IS BELL SYSTEM TECIIXICAL JOrRX.IE 



\aluc of X. Substitution of tliis expression and a normal error 

 function in Equation (2), yields upon integration '• the following 

 distribution /,.(A') of the observed \alues 



_,,,^ 1 -^-f. kofX ZHn 



a„\/27r L 2 \(To SaU J 



ao\/2 

 where 



(7) 



0-0 = \ Ot-\-o-e< (5) 



and 



ko=kT^. (8) 



We see that the distribution fo{X), Equation (7), of the observed 

 values is of the same form as that friX), Equation (6), of the true 

 values. The standard deviation of the errors of measurement <^e, 

 as in the previous case, has equal weight with the standard deviation 

 o'r in influencing the standard deviation ao of the observed values. 

 The degree of asymmetry of the observed distribution as measured 

 by the skewness ko is, however, less (Equation (8) ) than that of the 

 true distribution as measured by the skewness ^r of the true dis- 

 tribution. 



Now we can correct the observed distribution, Fig. 1, for the errors 

 of measurement, because we find that the observed frequencies, Fig. 1, 

 can be closely approximated by a function of the type defined by 

 Equation (7). Knowing that the law of error, Fig. 3, is normal we 

 conclude that the true distribution friX) must be a function of the 

 same type as fo{X) was found to be except that the true standard 

 deviation (^rwill be, from Equation (5), -x/al + al: and the true skewness 



i^ 7 will be, from Equation (8), -f ko. Now, ao and ko can be calcu- 



(T-p 



lated from the observed distribution. Fig 1, and '^e can be determined 

 by the data given in Fig. 3. 



Thus finding the values of <^r and kr and substituting them in 

 Equation (6), we have the function /-/(.Y) representing the true dis- 

 tribution which we started out to find. From this knowledge of 

 friX) we can now get the most probable frequencies of occurrence 

 of the different efficiencies. Subtracting these frequencies from those 

 observed and shown in Fig. 1, we get the corrections plotted in Fig. 6, 

 expressed as percentages of the observed frequencies. 



* This solution is also obtained by another method in Appendix 1. 



