22 BELL SYSTEM TECHNICAL JOURX.IL 



What is the most economical way of detejnnini^u the (luahty of 

 product within some predetermined range X±^X with a known 

 probabihty P, where X is the average quahty? Let us assume that: 



fli =cost of selecting each unit and making it available for measure- 

 ment, 

 a2=cost of making each measurement, 

 Wi = number of units selected, 

 n2 = number of measurements made on each unit. 

 o-i=standard deviation of the errors of observation. 

 (72 = 0-^= standard deviation of the true distribution /-/(A'). 



Let us take P = .9973. Then the ran^ge X±Sax includes 99.73 per 

 cent, of the observations, and hence AX = ^<rx- 



The average of W2 measurements made on one unit is the observed 

 value of the magnitude X for that unit, and this average has the 



standard deviation a E = —h. Hence, from the theory of the preced- 



ing section, the standard deviation of the observation is 



The standard deviation of the average of Wi observations is 



W2' 



o-Y = — 7^ and we find upon solving for Wi, 



W2 



Wl= 7, . 



Ox 



The cost of inspection is 



y=ai Hi+oo Hi nt, 

 and by customary methods this can be shown to be a minimum when 



0-2 \ ao 



The following values correspond to one practical case: 

 AX = .3 unit ai=$0.50 



cri-.3unit a2 = S0.02 



^2 = . 9 unit P=.9973 



