Bayesian Cpk Instrument
P(Cpk > 1.33), credible intervals, posterior distribution of capability.
No 1.5σ shift assumption. Monte Carlo via Normal-Inverse-Gamma conjugate.
Prior: Jeffreys (non-informative) · Ref: Gelman et al., BDA 3rd Ed.
No 1.5σ shift assumption. Monte Carlo via Normal-Inverse-Gamma conjugate.
Prior: Jeffreys (non-informative) · Ref: Gelman et al., BDA 3rd Ed.
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Specification Limits & Parameters
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Measurement Data
Accepts comma, space, tab, or newline delimiters. Paste from Excel, CSV, or any column of numbers.
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Traditional vs. Bayesian Estimate
Traditional (Point Estimate)
| Index | Value | Unit | Status |
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Bayesian (Posterior)
| Quantity | Value | Unit | Status |
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Posterior Distribution of Cpk
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Probability of Capability
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Summary Statistics
| Statistic | Value | Unit |
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Conditions at Analysis
What Is Bayesian Cpk?
Traditional Cpk gives you a single number — say, 1.34 — with no indication of how much to trust it. With 30 measurements, that 1.34 could easily be 1.1 or 1.6 in reality. Bayesian Cpk replaces the point estimate with a full probability distribution, so you know exactly how confident to be in your process capability.
Point Estimates vs. Posterior Distributions
A frequentist Cpk is calculated from the sample mean and standard deviation. It's a function of the data, nothing more. A Bayesian Cpk treats the process mean and variance as uncertain quantities, draws from their posterior distributions (using the Normal-Inverse-Gamma conjugate model), and computes Cpk for each draw. The result is a distribution of plausible Cpk values — not one number, but thousands.
This is what lets you ask questions like "What is the probability that Cpk exceeds 1.33?" and get a direct numerical answer (e.g., 72%). No 1.5σ shift assumption, no confidence interval gymnastics.
Credible Intervals and P(Cpk > Threshold)
- 95% credible interval — The range that contains the true Cpk with 95% posterior probability. Unlike a confidence interval, this is a direct probability statement about the parameter.
- P(Cpk > 1.00) — Probability the process is at least minimally capable (3σ level).
- P(Cpk > 1.33) — The standard "capable" threshold. Most automotive (IATF 16949) and aerospace (AS9100) standards require Cpk ≥ 1.33.
- P(Cpk > 1.67) — The "excellent" threshold. Some critical processes (safety-critical, medical devices under ISO 13485) target this level.
When to Use Bayesian Cpk
- Small samples — With fewer than 50 observations, traditional Cpk is unreliable. Bayesian Cpk honestly represents the uncertainty and avoids false confidence.
- Capability qualification — When you need to demonstrate P(Cpk > 1.33) ≥ 95% to a customer or auditor, the posterior probability is a direct answer.
- New process validation — Early production runs where data is limited but decisions must be made.
- Comparison with traditional — See the traditional Cpk calculator for point estimates, or learn about classical vs. Bayesian SPC.
Frequently Asked Questions
What is Bayesian Cpk?
Bayesian Cpk treats process capability as a probability distribution rather than a single number. Instead of "Cpk = 1.34," you get a posterior that says "median Cpk is 1.34 with a 95% credible interval of [1.18, 1.52], and there is a 62% probability that Cpk exceeds 1.33."
Why not just use traditional Cpk?
Traditional Cpk gives a point estimate with no uncertainty attached. A Cpk of 1.34 from 30 measurements could easily be 1.1 or 1.6 in reality. Bayesian Cpk quantifies that uncertainty directly and eliminates the 1.5σ shift assumption.
How many data points do I need?
Bayesian Cpk works with any sample size — even 10–15 observations give meaningful results. The credible interval width tells you how much to trust the result. With small samples, the interval is wide (honest about uncertainty). With 100+ observations, it tightens considerably.
What does P(Cpk > 1.33) mean?
P(Cpk > 1.33) is the posterior probability that your process is truly capable at the 4-sigma level, given your data. If P(Cpk > 1.33) = 0.85, there's an 85% chance your process meets the requirement — a direct answer, not a p-value that requires interpretation.
What prior does this use?
This calculator uses a non-informative (Jeffreys) prior: the standard reference prior for Normal mean and variance. With enough data (n > 20), the prior has negligible influence. The full SVEND platform supports informative priors from historical process data.
How does the Monte Carlo simulation work?
The calculator draws samples from the posterior distribution of (μ, σ) using the conjugate Normal-Inverse-Gamma model. For each (μ, σ) draw, it computes Cpk = min((USL − μ)/(3σ), (μ − LSL)/(3σ)). The collection of Cpk values forms the posterior distribution. 10,000 samples gives stable quantiles.