9.5 Control Chart Analysis: Common Cause, Special Cause, and Run Rules
Key Takeaways
- Common cause variation is inherent to the process and requires a system change; special cause variation is external and requires investigation of the specific event.
- Tampering means adjusting a stable process in response to common-cause variation, and it increases variation rather than reducing it.
- The most commonly used run rules are the Western Electric rules, with rule 1 being a single point beyond three sigma.
- Each additional run rule raises sensitivity to real shifts and simultaneously raises the false alarm rate.
- A chart with no reaction plan is a decoration; every rule must map to a documented out-of-control action.
The two kinds of variation
| Common cause | Special cause | |
|---|---|---|
| Also called | Chance, random, systemic | Assignable, sporadic |
| Source | Inherent in the process design and system | An event outside the usual system |
| Pattern on the chart | Random within the limits | A point outside a limit, or a non-random pattern |
| Share of total variation | Typically 80% to 95% in a stable process | The remainder |
| Who can fix it | Management, through a system change | Local personnel, at the process |
| Correct response | Do not react to individual points; change the system | Investigate the specific event and prevent recurrence |
The distinction matters because the two require opposite responses, and mixing them up produces the two classic errors:
- Tampering (Deming's Rule 1 error): treating common-cause variation as if it were special and adjusting the process. This is not merely wasted effort -- it demonstrably increases variation. The funnel experiment demonstrates it: adjusting after every observation roughly doubles the standard deviation of the output compared with leaving the process alone.
- Ignoring a signal (Rule 2 error): treating a genuine special cause as ordinary noise, so the event recurs.
A control chart's entire purpose is to make this distinction objectively rather than by judgment.
The Western Electric run rules
Zones are defined by dividing the distance between the centre line and each control limit into thirds: zone C is within $1\sigma$, zone B between $1\sigma$ and $2\sigma$, zone A between $2\sigma$ and $3\sigma$.
| Rule | Condition | Detects |
|---|---|---|
| 1 | 1 point beyond zone A (outside $3\sigma$) | Large sudden shift, outlier |
| 2 | 2 of 3 consecutive points in zone A or beyond, same side | Moderate shift |
| 3 | 4 of 5 consecutive points in zone B or beyond, same side | Small to moderate shift |
| 4 | 8 consecutive points on one side of the centre line | Small sustained shift |
Nelson's rules extend the set with four more patterns:
| Rule | Condition | Detects |
|---|---|---|
| 5 | 6 consecutive points steadily increasing or decreasing | Trend: tool wear, fouling, drift |
| 6 | 15 consecutive points in zone C (both sides) | Stratification, or limits computed too wide |
| 7 | 14 consecutive points alternating up and down | Over-adjustment, or two alternating sources |
| 8 | 8 consecutive points on both sides with none in zone C | Mixture of two distinct processes |
Reading the unusual ones
Rule 6, hugging the centre line, is counter-intuitive because the chart looks excellent. It usually means either the control limits were computed from data containing an extra source of variation (so they are too wide), or the subgroups are stratified -- for example, each subgroup deliberately contains one part from each of four machines, so between-machine variation is averaged away inside the subgroup instead of appearing between subgroups.
Rule 8, avoiding the centre line, is the mirror image and usually indicates a mixture: two processes with different means being charted together, such as two shifts, two suppliers, or two cavities of a mould.
Rule 7, systematic alternation, most often indicates over-adjustment -- an operator correcting after every measurement, which is tampering made visible.
The false alarm trade-off
Each rule has its own false alarm probability, and adding rules multiplies the risk of a false signal on a perfectly stable process.
| Rule set | Approximate in-control ARL | Comment |
|---|---|---|
| Rule 1 only | 370 | The classic 3-sigma chart |
| Rules 1-4 | about 92 | Considerably more sensitive to small shifts |
| Rules 1-8 | about 60 | Very sensitive; frequent false alarms |
Average run length of 60 means a stable process signals roughly every 60 subgroups for no reason. On a chart plotted hourly, that is a false alarm every day and a half.
The practical guidance:
- Use rule 1 alone plus rule 4 on high-volume charts where false alarms are expensive to investigate.
- Add rules 2, 3, and 5 where small shifts matter and investigation is cheap.
- Apply the full set only on critical characteristics.
- Decide the rule set in advance and write it into the control plan. Adding rules after seeing a pattern you dislike is not statistical process control.
The out-of-control action plan
A control chart without a documented reaction plan is a decoration. The OCAP specifies, for each signal:
| Element | Content |
|---|---|
| Trigger | Which rule fired, on which characteristic |
| Immediate containment | Stop, quarantine, segregate since the last good check |
| Who reacts | Named role, with the authority to stop |
| Investigation steps | An ordered checklist of the likely causes to check |
| Disposition | What happens to the material produced since the previous in-control point |
| Documentation | Where the reaction, the cause, and the corrective action are recorded |
| Escalation | Who is called and when, if the cause is not found within a defined time |
Interpreting a chart in practice
Work through the questions in order:
- Is the range or sigma chart in control? Always read the variation chart first. If the spread is unstable, the estimate of sigma used to build the X-bar limits is invalid, so the X-bar chart cannot be interpreted at all.
- Are there points outside the limits? Rule 1.
- Are there non-random patterns? Runs, trends, cycles, mixtures.
- Was the process stable over the period the limits were computed from? Limits calculated from unstable data are too wide and hide subsequent signals.
- Are the limits current? After a deliberate process change, recompute rather than continuing against limits that describe the old process.
Two final cautions. Specification limits never belong on a control chart -- the chart describes what the process does, the specification describes what is required, and the distribution of subgroup means is narrower than the distribution of individuals by a factor of $\sqrt{n}$. And a process in control is not necessarily capable: stability and capability are separate questions, answered by the control chart and the capability index respectively.
An operator adjusts a stable process after every measurement that falls away from target. What is happening, and what is the effect?
A control chart shows 15 consecutive points hugging the centre line within one sigma. Why is this a signal rather than an indication of excellent control?
When reading a paired X-bar and R chart, why must the R chart be interpreted first?