---
name: control-chart
model: sonnet
description: Build and interpret statistical process control (SPC) charts — distinguishing common cause from special cause variation using Western Electric rules, calculating control limits, and computing Cp and Cpk process capability indices — to determine whether a process is stable and whether it is capable of meeting specifications. Use when someone says "control chart", "SPC", "statistical process control", "is this process in control", "process capability", "Cp", "Cpk", "common cause", "special cause", "are these results normal or is something wrong", "control limits", "we're trying to understand if our variation is random", or shares process measurement data over time.
license: MIT
---

# Control Chart and Process Capability

The most expensive mistake in process management is treating common cause variation as if it were a special cause — adjusting a stable process in response to normal noise, which adds variation instead of reducing it. The second most expensive mistake is the reverse: ignoring a genuine signal because it "seems within range." Statistical process control (SPC) exists to make this distinction objectively. This skill builds control charts, applies the Western Electric detection rules, and calculates process capability.

## When to use / when not to

Use it when you have a series of process measurements over time and want to know whether the process is stable, whether something has changed, and whether the process is capable of meeting specification limits.

Don't use it to investigate why a special cause occurred (→ `root-cause`), to assess what failure modes could generate out-of-control signals (→ `fmea-builder`), or to calculate OEE losses from a process running outside its specification (→ `oee-from-csv`).

## The one concept that everything else depends on

**Common cause variation** is the inherent, random variation of a stable process — noise produced by the many small, consistent sources of variation that are always present. It is predictable in aggregate. You cannot eliminate common cause variation by reacting to individual points; you can only reduce it by changing the process itself.

**Special cause variation** is a signal — a pattern that could not plausibly arise from random noise. Something has changed: a new material lot, a different operator, equipment wear, a process shift. Special causes are investigated and removed. Reacting to common cause variation as if it were a special cause is called **tampering** — it adds variation and degrades the process.

**This distinction is the entire point of a control chart.** Every other feature — the limits, the rules, the capability indices — exists to support it.

## Chart selection

| Data type | Subgroup size | Chart |
|---|---|---|
| Continuous measurement (dimension, weight, time) | n = 1 | Individuals and Moving Range (I-MR) |
| Continuous measurement | n = 2–9 | X-bar and R |
| Continuous measurement | n ≥ 10 | X-bar and S |
| Count of defectives (pass/fail, % defective) | Variable | p-chart or np-chart |
| Count of defects per unit | Variable | c-chart or u-chart |

When in doubt with small datasets or individual measurements, default to I-MR. Say which chart you're using and why in one sentence.

## Control limits — calculation and meaning

Control limits are calculated from the data — they are not specification limits, and they are not set by management preference. Setting control limits equal to or near specification limits is a common and serious error; state this explicitly if the user suggests it.

**For I-MR charts:**

```
Moving Range (MR_i) = |X_i − X_{i−1}|  for i = 2 to n

X̄ = mean of all individual values
MR̄ = mean of all moving ranges

Upper Control Limit (UCL_X) = X̄ + 2.66 × MR̄
Lower Control Limit (LCL_X) = X̄ − 2.66 × MR̄  (floor at 0 if negative and data is non-negative)
UCL_MR = 3.267 × MR̄
LCL_MR = 0
```

**For X-bar and R charts** (use standard d2, A2, D3, D4 factors for the appropriate subgroup size — document the subgroup size used).

State the number of data points used to calculate control limits. Control limits calculated from fewer than 20–25 points are provisional — note this explicitly.

## Western Electric detection rules

These four rules detect special causes. A point or pattern meeting any rule is a signal — investigate it. Rule 1 comes from the Western Electric handbook (1956); Rules 2 to 4 are tests from Lloyd Nelson (1984). Practitioners often call the set "Western Electric rules," and this skill's report uses that heading.

**Rule 1 — Point beyond 3σ:** One point falls outside the 3-sigma control limit (above UCL or below LCL). The most fundamental rule. Probability of false alarm from random variation: 0.27%.

**Rule 2 — Nine consecutive on one side:** Nine or more consecutive points on the same side of the centerline (all above or all below X̄). Signals a sustained shift in the process mean.

**Rule 3 — Six consecutive trending:** Six or more consecutive points consistently increasing or consistently decreasing. Signals a drift — gradual wear, temperature change, material depletion, operator fatigue.

**Rule 4 — Fourteen alternating:** Fourteen or more consecutive points alternating up-down-up-down. Often signals over-adjustment (tampering), two alternating process streams (two machines, two shifts, two suppliers), or a measurement system problem.

Apply all four rules to every chart. Flag each violation with the rule number and the point(s) involved. Do not apply rules without stating which points triggered them.

**The tampering rule — never adjust a process showing only common cause variation.** If no Western Electric rule is triggered, the process is in statistical control. Adjusting it in response to an individual high or low point adds variation. State this plainly. If the user wants to reduce common cause variation, that requires a process improvement — not a reaction to individual data points.

## Process capability — Cp and Cpk

Capability indices are only meaningful for a **stable process** (in statistical control). Calculate capability only after confirming control. If the process is not in control, state that capability indices are not valid until special causes are removed.

```
Process sigma (σ) estimated from the control chart:
  For I-MR:  σ = MR̄ / 1.128
  For X-bar/R:  σ = R̄ / d2

Cp = (USL − LSL) / (6σ)
Cpk = min[(USL − X̄) / (3σ),  (X̄ − LSL) / (3σ)]
```

**Cp measures spread only** — whether the process variation fits within the specification window, assuming perfect centering. **Cpk measures centering** — whether the process is actually positioned within the window. Both are required. A process can have excellent Cp and poor Cpk if it is running off-center.

**Cpk thresholds:**

| Cpk | Interpretation |
|---|---|
| < 1.00 | Not capable — producing out-of-specification output |
| 1.00 – 1.33 | Marginal — technically capable but with little margin; vulnerable to shifts |
| > 1.33 | Capable — generally accepted as the minimum for production |
| > 1.67 | Highly capable — a common minimum for new safety-critical processes |

**When Cp >> Cpk:** the process has the spread to fit within spec but is running off-center. The improvement action is centering (adjust the target), not reducing variation.

**When Cp ≈ Cpk:** the process is centered but the spread is too large. The improvement action is reducing variation — which requires identifying and removing common causes, not reacting to individual points.

## How this skill works

**Open with one sentence the first time:** "Share the measurement data — values and sequence matter — along with the spec limits if you have them, and I'll build the control chart and tell you what it's showing."

1. **Read all data given.** Confirm data type (continuous/attribute), subgroup structure, and sequence.
2. **Calculate control limits** from the data, showing the formula and values used.
3. **Apply all four Western Electric rules.** Name every triggered rule and the specific points involved.
4. **State the stability conclusion explicitly:** in control (only common cause present) or out of control (special cause detected) — no ambiguity.
5. **Calculate Cp and Cpk** if spec limits are provided and the process is stable. If not stable, state capability analysis is deferred.
6. **State the capability conclusion** using the threshold table.
7. **Recommend the improvement path** — tamper warning if appropriate, centering recommendation if Cp >> Cpk, variation-reduction recommendation if Cp is low.
8. **Close** with the shared closing block, specific to this analysis.

Data-safety line: *"Share measurement values and sequence — leave out proprietary product names or customer-specific tolerances if those are sensitive; spec limits expressed as ± values are fine."*

## Quality bar

- Chart type is explicitly named with the rationale stated.
- Control limits are calculated from the data, not set to match spec limits — any suggestion to do otherwise is corrected.
- All four Western Electric rules are applied and each violation is identified by rule number and data point.
- The stability conclusion (in control / out of control) is stated explicitly — never left implicit.
- Cp and Cpk are both calculated when spec limits are available; neither is omitted.
- Cp and Cpk are interpreted together — the centering gap (Cp − Cpk) is called out when it is meaningful.
- Capability analysis is withheld or flagged as provisional if the process is not in statistical control.
- Any control limits calculated from fewer than 20 points are labeled provisional.
- The tamper warning is issued if the user proposes adjusting a process that shows only common cause variation.

## Toolkit contract

This skill keeps the ten promises in `docs/TOOLKIT-CONTRACT.md`. The ones that carry weight in every conversation, restated here so this file stands alone:

- **Before any paste**, in any setting, say: *"Leave out names of patients or employees, and anything your company treats as confidential; use roles or random codes, not names or initials."* In a healthcare or service setting add: *"Please don't paste protected health information — no patient identifiers, no chart numbers, no dates of birth or other dates finer than a year."*
- **Never invent a fact.** Anything not given, pasted, or reported is marked `[NEEDS GEMBA]` (go look, go ask) or `[ASSUMED — verify]`. Every number carries a source tag — `[observed]`, `[from system: …]`, `[user estimate]`, or `[NEEDS GEMBA]`. A number without a tag does not appear in the deliverable.
- **"Just draft it" always works.** Skip the questions, produce the deliverable now with placeholders where facts are missing, and point at the placeholders in the closing block. Never ask a question whose answer you won't use.
- **Mirror the user's vocabulary** once they have used it — unit / clinician / patient / turnaround, PDSA rather than PDCA — and never correct it.
- **Headcount framing.** If the ask is "how many people can we cut," answer the process or capacity question, then say once, plainly: *"These tools free capacity; what the organization does with freed capacity is a leadership decision."* No lecture, no refusal, and never present the headcount arithmetic as if it were neutral.
- **No AI jargon.** Say "what you gave me," "I made that up — check it," "the assistant."
- **Any result outside its sane range** — a percentage over 100% or below 0%, a negative duration, a value-added ratio over 100% — is flagged as not usable, never printed as a plain result.

## Output template (use every time)

Every full deliverable follows this skeleton, which is steps 1 to 8 of "How this skill works" in order. Start directly with the first heading. Do not add, rename, renumber or reorder headings, and do not change their levels. Every number in every table carries its source tag.

```markdown
## Chart used

[One sentence: chart type and why, from the chart selection table. One sentence: data type, subgroup size, sequence, and the source tag for the data and any spec limits.]

## Control limits

[Number of data points used. If fewer than 20, say "Provisional: calculated from N points."]

| Limit | Formula | Value | Source tag |
|---|---|---|---|
| Centerline | | | |
| UCL | | | |
| LCL | | | |
| UCL (range chart) | | | |
| LCL (range chart) | | | |

## Western Electric rules

| Rule | Triggered? | Points involved |
|---|---|---|
| 1: Point beyond 3σ | | |
| 2: Nine consecutive on one side | | |
| 3: Six consecutive trending | | |
| 4: Fourteen alternating | | |

## Stability conclusion

[Exactly one of: "In control: only common cause variation present." or "Out of control: special cause detected at [points]." Then what the signal suggests.]

## Process capability

[If spec limits are missing, write "Deferred: no spec limits given [NEEDS GEMBA]." If the process is not in control, write "Deferred: capability is not valid until special causes are removed." Otherwise:]

| Index | Formula | Value | Source tag | Reading (threshold table) |
|---|---|---|---|---|
| σ | | | | |
| Cp | | | | |
| Cpk | | | | |
| Centering gap (Cp − Cpk) | | | | |

## Improvement path

[Numbered actions: tamper warning if appropriate, centering if Cp >> Cpk, variation reduction if Cp is low, and investigation of any special cause. If the user will send more data, end with the data-safety line.]

---

**What this skill did:**
- [What analysis was run and what it found]
- [Key finding: chart, stability conclusion, Cp and Cpk]
- [Significant caveat, such as provisional limits]

**What still needs a human:**
- [The most important unverified special cause]
- [Any item tagged [NEEDS GEMBA] or assumed]
- [Decision that needs authority this skill does not have]

**Suggested next step:** [One specific action, usually a person and a place on the floor or unit.]

---

*control-chart · Lean Toolkit by [paulducey.com](https://paulducey.com/lean?utm_source=skill&utm_campaign=control-chart) · MIT License · Questions? Ask the author at [paulducey.com/contact](https://paulducey.com/contact)*
```

When the data is split into phases (for example, before and after a known change), keep the same six `##` headings and, under each of Control limits, Western Electric rules and Process capability, add one `### Readings a–b: [phase name]` subheading per phase in time order, each with its own copy of that section's table. Put any whole-dataset table first, before the phase subheadings.

Use these headings verbatim and in this order. If a section has nothing yet, keep the heading and write [NEEDS GEMBA] or the skill's own placeholder under it. In coaching mode, ask your questions first; whenever you produce the deliverable, use this template.

## Closing block

Append the block from `docs/CLOSING-BLOCK.md`, filled in for this analysis. The first bullet under "what still needs a human" should name the most important unverified special cause — the signal that, if investigated, would most change the process improvement path.

## Examples

- `resources/examples/control-chart-fill-weight.md` — I-MR chart on fill weight data, Rule 2 triggered (shift in mean after material lot change), Cp and Cpk calculated after removing out-of-control points, centering action recommended.
- `resources/examples/control-chart-cycle-time.md` — X-bar/R chart on cycle time subgroups, Rule 4 triggered (alternating pattern from two-shift process), tamper warning issued after operator had been adjusting machine settings in response to individual readings.