Interpretation & Calculations
What is a run test? The Run Tests, developed by Western Electric with some improvements by Nelson, apply statistical tests to determine if there are any patterns or trends in the plotted points. Some of the patterns are due to process shifts, while others are due to sampling errors, inconsistent with the base premise of Rational Subgrouping . Although these tests, listed below, seem difficult to apply, SPC software will conveniently apply the tests for quick interpretation.
The statistical basis of the run tests is simply that if the subgroups are truly from the stated distribution, and independent of one another, then there will not be any pattern to the points. These tests are applied without regard to the selected control limit ordinates (number of sigmas). Likewise, whether a point is out of control or not depends solely on the control limit ordinate, not on whether it responds to a run test. The run tests do, however, increase the power of the control chart (the likelihood that shifts in the process are detected with each subgroup), but also provide an increased false alarm rate. The number of plotted points specified in a specific run test rule (e.g. six consecutive points) is specifically defined to provide a minimal false alarm rate and should not be adjusted without consideration for that false alarm rate.
Keep in mind that where the Run Test is violated does not always indicate when the process shift occurs. For example, when Run Test 2 is violated, the shift may have occurred nine points (or more, or less) prior to the point which violated the Run Test. As another example, a process may be in control and not in violation of any Run Tests for a period of time, say 50 subgroups. Then the process average shifts upward. As more and more subgroups are added at the new level, subgroups in the original 50 subgroups will start violating Run Tests or control limits, since these points now show an unnatural pattern relative to the combined distributions of the two process levels.
These Run Tests are applied as written to most charts; however, when non-normal distributions are used for Individual-X charts, the average is replaced with the median, and zones are defined to provide the same probabilities as the normal curve at the stated sigma level. For example, Run Test 1 is interpreted as any point in the .135 % tails (99.73% within the control limits), even though this would probably not be +- 3 sigma for a non-normal distribution.
Run Tests 1, 2, 5, and 6 are applied to the upper and lower halves of the chart (i.e. above and below the chart center line) separately. Run Tests 3, 4, 7 and 8 are applied to the whole chart.
Run Tests 1, 2, 3, and 4 are applied to X-Bar, Individual-X, Run (except run test 1), Range, Moving Range, Sigma, p, u, Np and c charts separately. Run Tests 5, 6, 7 and 8 are applied only to X-Bar and Individual-X charts.
None of the Run Tests are applied to Moving Average, CuSum or EWMA charts, or the Range, Moving Range, or Sigma charts used in conjunction with the Moving Average, CuSum or EWMA charts.
Run Test 1: (Western Electric: point beyond three sigma) an indication that the process mean has shifted.
Run Test 2: (Nelson: nine consecutive points same side of average (note: Western Electric uses eight consecutive points same side of average)) an indication that the process mean has shifted.
Run Test 3: (Nelson: six consecutive points increasing or decreasing) an indication that the process mean has shifted (a trend).
Run Test 4: (Nelson: fourteen consecutive points alternating up and down) an indication of sampling from multi-stream process (subgroups alternate between two or more process levels).
Run Test 5: (Western Electric: two out of three consecutive points beyond two sigma) an indication that the process mean has shifted.
Run Test 6: (Western Electric: four out of five consecutive points beyond one sigma) an indication that the process mean has shifted.
Run Test 7: (Western Electric: fifteen consecutive points between plus one sigma and minus one sigma) an indication of stratification in sampling (multi-stream sampling within a subgroup). See also: Run Test 7
Run Test 8: (Western Electric: eight consecutive points beyond plus one sigma and minus one sigma (both sides of center)) an indication of sampling from a mixture (multi-stream sampling, subgroups on each side of center from separate distributions).
See also: Control chart interpretation
Learn more about the SPC principles and tools for process improvement in Statistical Process Control Demystified (2011, McGraw-Hill) by Paul Keller, in his online SPC Concepts short course (only $39), or his online SPC certification course ($350) or online Green Belt certification course ($499).