## Example 1 pChart Analysis for Fraction Defective during Tapping

Consider a tapping operation on a P/M bracket. Suppose the measures of quality conformance of interest are the presence of threads (generally absent due to parts being inadvertently transferred into the finished parts bin without processing) and the size of the tapped threads as measured with a go/no-go plug (generally defective due to cutting tool wear).

To establish the control chart, rational samples of size n = 50 parts are drawn from production periodically (perhaps, each shift), and the sampled parts are inspected and classified as either defective (from either or both possible defects) or nondefective. The number of defectives, d, is recorded for each sample. The process characteristic of interest is the true process fraction defectivep\ Each sample result is converted to a fraction defective:

The data (fraction defective p) are plotted for at least 25 successive samples of size n = 50. The individual values for the sample fraction defective, p, vary considerably, and it is difficult to determine from the plot at this point if the variation about the average fraction defective, P. is solely due to the forces of common causes or special causes.

Control Limits for the p-Chart. It can be shown that for random sampling, under certain assumptions, the occurrence of the number of defectives, d, in the sample of size n is explained probabilistically by the binominal distribution. Because the sample fraction defective, p, is simply the number of defectives, d, divided by the sample size, n, the occurrence of values for p also follows the binominal distribution. Given k rational samples of size n, the true fraction defective, p', can be estimated by:

Equation 3 is more general because it is valid whether or not the sample size is the same for all samples. Equation 2 should be used only if the sample size, n, is the same for all k samples.

Therefore, given P. the control limits for the /?-chart are then given by:

Thus, only /Jhas to be calculated for at least 25 samples of size n to set up a p-chart. The binomial distribution is generally not symmetric in quality control applications and has a lower bound of p = 0. Sometimes the calculation for the lower control limit may yield a value of less than 0. In this case, a lower control limit of 0 is used.

c-Chart: Analysis for Number of Defects. The p-chart deals with the notion of a defective part or item where defective means that the part has at least one nonconformity or disqualifying defect. It must be recognized, however, that the incidence of any one of several possible nonconformities would qualify a part for defective status. A part with ten defects, any one of which makes it defective, is on equal footing with a part with only one defect in terms of being defective.

Often it is of interest to note every occurrence of every type of defect on a part and to chart the number of defects per sample (c). A sample may only be one part, particularly if interest is focusing on final inspection of an assembled product, or inspection may focus on one type of defect or multiple defects.

The probability law that governs the incidence of defects is known as the Poisson law or Poisson probability distribution, where c is the number of defects per sample. It is important that the opportunity space for defects to occur be constant from sample to sample. The Poisson distribution defines the probability of observing c defects in a sample where C is the average rate of occurrence of defects per sample.

Construction of c-Charts from Sample Data. The number of defects, c, arises probabilistically according to the Poisson distribution. One important property of the Poisson distribution is that the mean and variance are the same value. Then given c\ the true average number of defects per sample, the 3ij limits for the c-chart are given by:

Note that the standard deviation of the observed quantity c is the square root of c'. The Poisson distribution is a very simple probability model, being completely described by a single parameter c'.

When c' is unknown, it must be estimated from the data. For a collection of k samples, each with an observed number of defects c, the estimate of c' is:

Therefore, trial control limits for the c-chart can be established, with possible truncation of the lower control limit at zero, from:

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