Western Electric rules

The Western Electric Rules are general rules for detecting "out-of-control" or non-random conditions for data plotted on a run chart (plot of data as a function of time). Statistically calculated upper and lower control limits, when placed on a run chart, transform it into a control chart. The upper and lower control limits are located plus and minus 3 standard deviations respectively from the mean value or centerline of the plotted data. Locations of the data points and their trends with respect to the control limits and centerline value indicate whether or not to flag the possibility that a non-random signal in the data is present, with respect to the rules.

Design
The Western Electric Rules were designed under the guidance of a committee appointed by the engineering staff of the manufacturing division of the Western Electric Company. The rules appeared in the first edition of the Western Electric Handbook, in 1956. This book has evolved into the AT&T Statistical Quality Control Handbook.

The rules are based on the idea of splitting the chart into areas above and below the centerline (the point estimate of the central tendency of the process), and into zones on each side of the centerline. The zones are sometimes referred to as Zone A through Zone C, and sometimes as the one sigma zone, the two sigma zone, etc. Zone C is the region within 1 sigma of the centerline. Zone B is the region between 1 and 2 sigma of the centerline. Zone A is the region from 2 to 3 sigma from the centerline.

Basic rules
There are 4 basic rules with respect to how the data is situated on a control chart to indicate if it is not in statistical control (i.e. special causes of variation are present instead of random sources). These are as follows:


 * Rule 1: Any single data point falls outside the 3-sigma limit from the centerline (i.e., any point falls outside Zone A, beyond either the upper or lower control limit);


 * Rule 2: Two out of three consecutive points fall beyond the 2-sigma limit (in zone A or beyond), on the same side of the centerline;


 * Rule 3: Four out of five consecutive points fall beyond the 1-sigma limit (in Zone B or beyond), on the same side of the centerline;


 * Rule 4: Nine consecutive points fall on the same side of the centerline (in Zone C or beyond);

 If the data satisfy any of these conditions then it can be said that special cause(s) of variation in the data may be present. Each of the rules has about the same probability of occurrence under the assumption that the process is under control and the measured quantity is normally distributed (approximately 3 in 1000). This still means that any long-running control chart has a significant probability of false positives.

These tests apply to X-bar charts where the data are expected to behave symmetrically about the centerline. There are additional guidelines for control charts where the control limits are not symmetrical, as one would expect for an R chart or a p chart.

The above are considered patterns of instability in the Handbook. The Handbook recommends watching for additional patterns too. These patterns are looking for the features: stratification, mixture, systematic variable, tendency of one chart to follow another, and trends. Note that we look at both halves of a control chart when we look for these patterns, which is a change from the concepts above.

Deviations and trends
If fifteen consecutive points fall within one standard deviation of the centerline, on either side, this may indicate stratification. This means that the sampling is being done in some systematic way that is yielding a smaller variability within samples than we saw when we constructed the control chart originally. This is sometimes listed as a 'fifth rule' but is not a part of the original Western Electric Handbook's four tests.

If the chart shows eight consecutive points on *both* sides of the centerline, with none of the points falling in Zone C, then this is usually taken as an indication of sampling from a mixture.

If a long series of points alternate up-down-up-down without any interruption, then this is a symptom of a systematic variable. A pattern like this is often created by the way in which the data are grouped together to form sequence points. The original Handbook did not specify a length to check for when looking for such a sequence.

The tendency of one chart to follow another is only relevant when multiple control charts are being plotted. It is a situation where two or more variables show the same patterns in their control charts.

The Handbook says that trends may be "indicated by: (a) x's on one side of the chart followed by x's on the other. (b) a series of consecutive points without a change in direction." Sometimes a "sixth rule" is used, where one looks for a trend of six consecutive points increasing or decreasing.