366 BELL SYSTEM TECHNICAL JOURNAL 



n = number inspected for a given type of defect during the month. 



( — J = expected demerits per unit. 



. , . „ 1 r^ 1- Expected Demerits per Unit 



le = mdex for Expected (Juahty = -5 ^5 — ■ . ^ r- ^T^ ' 



*^ Base Period Demerits per Unit 



Re = rate for Expected QuaHty. 

 Rj^ = control Hmit value of rate. 



Base Period Demerits per Unit 

 a = standard (root mean square) deviation. 



The rate for Expected Quality is 



Re = 10(1 - le). (1) 



To find the control limits for the rate, first determine its standard 



deviation, cr^^. 



a,^ = 10(r,^. (2) 



The index for Expected Quality is given by 



I^^ k( — -]-'^ + etc. for all types of defects ) , (3) 



where d,\, di, etc. = expected number of defects per month for defects of 

 type 1, 2, etc., and the subscripts 1, 2, etc., refer generally to the several 

 types of defects.'^ 



To find 0-/^, the standard deviation of the index, the J's are considered 

 as independent variables subject to sampling variations. le is then a 



linear function of du d^, etc., with the constant coerncients , , 



etc. Hence 



The values of cd,, 0-^^, etc., are evaluated by the following consideration. 

 Assume that a sample of size N is drawn from a source for which 

 the probability of occurrence of a defect is p. The expected number 

 of defects in the sample is pN, and the standard deviation of the 

 expected number is -Vp(l — p)N. If p is small, the factor (1 — p) 



" In carrying out the computations of rates and control limits, it is convenient to 

 group together all defects having the same "weight" {w) and the same "number 

 inspected" (n), and to let the subscripts 1, 2, etc., of the equations refer to these groups. 



