14 



BELL SYSTEM TECHNICAL JOURNAL 



bell-shaped distribution so familiar in the theory of errors. We often 

 find, as we do in this case, that the observed distribution can be closely 

 approximated by a function /eCa:) of the form 



fE{x)dx = — ^7=r e 2(72 dx, 

 (TV 27r 



(1) 



where fE(x)dx is the probability that an error .r will lie within the 

 interval x to x+dx, a is the root mean square or standard deviation, 

 X is the arithmetic mean value and (X-X) is the deviation x. The 



7 

 200 



180 



160 



140 



lao 



100 ■ 

 80 

 60 

 40 

 20 

 



•1.0 



.5 .5 



EFFICIENCY 



Fig. 2 — Typical form of distribution of errors of measurement. Chart showing 

 number of measurements on a single transmitter versus efficiency 



function Je{x) is referred to in the literature as the normal law of 

 error. If we try to fit such a curve to the deviations- 5;iven in Fig. 2, 

 we obtain the results shown in Fig. 3. This figure is the same as 

 Fig. 2 except for the addition of the smooth normal curve of error 

 calculated for the observed data. Without further consideration, we 

 shall assume the law of error to be normal and hence of the form 

 indicated by Equation (1). 



Finding the True Distribution friX) 



We have next to consider the choice of the function to represent 

 the true distribution /r(X). Often we have reason to believe that this 



2 If the average of the observed values of the 500 observations of efficiency given 

 in Fig. 3 is assumed to be the true value of the efficiency of the transmitter, then the 

 deviation of an observed Vcdue from this mean is also the error of this observed value. 

 We shall use the terms "error" and "deviation" interchangeably in this sense. 



