Foundations of statistical quality control

Abstract

The origins of statistical quality control are first reviewed relative to the concept of statistical control. recent Bayesian approach developed at ATT namely i) variation due to chance causes (called common causes by Deming, 1986); ii) variation due to assignable causes (called special causes by Deming, 1986). Chance causes are inherent in the system of production while assignable causes, if they exist, can be traced to a particular machine, a particular worker, a particular material, etc. According to both Shewart and Deming, if variation in product is only due to chance causes, then the process is said to be in statistical control. Nelson (1982) describes a process in statistical control as follows: A process is said to have reached a state of statistical control when changes in Now at George Washington University, Washington, D.C. 20006. This research was partially supported by the U.S. Air Force Office of Scientific Research (AFOSR-90-0087) to the University of California at Berkeley. This content downloaded from 157.55.39.111 on Wed, 03 Aug 2016 04:54:04 UTC All use subject to http://about.jstor.org/terms 100 R.E. Barlow k T.Z. Irony measures of variability and location from one sampling period to the next are no greater than statistical theory would predict. That is, assignable causes of variation have been detected, identified, and eliminated. Duncan (1974) describes chance variations: If chance variations are ordered in time or possibly on some other basis, they will behave in a random manner. They will show no cycles or runs or any other defined pattern. No specific variation to come can be predicted from knowledge of past variations. Duncan, in the last sentence, is implying statistical independence and not statistical control. Neither Shewhart nor Duncan have given us a mathematical definition of what it means for a process to be in statistical control. The following example shows that statistical independence depends on the knowledge of the observer and, therefore, we think it should not be a part of the definition of statistical control. Example The idea of chance causes apparently comes from or can be associated with Monte Carlo experiments. Suppose I go to a computer and generate ? random quantities normally distributed with mean 0 and variance 1. Since I know the distribution used to generate the observed quantities, I would use a iV(0,l) distribution to predict the (n+l)st quantity yet to be generated by the computer. For me, the process is random and the generated ? random quantities provide no predictive information. However, suppose I show a plot of these ? numbers to my friend and I tell her how the numbers were generated except that I neglect to tell her that the variance was 1. Then for her, zn+1 is not independent of the first ? random quantities because she can use these ? quantities to estimate the process variance and, therefore, better predict xn+iWhat is interesting from this example is that for one of us the observations are from an independent process while for the other the observations are from a dependent process. But of course (objectively) the plot looks exactly the same to both of us. The probability distribution used depends on the state of knowledge of the analyst. I think we both would agree however that the process is in statistical control. All authors seem to indicate that the concept of statistical control is somehow connected with probability theory although not with any specific probability model. We think de Finetti (1937, 1979) has given us the concept which provides the correct mathematical definition of statistical control. Definition: Statistical control We say that a product process is in statistical control with respect to some measurement variable, x, on units 1, 2,...,? if and only if in our judgement p(xv 22,..,i?) = ?(???, xi2,...,xin) for all permutations {ix, ?2,...,in} of units {1, 2,...,n}. That is, the units are This content downloaded from 157.55.39.111 on Wed, 03 Aug 2016 04:54:04 UTC All use subject to http://about.jstor.org/terms