Shewhart control chart rules

Tests for special-cause variation determine when a process needs further investigation.

There are various tests that can be used in conjunction with a control chart to identify special-cause variation:
Test Rule Problem indicated
1 1 point is outside the control limits. A large shift.
2 8/9 points on the same size of the center line. A small sustained shift.
3 6 consecutive points are steadily increasing or decreasing. A trend or drift up or down.
4 14 consecutive points are alternating up and down. Non-random systematic variation.
5 2 out of 3 consecutive points are more than 2 sigmas from the center line in the same direction. A medium shift.
6 4 out of 5 consecutive points are more than 1 sigma from the center line in the same direction. A small shift.
7 15 consecutive points are within 1 sigma of the center line. Stratification.
8 8 consecutive points on either side of the center line with not within 1 sigma. A mixture pattern.

You should choose tests in advance of looking at the control chart based on your knowledge of the process. Applying test 1 to a Shewhart control chart for an in-control process with observations from a normal distribution leads to a false alarm once every 370 observations on average. Additional tests make the chart more sensitive to detecting special-cause variation, but also increases the chance of false alarms. For example, applying tests 1, 2, 5, 6 raises the false alarm rate to once every 91.75 observations.

Most tests beyond test 1 are only appropriate when trying to bring a process under control. Tests 2, 3, 5, and 6 detect small shifts once a process is under control although it is often preferable to using a combination of a Shewhart chart with test 1 for detection of large shifts and an EWMA or CUSUM chart for detecting smaller shifts and trends.

It can be useful to add zones at ±1, 2, and 3 sigmas to the control chart to help interpret patterns. For control charts with unequal subgroup sizes, the center line, control limits and zones may vary. In this case, the tests apply to a standardized control chart where the points are the number of standard deviation units from the center line. Such a control chart has a constant center line at 0, and upper and lower control limits of +3 and -3 respectively making patterns easier to spot.

Tests 1, 5, 6, 2 are defined by the Western Electric CO (1958) as the original 4 rules. Tests 1-8, with the modification of test 2 to from 8 to 9 points, are defined by Lloyd S. Nelson (1984). Variations of these eight rules with different run lengths and rule ordering are recommended by various other authors, one of the most popular in the book by (Douglas C. Montgomery, 2012).