Nelson 8 rules
Overview
The Nelson Rules are a set of statistical rules used with control charts to detect non-random patterns in process data. They help identify potential process instability even when data points remain within the traditional control limits.
The rules evaluate observations relative to the process mean (centre line), the standard deviation (σ) and the upper and lower control limits (typically ±3σ).
Cloud QC uses the CRM’s certified value and standard deviation when “Use CRM provided statistics” is ticked; otherwise it uses the mean and standard deviation of the results on the chart. Tick “Run Nelson 8 rules” on the Shewhart chart for a PASS/FAIL table of the rules; click a rule in the table to highlight its points.
A violation of any Nelson Rule indicates that the process should be investigated for a potential special cause.
When to use
Nelson Rules are commonly used for statistical process control (SPC), laboratory quality monitoring, and analytical chemistry and assay control.
Control zones
| Zone | Range |
|---|---|
| Zone C | Within ±1σ |
| Zone B | Between 1σ and 2σ |
| Zone A | Between 2σ and 3σ |
| Out of control | Beyond ±3σ |
The eight rules
| Rule | Condition | Interpretation |
|---|---|---|
| 1 | One point beyond 3σ: xi > μ + 3σ or xi < μ − 3σ | A very unlikely event under normal variation, suggesting a special cause |
| 2 | Nine consecutive points on the same side of the mean | A sustained process shift |
| 3 | Six consecutive points steadily increasing or decreasing | Gradual drift in the process |
| 4 | Fourteen consecutive points alternating up and down (e.g. 101, 99, 102, 98, 103, 97 …) | Over-adjustment or process tampering |
| 5 | Two of three consecutive points beyond 2σ on the same side | Early indication of a process shift |
| 6 | Four of five consecutive points beyond 1σ on the same side | Strong evidence of a process mean change |
| 7 | Fifteen consecutive points within ±1σ | Reduced variation, data stratification, an incorrect sampling method or calculation errors |
| 8 | Eight consecutive points outside ±1σ (none in Zone C), with points on both sides of the mean | Unusually high variation or a mixture of different process populations |
Worked example
Mean μ = 100 and standard deviation σ = 2, so ±1σ = 102 and 98, ±2σ = 104 and 96, and ±3σ = 106 and 94.
| Obs | Value | Zone | Nelson rule triggered |
|---|---|---|---|
| 1 | 100.5 | C | None |
| 2 | 101.2 | C | None |
| 3 | 102.8 | B | None |
| 4 | 103.1 | B | None |
| 5 | 104.6 | A | None (one point beyond 2σ) |
| 6 | 105.0 | A | Rules 3, 5 and 6 triggered |
| 7 | 105.4 | A | Rules 3, 5 and 6 continue |
| 8 | 105.7 | A | Rules 3, 5 and 6 continue |
| 9 | 106.3 | Above 3σ | Rules 1 and 2 triggered; 3, 5 and 6 continue |
| 10 | 106.8 | Above 3σ | Rules 1, 2, 3, 5 and 6 continue |
- Rule 5: of Observations 4–6 (103.1, 104.6, 105.0), Observations 5 and 6 are above +2σ (104), so Rule 5 is violated.
- Rule 6: Observations 5–9 all exceed +1σ (102). Rule 6 is first met earlier, at Observation 6: four of Observations 2–6 are above 102.
- Rule 1: Observation 9 (106.3) and Observation 10 (106.8) both exceed the +3σ limit of 106, so Rule 1 is violated.
- Rule 3: Observations 1–6 rise steadily from 100.5 to 105.0 (six points in a row increasing), so Rule 3 is violated at Observation 6 and continues to Observation 10.
- Rule 2: Observations 1–9 are all above the mean of 100 (nine points in a row on the same side), so Rule 2 is violated at Observation 9 and continues at Observation 10.
Summary
The Nelson Rules provide a structured method for detecting unusual process behaviour that may not breach standard control limits. Each rule identifies a different pattern such as shifts, trends, cycles, excessive clustering, or increasing variability. Applying the rules alongside traditional Shewhart control charts improves sensitivity to early process changes and helps distinguish common-cause variation from special-cause variation.