Module 17 · Control

Statistical process control II

Module 16 built the charts for measured data with rational subgroups. This module covers everything else you will meet on a shop floor: counts of rejects and defects, daily proportions on thousands of units, small drifts that a Shewhart chart takes weeks to notice, job shops that never make twenty of anything, characteristics that cannot be normal, and the moment a chart signals and someone has to decide what to do.

Learning objectives

• Build p, np, c and u charts with the correct limits, including limits that move with the sample size, and choose between them.
• Recognise over-dispersion on a high-volume p chart and apply Laney's p′ correction.
• Select the right chart for a given data type, subgroup structure and question, using the selection tree.
• Set up an EWMA and a tabular CUSUM chart, explain their parameters, and say when they beat a Shewhart chart.
• Chart a short-run, mixed-part process with a deviation-from-nominal chart, and a non-normal characteristic without fooling yourself.
• React to a signal in the right order, and handle the case of a process that is out of control while every part is within specification.

Before you start

Three questions on Modules 1 and 16. They do not count toward completion.

Why this matters

Most of the data a plant collects every day is not a micrometer reading in a subgroup of five. It is a count: units rejected at a leak test, pores on a radiograph, defects found by optical inspection, cores returned. Those counts deserve a chart with limits that respect how counts vary, and they get one in the first half of this module. The second half is about the situations where the Shewhart chart of Module 16 is the wrong tool: small sustained drifts, which a three-sigma chart takes tens of points to see; high-volume proportions, whose binomial limits are far too tight; short runs of many part numbers; and characteristics that are bounded or skewed by physics.

Two of the disagreements in this module are documented ones. On charts for non-normal data, the AIAG & VDA SPC manual states that the charts Shewhart developed require normally distributed data, and recommends larger subgroups, Pearson charts for skewed distributions and extended charts for multimodal ones;^[4] Wheeler holds that three-sigma limits bracket nearly all of any homogeneous histogram and that no normality check or transformation is needed before charting.^[15] Both are right about something, and the flatness example in this module shows exactly what each is right about. On over-dispersion, the classic p chart's limits assume that the only variation in a day's proportion is binomial sampling, which for thousands of units a day is almost never true; Laney's 2002 correction is now standard in software and the example here shows why.^[5] The EWMA and CUSUM charts have their own history, Roberts in 1959 and Page in 1954, and the NIST handbook's worked EWMA example is reproduced by this course's harness as a published check.^[8][11][10]

The following is an illustrative failure with invented details, not a real program. A connector line tests five thousand units a day. Somebody puts the daily reject proportion on a p chart. The binomial limits for n = 5,000 are a fraction of a percentage point wide; a third of the days fall outside them; the line leader is asked to explain each one; after two weeks the chart is still on the wall and nobody looks at it. The proportion really does vary from day to day, with the material lot, the ambient humidity and the test fixture, and that variation is the process's routine variation at the scale of a day. The chart was told the wrong noise. Worked example 3 fixes it.

Attribute charts: p, np, c, u

Attribute charts come in two families, and the first decision is which one you have. If each unit is classified as conforming or not, you are counting defectives (nonconforming units), the count is binomial, and the charts are p (proportion, sample size may vary) and np (count, sample size constant). If you are counting defects (nonconformities) on a unit or an area, where one unit can carry several, the count is Poisson, and the charts are c (count per constant inspection unit) and u (defects per unit, inspection units may vary).^{[1][4], [2], [3]} The limits follow from the two distributions, and every one of them is a centre line plus and minus three standard deviations of the plotted statistic, exactly as in Module 16.

Three things about attribute charts that are not true of variables charts. They cannot warn before nonconformities occur: a chart of zeros has nothing to say, which is why the manual observes that attribute charts do not serve a zero-defect strategy.^[4] They need large samples to be useful: with p̄ = 2 % and n = 50 the expected count per sample is one, and the chart can only distinguish very large changes. And the plotted statistic is discrete, so the lower limit is often zero and a fall in the defect rate, which is the change you most want to see, is hard to detect; a run below the centre line (rule 4) is often the only evidence of improvement.

Worked example 1: a p chart that catches a supplier change

The data below is a constructed example generated by a seeded script, not a real production run. Setting: pressed fittings leak-tested 100 %, one lot per shift for 30 shifts, lot sizes 180 to 260; a change of seal supplier took effect at lot 21.

p̄ = 271 / 6,501 = 0.0417. The standard deviation of a lot proportion depends on the lot: for n = 180 it is √(0.0417 × 0.9583 / 180) = 0.0149 and the limits are −0.0030 (set to zero) and 0.0864; for n = 260 the limits are 0.0045 (zero) and 0.0789. The lower limit is at or below zero for every lot, so this chart can never signal an improvement by rule 1. Lot 23 (21 rejects in 200, p = 0.1050) is beyond its limit of 0.0841: the first signal, two lots after the change. Rule 2 also fires at lots 23 and 25, and rules 3 and 4 at lots 29 and 30 as the run above the centre line builds. The first 20 lots ran at 3.00 % and the last ten at 6.52 %.

Two lessons. The chart caught a doubling of the reject rate within two or three lots, which is what a p chart is for. But look at the centre line: p̄ was computed from all 30 lots, so it already contains the higher rate, and the limits are wider than the first 20 lots alone would give. In use, the limits would have been established from the first 20 lots and frozen (the calculator's freeze option shows this), and the signal would have come sooner and more clearly. The response to the signal is in the last section of this module; the short version is that "new seal supplier at lot 21" should be written on the chart before anyone starts a root-cause investigation.

Worked example 2: a c chart with one special cause, and the np and u charts

Constructed data, not a real production run. Setting: gas pores counted on the radiograph of a die-cast housing, one casting per hour for 30 hours.

c̄ = 150 / 30 = 5.000 pores per casting; √c̄ = 2.236; UCL = 5.000 + 3 × 2.236 = 11.708; the lower limit, −1.708, is negative and set to zero. Casting 17, with 14 pores, is beyond the limit; no run rule fires elsewhere. One special cause (the log recorded a die-spray interruption that hour), documented and excluded, and the chart continues.

Two more constructed sets, shown for the limits only. An np chart of rejects in fixed samples of 200 connectors from an automatic gauge, 25 hourly samples: np̄ = 8.24, p̄ = 0.0412, √[np̄(1 − p̄)] = 2.811, UCL = 8.24 + 8.43 = 16.67, LCL −0.19 → 0; stable, no rule fires. A u chart of solder defects found by optical inspection on lots of 8 to 15 boards, 25 lots: ū = 431 / 296 = 1.4561 defects per board; for a lot of 8 boards the limits are 0.1762 to 2.7360 and for 15 boards 0.5214 to 2.3908, moving with the lot size; stable. The u chart's lower limits are positive here because ū × n is large enough, so this chart could signal an improvement.

Over-dispersion and Laney's p′ chart

The p chart's limits assume that if the true proportion were constant, a day's proportion would vary only by binomial sampling: standard deviation √[p̄(1 − p̄)/n]. For n in the thousands that standard deviation is tiny, and any genuine day-to-day variation in the true proportion, which every real process has, puts most days outside the limits. The chart is not detecting special causes; it is detecting that the day is the wrong subgroup for binomial noise. Laney's 2002 correction measures the actual day-to-day variation and scales the limits by it.^[5]

Worked example 3: a high-volume line

Constructed data, not a real production run. Setting: an automated end-of-line test on a high-volume connector line, 4,000 to 6,000 units a day for 25 days; the data were constructed with a true daily proportion that varies from day to day around 2 %, as material lots change.

p̄ = 2,455 / 126,762 = 0.01937, that is 1.94 %. The binomial standard deviation of a day's proportion is about 0.00195; the classic p chart's limits for day 1 (n = 5,070 on average) are 0.0135 to 0.0252, and 7 of the 25 days fall outside their limits: days 10, 11, 12, 13 and three more. The observed standard deviation of the 25 daily proportions is 0.00528, 2.7 times the binomial value. The days vary far more than sampling explains.

The z-scores make it explicit: day 11 has p = 0.0298 and z = 5.74. The moving range of the z-scores averages 2.733, so σ_z = 2.733/1.128 = 2.423. The p′ limits for day 1 become p̄ ± 3 × 0.00196 × 2.423 = 0.0051 to 0.0336, and no day is outside them; no runs rule fires on the z-scores either. The line is stable at the scale of a day, with a day-to-day variation about 2.7 times what a binomial would give. That is the correct description, and it is also a finding: the routine variation of this process includes something that changes daily, which a Module 10 root-cause study or a Module 13 experiment might reduce. The p′ chart did not hide the variation. It stopped calling it a special cause 25 times a month.

Use the p′ chart when subgroups are large (hundreds to thousands) and the classic chart signals on most points; check σ_z first, because a value near 1 means the classic chart was right. Do not use it to silence a chart on small samples, where over-dispersion cannot be estimated from a moving range, and do not use it to hide a genuine trend: a run of z-scores rising over weeks is still a signal, and the p′ chart keeps the runs rules to find it.

Which chart? A selection tree

Two questions the tree does not show but which come first: is the measurement system adequate (Module 4), and is the subgroup rational (Modules 5 and 16)? No chart survives a bad answer to either. And one question that comes last: what will be done when the chart signals? A chart without a reaction plan is decoration (Module 18).

Charts with memory: EWMA and CUSUM

A Shewhart chart judges each point on its own; that is what makes it fast on large changes and slow on small ones (Module 16, Table 5: a one-sigma shift takes 44 points on average with rule 1). Two charts accumulate evidence across points instead. Roberts' exponentially weighted moving average (1959) plots a running average that weights recent points most;^[8] Page's cumulative sum (1954) accumulates the deviations from target above and below a small allowance and signals when the accumulation exceeds a decision interval.^[11] Both detect a half-sigma to one-sigma shift in a fraction of the Shewhart chart's time, and both are slower on a large jump, which is why the AIAG & VDA manual recommends running a Shewhart chart alongside.^[4] ISO 7870-4 and 7870-6 standardise the two charts.^[16]

Worked example 4: a 0.7σ drift on three charts

Constructed data, not a real production run. Setting: the etch depth of a wafer in µm from a profilometer to 0.1 µm, one wafer per hour for 60 hours; a gas-flow controller drifted from hour 31. The data were constructed with a standard deviation of 2.0 µm and a shift of 1.4 µm, 0.7 standard deviations, from reading 31; the first 30 readings are the baseline for every chart.

Baseline. From the first 30 readings, target = 49.167 µm and σ̂ = MR̄/1.128 = 1.590 µm. The readings after hour 30 average 51.38 µm against 49.17 before, an observed shift of 2.21 µm, about 0.88 σ̂.

Individuals chart (limits from all 60 readings, 44.79 to 55.75 µm): rule 1 never fires in the 30 shifted readings. Rules 2 and 3 fire at readings 21 and 23, before the drift began, which on this record is a false alarm and, in real life, a reminder that the story of "a drift from hour 31" is itself a hypothesis the chart cannot confirm.

EWMA, λ = 0.2, L = 3: the limits start narrow (target ± 0.954 at the first point, since (1 − 0.8²) = 0.36) and settle at target ± 3 × 1.590 × √(0.2/1.8) = target ± 1.590, that is 47.58 to 50.76 µm. The first EWMA value is 0.2 × 50.0 + 0.8 × 49.167 = 49.333. After the drift the EWMA climbs: 48.86, 49.22, 50.17, 50.33 at readings 31, 33, 35 and 37, and crosses the upper limit at reading 38, 8 readings after the change, staying above it for most of the rest of the record.

CUSUM, k = 0.5, h = 4 (in σ̂ units: an allowance of 0.79 µm and a decision interval of 6.36 µm): C⁺ sits at zero or near it through the baseline (it is 0.15 at reading 30), then accumulates: 2.23 at reading 35, 2.97 at 37, and 5.00 at reading 38, beyond h = 4: a signal 8 readings after the change, the same hour as the EWMA. C⁻ never exceeds 0.5h.

Eight readings to catch a 0.7σ shift, against a Shewhart chart that had not caught it after 30. That is the case for memory charts, and the price is visible in the same record: the EWMA's first values are dominated by the target and would be slow to see a jump at hour 3, and neither chart says anything about the shot-to-shot spread. Run them beside a Shewhart chart, not instead of one. A useful by-product of the CUSUM: counting back from the signal to where C⁺ last touched zero (reading 34 here) estimates when the shift began, which is where the investigation should start.

A published check: the NIST EWMA example

The NIST handbook's EWMA example (20 values, λ = 0.3, target 50, σ = 2.0539) gives asymptotic limits of 47.4116 and 52.5884 (3 × 2.0539 × √(0.3/1.7) = 2.5884 either side of 50) and an EWMA sequence starting 50.60, 49.52; the course's harness reproduces the limits and every value of the sequence the handbook prints.^[10]

Short runs: the deviation-from-nominal chart

A job shop that turns three part numbers on one lathe, ten of each in a mixed order, never has twenty subgroups of one part to set limits from. If the process is the same (same machine, same tooling, same operator, same routine variation) and only the nominal differs, chart the deviation of each part from its own nominal, x − nominal, on one I-MR chart. This is the deviation-from-nominal (DNOM) technique of ISO 7870-8 and of Wheeler's Short Run SPC, and the AIAG & VDA manual points to that standard for the high-mix, low-volume case.^[13][14][4] The assumption to check is that the routine variation is the same for every part number; if a 32 mm diameter varies twice as much as a 20 mm one, divide each deviation by its part's own σ̂ instead and chart the standardised value (Wheeler's Zed chart), which needs a σ̂ per part number from history.

Worked example 5: three diameters on one lathe

Constructed data, not a real production run. Setting: three part numbers with nominal diameters 20.00, 25.00 and 32.00 mm turned on one lathe in the order the jobs arrived, 30 parts, micrometer to 0.001 mm.

The deviations have mean 0.0006 mm and MR̄ = 0.0125 mm, so σ̂ = 0.0111 mm and the I chart limits are −0.0326 to 0.0338 mm; the MR limit is 0.0408 mm. No deviation or moving range crosses a limit. Rule 3 (four of five beyond 1σ) fires at part 15: parts 11, 13, 14 and 15 all ran 0.011 to 0.018 mm small, two P25 and two P32 parts in a row, which is the kind of pattern worth a glance at the tool (a chipped insert cuts small on every part number). Per part number, the deviations average −0.0017, −0.0009 and 0.0035 mm for P20, P25 and P32 with standard deviations of 0.0107, 0.0103 and 0.0113 mm; the largest is 1.10 times the smallest, close enough to pool. Had P32's spread been double the others', the chart would need standardising.

The DNOM chart also answers a question the three separate charts could not: the lathe as a process, across everything it makes, is stable with a routine standard deviation of about 0.011 mm. That is the number to compare with each part's tolerance (Module 7), and the number the next job's quotation should be based on.

Charts for non-normal data

Module 8 met the flatness data, 200 values from a lognormal process bounded at zero, and saw that a normal-based capability index was wrong by orders of magnitude. What does an individuals chart do with the same data?

Constructed data: the Module 1 flatness values, µm, in production order. On the raw values, x̄ = 7.875, MR̄ = 3.711, σ̂ = 3.289, and the limits are −1.99 (impossible for a flatness) and 17.74 µm. Three readings exceed the upper limit (112, 168 and 181), 1.5 % of the record against the 0.27 % a normal process would give, about 6 times the expected false-alarm rate, all of them in the long tail; rules 2, 3 and 4 fire as well at 121, 133 and 135 to 137, and seven moving ranges cross their limit. On ln x the limits are 0.715 to 3.226, which is 2.0 to 25.2 µm, one reading (181) is beyond, one moving range, and the same runs rules still fire around 121 to 137.

Read this against the two positions quoted at the start. The manual is right that a symmetric three-sigma chart on a strongly skewed characteristic gives more tail signals than the nominal rate, and that a lower limit below zero is meaningless; its remedies, larger subgroups (X̄ charts of skewed data are close to normal for n of 4 or 5, Module 1), a transformation, or limits from a fitted skewed distribution, all work here.^[4] Wheeler is right that the raw chart still bracketed 197 of 200 values, that the excursions are in the direction the physics predicts, and that no transformation was needed to see the runs around 121 to 137, which are the more interesting feature: something happened over those fifteen parts that both charts agree on.^[15] The practical rule for a Green Belt: chart bounded, skewed characteristics on subgroup means when you can; when you must chart individuals, expect a few extra tail signals and read them with the physics in mind, and use a log scale when the tail signals drown the runs.

Reacting to a signal

A chart signal is a question, not a verdict, and the order of the answer matters.

1. Check the measurement and the entry. A transposed digit, a gauge left on the wrong range, a part measured twice. Half of first signals end here.
2. Note what changed. Time, operator, lot, tool change, maintenance, ambient. Write it on the chart. The seal supplier of worked example 1 goes here, before any analysis.
3. Contain. If the signal could mean nonconforming product, hold what was made since the last good point and sort or re-test as the control plan says (Module 18).
4. Find the cause with the Module 10 tools, on the process, not on the chart.
5. Correct and verify. Remove the cause, then show on the chart that the process has returned: several points inside the limits with no pattern, not one.
6. Decide about the limits. A documented, removed special cause is excluded from the limit calculation; an unexplained signal stays in the data. Limits change only after a deliberate, verified process change.
7. Record the signal, the cause and the action. The record is what a later engineer, or a customer audit under IATF 16949, will ask for (Module 18).

Every one of these steps belongs in a written reaction plan (an out-of-control action plan) attached to the chart, so that the operator on the night shift does not have to invent it. What the plan must never say is "adjust toward nominal": that is the funnel experiment again.

Out of control, but every part within specification

This is the third and last treatment of the limits question in this course, and it is the case that confuses production most. A p chart signals a doubled reject rate that is still under the customer's ceiling; an X̄ chart signals a shift of 0.010 mm inside a ±0.025 mm tolerance; a CUSUM says the etch depth has drifted by 1.4 µm on a process whose specification is ±10 µm. Nothing is nonconforming. Is there a problem?

Yes, and it is a different problem from a nonconforming part. The chart has detected that the process is not the one whose capability was established. Its future output is no longer predictable from the study; the next shift of the same size may cross the specification; the cause, whatever it is, is now in the process and will stay there until found. The right response is the seven steps above, at the urgency the control plan assigns, and the wrong responses are the two obvious ones: to ignore the signal because the parts are good, and to adjust the process back by the amount of the shift without knowing why it moved. The AIAG & VDA manual's framing is useful: a process can be stable and not capable (Module 7's case, a capability problem for engineering), capable and not stable (this case, a control problem for the line), or, the goal, both.^[4] Specification limits tell you about the part in your hand. Control limits tell you about the parts you have not made yet.

Common mistakes

1. A p chart on tiny samples. Consequence: expected counts below one, a chart of zeros and an occasional 1 that means nothing. Fix: samples large enough for np̄ of at least 5, or an np chart on a fixed large sample, or a variables measurement instead.
2. A c chart for defectives, or a p chart for defects. Consequence: the wrong distribution and the wrong limits. Fix: units conforming or not → binomial (p, np); defects counted on units → Poisson (c, u).
3. Binomial limits on thousands of units a day. Consequence: most days flagged, chart ignored. Fix: check σ_z; use the p′ chart when it is well above 1.
4. An EWMA with no Shewhart chart beside it. Consequence: a large sudden jump is seen late. Fix: run both; the Shewhart chart owns the big changes.
5. CUSUM parameters chosen by default. Consequence: k tuned to the wrong shift, h giving false alarms every few dozen points. Fix: k = half the shift that matters, h = 4 or 5, and check the in-control ARL against the tables.
6. A DNOM chart pooling parts with different spreads. Consequence: the loose part's variation sets limits that the tight part can never violate. Fix: compare the per-part standard deviations; standardise when they differ.
7. Transforming data before checking homogeneity. Consequence: the runs that both charts of the flatness example saw are attributed to "non-normality". Fix: look for the pattern first; transform for the tails, not for the runs.
8. "The parts are in spec, so ignore the signal." Consequence: an unexplained shift stays in the process until it is large enough to make scrap. Fix: the seven steps, at the urgency the control plan assigns.

Exercises

Exercise 1: defects per joint on brazed cores of different sizes

Constructed data, not a real production run. Setting: braze-joint defects found by pressure decay and dye check on heat-exchanger cores, one core per shift for 20 shifts; the cores are of different models with 40 to 80 joints each.

Tasks. (a) Which chart, and why not a c chart? (b) Compute ū and the limits for the smallest and largest cores. (c) Which core signals, and how would you check that it is not the model rather than the shift? (d) After the cause is found and fixed, what should the limits be based on?

Exercise 2: a half-sigma shift, two charts

Constructed data, not a real production run. Setting: plating thickness in µm by XRF to 0.01 µm, one panel per hour for 40 hours; a bath concentration change from hour 21. The data were constructed with a shift of half a standard deviation from reading 21; the first 20 readings are the baseline.

Tasks. (a) Build the I-MR chart on all 40 readings; does it signal? (b) Build an EWMA with λ = 0.2 and L = 3 from the first 20 readings; when does it signal? (c) Why does the EWMA see what the I chart does not? (d) Would you act on the EWMA signal if the specification were ±0.5 µm?

Quiz

Ten questions. Score 70 % or more to mark the module complete on this device.

Key takeaways

• Defective units are binomial (p, np charts); defects are Poisson (c, u charts). Every limit is the centre ± 3 standard deviations of the plotted statistic, and the limits move with the sample size on p and u charts. They also cannot warn before nonconformities occur, they need large samples, and a fall in the rate usually shows only as a run below the centre line.
• On thousands of units a day the binomial limits are far too tight. Compute σ_z from the moving range of the z-scores; if it is well above 1, use Laney's p′ chart.
• Choose the chart by data type, subgroup structure and the shift size that matters; Figure 1 is the tree. Measurement system and rational subgrouping come before any of it.
• EWMA (λ ≈ 0.2, L ≈ 3) and CUSUM (k = half the shift, h = 4 to 5) detect a half- to one-sigma shift in a fraction of a Shewhart chart's time and are slow on large jumps; run them beside a Shewhart chart.
• Short runs: chart the deviation from nominal across part numbers on one I-MR chart when the routine variation is the same; standardise when it is not.
• Bounded, skewed characteristics give extra tail signals on a symmetric individuals chart; chart subgroup means or a log scale, and read tail signals with the physics in mind. Runs are the signal both views agree on.
• A signal with every part in specification is still a signal: the process is no longer the one whose capability was established. Follow the reaction plan; never adjust toward nominal.

References

All web sources accessed 2026-09-09 or 2026-09-10. Sources marked "secondary" or "catalogue" were not read in full by the course author; the claim is taken from the part shown. Standards and books are paraphrased.

1. NIST/SEMATECH. "6.3.3. What are Attributes Control Charts?" and "6.3.3.2. Proportions Control Charts." e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc33.htm
2. NIST/SEMATECH. "6.3.3.1. Counts Control Charts." e-Handbook of Statistical Methods. The c chart and its Poisson basis. https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm
3. NIST/SEMATECH. "6.3.3.2. Proportions Control Charts." e-Handbook of Statistical Methods. The p chart formulas. https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm
4. AIAG & VDA. Statistical Process Control (SPC) Manual: Process Management, Performance and Capability, Control Charts, 1st ed. AIAG and VDA QMC, February 2026 (AIAG catalogue SPCAV-1). Sections 10.3.2.5 to 10.3.2.9 (pp. 88 to 92: charts with memory, non-normal characteristics, short runs), 10.3.5.4 to 10.3.5.5 (pp. 111 to 116: CUSUM with its ARL example, EWMA and the recommendation to run a Shewhart chart alongside), 10.3.6 (pp. 117 to 118: attribute charts, the binomial and Poisson basis, np̂(1 − p̂) ≥ 9, sample-size fluctuation below 25 %). https://www.aiag.org/training-and-resources/manuals/details/SPCAV-1 (licensed copy read by the author)
5. Laney, D. B. "Improved Control Charts for Attributes." Quality Engineering 14(4):531 to 537, 2002. https://www.tandfonline.com/doi/abs/10.1081/QEN-120003555 (abstract read; the p′ and u′ charts for over-dispersed data)
6. Arafah, M. "Using the Laney p′ Control Chart for Monitoring COVID-19 Cases in Jordan." Journal of Healthcare Engineering, 2022. https://pmc.ncbi.nlm.nih.gov/articles/PMC9553754/ (open access; the source read for the z-score, σ_z = MR̄/1.128 and limit formulas as implemented here)
7. Minitab, LLC. "Methods and formulas for Laney P′ Chart." Minitab Statistical Software help. https://support.minitab.com/.../laney-p-chart/methods-and-formulas/ (the same construction as implemented in software; page read, formulas rendered as images)
8. Roberts, S. W. "Control Chart Tests Based on Geometric Moving Averages." Technometrics 1(3):239 to 250, 1959. https://www.tandfonline.com/doi/abs/10.1080/00401706.1959.10489860 (catalogue; origin of the EWMA chart)
9. Lucas, J. M., and Saccucci, M. S. "Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements." Technometrics 32(1):1 to 12, 1990. https://www.tandfonline.com/doi/abs/10.1080/00401706.1990.10484583 (catalogue; the ARL tables for choosing λ and L)
10. NIST/SEMATECH. "6.3.2.4. EWMA Control Charts." e-Handbook of Statistical Methods. Formulas, the typical λ of 0.2 to 0.3, and the worked example reproduced by this course. https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc324.htm
11. Page, E. S. "Continuous Inspection Schemes." Biometrika 41(1/2):100 to 115, 1954. https://doi.org/10.1093/biomet/41.1-2.100 (catalogue; origin of the CUSUM)
12. NIST/SEMATECH. "6.3.2.3. Cusum Control Charts" and "6.3.2.3.1. Cusum Average Run Length." e-Handbook of Statistical Methods. The tabular CUSUM and the k = δ/2, h = 4 or 5 guidance. https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc323.htm
13. ISO 7870-8:2017. Control charts. Part 8: Charting techniques for short runs and small mixed batches. International Organization for Standardization. https://cdn.standards.iteh.ai/samples/67410/.../ISO-7870-8-2017.pdf (scope and structure read from the public preview)
14. Wheeler, D. J. Short Run SPC. SPC Press, 1991. https://www.amazon.com/Short-Run-SPC-Donald-Wheeler/dp/0945320124 (catalogue; difference and Zed charts)
15. Wheeler, D. J. "Problems with Skewness and Kurtosis, Part Two." Quality Digest, 2 August 2011 (SPC Press manuscript 232). https://www.spcpress.com/pdf/DJW232.pdf
16. ISO 7870-4:2021, Control charts. Part 4: Cumulative sum charts; ISO 7870-6:2016, Control charts. Part 6: EWMA control charts. International Organization for Standardization. https://www.iso.org/standard/74101.html ; https://www.iso.org/standard/40173.html (secondary: existence and scope from catalogue listings; not read)

Further reading

• Montgomery, D. C. Introduction to Statistical Quality Control, 8th ed. Wiley, 2019. Chapters on attribute charts, CUSUM and EWMA with ARL tables and design guidance.
• Module 18 for control plans and reaction plans; Module 8 for the flatness data's capability; Module 16 for the Shewhart charts and their false-alarm arithmetic.