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Defect Maps by the Numbers

Where defects fall on a part, whether they are random or clustered, and what their pattern says about the cause, the yield and the inspection

How to map where defects fall on parts, test whether they are random or clustered, and use the pattern to find causes, predict yield, chart counts and plan inspection.

11 chapters · About 110 minutes · Free to read · Intermediate quality and process engineering

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KEVOS® LibraryMANUFACTURING↑→↓←VALUEDefect Maps bythe NumbersKEVOS® ORIGINAL BOOK11 CHAPTERS

About this book

Most factories count their defects. Far fewer record where on the part each one was. Yet the position of a defect is often the best clue to its cause: dirt that falls from a hook lands below the hook, a contaminated spray gun leaves craters along its path, and a cold corner in a mould leaves porosity in the same place on every casting. The pattern of positions also decides how defects affect yield and how they should be charted and inspected, because defects that cluster behave very differently from defects scattered at random.

This book teaches the arithmetic of defect positions, which statisticians call spatial point pattern analysis, for practical use on the shop floor. It starts with recording defects and turning counts into densities, then sets out what random scatter looks like and how to test for it with quadrat counts, nearest-neighbour distances and Ripley’s K function. It goes on to clustered defects and their effect on yield, uneven defect density across a part, overlapping spots and coverage, and finally charting, sampling and proving that a change has worked.

It is for quality and process engineers, production supervisors and inspectors who deal with surface, coating, casting, welding or assembly defects. It assumes algebra, logarithms and basic probability, and the idea of a control chart. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, and ends with an exercise and its worked answer.

The running case

Every chapter follows one fictional plant. This is an illustration. Plant E makes steel electrical enclosures and powder-coats its own switchboard doors on a conveyorised line: the doors are hung from two hooks through holes in their top edge, pass through pretreatment, a dry-off oven, an automatic powder booth with manual touch-up, and a curing oven. Each door is 800 mm wide and 1,200 mm high, and its outer face, 0.96 m², is a visible surface on which no defect is accepted. A door with any defect on its outer face is sanded and recoated.

After the line was moved into a new building, the share of doors needing rework rose to about one in four. The plant’s quality engineer is asked to find out why and to fix it. The engineer starts by mapping where the defects are.

The plant, its line and every count, position and cost are invented. None describes KEVOS® work, a real company or a real supplier’s product.

How to use the book

Read Chapters 1 and 2 first: they set up the counts, densities and Poisson model that everything else uses. Chapters 3 to 6 ask whether defects are random or clustered, in increasing depth. Chapters 7 to 9 deal with the consequences: yield, uneven density and overlapping defects. Chapters 10 and 11 turn the analysis into charts, inspection plans and a proven improvement.

KEVOS® tools are linked where they perform a step of the method, with a note of exactly what each one calculates and where its model stops. Where a tool solves a related problem rather than the chapter’s own, the text says so.

Source limits and professional responsibility

The methods in this book are the established ones of spatial statistics and quality control: the Poisson process, quadrat and distance tests, the K function with Monte Carlo envelopes, clustered yield models and attribute control charts. Each chapter states their assumptions, chiefly that defects can be treated as points and that the area inspected is known. Standards are named by number and scope only.

The statistical tests here help to find causes and to decide when evidence is strong enough to act. They do not replace engineering judgement about the process, the safety of the people working on it, or the requirements of a customer’s specification. This book is educational; real acceptance criteria, sampling plans and process changes should be agreed with the people responsible for them.

Who this book is for

Quality and process engineers, production supervisors and inspectors who count surface, coating, casting or assembly defects and need to find their causes and predict their cost.

What you will learn

  • Record defects with their positions and turn counts on parts into defect densities and Poisson probabilities.
  • Test whether defects are spread at random, using quadrat counts, the index of dispersion and nearest-neighbour distances.
  • Read Ripley's K function with simulation envelopes to find the scale at which defects cluster.
  • Predict first-pass yield from defect density with Poisson and clustered (negative binomial) models, and judge the effect of a smaller part or a cleaner process.
  • Chart defect counts without false alarms from clustering, plan sampling inspection and prove that a process change reduced defects.

Inside this book

Chapters

  1. 01Why defect locations matterWhy a tally of defect types hides causes that a map of defect positions reveals, how to record positions on parts, the kinds of defect map, and what rework is costing.
  2. 02Defect density and the Poisson countTurning defect counts into a density per area, the Poisson model for counts on parts, comparing the counts it predicts with those found, and why defects per part misleads across part sizes.
  3. 03What random looks likeWhat complete spatial randomness means, why random patterns look clumpy and gappy to the eye, how to simulate random defects for comparison, and how chance creates convincing false clusters.
  4. 04Quadrat counts and the dispersion testDividing a part into cells, counting defects in each, testing the counts against randomness with the index of dispersion and chi-square, choosing cell size, and finding which cells matter.
  5. 05Nearest-neighbour distancesTesting whether defects on one part are clustered or regular from each defect's distance to its nearest neighbour: the Clark–Evans ratio, the G function, edges and the choice of region.
  6. 06Ripley's K and the scale of clusteringUsing Ripley's K function and its L transform to see clustering or regularity at every distance at once, testing against simulation envelopes, and reading the scale of clusters from the curve.
  7. 07Clustered defects and yieldWhy clustering between parts raises first-pass yield at a given defect density, the negative binomial yield model and its clustering parameter, and yield for new parts and processes.
  8. 08Uneven density and zonesDescribing defect density that varies across a part, estimating it by cells or smoothing, predicting the effect of removing one source, and why uneven density can masquerade as clustering.
  9. 09Overlapping spots and coverageWhat happens when random spots have size and overlap: the Boolean model, why counts of merged defects stop rising, and how coverage by random impacts grows in blasting and peening.
  10. 10Charting and sampling defect countsCharting daily defect counts with c and u charts, why clustering between parts causes false alarms and how to widen the limits, how inspection misses defects, and planning sampling of outgoing parts.
  11. 11From map to cause and proofTurning map evidence into changes, planning a before-and-after trial, proving the defect rate fell, using a fresh map to credit each change correctly, and keeping the gain.

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.

  • K. R. Mecke and D. Stoyan (eds), Lecture Notes in Physics 554, Springer, Berlin (ISBN 3 540 67750 X). Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation (2000). Supplied source, in copyright. Read only as a subject map: its contents and preface name point processes, second-order characteristics, edge corrections, the Boolean model and Monte Carlo methods as shared ground between physics and statistics. No text, equations, figures, examples or exercises are reproduced or adapted; the manufacturing treatment, running case and numbers here are original.
  • P. J. Diggle, Arnold, London. Statistical Analysis of Spatial Point Patterns, second edition (2003). Standard text for complete spatial randomness, quadrat and distance methods and Monte Carlo tests, used in Chapters 3 to 6.
  • J. Illian, A. Penttinen, H. Stoyan and D. Stoyan, Wiley, Chichester. Statistical Analysis and Modelling of Spatial Point Patterns (2008). Background for the K and L functions, edge effects, cluster processes and inhomogeneous intensity in Chapters 6 and 8.
  • P. J. Clark and F. C. Evans, Ecology, vol. 35, pp. 445–453. Distance to nearest neighbor as a measure of spatial relationships in populations (1954). Original paper for the nearest-neighbour ratio and its test statistic, used in Chapter 5.
  • B. D. Ripley, Journal of the Royal Statistical Society Series B, vol. 39, pp. 172–192. Modelling spatial patterns (1977). Source of the K function used in Chapter 6.
  • C. H. Stapper, F. M. Armstrong and K. Saji, Proceedings of the IEEE, vol. 71, pp. 453–470. Integrated circuit yield statistics (1983). Review of Poisson and clustered (negative binomial) yield models, the basis of Chapter 7's treatment of yield with clustered defects. No data are reproduced.
  • S. N. Chiu, D. Stoyan, W. S. Kendall and J. Mecke, Wiley, Chichester. Stochastic Geometry and Its Applications, third edition (2013). Background for the Boolean model and coverage relations in Chapter 9.
  • D. B. Laney, Quality Engineering, vol. 14, pp. 531–537. Improved control charts for attributes (2002). Source of the adjustment of attribute chart limits for overdispersion described in Chapter 10.
  • National Institute of Standards and Technology. NIST/SEMATECH e-Handbook of Statistical Methods: attributes control charts (2012). Public reference for c and u charts and their Poisson basis, used in Chapter 10.
  • Joint Committee for Guides in Metrology (BIPM). Evaluation of measurement data: Guide to the expression of uncertainty in measurement (JCGM 100:2008) (2008). Background for treating inspection as a measurement with its own detection probability in Chapter 10.
  • International Organization for Standardization; SAE International. ISO 2859-1 (sampling procedures for inspection by attributes), SAE J2277 (shot peening coverage determination) and ISO 8501-1 (visual assessment of surface cleanliness before painting) (current editions). Named by number and scope only in Chapters 9 and 10. Nothing is quoted; the sampling plans in the book are designed from first principles with the KEVOS® tools.

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