The analysis of a sequence of small samples taken at regular intervals from the output of an industrial production process, with the aim of ensuring that the output meets its required specification.
Standard statistical procedures expect large samples whereas here the samples are small. The solution suggested by Shewhart was the use of a control chart (Shewhart chart). The basis of this chart is a simple plot of the successive sample means, though often a simultaneous parallel chart of some measure of spread is maintained. The purpose of the chart is to give a quick visual indication of any trends in the production process (e.g. a drift in the mean), or an increase in variability (possibly a consequence of some problem in part of the process), without the need for advanced statistics. The diagram illustrates a combined mean chart (x̄- chart) and range chart (R-chart).
On the x̄-chart are two pairs of lines: warning lines and action lines. If any sample mean lies outside an action line, or if a succession of sample means lie outside the warning lines then production is stopped. One set of rules to have found favour are the Western Electric Rules which state that the process should be judged out of control if any of the following occur:
In some applications of quality control the discovery of a value outside the warning line will be a signal to inspect more often—this is
tightened inspection (as opposed to
normal inspection).
An alternative to the control chart is the cumulative sum chart (cusum chart). If x̄j denotes the j th sample mean and m denotes the expected mean of the production process, then, on the cusum chart, is plotted against k. If production is normal then the plot will be roughly horizontal, whereas a trend indicates a departure from the expected mean. See acceptance sampling.