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Statistical Control Charting
Published in John Gaylord, Factory Information Systems, 2020
A p-chart is a time-plot of values of p (usually rates of defective product), which are computed from successive groups of binary data (pass/fail, accept/reject, or go/no-go), plus control limits added to help determine whether the process resulting in the binary data is in control with respect to the p level. Basically, a p-chart is used to answer the question, “Is the level of defectives (or defects) that we are currently experiencing simply due to random variation, or is it significantly higher than normal?” An example used frequently by the author is the simple but surprising case of a part-making process running 10% defective (p = 0.1). Suppose parts are produced in lots of 10. What is the minimum number of parts in a lot of 10 which would signal (with high certainty) that the production process is out of control, and thus should be fixed before more parts are produced? I asked this question of people at all levels from operator to middle-level manager in a manufacturing plant. I had a custom-made sampling device to simulate the production of parts at the 10% defective level. How would the reader answer the question? Is 4 out of 10 high enough to indicate out of control? Should we demand more, say, 7 out of 10? The answer is given exactly by the binomial distribution, used in p-charting to compute the control limits. This requires that we believe the binomial distribution assumptions, as discussed in Section 2.4.1. They usually are valid, so let us compute in Table 5.1 the likelihood of seeing k defectives out of 10 parts, given a historical defective rate of 10%.
Quality in Production—Process Control I
Published in K. S. Krishnamoorthi, V. Ram Krishnamoorthi, Arunkumar Pennathur, A First Course in Quality Engineering, 2018
K. S. Krishnamoorthi, V. Ram Krishnamoorthi, Arunkumar Pennathur
The P-chart is also known as the “fraction defectives” or “fraction nonconforming chart,” because it is used to monitor and control the fraction produced in a process that is defective or nonconforming. The terms “defective” and “nonconforming” need explanation first. The term “defective” was the original name used in SPC literature for a unit of a product that did not meet the specified requirements, and so the P-chart, used to control the proportion defectives in a process, was known as the fraction defectives chart. The term “nonconforming” became more commonly used in the literature in place of the term “defective”; however, since around 1980, when liability suits and awards became more common. The term “defective” was thought to convey a negative connotation, as if such a unit was unsafe or dangerous to use. Therefore, the term “nonconforming,” because it simply means that the unit in question is not meeting the chosen specification, became the accepted term. We use the term “defective” here for its easy readability but it is used to mean the same as “nonconforming.”
X̄ and R Charts
Published in Roger W. Berger, Thomas Hart, Statistical Process Control, 2020
There are four primary ways to chart attribute data. First, for samples of not necessarily equal size the p chart is used to track the proportion of units nonconforming. Second, for samples of constant size the np chart is used to indicate the number of units nonconforming per sample. Next, for samples of constant size the c chart can be used to follow the number of defects per lot. Finally, the u chart is used to note the number of defects per unit from samples not necessarily of constant size. The next section will introduce and briefly cover the application of each of these charts.
Monitoring proportions with two components of common cause variation
Published in Journal of Quality Technology, 2022
Rob Goedhart, William H. Woodall
The p-chart is used to detect special causes by monitoring the fraction of nonconforming products. An important assumption for the construction of this chart is that the data should follow a binomial distribution with a constant in-control parameter. While violation of this assumption does not necessarily lead to issues directly, performance issues of the chart become more noticeable if sample sizes are very large, as recognized by Alwan and Roberts (1995), Heimann (1996), Laney (2002) and Woodall (1997), among others. As an illustrative example we consider nonconformance data from 20 consecutive weeks of hospital emergency department data records, as available in Table 7.7 of Montgomery (2020) and displayed in our Table 1. In Figure 1 the fraction nonconforming is plotted, along with the classic p-chart control limits (see e.g. Montgomery (2020)).
Digital Twin simulation models: a validation method based on machine learning and control charts
Published in International Journal of Production Research, 2023
Carlos Henrique dos Santos, Afonso Teberga Campos, José Arnaldo Barra Montevechi, Rafael de Carvalho Miranda, Antonio Fernando Branco Costa
The p-chart control limits, called ‘Upper control limit’ (UCL) and ‘Lower control limit’ (LCL), and the Center Line (CL) depend on the proportion ‘p’ of a characteristic evaluated considering a given dataset. In this case, details of their calculations can be consulted in the work of Abbas et al. (2019). It is important to highlight that a control chart study is divided into two main phases. Phase I corresponds to the UCL, LCL, and CL determination, while phase II is dedicated to process monitoring (Chukhrova and Johannssen 2019). Figure 3 illustrates a typical p-chart.