THE L3 SYSTEM — QUALITY CONTROL REQUIREMENTS 961 



tributions.* Since a primary objective is to limit the displacement of the 

 process average, X', from the nominal value, Nj this parameter will be 

 used as the independent variable against which the performance of the 

 methods is computed. The curves to be shown will be referred to as ''op- 

 erating characteristic curves" or ''OC curves" for the respective criteria 

 of these methods, a termf that is applied generally to the related "prob- 

 ability of acceptance" curves for sampling inspection plans. 



4.1 OPERATING CHARACTERISTICS OF THE CONTROL CHART METHOD 



First, let us assume that we have a process the output of which has a 

 Normal distribution with a standard deviation, o-', equal to 0.3^, which 

 is the same as the standard value, a'^, used in setting the specified limits. 

 For this case the limits N zL A are A^ =b S}^a'. Ideally, the process av- 

 erage, X', should be centered at N, but by design it is agreed that a dis- 

 placement of X' by an amount 0.1^ from N should be acceptable. Con- 

 ceivably the displacement could well be considerably larger than this, 

 and the question arises as to how the two criteria of the control chart 

 method will function for different magnitudes of displacement. With the 

 model assumed, it is possible to work out an analytic answer to this 

 question. For example, for any given displacement of X', from N, the 

 resulting formula for Pi , the probability of meeting Criterion I, is: 



Pi = [(1 -Pt - PI)" - (Pty - (Pr)'][(l - P4)'] (1) 



and the formulaj for Pu , the probability of meeting Criterion II is 



Pii = [Po + pt\i - (Pt + Pty\ 



+ Pt\i- {PI + PTf\ + Pt\ (1 - Ft - P^? - {Ptf\ (2) 

 + P7J {\-Pt- PT)' - (Pr)'[][l -P* + P4(l - P4)'] 



* For other distributions this limitation is unimportant for those portions of 

 the criteria that relate to sample averages since averages of samples from a non- 

 Normal universe may ordinarily be considered to be distributed Normally. See 

 W. A. Shewhart, Economic Control of Quality of Manufactured Product, D. Van 

 Nostrand Co., New York, 1931, pp. 180-184. But due among other things to lack 

 of independence of X and R for skew distributions, the results given here should 

 be considered only as reasonably close approximations for the degrees of non- 

 Normality that may be encountered in practice, 



t A term first used in the late 1930's by Capt. H. H. Zornig, of the Ballistic 

 Research Laboratories, at Aberdeen Proving Ground. 



X Pii is an unconditional probability in the sense that it does not involve the 

 condition that previously there was a sequence of 7 samples satisfymg Criterion 

 I and no intervening sequence of 7 samples not satisfying Criterion II. Pu has 

 been used as an approximation to, although possibly somewhat less than, the 

 corresponding conditional probability. A similar consideration applies to Pi . 



