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Control Charts by the Numbers
Variation as a random process: Shewhart, individuals, CUSUM and EWMA charts, run lengths, counts, drift and capability, worked through one turning cell
Design, read and trust control charts: subgroup statistics, false alarms and run lengths, CUSUM and EWMA for small shifts, attribute charts, autocorrelation, tool wear and capability.
Start readingAbout this book
A control chart is a simple picture with a demanding job. It plots a measurement over time against limits drawn from the process’s own variation, and it has to say, as early as possible, when something has changed, without crying wolf so often that people stop looking. Every design choice on a chart, from the subgroup size to the type of chart to the width of its limits, is a trade between those two errors, and every trade can be calculated. The calculations come from the theory of random processes: a measurement is a random variable, a subgroup mean obeys the central limit theorem, a cumulative sum is a random walk, an exponentially weighted average is a process with a fading memory, and the time until a chart signals is a first-passage time.
This book works through those ideas one chart at a time. It starts with variation itself and the X̄–R chart, moves to individuals charts and to the run lengths that measure every chart’s performance, then to the CUSUM and EWMA charts that catch small sustained shifts. It covers charts for defect counts and proportions, the autocorrelation and tool-wear drift that break the textbook assumptions, and finishes with capability, quoted honestly with its confidence bound.
It is for quality, manufacturing and process engineers, setters and supervisors, and students who measure parts and want to act on the numbers correctly. It assumes averages, standard deviations and a first acquaintance with the normal distribution. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, links the KEVOS® tool that performs the chapter’s calculation, and ends with an exercise and its worked answer.
The running case
Every chapter follows one fictional cell. This is an illustration. A hydraulics manufacturer turns valve spools, part SP-12, on a CNC lathe with an in-process gauge. The critical feature is a land diameter of 12.000 mm with a tolerance of ±0.010 mm: 11.990 to 12.010 mm.
- The process: a cycle of 45 seconds, 40 spools every half hour, two shifts a day.
- Sampling: five consecutive spools measured every half hour, a subgroup of five, 32 subgroups a day.
- Baseline: 25 subgroups from a stable period gave a grand average of 12.0010 mm and an average range of 0.0051 mm, a within-subgroup standard deviation of about 0.0022 mm.
- Other characteristics: case hardness measured once per heat-treatment batch, burrs found at visual inspection, and scratches counted on trays of finished spools.
The cell, its data and any costs are invented. None describes KEVOS® work or a real business. Measurements are in millimetres throughout.
How to use the book
Read Chapters 1 to 4 in order: they set out the random-variable view of a measurement, the two basic charts and the run lengths by which every chart is judged. Chapters 5 and 6, on CUSUM and EWMA, belong together. Chapters 7 to 9 can be read in any order after Chapter 4.
The run lengths in the book were calculated by the Markov-chain methods of Brook and Evans for the CUSUM and of Lucas and Saccucci for the EWMA, and checked against values widely published for the same chart designs. Values are given to more figures than the data could justify, so that a reader can follow each step.
Source limits and professional responsibility
The charts in this book assume measurements that are independent and roughly normal, from a measurement system that is fit for purpose; Chapter 8 shows what happens when the first assumption fails, and the companion books on inspection and measurement deal with the last. Real processes need their own baselines, and the chart designs here are starting points, not prescriptions.
A control chart says whether a process has changed; it does not say whether parts meet their specification, and it does not replace inspection required by a customer or a regulation. This book teaches how to design and read control charts. It is educational and is not a quality plan for any product.
Who this book is for
Quality, manufacturing and process engineers, machinists and setters, production supervisors and students who need to set up control charts, read them correctly and decide when a process needs attention.
What you will learn
- Treat a measured characteristic as a random variable and predict out-of-tolerance parts from its mean and standard deviation.
- Set up X̄–R and individuals charts from a stable baseline and explain why their limits sit where they do.
- Calculate average run lengths for false alarms and for real shifts, and choose a subgroup size and chart to match the shift that matters.
- Design and read CUSUM and EWMA charts for small sustained shifts, and p and c charts for counts and proportions.
- Recognise autocorrelation and tool-wear drift, and quote capability with its confidence bound only for a process in control.
Inside this book
Chapters
- 01Variation is a random variableA measured dimension as a random variable: mean, standard deviation and the normal distribution, how independent sources of variation add, subgroup means, and predicting parts out of tolerance.
- 02Shewhart X̄ and R chartsCommon and special causes, rational subgroups, X̄ and R limits from a stable baseline, where the constants come from, reading a day of SP-12's chart, and why tolerance limits do not belong on it.
- 03Individuals and moving rangesCharting one reading per period: the moving range estimate of σ, individuals limits, why an outlier can hide itself, setting a clean baseline, and the extra care that single readings need.
- 04Run lengths: false alarms and detectionThe run length as a random variable: average run length for false alarms and real shifts, subgroup size against sampling interval, why 2σ limits backfire, and run lengths as first-passage times.
- 05CUSUM: a random walk that remembersThe tabular CUSUM as a random walk with drift: reference value and decision interval, its run lengths, reading where a shift began, running it beside an X̄ chart, and a CUSUM that was never reset.
- 06EWMA: a chart with a fading memoryThe exponentially weighted moving average as a smoothed, mean-reverting statistic: choosing λ and L, its limits and run lengths, comparison with the CUSUM, and the wrong standard deviation.
- 07Charts for counts and proportionsDefectives and defects as binomial and Poisson random variables: p and c chart limits, why they are asymmetric, limits that change with sample size, missing lower limits, and when to use a u chart.
- 08Autocorrelation, drift and tool wearWhen readings are not independent: autocorrelation and the false alarms it causes, sampling far enough apart, residual charts, tool-wear drift as a random walk with drift, and compensation.
- 09Capability from a stable processCp, 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.
Apply This in Practice
Use these KEVOS® tools to try the method from this reading with your own numbers.
Applies this method
c-Chart Check (Defects per Unit, Poisson Control Limits, Counts Out of Control)
Sets Poisson c-chart limits for equal inspection units, the count that would signal and which units fall outside the limits.
Applies this method
Capability Confidence Check (Cp, Cpk and Their Lower Confidence Bounds, Sample Size)
Gives Cp and Cpk with their lower confidence bounds and the number of parts needed to prove a required Cpk.
Used in Capability from a stable process
Applies this method
CUSUM Control Chart Check (Tabular Upper and Lower Sums, Decision Interval, First Signal)
Runs the tabular CUSUM on up to twenty values: reference value, upper and lower sums, first signal and where the shift probably began.
Applies this method
EWMA Control Chart Check (Weighted Moving Average, Control Limits, Small Shifts)
Calculates the EWMA from the target, its steady-state limits for a chosen λ and width, and the first reading at which it leaves them.
Applies this method
Individuals Control Chart Check (Moving Range, Control Limits, Points Out of Control)
Sets individuals chart limits from the average moving range for up to twelve readings and flags readings outside them.
Used in Individuals and moving ranges
Applies this method
Normal Distribution Out-of-Tolerance
Gives the z value to each limit, parts per million out of tolerance, expected count and yield for a normal characteristic from its mean and standard deviation.
Used in Variation is a random variable
Applies this method
p-Chart Check (Fraction Defective, Control Limits, Samples Out of Control)
Sets binomial p-chart limits for samples of equal size, lists samples outside them and the defectives per sample that would signal.
Applies this method
Process Capability Check (σ from R̄, Cp, Cpk, Rejects, Limits)
Estimates σ from R̄, gives X̄ and R limits, Cp, Cpk and predicted parts per million, checked against a required Cpk.
Used in Capability from a stable process
Applies this method
X-bar and R Control Chart Limits
Calculates X̄ and R chart centre lines and control limits from the grand average, average range and subgroup size, with the estimated process σ.
Used in Shewhart X̄ and R charts
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 30 October 2026.
- D. S. Lemons, Johns Hopkins University Press. An Introduction to Stochastic Processes in Physics (2002). In copyright. Used only as a map of the probability behind the book: random variables, the variance sum and central limit theorems, random walks with drift, the Ornstein–Uhlenbeck process and autocorrelation. No text, structure, figures, tables or problems are reproduced.
- National Institute of Standards and Technology. NIST/SEMATECH e-Handbook of Statistical Methods, Chapter 6: Process or Product Monitoring and Control (2012). Open reference for control charts for variables and attributes, process capability and exponential smoothing, used throughout.
- W. A. Shewhart, Van Nostrand. Economic Control of Quality of Manufactured Product (1931). Named in Chapter 2 as the origin of the control chart and its three-sigma limits.
- E. S. Page, Biometrika 41, 100–115. Continuous inspection schemes (1954). Named in Chapter 5 as the origin of the cumulative sum chart.
- S. W. Roberts, Technometrics 1, 239–250. Control chart tests based on geometric moving averages (1959). Named in Chapter 6 as the origin of the exponentially weighted moving average chart.
- J. M. Lucas and M. S. Saccucci, Technometrics 32(1), 1–12. Exponentially weighted moving average control schemes: properties and enhancements (1990). Named in Chapters 4 and 6 for the Markov-chain method of calculating EWMA run lengths, which the book uses with its own calculations.
- D. Brook and D. A. Evans, Biometrika 59(3), 539–549. An approach to the probability distribution of cusum run length (1972). Named in Chapters 4 and 5 for the Markov-chain method of calculating CUSUM run lengths. The book's run lengths are its own calculations, checked against values widely published for the same designs.
- D. C. Montgomery, John Wiley & Sons. Introduction to Statistical Quality Control (current edition). Named as a standard text on control chart design and capability analysis. No text or tables are reproduced.
- International Organization for Standardization. ISO 7870 Control charts (series) and ISO 22514 Statistical methods in process management: Capability and performance (series) (current editions). Named by number and scope in Chapters 2 and 9. Paywalled; no clauses or values are quoted.