KEVOS® MAKEOS · Quality
Capability Confidence Check (Cp, Cpk and Their Lower Confidence Bounds, Sample Size)
From a capability study of this many parts, is the process proven capable at my confidence level, or does the sample only suggest it?
Capability indices with lower confidence bounds: Cp and Cpk from the sample mean, standard deviation and specification limits, the one-sided lower confidence bound on Cp from the chi-square distribution and on Cpk from its normal approximation (NIST e-Handbook), the sample size at which the same Cpk would be proven, against the Cpk you require.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Number of parts measured | parts | Number of parts in the capability study (NIST advises at least 25). |
| Sample mean | Mean of the measurements, in the unit of the specification. | |
| Sample standard deviation | Sample standard deviation of the measurements. | |
| Lower specification limit | Lower specification limit. | |
| Upper specification limit | Upper specification limit. | |
| Confidence level (one-sided) | choice | Confidence you need that the true capability is at least the lower bound. |
| Cpk required | Capability your customer or quality system requires. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Number of parts measured
- What it is
- The number of parts measured in the capability study, which sets how confident the study can be about the true capability.
- Units and format
- Enter a number in parts: parts. The form accepts 5 parts to 100000 parts. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 parts as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More parts tighten the confidence bounds towards the sample values.
- Get it right
- NIST advises at least 25 parts for these bounds; 100 or more for a full capability study.
What this means: Sample mean
- What it is
- The average of the measurements in the study, which sets how well centred the process is between the specification limits.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10.004 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moving it towards either limit lowers Cpk and its bound.
- Get it right
- Use measurements from a stable process; a mean from a drifting process misleads.
What this means: Sample standard deviation
- What it is
- The sample standard deviation of the measurements, the spread of the process.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.008 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Cp, Cpk and their bounds fall in proportion as the spread rises.
- Get it right
- Use the overall sample standard deviation of individual parts, dividing by n − 1.
What this means: Lower specification limit
- What it is
- The lowest acceptable value of the characteristic, the lower end of the tolerance.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 9.95 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Lowering it widens the tolerance and raises the indices.
- Get it right
- Use the drawing limit, not an internal tightened limit, unless that is what you are judging.
What this means: Upper specification limit
- What it is
- The highest acceptable value of the characteristic, the upper end of the tolerance.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10.05 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raising it widens the tolerance and raises the indices.
- Get it right
- It must be above the lower limit, in the same unit as the measurements.
What this means: Confidence level (one-sided)
- What it is
- The confidence you need that the true capability is at least the lower bound, as a one-sided level.
- Units and format
- Choose one of 3 options from the list: 90%; 95%; 99%.
- Example
- The form opens with “95%” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- A higher confidence lowers both bounds.
- Get it right
- 95 % is common; 99 % needs larger samples to prove the same capability.
What this means: Cpk required
- What it is
- The capability index your customer or quality system requires, which the lower confidence bound on Cpk must reach.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.33 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the pass or fail line and the sample size needed.
- Get it right
- Take it from the customer requirement or control plan; this tool does not set a value.
Method and formulas
- Cp = (USL − LSL) ÷ 6s; Cpk = min(USL − x̄, x̄ − LSL) ÷ 3s.
- Lower bound on Cp = Cp √(χ²_{α, n−1} ÷ (n − 1)), with χ²_{α} the lower α quantile; lower bound on Cpk ≈ Cpk − z_{1−α} √[1 ÷ (9n) + Cpk² ÷ (2(n − 1))] (NIST e-Handbook 6.1.6).
- Sample size to prove: the smallest n (from 25, the approximation’s lower limit) for which the Cpk lower bound reaches the requirement at the same Cpk.
- Example: Cpk 1.9167 with a 1.5889 lower bound at 95 % confidence.
Limitations
- Assumes a stable, normally distributed process; check control charts and normality first, or the indices mean little.
- The Cpk bound is an approximation that NIST gives for samples of about 25 or more.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Judge whether a 30-piece capability study really proves Cpk 1.33
- Plan how many parts to measure in a capability study
- Compare suppliers’ capability claims fairly
Who it is for
- Quality engineers
- Supplier quality staff
- Process engineers
What you get
- Cp and Cpk from the sample
- Lower confidence bounds on Cp and Cpk
- Sample size that would prove the same Cpk
Interpreting the result
- A small study can show a high Cpk by chance: with 30 parts a sample Cpk of about 1.7 is needed to prove 1.33 at 95 % confidence.
References
- NIST/SEMATECH e-Handbook of Statistical Methods, 6.1.6 What is Process Capability? (confidence bounds for Cp and Cpk) — https://www.itl.nist.gov/div898/handbook/pmc/section1/pmc16.htm
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 128–133. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Explains the concept · Library collection
Proving a process is ready for production: measurement, stability, capability and PPAP-style evidence
In Product Development and Launch
What it covers: Why good samples do not prove a process is ready to scale, and how measurement checks, control charts, capability indices and a traceable evidence chain show it can repeat.
Method explained here · KEVOS® original book
Testing bonded joints
In Adhesive Bonding by the Numbers
What it covers: Lap shear coupons that travel with production, reading cohesive and interfacial failures, never discarding low results, capability against a minimum strength, and inspecting parts.
Method explained here · KEVOS® original book
Continuous blending: feeders, fluctuations and back-mixing
In Blending Bulk Solids by the Numbers
What it covers: How feeder fluctuations become composition variation, how an extruder's back-mixing filters them by their time scale, how refill floods and slow drifts get through, and how capability is proved.
Method explained here · KEVOS® original book
Capability from a stable process
In Control Charts by the Numbers
What it covers: Cp, Cpk and predicted rejects from a stable process, capability against performance, the confidence bound on a capability index and how many parts a capability claim needs.
Method explained here · KEVOS® original book
Thresholds, false calls and acceptance decisions
In Eddy-Current Inspection by the Numbers
What it covers: Probability of detection and false calls, guard bands from measurement uncertainty, capability with confidence bounds, lot acceptance, and testing a suspect reading before discarding it.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.