12.4 Statistical Process Control, Process Capability & Acceptance Sampling
Key Takeaways
- Control limits come from the process (typically the mean plus or minus three standard deviations); specification limits come from the customer — confusing the two is the most common SPC error on Exam 2.
- Cp measures potential capability from spread alone; Cpk measures actual capability including centering, so Cpk can never exceed Cp.
- A common industry benchmark treats Cpk of 1.33 as the minimum capable process and Cpk of 2.00 as Six Sigma capability.
- Common cause variation is inherent to the system and requires management action; special cause variation is assignable and requires local investigation — reacting to common cause is tampering.
- Acceptance sampling accepts or rejects a lot without changing it; producer's risk (alpha) is rejecting a good lot and consumer's risk (beta) is accepting a bad lot.
Statistical Process Control, Process Capability & Acceptance Sampling
Exam 2 does not ask you to construct a control chart by hand, but it does ask you to read one, interpret capability indices, and decide what a buyer should do about them. These are the quantitative items in the Quality Management section, and they are highly learnable.
Common Cause vs. Special Cause Variation
Every process varies. The discipline is knowing which kind of variation you are looking at.
| Common cause (chance / random) | Special cause (assignable) | |
|---|---|---|
| Source | Inherent in the system as designed | External, specific, identifiable event |
| Pattern | Stable, predictable, within control limits | Point outside limits or a non-random pattern |
| Example | Normal tool wear, ambient temperature drift, small operator variation | A broken die, an untrained relief operator, a new raw-material lot |
| Who fixes it | Management — the system must be redesigned | Local operator or supervisor — find and remove the cause |
| Process state | "In control" or stable | "Out of control" |
Tampering is Deming's term for adjusting a process in response to common cause variation. It reliably makes output worse, and the exam rewards recognizing it: a scenario where an operator tweaks a setting after every slightly-off reading, and variation grows, is describing tampering.
Control Charts
A control chart plots a process statistic over time against a centre line and upper and lower control limits (UCL / LCL), conventionally set at the process mean plus or minus three standard deviations.
The single most tested distinction in this topic: control limits are calculated from the process itself and describe what the process actually does. Specification limits (USL / LSL) come from the customer or the design and describe what the process is required to do. Control limits never appear on a control chart because someone chose them, and specification limits are never used to judge statistical control. A process can be perfectly in control and still produce entirely out-of-specification parts.
Chart Selection
| Data type | Chart | Use |
|---|---|---|
| Variable (measured on a continuous scale) | X-bar and R | Subgroup averages and ranges; the workhorse for dimensions, weights, temperatures |
| Variable | X-bar and s | Larger subgroups, using standard deviation instead of range |
| Variable | Individuals and moving range (I-MR) | Subgroup size of one, such as batch chemistry |
| Attribute (counted pass/fail) | p chart | Proportion defective, variable sample size |
| Attribute | np chart | Number defective, constant sample size |
| Attribute | c chart | Count of defects per unit, constant opportunity area |
| Attribute | u chart | Defects per unit, variable opportunity area |
Out-of-Control Signals
Beyond a single point outside the control limits, classic non-random patterns include a run of seven or more consecutive points on one side of the centre line, a trend of seven or more consecutively increasing or decreasing points, obvious cycles, and hugging of the centre line (which often indicates the data has been manipulated or the wrong subgrouping was used).
Process Capability: Cp and Cpk
Capability compares the voice of the customer (specification width) against the voice of the process (natural variation).
- Cp answers: could this process fit inside the specification if it were perfectly centred? It measures spread only.
- Cpk answers: does it fit, given where the process mean actually sits? It measures spread and centering.
- Therefore Cpk ≤ Cp always, and Cpk = Cp only when the process is perfectly centred.
| Cpk | Interpretation |
|---|---|
| < 1.00 | Not capable; the process is producing out-of-specification parts |
| 1.00 | Marginal; spec limits sit exactly at 3 sigma |
| 1.33 | Common contractual minimum for a capable process (4 sigma) |
| 1.67 | Strong capability (5 sigma) |
| 2.00 | Six Sigma capability |
Worked Capability Calculation
Scenario: A supplier machines a bushing with a specification of 25.00 mm ± 0.15 mm. A capability study of the stable process reports a mean of 25.04 mm and a standard deviation of 0.03 mm.
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Specification limits: USL = 25.15 mm, LSL = 24.85 mm. Specification width = 0.30 mm.
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Cp:
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Cpk — evaluate both sides:
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Interpret and act. Cp of 1.67 says the process is tight enough; Cpk of 1.22 says it is off-centre toward the upper limit and falls below the 1.33 contractual minimum. The correct supply management action is not to demand tighter tolerances or more inspection — it is a centering adjustment to move the mean from 25.04 toward the 25.00 nominal. Recentring alone would raise Cpk to equal Cp at 1.67 with no change in process variation and no capital spend.
Exam trap: whenever Cp is comfortably above target but Cpk is below it, the diagnosis is always centering, never spread. Recommending variation-reduction capital investment in that scenario is the distractor.
Acceptance Sampling
Acceptance sampling inspects a random sample from a lot and uses the result to accept or reject the whole lot. It is a QC (detection) activity and a fundamentally defensive tool — it sorts good lots from bad ones but improves nothing.
Key Terms
- AQL (Acceptable Quality Limit): the worst tolerable process average that will still be accepted most of the time. It is a producer-oriented benchmark.
- LTPD (Lot Tolerance Percent Defective) / RQL: the poor quality level the consumer wants rejected most of the time.
- Producer's risk (α): probability that a good lot is rejected. Conventionally about 5%.
- Consumer's risk (β): probability that a bad lot is accepted. Conventionally about 10%.
- Operating Characteristic (OC) curve: plots the probability of acceptance against actual lot quality. Larger sample sizes steepen the curve and discriminate better; changing the acceptance number shifts the curve.
- AOQL (Average Outgoing Quality Limit): the worst average outgoing quality after rectifying inspection.
Sampling Plan Types
- Single sampling: one sample, one accept/reject decision.
- Double sampling: a small first sample may accept, reject, or trigger a second sample.
- Sequential sampling: units inspected one at a time until a decision boundary is crossed; the most efficient in average sample size.
- Skip-lot sampling: inspection frequency reduced for suppliers with a proven record — the reward mechanism in a supplier certification program.
- 100% inspection: the fallback. It is expensive, slow, and — because of inspector fatigue — typically only about 80% effective, which is why the exam rarely treats it as the right long-term answer.
Dodge-Romig and ANSI/ASQ Z1.4 (attributes) and Z1.9 (variables) are the classical published sampling-plan systems. Z1.4 is the descendant of MIL-STD-105E.
Turning Statistics into Contract Language
The procurement deliverable from all of this is a quality clause that states specific, auditable obligations:
- Capability requirement: "Supplier shall demonstrate and maintain Cpk ≥ 1.33 on all designated key characteristics, evidenced by monthly capability reports."
- Control requirement: "Supplier shall maintain SPC on key characteristics and notify the buyer within 24 hours of any out-of-control condition affecting delivered product."
- Sampling and documentation: the accepted sampling plan, AQL, and required documents — certificate of conformance, certificate of analysis, first-article inspection report.
- Change control: no change to process, tooling, sub-tier supplier, or location without written approval and re-qualification.
- Right of access: second-party audit rights extending to sub-tier suppliers.
- Escalation: defined SCAR and 8D timelines, containment obligations, and cost-recovery terms for sorting, rework, and premium freight.
A supplier delivering documented Cpk of 1.67 on key characteristics earns skip-lot or dock-to-stock status, which removes incoming inspection cost — the concrete financial argument for supplier quality development.
A capability study on a stable supplier process reports Cp = 1.80 and Cpk = 1.05 against a contractual minimum of Cpk 1.33. What is the correct diagnosis and supply management response?
During a supplier review, a buyer states that the supplier's process is unacceptable because several individual measurements fell outside the upper and lower control limits shown on the X-bar chart, even though every part met the drawing tolerance. What error is the buyer making?
In an acceptance sampling plan, what do producer's risk and consumer's risk describe?