THE ASYMMETRICAL DISTRIBUTION 9 



sample is actually reasonably large. This situation has been analyzed and 

 the distribution which governs it has been given the name of the French 

 mathematician Poisson. The Poisson Distribution is 



— a n 

 r> i \ e a 



n 



where e is the base of natural logarithms, and equals 2.71828. This 

 represents the probability of finding precisely n of something if the 

 average is a. If we were talking the language of deviations, we would 

 be talking about the quantity n — a. Here, however, the least we can 

 have is a equal to zero, so that we do not have very many negative 

 deviations: if the average were 2, then we would have negative deviations 

 of only —1 and —2. Thus this distribution is basically asymmetrical. 



As an illustration of the use of this distribution, suppose we distribute 

 100 spores into 100 test tubes at random. The probability of finding any 

 given spore in any given test tube has an average value of 1/100, but 

 the probability of finding some spore is quite appreciable. Indeed, the 

 average number of spores per tube is 100/100, or 1. The Poisson formula 

 says that if the average is 1, then, for example, the probability of finding 

 precisely 2 spores in a tube is 



P 2 (D = 6 ' 2 * ^ = 0.18, 



as we can determine from tables or from direct computation using the 

 numerical value of e. 



One of the most important uses of the formula has to do with finding 

 the zero class— the probability of finding none if the average is a. We 

 know intuitively that if we tossed 100 spores randomly into 100 test tubes, 

 there would be a substantial number of tubes having no spores. The 

 formula gives the zero class as 



Po(a) = e~ a . 



In the illustration above, e" 1 equals 0.37. In 377c of the tubes there will 

 be no spores at all. 



The zero class formula is just as useful in the inverse way. If we 

 measure the zero class, we can use the formula to obtain the average. If, 

 for example, we found 2% of the test tubes without spores, we would solve 

 the equation 



0.02 = e~ a . 

 A look at some tables gives the result for a as a little less than 4 spores 



