreement between the ob- 

 served and the theoretical distribution is, in general, fairly good, and 

 the applicability of the Poisson summation to a great variety of data 

 is clearly indicated. The practical importance of some of these 

 cases has been discussed above. 



The use of the probability curves showing Poisson's exponential 

 summation in place of double-entry tables as a source of data is 

 shown to be simple, and their convenience as a background for plotting 

 and comparing frequency-distributions is illustrated by Figs. 4 and 

 6-9. The new chart with a logarithmic scale for a (Fig. 5) is con- 

 venient in comparing distributions of different averages. It also 

 shows the complete set of curves up to a = 30 instead of only to a = 15, 

 and it makes it possible to read with considerable accuracy values of 

 the variables in the range 0.1 ^a^ 2, which is not clearly shown in 

 Fig. 1 or Fig. 2. 



