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Process Capability: Engineering and Statistical Issues

The sample size here will generally be sufficiently large to skip the division by c₄, the “de-biasing” constant.

A comparison of the two process standard deviation estimates allows for an assessment of the model. If the model is correct, these two estimates should be about equal. If the procedurę 2 estimate is substantially larger than the procedurę 1 estimate, then this indicates that the ‘long run” variability is larger than the “short run” variability.

A decision must be madę on the reporting of the information. Again, we emphasize that, at this writing, both methods appear to be acceptable (according the proposed standard) for representing capability and the decision should be madę in the context of application. The procedurę 1 estimate indicates the potentially achievable capability, while the procedurę 2 estimate takes into account other sources of variability. It is unlikely that the procedurę 2 estimate will be substantially smaller than the procedurę 1 estimate.

The concept of rational sampling arises because the variation within the sample is different from that between the samples. To understand this, one needs to understand the process. Many products are produced on multiple-headed equipment or in batches. Examples include filling machines for liquids or solids, such as shampoo or laundry detergent. Similar situations are present in semiconductor manufacture, where wafers are often processed in fumaces or deposition units. The notion of a rational sample is based on information conceming the processing equipment. In other situations, where batching is not apparent, we often notę that the ‘fchort run” variability is substantially smaller than the ‘long run” variability. This can be due to many causes, including those involving materials and people.

If data are collected in rational samples, a model morę appropriate than [1] might be:

Observation + Mean + Noise (Between Samples) + Noise (Within Samples)

The within-sample noise includes the measurement variability which could also be introduced as a separate source of variation in a morę complex model. The model suggests that there are (at least) two components of variability. That is, the variability, ct² , of an observation can be represented as the sum of the between-samples variability, ct² (Between Samples), and the within-samples variability, ct² (Within Samples). That is,

a² (Observation) = a² (Between Samples) + a² (Within Samples)