RELIABILITY OF IIOLDIXC TIME MEASUREMENTS 389 



of the switch counl iiK-thod of estimatiii)^ a\'er;i^c liohhn^ times. Su])|)f)st' 

 we have completed a succession of r + 1 switch counts sj)acc(l uniformly 

 by the interval /. 'I'he period covered will then be ri units lonj^, and we 

 shall assume that the call count has covered exactly the same total interval 

 so as to eliminate certain of the correction difficulties described under IV-1) 

 above. Suppose the first switch count (at the beginning of the period) 

 showed m calls up, the last switch count (at the exact end of the period) 

 showed w calls up, and the sum of all of the r -\- 1 counts, including the first , 

 totalled the number s. 



As we found in equations (5) and (6) the average correction in the number 

 5 to be made on account of calls from the previous period being switch 

 counted is 



_ _ m 



8m — Z- . 



\ — e '<■ 

 and the standard deviation in the correction here is 



/— e 2t 



Cm = V W 



\ — e i 



At the end of the hour the corresponding correction was found in equations 

 (10) and (11) to have an average of 



we t 

 s,„ = 



\ - e t 



e 2« 



and a standard deviation of 



(Tw = V W ,• • 



1 - e"'' 



These are quite independent corrections if the observation period is several 

 times the average holding time, the usual case. Then our best estimate of 

 the number of switch counts we would have obtained if all (and only) 

 those associated with the r calls originated in the period had been counted is, 



s' = s - .5„, + Su, , (28) 



and the standard deviation of s' is 



cr.,' = V o-„, + Gu,. (29) 



