CORRECTION OF DATA FOR MEASUREMENT 13 



desire to find the most probable numbers of transmitters within the 

 different interv'als indicated in Fig. 1. 



Analytical Statement of Problem 



Let us assume that the most probable number of transmitters within 

 the interval of efficiency from X to X-{-dX is fr{X)dX. It is this 

 function friX) that we want to find. Similarly let us assume that 

 there is some function foiX) such that fo{X)dX gives the observed 

 number of transmitters appearing to have efficiencies within the 

 interval X to X-\-dX where the measurements are made by a method 

 wherein the probability of making an error within the interval x to re 

 -\-dx is fEix)dx. It is reasonable to expect that, if two of these func- 

 tions are known, the third can be easily determined. We shall pro- 

 ceed to show that this is the case. Let us first find the law of error 

 experimentally. 



Finding the Law of Error 



The problem is to determine the chance of making an error of a 

 given magnitude in measuring the efficiency of any transmitter. 

 Naturally, the only way of doing this is to make a series of measure- 

 ments on a single transmitter from which we can determine the 

 observed frequency of occurrence of measurements which differ from 

 the average by some fixed amount, and thus find what percentage of 

 the total number of measurements may be expected to fall within 

 any given range on either side of the average. Common sense and 

 intuition may tell us that we may expect to find a large percentage 

 of the measurements within a narrow range on either side of the 

 average, that there wall be just as many measurements greater than 

 the average by a certain amount as there are less than the average 

 by the same amount, and that large deviations from the average may 

 be expected to occur with less frequency than small deviations. 

 Suppose we make 500 observations of the efficiency of a single trans- 

 mitter and find the distribution given in Fig. 2. Just as we might 

 have expected, the observed values of the efficiency of the transmitter 

 are grouped symmetrically about the average of all the observed 

 values. We see that the maximum deviation between observations 

 on a single transmitter is quite large (33%) compared with the actual 

 maximum differences observed between the efficiencies of the trans- 

 mitters. 



The results reproduced in Fig. 2 suggest that the deviations for the 

 case in hand are distributed in a manner closely approximating the 



