962 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1953 



where each of the above P symbols designates the probability that a 

 sample average (average of a random sample of 5 units) will fall in the 

 band associated with that s3nnbol in the following tabulation: 



Symbol 



Pt 

 Pt 



Po 



Band 



Above +0.5A 

 +0.5A to +0.1A 

 +0.1A to -O.IA 

 -O.IA to -0.5A 

 Below -0.5A 



and P4 = probability that a sample range (range of a random sample 

 of 5 units) will fall above its upper control limit, 1.48A. (P4 = 0.0044 for 

 (t' = 0.3 A.) 



The OC curves for Criterion I and Criterion II computed from these 

 formulas are given in Fig. 7(a), when a-' = 0.3 A. These show how the 

 control chart method serves on a probability basis as a band-pass filter 

 for a manufacturing process, permitting the introduction of and allowing 

 the continuance of sampling so long as the process average is maintained 

 reasonably close to the nominal, and imposing an increasingly higher 

 barrier to acceptance by sampling when the displacement of the process 

 average from N is increased, thus forcing the use of the three-cell method. 

 Examination of these curves shows that Criterion I is more stringent 

 than Criterion II, as it should be. 



Suppose for the moment that both Criterion I and Criterion II omitted 

 condition (c) relating to seven successive sample averages. The OC curves 

 in this case are shown in Fig. 7(b) with the designation "I, II, less c." 

 It is seen that this requirement relating to seven successive averages is 

 most important. Without it the criteria, particularly Criterion II, would 

 be very ineffective in controlling excursions of the process average. 



A second question of importance is: "What happens if the specified 

 dispersion limits {A limits) are improperly set or if the process dispersion 

 changes to produce this effect?" In other words, what are the operating 

 characteristics of the procedure when a' 9^ 0.3A? This is shown in Figs. 

 7(c) and 7(d) for Criterion I and Criterion II respectively, where the 

 values of process standard deviation (o-') are expressed as fractional 

 values of A . 



4.2 OPERATING CHARACTERISTICS OF THE BATCH METHOD 



For the batch method the procedure is essentially a lot acceptance 

 prricedure and except for permitting an occasional failure of the sample 



