60 BELL SYSTEM TECUMC.U. JOURNAL 



logarithms of the intensities is given in the second column of the 

 table in Fig. G. The smooth line is the normal curve based upon the 

 observed value of standard deviation. The distribution of the 

 logarithms of the intensities is normal." The arithmetic mean of 

 the logarithms is the most probable. Therefore, the distribution 

 of intensities is decidedly skew, and the geometric mean intensity is 

 the most probable. Here, then, is an excellent example in which 

 it is highly probable that the distribution of the causes is random 

 and normal, but in which the resultant elTect i> nut a liiu-ar function 

 of the number of causes.'' 



Can We Ever E.xpect to Find \ Nor.m.-\l Distribl'tion 



IN N.\Tl'RE? 



The answer is aftirmalive. If the resultant effect of tiie inde- 

 pendent causes is proportional to their number, the distribution 

 rapidly approaches normality as the number of causes is increased 

 ev-en though p=l=Q^- 



To show this, let us assume that the \ariation in a physical quan- 

 tity is produced by 100 causes, and that each cause produces the 

 same efifect Aa;. Also, let us assume the probability p to be 0.1, that 

 each cause produces a positive effect. The distribution of 0, 1 , 2, . . . n 

 successes in 1000 trials is given by the terms of the expansion 1000 

 (.9-f .l)'"". Ob\iously such a distribution is skew, p is certainh- 

 not equal to q, and n is far from being infinite. If the normal law 



" In fact this is an exceptionally close approximation to the normal law. 

 This will be more evident after we have considered the methods for measuring 

 the goodness of fit as indicated by the other calculations given in this figure. 

 For the present it is sufficient to know tlut approximately 75 times out of 100 

 we must expect to get a system of observations which differ as much or more 

 from the theoretical distril)Ution calculated from the normal law than the ob- 

 served distribution dillfcrs therefrom in this case. The fact that the second 

 approximation does not fit the observed distribution as well as the normal — 

 i.e. the measure of probability of fit P is less — indicates that the observed value 

 of the skcwness k is not significant. 



" These results are of particular interest to telephone engineers. The fact 

 that the distribution of the logarithms of the intensities is normal is consistent 

 with the assumption of Fechner's law which states that the sensation is pro- 

 portional to the logarithm of the stimulus. ITie range of variation (that is, 

 A' =*= 3 <T ) in different observers' estimates of the sound intensity required to 

 produce the minimum audible sensation is approximately 20 miles. The range 

 of error of estimate depends upon the intensity of sound and decreases as the 

 sound energy level increases. Thus for the average level which prevails for 

 transmission over the present form of telephone system in a three mile loop 

 common battery circuit it is less than 9 miles. Even at this intensity, however, 

 it is obvious that although scarcely any observers will differ in their estimates 

 by more than 9 miles, 50% of them will diflfer by at least 2 miles. These 

 results also furnish experimental basis for the statement made in the beginning 

 of tliis paper: lliat is, the variations introduced in the method of measurement 

 of transmitter efficiencies are large in comparison with the average efficiency. 



