Run rules¶

Control limits catch a single point thrown far out — a large, abrupt shift. They are deliberately blind to smaller, sustained patterns: a slow drift, a process that has settled on the wrong side of center, a creeping increase in spread. Run rules (also called zone tests or tests for special causes) close that gap by examining the pattern of points relative to the center line and the standard-deviation zones:

Zone	Distance from center line
A	between \(2\sigma\) and \(3\sigma\)
B	between \(1\sigma\) and \(2\sigma\)
C	within \(1\sigma\)

mfgQC ships two rule sets — Nelson (the default) and Western Electric — and reports every violation as a structured Violation object, not just a flag on a plot:

Field	Meaning
`point`	1-based index of the point that completes the pattern
`value`	the plotted value at that point
`chart`	`"location"` (the X̄ / individuals panel) or `"dispersion"` (R / S / MR)
`rule`	the rule id, e.g. `nelson_2`, `western_electric_4`
`description`	a plain-language statement of the criterion

Because the result object carries them structurally, a frontend or report builder consumes result.to_dict() or iterates result.violations — it never parses the report text. See Control charts for the chart families these rules run on.

Selecting the rule set¶

The rule set is chosen with the rules= argument on control_chart:

qc.control_chart(rules="nelson")            # default
qc.control_chart(rules="western_electric")

Any other value raises ValueError. The chosen set is recorded in the result's provenance step (params["rules"]), so an archived analysis records which tests it was judged against.

Rule 1 is shared

Both sets begin with the same headline test — one point beyond \(3\sigma\) — emitted as nelson_1 or western_electric_1 depending on the set. Everything after Rule 1 differs.

The Nelson rules¶

rules="nelson" applies the following. Each rule's "point" is the index that completes the pattern (e.g. the 9th point of a run, not the 1st).

Rule id	Detects	Exact criterion as implemented
`nelson_1`	A gross, single-point excursion	One point with \(\lvert z\rvert > 3\) (beyond \(3\sigma\))
`nelson_2`	A sustained shift in the mean	Nine points in a row on the same side of the center line
`nelson_3`	A trend / drift	Six points in a row, each strictly greater than (or each strictly less than) the one before
`nelson_4`	Over-control / systematic oscillation	Fourteen points in a row strictly alternating up and down
`nelson_5`	A shift, caught faster than Rule 2	Two out of three consecutive points beyond \(2\sigma\) on the same side, with the completing point itself beyond \(2\sigma\)
`nelson_6`	A smaller sustained shift	Four out of five consecutive points beyond \(1\sigma\) on the same side, with the completing point itself beyond \(1\sigma\)
`nelson_7`	Stratification (variance too small)	Fifteen points in a row all within \(1\sigma\) (Zone C, either side)
`nelson_8`	A mixture / bimodal pattern	Eight points in a row all beyond \(1\sigma\) (none in Zone C), on either side

Here \(z = (x - \text{CL})/\sigma\) is the standardized distance of a plotted point from the center line.

Implementation details worth knowing

Rules 5 and 6 require the completing point to clear the threshold itself. The code (_k_of_m_beyond) demands abs(z[i]) > thresh for the point that closes the window, not merely "\(k\) of the last \(m\)." This is slightly stricter than a literal "2 of 3 in Zone A" reading and prevents an in-control point from inheriting a signal.
The "\(k\) of \(m\) same side" count is taken per side, not netted: a window with two points above \(2\sigma\) and one below does not cancel — it is the count on the high side (\(\ge k\)) or the low side (\(\ge k\)) that triggers.
Rule 1 takes priority on shared points. If two rules would flag the same index, the first one assigned wins (the engine uses setdefault), and Rule 1 is assigned first. One index yields at most one Violation.

The Western Electric rules¶

rules="western_electric" applies four rules. The first three are the classic zone tests; the fourth is the run test.

Rule id	Detects	Exact criterion as implemented
`western_electric_1`	A gross, single-point excursion	One point beyond \(3\sigma\) (\(\lvert z\rvert > 3\))
`western_electric_2`	A shift toward one side	Two out of three consecutive points beyond \(2\sigma\) on the same side, with the completing point itself beyond \(2\sigma\)
`western_electric_3`	A smaller sustained shift	Four out of five consecutive points beyond \(1\sigma\) on the same side, with the completing point itself beyond \(1\sigma\)
`western_electric_4`	A sustained shift in the mean	Eight points in a row on the same side of the center line

The same per-side counting and "completing point clears the threshold" details from the Nelson section apply to western_electric_2 and western_electric_3 — they are the same engine (_k_of_m_beyond with \((k,m,\text{thresh}) = (2,3,2.0)\) and \((4,5,1.0)\)). The only differences between the two sets are: Nelson adds the trend, alternation, stratification and mixture tests (Rules 3, 4, 7, 8) and uses a 9-point run for the same-side test, whereas Western Electric uses an 8-point run.

How the engine detects patterns¶

All rules run off two derived series computed once per chart panel:

\(z = (x - \text{CL})/\sigma\) — the standardized distance, used by every zone test.
\(\text{sign}(x - \text{CL})\) — used by the same-side run tests.

For the location panel, \(\sigma\) is recovered from the limits as \(\sigma = (\text{UCL} - \text{CL})/3\), so the zones are always exactly thirds of the control band regardless of the chart family. The helpers are:

_runs_same_side(signs, length) — Rules 2 / 4 (WE): a window of length consecutive points all with the same sign.
_trend(x, length) — Rule 3 (Nelson): a window whose successive differences are all positive or all negative (strict monotonicity; equal consecutive values break the trend).
_alternating(x, length) — Rule 4 (Nelson): a window with every successive difference non-zero and the sign of each difference flipping relative to the last.
_k_of_m_beyond(z, k, m, thresh) — Rules 5 / 6 (Nelson) and 2 / 3 (WE).

Guardrail: rules are skipped when σ is not usable

If the recovered \(\sigma\) is non-positive or non-finite, _apply_rules returns no violations rather than dividing by zero or emitting garbage. This happens, for example, when every subgroup range is identical to zero. The control limits still print; the zone tests simply have nothing meaningful to test against.

Where each panel runs the rules¶

The full run-rule engine runs on the location panel (X̄ / individuals / standardized). The dispersion panel (R, S, or moving-range) is checked with Rule 1 only — a single point beyond its control limits — so that an inflated subgroup range surfaces even when the subgroup mean stays inside the X̄ limits. On the dispersion panel the violation is described as one point beyond control limits.

One-sided dispersion and attribute limits

For small subgroups the lower limit on the R, S, and moving-range panels is zero (the constants \(D_3\) / \(B_3\) are zero), so that panel is effectively one-sided — only an over-dispersed subgroup can violate Rule 1. The same clamping applies to attribute charts, where the lower control limit is clipped at zero (np.clip(cl - 3*sd, 0, None)); a count or proportion can never be flagged for going below zero. Attribute charts (p, np, c, u) run only the per-point beyond-limits test (their limits vary point-to-point with the sample size), not the zone/run pattern rules.

Worked example¶

A process running on target for the first twelve subgroups, then shifting up by about one standard deviation. The shift is too small to be an obvious \(3\sigma\) excursion every time, but the run and zone rules catch it. The data are five measurements per lot across twenty lots, so mfgQC infers an X̄-R chart.

import numpy as np, pandas as pd, mfgqc

rng = np.random.default_rng(11)
parts = []
for lot in range(1, 21):
    center = 50.0 if lot <= 12 else 50.9        # sustained shift after lot 12
    parts.append(pd.DataFrame({"x": rng.normal(center, 0.5, 5), "lot": lot}))
df = pd.concat(parts, ignore_index=True)
df["x"] = df["x"].round(3)

qc = mfgqc.load(df, measure="x", subgroup="lot", subgroup_size=5)
print(qc.control_chart(rules="nelson").report())

Control Chart: xbar_r (inferred); rules=nelson
==============================================
Xbar: CL=50.372  UCL=50.963  LCL=49.782
R: CL=1.0233  UCL=2.1634  LCL=0

Out-of-control signals: 14
  point 5 (location): nelson_1 - one point beyond 3 sigma
  point 6 (location): nelson_1 - one point beyond 3 sigma
  point 7 (location): nelson_6 - four of five points beyond 1 sigma (same side)
  point 8 (location): nelson_6 - four of five points beyond 1 sigma (same side)
  point 9 (location): nelson_2 - nine points in a row on one side of CL
  point 10 (location): nelson_1 - one point beyond 3 sigma
  point 12 (location): nelson_5 - two of three points beyond 2 sigma (same side)
  point 14 (location): nelson_5 - two of three points beyond 2 sigma (same side)
  point 15 (location): nelson_1 - one point beyond 3 sigma
  point 16 (location): nelson_1 - one point beyond 3 sigma
  point 17 (location): nelson_1 - one point beyond 3 sigma
  point 18 (location): nelson_6 - four of five points beyond 1 sigma (same side)
  point 19 (location): nelson_1 - one point beyond 3 sigma
  point 20 (location): nelson_5 - two of three points beyond 2 sigma (same side)

Assumption checks:
  [FAIL] independence (lag-1 autocorrelation): r=0.69, p=0.00203; n=20 [low power]

Recommendations:
  - Observations are autocorrelated (lag-1 r=0.69); control limits are unreliable - consider a time-series chart (e.g. EWMA).

The same data under Western Electric flags the same points; only the rule labels change (and western_electric_4 replaces nelson_2, firing one point sooner — an 8-point run instead of 9):

print(qc.control_chart(rules="western_electric").report())

Control Chart: xbar_r (inferred); rules=western_electric
========================================================
Xbar: CL=50.372  UCL=50.963  LCL=49.782
R: CL=1.0233  UCL=2.1634  LCL=0

Out-of-control signals: 14
  point 5 (location): western_electric_1 - one point beyond 3 sigma
  point 6 (location): western_electric_1 - one point beyond 3 sigma
  point 7 (location): western_electric_3 - four of five beyond 1 sigma (same side)
  point 8 (location): western_electric_3 - four of five beyond 1 sigma (same side)
  point 9 (location): western_electric_3 - four of five beyond 1 sigma (same side)
  point 10 (location): western_electric_1 - one point beyond 3 sigma
  point 12 (location): western_electric_2 - two of three beyond 2 sigma (same side)
  point 14 (location): western_electric_2 - two of three beyond 2 sigma (same side)
  point 15 (location): western_electric_1 - one point beyond 3 sigma
  point 16 (location): western_electric_1 - one point beyond 3 sigma
  point 17 (location): western_electric_1 - one point beyond 3 sigma
  point 18 (location): western_electric_3 - four of five beyond 1 sigma (same side)
  point 19 (location): western_electric_1 - one point beyond 3 sigma
  point 20 (location): western_electric_2 - two of three beyond 2 sigma (same side)

Read the assumption block alongside the signals

The independence check failed here because a sustained shift is a form of serial dependence — successive points stay on the same side. That is the special cause the rules are flagging, not an artifact. For genuinely autocorrelated in-control processes, the recommendation to use a time-series chart (EWMA) is the point: the standard \(3\sigma\) limits and these run rules both assume independent points.

Choosing between the sets¶

Western Electric is the older, leaner set: Rule 1 plus the three zone tests plus an 8-point run. It is the historical baseline and what many quality manuals and audits expect.
Nelson extends it with explicit tests for trend (Rule 3), over-control oscillation (Rule 4), stratification (Rule 7) and mixtures (Rule 8), and lengthens the same-side run to 9.

More tests means a higher chance of a false alarm on a genuinely in-control process; fewer tests means slower detection of subtle special causes. mfgQC does not choose for you and does not blend the sets — it applies exactly the set you name and tells you which rule fired. See Choosing a control chart for picking the chart the rules run on.

Source standards¶

The four zone/run tests follow Western Electric Co., Statistical Quality Control Handbook (1956) — the original "Western Electric rules."
The eight-rule set follows Nelson, L. S. (1984), "The Shewhart Control Chart — Tests for Special Causes," Journal of Quality Technology 16(4).

Both are listed in the Bibliography. The control-chart constants that set the limits the zones are measured against come from Montgomery; see Control charts.