RELIABILITY OF IIOLDI.XG TIME M EASE REM ENTS 397 



average length / is accurately known for the n calls, it may not exactly or 

 even closel}' coincide with the true average of the universe of calls of which 

 the n are presumed to form a random sample. The errors we have just 

 studied and those described in section III must now be combined to give 

 us a measure of the overall accuracy of the switch count method. 



Equation (2), when aj^plied to the exponential holding limes we are here 

 concerned with, gives us for the standard error of the average in a sample 



Csniiii'iiMn — '~T— ~ — /— • (38) 



\/ n v;/ 



This error is independent of that represented by ai' , so we may determine 

 the overall (oa) standard error by 



(Toa = V O-gampling -\- CT't' ■ W^/ 



We shall be particularly interested in knowing how much the value of ai' 

 contributes to the overall standard deviation, a^a- This may be conveni- 

 ently expressed by writing the ratio 



(foa , / (T7t 



"i ^ a T- "" 4/ 1 + 



O'sampline y ^2 



"sampling 



2i i -I , 

 t T 



r(2+j)^ ^-2 + 51(1 -e 1) 



/ 1 + iu^ LA V yV I . (40) 



y (i-.-o= 



Now it is readily seen from (40) that q depends on /, i and T. Hence if T 

 is held constant we may plot curves between g, i and / as shown on Fig. 19. 

 What is more, if T is varied, say increased by a factor k, equation (40) 

 shows that if i and / are also increased by the same factor the values of q 

 resulting may still be read directly from Fig. 19. 



For example: If approximately 100-second exponential calls are to be 

 switch counted for an hour with observation intervals of 120 seconds, we read 

 on Fig. 19 that the overall standard error (or P = .50, .90, .99, etc. error) in 

 estimating the true average holding time is q — 1.134 times the basic 

 standard error resulting from taking a random sample of n calls from a very 

 large universe of calls. That is to say, the residual sampling error present 

 in a stop watch measurement of the n calls is increased by 13.4 per cent due 

 to our resort to switch count methods. 



If a continuous period oi T = 2 hours (i.e. ^ = 2) is switch counted in 

 just the same manner and under the same conditions, we should now read 

 on the i = 50 seconds curve opposite / = 60 seconds, giving us an increase 

 over the basic sampling error of 12.4 per cent. This meets one's common 



