How to use the control chart run rules detector
- Paste your measurements in time order, separated by commas, spaces or line breaks.
- Enter the center line and sigma of the process from a stable baseline, or choose to estimate them from the pasted data.
- Read how many points complete a pattern under each of the four rules.
- Look at the table of flagged points to see the value, how many sigmas it sits from the center and which rules it breaks. The CSV lists every point.
Formula
z = (value - center line) / sigma
rule 1: |z| > 3
rule 2: this point and at least one of the two before it have z > 2 (or all < -2) on the same side
rule 3: this point and at least three of the four before it have z > 1 (or all < -1) on the same side
rule 4: this point and the seven before it are all on the same side of the center line
These are the four Western Electric rules in their usual form. A point exactly on a boundary is not counted as beyond it, and a signal is shown on the point that completes the pattern.
Planted signals in 22 measurements
The center line is 10 and sigma is 0.2. Point 6 is 10.9, which is 4.50 sigma above the center, so it breaks rule 1. Point 8 (10.5, +2.50) has point 6 in its last three and point 9 (10.45, +2.25) has point 8, so both complete rule 2.
Points 13 to 20 all sit just above the center line, which is eight in a row, so point 20 breaks rule 4. Point 21 is 9.35, 3.25 sigma below the center, so it breaks rule 1 again. That is five points flagged out of 22.
What the rules detect
A single point beyond three sigma is rare if the process is stable, which is why rule 1 is the classic signal. The other rules look for patterns that are unlikely by chance even when no single point is extreme: two of three points far out on one side, four of five moderately out, and a long run on one side of the center. Each pattern points to a shift in the process, not to bad luck.
Because every rule has its own chance of a false alarm, using all four raises the overall rate of false signals. Treat a signal as a reason to look, not as proof of a cause. For the limits themselves, use the Control Chart Limits Calculator.
Choosing the center line and sigma
The center line and sigma should come from a period when the process was known to be stable. If you estimate them from the same data you are checking, a shift or trend widens the standard deviation and can hide the signals, so the page warns you when you choose that option. The sigma here is the standard deviation of individual values, not of sample averages.
To check the spread of a sample, see the Standard Deviation Calculator, and for single unusual values, the Outlier Calculator. To compare the process with specification limits, use the Process Capability Calculator.
Assumptions and limits
- Values are individual measurements in time order, and each one is compared with the same center line and sigma.
- The estimate from data uses the sample standard deviation of all the values and needs at least 8 of them.
- Zones are strict: a value exactly 1, 2 or 3 sigma from the center is not beyond that zone.
- Only the four Western Electric rules are checked. Other rule sets add or change rules.
- Up to 500 values are accepted, and nothing typed here leaves your browser.
Frequently asked questions
Why is a signal shown only on the last point of a pattern?
The pattern is not complete until its last point arrives. Marking that point tells you when you could first have noticed the signal, and avoids marking points that were fine on their own.
Should I use the data to estimate sigma?
Only if you have nothing better. A sigma from a stable baseline period is more reliable, because a shift in the data itself would otherwise inflate it and hide the signal.
Does a signal mean the process is out of control?
It means the pattern is unlikely if the process is stable at the center and sigma you entered. With four rules there will be occasional false alarms, so investigate before acting.
Why does a point on the line not count?
The rules say beyond a boundary, so a value equal to the boundary is within it. This also avoids rounding noise from decimals that sit on a boundary.