Quality Control Introductory Overview - Out-of-Control Process: Runs Tests

As mentioned earlier in the introduction, when a sample point (e.g., mean in an X-bar chart) falls outside the control lines, one has reason to believe that the process may no longer be in control. In addition, one should look for systematic patterns of points (e.g., means) across samples, because such patterns may indicate that the process average has shifted. The Quality Control module will (optionally) perform the standard set of tests for such patterns; these tests are also sometimes referred to as AT&T runs rules (see AT&T, 1959) or tests for special causes (e.g., see Nelson, 1984, 1985; Grant and Leavenworth, 1980; Shirland, 1993). The term special or assignable causes as opposed to chance or common causes was used by Shewhart to distinguish between a process that is in control, with variation due to random (chance) causes only, from a process that is out of control, with variation that is due to some non-chance or special (assignable) factors (cf. Montgomery, 1991, p. 102).

As the sigma control limits discussed earlier, the runs rules are based on "Statistical" reasoning. For example, the probability of any sample mean in an X-bar control chart falling above the center line is equal to 0.5, provided 1) that the process is in control (i.e., that the center line value is equal to the population mean), 2) that consecutive sample means are independent (i.e., not auto-correlated), and 3) that the distribution of means follows the normal distribution. Simply stated, under those conditions there is a 50-50 chance that a mean will fall above or below the center line. Thus, the probability that two consecutive means will fall above the center line is equal to 0.5 times 0.5 = 0.25.

Accordingly, the probability that 9 consecutive samples (or a run of 9 samples) will fall on the same side of the center line is equal to 0.59 = .00195. Note that this is approximately the probability with which a sample mean can be expected to fall outside the 3-times sigma limits (given the normal distribution, and a process in control). Therefore, one could look for 9 consecutive sample means on the same side of the center line as another indication of an out-of-control condition. Refer to Duncan (1974) for details concerning the "Statistical" interpretation of the other (more complex) tests.

The Quality Control module will perform several such tests for special causes (runs tests) for X-bar, R, S, C, U, P, and Np charts, and highlight out-of-control runs in the chart.

Note: the Statistica S chart differs from the Montgomery formulas, even though the Montgomery text is referenced in Statistica Help. This discrepancy occurs because C4 is calculated more precisely in Statistica than by Montgomery, who used lookup tables.

The following is a description of the runs tests.

Zone A, B, C
Customarily, to define the runs tests, the area above and below the chart center line is divided into three "zones."

By default, Zone A is defined as the area between 2 and 3 times sigma above and below the center line; Zone B is defined as the area between 1 and 2 times sigma, and Zone C is defined as the area between the center line and 1 times sigma.

9 points in Zone C or beyond (on one side of central line)
If this test is positive (i.e., the pattern of 9 consecutive points on the same side of the center line is detected), then the process average has probably changed. Note that it is assumed that the distribution of the respective quality characteristic in the plot is symmetrical around the mean. This is, for example, not the case for R charts, S charts, or most attribute charts. However, this is still a useful test to alert the quality control engineer to potential shifts in the process. For example, successive samples with less-than-average variability may be worth investigating, since they may provide hints on how to decrease the variation in the process.
6 points in a row steadily increasing or decreasing
This test signals a drift in the process average. Often, such drift can be the result of tool wear, deteriorating maintenance, improvement in skill, etc. (Nelson, 1985).
14 points in a row alternating up and down
If this test is positive, it indicates that two systematically alternating causes are producing different results. For example, one may be using two alternating suppliers, or monitor the quality for two different (alternating) shifts.
2 out of 3 points in a row in Zone A or beyond
This test provides an "early warning" of a process shift. Note that the probability of a false-positive (test is positive but process is in control) for this test in X-bar charts is approximately 2%.
4 out of 5 points in a row in Zone B or beyond
Like the previous test, this test may be considered to be an "early warning indicator" of a potential process shift. The false-positive error rate for this test is also about 2%.
15 points in a row in Zone C (above and below the center line)
This test indicates a smaller variability than is expected (based on the current control limits).
8 points in a row in Zone B, A, or beyond, on either side of the center line (without points in Zone C)
This test indicates that different samples are affected by different factors, resulting in a bimodal distribution of means. This may happen, for example, if different samples in an X-bar chart where produced by one of two different machines, where one produces above average parts, and the other below average parts.