10 THE MATHEMATICAL TREATMENT OF DATA 



per cell. The following table for the zero class may be of some use: 



a = 1 2 3 4 5 



p (a) = 0.37 0.14 0.05 0.018 0.0067 



The Poisson distribution turns out to have another interesting and use- 

 ful property. If the standard deviation of things satisfying this distribu- 

 tion is calculated, it is found that 



s 2 = a. 



This remarkable result permits obtaining the standard deviation from 

 the average alone. Since the average can be obtained from the zero class, 

 it becomes an exceedingly simple matter to obtain the standard deviation. 

 In the illustration above, where there were 2% of the tubes unoccupied by 

 spores, we not only know from the little table that the average occupancy 

 is about 4, but also that the standard deviation is close to 2. 



This property is frequently used in the design of experiments. Suppose 

 we wish to determine the number of spores to an accuracy of, say 5%. 

 We have only to solve the relation 



* = ^ = 0.05. 

 a a 



The result is a equals about 400. So we know that we must count until a 

 total of four hundred spores have been tallied in order to know the num- 

 ber 400 to an uncertainty of 5%. 



COMPUTATION OF ERRORS IN COMPOUND QUANTITIES 



It frequently happens that the desired quantity cannot be measured 

 directly, but can be determined from two or more other measurements. 

 For example, the molecular weight of a particle can be determined by a 

 combination of sedimentation and diffusion data according to the follow- 

 ing formula: 



s RT 



M 



D \ (di/d p ) 



where M is the molecular weight of the particle of density d p , suspended 

 in a liquid of density d l} at a temperature T, and R is the gas constant. 

 Respectively, 1) and s are the coefficients of diffusion and sedimentation, 

 and are obtained experimentally. For our purposes, the desired quantity, 

 M, is given in terms of the complex ratio of 5 measured quantities. If 

 each of these has its own standard deviation, what is the standard devia- 

 tion of M? 



