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Process Capability: Engineering and Statistical Issues

Cramer-von Mises and Anderson-Darling tests. The %² goodness-of-fit test is somewhat problematic sińce, like using a histogram, different conclusions can be drawn from the same set of data depending simply on one's choice for the data intervals or bins and their widths. For further details on these procedures, see d'Agostino and Stephens (1986).

Standard Errors of Process Capability Indices

We will denote the estimates of the process mean and standard deviation as X and S, respectively. Formally, we notę that to obtain an unbiased estimate of a based on a sample standard deviation reąuires dividing S by the c_Ą correction factor. Since c₄ approaches 1 rapidly with increasing n,

we will not concem ourselves with this issue at this point. However, we further notę that the use of unbiased estimates of the process parameters does not necessarily yield an unbiased estimate for the capability index. Hence, we will calculate our estimates of the process capability indices by simply substituting estimates of the process parameters into the formulas for the process capability indices given above. We will denote the estimated indices as C_p, C_pk, etc.

Kushler and Hurley (1992) summarized the standard errors of many capability indices. The table below provides a summary of the standard errors of the process parameters and some of the capability indices. We present these in a form that we think is helpful in understanding them (Table 1).

These approximated standard errors are useful in deciding whether a specified sample size is sufficient to provide the desired precision for reporting a capability index. For example, physical scientists often report their results and give error bounds of one or two standard errors to represent the potential variability. Limits on Shewhart control charts are set at ±3 standard errors.

Estimates of Process Capability Indices are Random Quantities

To further illustrate the notion that estimated process capability indices are random quantities, consider the following study based on a Monte Carlo simulation. Figures 5 and 6 are each histograms of 1000 Ć_p values

obtained through Monte Carlo simulation. A total of N = 1000 samples, each of size n=30, were generated from a Computer simulation of a stable normal process. A C_p value was recorded for each of the 1000 samples. The two processes that were simulated had true C_p values of 1.00 and 1.33.