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Accept or Reject by the Numbers

Inspection decisions as inference: base rates, measurement error, guard bands, repeat checks, combined evidence, thresholds and lot sampling, priced in dollars

Make better accept and reject decisions: Bayes' rule, readings near the limits, cost-based guard bands, gauge studies, repeat checks, combined evidence, thresholds and sampling.

10 chapters · About 100 minutes · Free to read · Intermediate quality and manufacturing engineering

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KEVOS® LibraryMANUFACTURING0.60.40.60.4Accept or Rejectby the NumbersKEVOS® ORIGINAL BOOK10 CHAPTERS

About this book

Every inspection ends in a decision: accept or reject, pass or hold, ship the lot or sort it. The decision is made from evidence that is never perfect. Gauges have uncertainty, cameras make false calls, operators miss things, samples are small. And the evidence is never the only information available: the process’s own record says how likely a part is to be bad before anyone measures it. Good decisions combine the two, and they weigh the cost of rejecting a good part against the cost of shipping a bad one.

This book treats inspection as inference. It starts with Bayes’ rule and the base rate, works through readings near the tolerance limits, guard bands set from costs, gauge studies, repeat readings and majority votes, combining several checks, setting automatic inspection thresholds and learning a process’s defect rate, and finishes with lot sampling and the design of a whole inspection plan. The mathematics is the same that information theory and statistical physics use to decode noisy signals, applied to parts instead of bits.

It is for quality engineers, inspectors, metrologists, production engineers and supervisors. It assumes familiarity with tolerances, the normal distribution and a spreadsheet. Each chapter works the running case with real numbers, includes one plausible mistake and how it was caught, links the KEVOS® tool that does the chapter’s calculation where one exists, and ends with an exercise and its worked answer.

The running case

Every chapter follows one fictional part. This is an illustration. A CNC turning shop in Adelaide, South Australia, makes a steel bush, BU-25, for a hydraulic cylinder maker, about 2,000 a shift.

  • The characteristic: a Ø25 H7 bore, 25.000 to 25.021 mm. The turning process runs with a mean of 25.0125 mm and a standard deviation of 3.5 µm, a little above the middle of the tolerance.
  • The gauge: every bush’s bore is measured by an air gauge with an expanded uncertainty of 2.26 µm at k = 2, about 6 seconds a reading. A bench gauge with a standard uncertainty of 0.30 µm is available for difficult cases, about 60 seconds a reading.
  • The camera: a vision system checks every bore for chatter marks, a surface defect found on about 0.6% of bushes. At its current setting it catches 95% of them and calls 2% of good bushes defective.
  • The costs: a bad bush that reaches the customer costs about A$400 in sorting, freight and goodwill; a good bush rejected costs about A$6.50 in material and machining. Operator time costs A$80 an hour. All excluding GST.
  • The customer: checks incoming lots by attributes, with a plan of 132 bushes and an acceptance number of 3.

The shop, the customer, the bush and every number about them are invented for the case and do not describe KEVOS® work or any real business or product.

How to use the book

Read Chapter 1 first: Bayes’ rule and the base rate are the foundation of everything after. Chapters 2 to 5 concern measured characteristics and the gauge; Chapters 6 and 7 concern pass–fail checks and automatic inspection; Chapter 8 learns the defect rate from data; Chapter 9 decides about lots; Chapter 10 puts the pieces together into an inspection plan.

Every calculation in the book was checked by running it, and the false-accept and false-reject rates were calculated by integrating over the process distribution. The KEVOS® tools linked from the chapters perform the same calculations with the reader’s own data.

Source limits and professional responsibility

The models in this book are deliberately simple: normal process distributions, normal measurement errors and independent checks. Real processes drift, real errors correlate and real defects cluster, and each chapter notes where that matters. Contractual acceptance rules, customer specifications and the standards they call up govern real decisions, and in safety-critical or regulated products, acceptance decisions must follow the applicable standard and be approved by qualified people. This book is educational. It is not an acceptance procedure for any product, and a competent person must approve real ones.

Who this book is for

Quality engineers, inspectors, metrologists, production engineers and supervisors who set acceptance limits, choose gauges and inspection methods, tune automatic inspection, or decide what to do with parts and lots near the line.

What you will learn

  • Use Bayes' rule with a process's own defect rate to work out what an inspection result really means for a part.
  • Set acceptance limits from measurement uncertainty and the costs of false accepts and false rejects, and judge whether a gauge is good enough.
  • Choose between single readings, repeat readings, averages and majority votes, and combine the evidence of several imperfect checks.
  • Set an automatic inspection threshold for the shop's own defect rate, and learn that rate from inspection and audit data.
  • Design lot sampling and two-stage inspection plans, and know when improving the process beats improving the inspection.

Inside this book

Chapters

  1. 01Inspection as inferenceTreat an inspection result as evidence: false accepts and false rejects, the base rate from process capability, Bayes' rule in odds form, and why reject bins hold mostly good parts.
  2. 02Readings near the limitsWhat a reading near a tolerance limit says about true size: the uncertainty budget tool, test uncertainty ratio, combining the reading with the process, and false accept and reject rates.
  3. 03Guard bands and the cost of errorsSet acceptance limits inside the tolerance: the one-uncertainty guard band in the KEVOS® tool, cost-based limits from the costs of each error, sensitivity to costs, and contract rules.
  4. 04How good is the gauge?Judge a gauge three ways: a Type 1 study of repeatability and bias with the KEVOS® gauge check, the test uncertainty ratio, and the money a better gauge would save in wrong decisions.
  5. 05Measuring again and majority votesCompare repeat-measurement policies: re-measuring rejects, measuring until a pass, averaging, majority votes and re-measuring only near the limit, with the cost of each per shift.
  6. 06Combining several checksCombine several imperfect checks with likelihood ratios: the evidence in each pass or fail, posteriors for every pattern, cost-based rules, checking only disagreements, and correlated checks.
  7. 07Setting an automatic inspection thresholdSet an automatic check's reject threshold: score distributions, the trade-off curve, the cost-based threshold, how the defect rate moves it, review bands and the cost of a wrong prior.
  8. 08Learning the defect rateEstimate and update a defect rate from audits: a p-chart with the KEVOS® tool, Bayesian updating with a beta prior, separating special causes, and correcting an imperfect reject rate.
  9. 09Accepting or rejecting a lotDecide about lots by sampling: designing a plan for producer's and consumer's risk, operating characteristics, lot acceptance from process capability, and what a sample adds to prior data.
  10. 10Designing the inspection planBuild the inspection plan: two-stage gauging, a camera review band, bench capacity, reaction plans and records, and the comparison that matters most, inspecting better against making better.

Apply This in Practice

Use these KEVOS® tools to try the method from this reading with your own numbers.

References and source notes

The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 31 October 2026.

  • H. Nishimori, Oxford University Press (International Series of Monographs on Physics 111). Statistical Physics of Spin Glasses and Information Processing: An Introduction (2001). In copyright. Used only as a map of the ideas behind Chapters 1, 5, 6 and 7: Bayesian decoding of noisy signals, the gain from assuming the true noise level and prior, majority voting as the simplest error-correcting code, and per-bit against whole-message decisions. No text, structure, figures, tables or problems are reproduced, and all cases and calculations are KEVOS® originals.
  • T. Bayes, communicated by R. Price, Philosophical Transactions of the Royal Society of London 53, 370–418. An essay towards solving a problem in the doctrine of chances (1763). Named in Chapter 1 as the origin of the rule for updating a probability with evidence, used throughout the book.
  • Joint Committee for Guides in Metrology, published by BIPM. JCGM 100:2008, Evaluation of measurement data: Guide to the expression of uncertainty in measurement (GUM) (2008). Named in Chapter 2 as the basis of the uncertainty budget, combined and expanded uncertainty used there and in the KEVOS® uncertainty tools. The guide is free from the BIPM publications page.
  • Joint Committee for Guides in Metrology, published by BIPM. JCGM 106:2012, Evaluation of measurement data: The role of measurement uncertainty in conformity assessment (2012). Named in Chapters 2 and 3 as the guidance that treats acceptance decisions with a prior distribution of the measured quantity, guard bands and the risks of false acceptance and rejection, the approach this book applies. No text or figures are reproduced.
  • International Organization for Standardization. ISO 14253-1, Geometrical product specifications (GPS): Inspection by measurement of workpieces and measuring equipment, Part 1: Decision rules for verifying conformity or nonconformity with specifications (current edition). Named by number and scope only in Chapter 3. Contracts often call up its decision rules; read the edition the contract names. No values are reproduced.
  • International Organization for Standardization. ISO 2859-1, Sampling procedures for inspection by attributes, Part 1: Sampling schemes indexed by acceptance quality limit (AQL) for lot-by-lot inspection (current edition). Named by number and scope only in Chapter 9. The plans in this book are designed from the binomial distribution by the KEVOS® tools, not taken from the standard's tables.
  • National Institute of Standards and Technology. NIST/SEMATECH e-Handbook of Statistical Methods (n.d.). Public reference for the attribute control charts and acceptance sampling methods used in Chapters 8 and 9.
  • W. J. Rogan and B. Gladen, American Journal of Epidemiology 107(1), 71–76. Estimating prevalence from the results of a screening test (1978). Source of the correction applied in Chapter 8 to estimate a true defect rate from the results of an imperfect inspection.
  • C. E. Shannon, Bell System Technical Journal 27(3), 379–423. A mathematical theory of communication (1948). Named in Chapter 5 as the foundation of the theory of reliable decisions from noisy signals, of which repeated readings and majority votes are the simplest cases.

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