These tests are designed to detect patterns measurement (e.g., sample
means) that may indicate that the process is out of control. In quality
control charting, when a sample point (e.g., a mean in an X-bar chart)
falls outside the control lines, you have reason to believe that the process
may no longer be in control. In addition, you 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 Charts 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 (

As the sigma control limits for quality control charts, 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.

For additional information, see Runs Tests and Assignable Causes and Actions.