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Western Electric rules
Decision rules for interpreting control-chart data
Decision rules for interpreting control-chart data
The Western Electric rules are decision rules in statistical process control for detecting out-of-control or non-random conditions on control charts.{{Citation | publication-date = 2005
Motivation
The rules attempt to distinguish unnatural patterns from natural patterns based on several criteria:{{Citation The absence of points near the centerline is identified as a mixture pattern.
The absence of points near the control limits is identified as a stratification pattern.
The presence of points outside the control limits is identified as an instability pattern.
Other unnatural patterns are categorized as systematic (autocorrelative), repetition, or trend patterns.
This classification divides the chart of observations into zones, measured in units of standard deviation (σ) between the centerline and control limits, as follows:
| Zone | Region |
|---|---|
| Zone A | Between 2σ from the centerline and the control limit (3σ) |
| Zone B | Between 1σ and 2σ from the centerline |
| Zone C | Within 1σ of the centerline |
Zones A, B, and C are sometimes called the three sigma zone, two sigma zone, and one sigma zone, respectively.
Zone rules
An important aspect of the Western Electric rules are the zone rules, designed to detect process instability, and the presence of assignable causes.{{Citation Data sets of observations are appraised by four basic rules, that categorize the occurrence of data samples in a set of zones defined by multiples of the standard deviation.
| Description | Chart example | |
|---|---|---|
| Rule 1 | Any single data point falls outside the 3σ-limit from the centerline (i.e., any point that falls outside Zone A, beyond either the upper or lower control limit) | [[File:Rule 1 - Western electric control chart.svg]] |
| Rule 2 | Two out of three consecutive points fall beyond the 2σ-limit (in zone A or beyond), on the same side of the centerline | [[File:Rule 2 - Western electric control chart.svg]] |
| Rule 3 | Four out of five consecutive points fall beyond the 1σ-limit (in zone B or beyond), on the same side of the centerline | [[File:Rule 3 - Western electric control chart.svg]] |
| Rule 4 | Eight consecutive points fall on the same side of the centerline (in zone C or beyond) | [[File:Rule 4 - Western electric control chart.svg]] |
These rules are evaluated for one side of the center line (one half of the control band) at a time (e.g., first the centerline to the upper control limit, then the centerline to the lower control limit).
Data satisfying any of these conditions as indicated by the control chart provide the justification for investigating the process to discover whether assignable causes are present and can be removed. Note that there is always a possibility of false positives: Assuming observations are normally distributed, one expects Rule 1 to be triggered by chance one out of every 370 observations on average. The false alarm rate rises to one out of every 91.75 observations when evaluating all four rules.{{Citation
Asymmetric control limits
The zone rules presented above apply to control charts with symmetric control limits. The handbook provides additional guidelines for control charts where the control limits are not symmetrical, as for R charts and p-charts.
For \bar x and R charts (which plot the behavior of the subgroup range), the Handbook recommends using the zone rules above for subgroups of sufficient size (five or more). For small sample subgroups, the Handbook recommends:
| Small-sample R chart rules |
|---|
| Any single data point falling above the +3σ limit |
| Two consecutive points falling above the +2σ-limit (in the upper zone A or above) |
| Three consecutive points falling above the +1σ-limit (in the upper zone B or above) |
| Seven consecutive points falling above the centerline (in the upper zone C or above) |
| Ten consecutive points falling below the centerline (in the lower zone C or below) |
| Six consecutive points falling below the -1σ-limit (in the lower zone B or below) |
| Four consecutive points falling below the -2σ-limit (in the lower zone A) |
For other control charts based on skewed distributions, the Handbook recommends:
- Finding the probabilities associated with each zone
- Devising tests that should result in the desired sensitivity to the presence of assignable causes
Other unnatural patterns
The handbook also identifies patterns that require consideration of both the upper and lower halves of the control chart together for identification:
| Pattern | Description | Rule |
|---|---|---|
| Stratification | Variation small relative to control limits | Fifteen consecutive points fall within ±1σ (zone C) |
| Mixture | The tendency to avoid the centerline | Eight consecutive points on both sides of the centerline with no points falling in zone C. |
| Systematic | Negative autocorrelation—a long series of observations that alternate high-low-high-low | (No rule given) |
| Repetition | The tendency of one chart to follow the same pattern as its predecessor | (No rule given) |
| Trend | Sustained drift or shift in the long-term mean | A series of out-of-control points in the lower zones followed by a series of out-of-control points in the upper zones or vice versaA series of points without a change in direction |
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
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