Western Electric run rules

Four pattern rules that flag non-random behaviour on a Shewhart chart even when no point crosses the 3-sigma limits.

z_i = \frac{x_i - \text{CL}}{\sigma_{\text{zone}}}, \qquad \sigma_{\text{zone}} = \frac{\text{UCL} - \text{CL}}{3}

Zones used by the rules

$x_i$: plotted value (subgroup mean, proportion, count)
$\text{CL}, \text{UCL}$: centre line and upper control limit
$z_i$: position of the point in sigma units: zone C within 1, B between 1 and 2, A between 2 and 3

When to use

To increase a control chart's sensitivity to small sustained shifts. The four rules:

one point beyond 3σ (outside the limits);
2 of 3 consecutive points beyond 2σ on the same side;
4 of 5 consecutive points beyond 1σ on the same side;
8 consecutive points on the same side of the centre line.

The Nelson rules add patterns such as 6 points steadily increasing or decreasing and 14 alternating up and down.

Assumptions

Same as the chart: independent points, approximately symmetric distribution around the centre line in control.
The engine flags the point that completes each pattern; for rules 2 and 3 that point must itself be beyond the zone boundary. Points exactly on the centre line break a same-side run.

How to interpret

A flag is a signal to look for a cause (new material, a shift change, a worn tool), not proof of one.
Each added rule raises the false-alarm rate; with all four, false alarms are noticeably more frequent than with rule 1 alone.

Common mistakes

Applying zone rules to autocorrelated data (runs are expected) or to strongly skewed attribute charts with truncated limits.
Inventing new rules after seeing the chart.

Example

Centre 0, limits ±3 (σ_zone = 1), values 0.5, −0.4, 2.3, 0.1, 2.4, 0.2, 1.2, 1.4, 1.1, 1.6. No point is beyond 3 (rule 1 silent). Rule 2 fires at point 5 (2.3 and 2.4 among points 3–5); rule 3 fires at points 9 and 10 (four of the last five above 1); rule 4 fires at point 10 (points 3–10 all above the centre line).