618 BELL SYSTEM TECHNICAL JOURNAL 



resistances with a return loss substantially the same at all frequencies. 

 Appendix V derives the distribution of active singing points for these 

 cases. 



For exact results, the modification of the return loss due to the near- 

 end and far-end equipment should be added before combining the 

 end path. In the practical case, however, only the lower values of 

 return loss are generally significant, and the approximate method 

 outlined above may generally be used without serious error. 



Minimum Value of a Group of Singing Points 



When a large number of independent measurements are taken, say 

 of singing points on different pairs at a given repeater station, it is 

 often of interest to know whether the maximum or minimum in that 

 group represents some special condition, say, definite circuit trouble, 

 or is merely a case where the various components happen to have 

 added up or subtracted by pure chance. 



In the case of singing points, the question of interest may usually 

 be stated as follows: In " n " similar measurements at a given place, 

 what is the probability that the smallest value will be more than x db 

 below the true average measurement, assumed known? 



If the fraction of cases which will be greater than a value y db below 

 the average is P, the probability that " « " values will all be greater 

 than this value is P'\ In other words Pm{n, y) = Pm = P" is the 

 probability that no one in " « " values will be as much as y db below 

 the true average. 



For example, if the singing point distribution is 25 + Sq{2), the 

 chance that any given value selected at random will be as low as 

 25 — 4.6 = 20.4 db is one in a hundred. However, the chance that 

 the lowest one of 20 measurements will be that low is 1 — (0.99)^" 

 = 0.1819 or almost one in five. If we substitute V^ = "^ — Pm and 

 F = 1 - P, we may write that 1 - F^ = (1 - 7)". 



F„ = 1 - (1 - Vy = 1 - 1 -f nF - !^(^_=J) F^ + . . . 



= nV 2! + 3! ^~ ' 



And for small values of F, Vm = nV. In the above case, therefore, if 

 we wanted to compute a value of singing point which would be lower 

 than all the 20 measurements nine times out of ten, we may compute 

 V = (0.1/20) = 0.005, and read on the normal distribution Sq(2) 

 that this corresponds to about 5 db below the average; i.e., to 

 25 - 5 = 20 db. The more e.xact formula would be that 0.9 = P^o 



