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Unformatted text preview: STATISTICAL FOUNDATIONS OF CONTROL CHARTS Control charts are defi ned by the center line, upper control limit, and lower control limit. These values are related to the expected value and variance of the statistics plot ted on the charts. In Chapter 13, the upper and lower control limits were speci- fi ed through the use of certain constants given in Appendix B. This section shows how these factors are developed and discusses the statistical basis for the rules used to in terpret control charts. Variables Control Charts When a process is in control, the distribution of individual measurements for variables data is assumed to have a mean m and a variance s 2 x . If a sample of size n is cho- sen, the sampling distribution of x will also have a mean m , but will have a variance s 2 x 5 s 2 x / n . If the original distribution of individuals is normal, the sampling distri- bution of averages will also be normal. If not, the central limit theorem states that the sam pling distribution of averages will be approximately normal for large sample sizes. Because control chart samples are usually small (n 5 4 or 5), the central limit theorem does not always apply. However, normality is usually assumed in develop- ing vari ables control charts. Under this assumption, 100 1 1 2 a 2 percent of the sample means fall between m 2 z a /2 s x and m 1 z a /2 s x ; these values become the lower and upper control lim- its. A value of z a /2 5 3 gives a six-standard deviation range with a /2 5 0.0014. Thus, only about 0.3 percent of the sample observations will be expected to fall out- side these limits. If the process is in control, the likelihood that a sample will fall outside the control lim its is extremely small. On the other hand, if the true mean has shifted, this probabil ity will be much larger. This reasoning is the theoretical basis for assigning three-sigma control limits. The value of z a /2 can, of course, be chosen arbitrarily. In the United States, the value of 3 is commonly accepted. In England, however, z a /2 is selected by fi rst setting the probability of a Type I errorusually chosen as a /2 5 0.001. Thus, z 0.001 5 3.09 is commonly used to establish control limits. Such limits are called probability limits. R- Chart The range is used as a substitute for the standard deviation primarily be- cause of its simplicity. As noted in Chapter 13, the factor d 2 in Appendix B is used to relate the range to the actual process standard deviation. The factor d 2 is determined as follows. Consider an experiment in which samples of size n are drawn from a nor mal distribution having a known standard deviation s x . If the range R of each sam ple is computed, the distribution of the statistic R / s x can be determined. The expected value of this statistic is the factor d 2 , that is E 1 R / s x 2 5 d 2 or, because R is a random variable and s x is known, E 1 R 2 / s x 5 d 2 This experiment can be performed for each...
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