Module 16 · Control

Statistical process control I

A control chart is a running record of a process with two lines drawn on it that say how far the process wanders when nothing has changed. Everything about it, the constants, the rules, the false alarms, follows from that one idea. This module builds the three variables charts by hand from printed data, states the run rules exactly, computes what each rule costs in false alarms, and returns to the question of how often to sample.

Learning objectives

• Explain Shewhart's distinction between common and special causes and what a control chart does with it.
• State the difference between control limits and specification limits and why they never appear on the same chart.
• Build X̄-R, X̄-S and I-MR charts from raw data with the tabulated constants, and choose the right one for the data you have.
• Apply the Western Electric and Nelson rules exactly as written, and read a pattern (a shift, a trend, a special cause) from the chart.
• Quantify the false-alarm rate and average run length of a chart with and without runs rules, and choose rules deliberately.
• Decide how often to sample and how to form subgroups so that the chart answers the question you are asking.

Before you start

Three questions on Modules 2, 5 and 7. They do not count toward completion.

Why this matters

On 16 May 1924 Walter Shewhart, a physicist at Western Electric's Bell Telephone Laboratories group, sent his manager a one-page memo with a small diagram attached. The diagram was the first control chart. The memo's point was that some variation in a manufactured product is due to chance and is to be expected, and some is due to assignable causes that can be found and removed; the chart was a way to tell the two apart in the running record of a process.^[1] Shewhart worked the idea out in his 1931 book, where the choice of three-sigma limits is justified on economic grounds, as a balance between looking for causes that are not there and missing causes that are, rather than by any probability distribution.^[2] A century on, the chart is unchanged, and the argument about what it means is still going: Woodall's 2000 review of the controversies in SPC lists the hypothesis-testing view of a control chart against Shewhart's own, the role of statistical theory, and the gap between researchers and practitioners as live questions.^[9] This module teaches the chart as Shewhart intended it, an operational rule for telling noise from signal, and gives the probability arithmetic where it helps, which is in choosing rules.

The following is an illustrative failure with invented details, not a real program. A machining cell charts a bore diameter. Somebody helpful draws the drawing limits, ±0.025 mm, on the chart as red lines, "so the operator can see the spec". The operator, seeing a reading move toward a red line, adjusts the offset. The next reading moves back past nominal, so the operator adjusts again. Within a week the measured variation has roughly doubled and the cell is producing more scrap than before the chart existed. Module 2's funnel experiment showed the arithmetic: adjusting a stable process to every deviation inflates its variance. The chart did not fail. It was never a control chart; it was a specification chart, and a specification chart tells an operator to tamper.

Common causes, special causes, and what the chart does

Module 2 introduced the vocabulary in words. Here it becomes operational. Shewhart's chance causes, Deming's common causes, are the many small influences that are always present: the play in a bearing, the grade of the material within its specification, the temperature of the shop, the operator's ordinary variability. Their sum is the process's routine variation, and it is the same from hour to hour as long as the system is the same. Assignable or special causes are events: a tool chips, a lot of material is different, a fixture works loose, a new operator misreads a setting. They are not part of the system and their effect is a change in the record.

A control chart makes the distinction with a rule. Take the process's routine variation, estimated from small samples taken close together in time (the rational subgroups of Module 5), and draw limits at three of those standard deviations either side of the centre. A point outside the limits, or a pattern inside them that routine variation would almost never produce, is the signal to look for a special cause. Everything else is noise, and reacting to noise is tampering. The two mistakes are not symmetric in cost, and Shewhart set the limits where he did because in his experience three sigma made both mistakes rare enough to live with.^[2]

Control limits are not specification limits

This is the second of three places in the course where this point is made, because it is the most common error in industry (Module 7 made it first; Module 17 will make it again). The two kinds of limit come from different sources, describe different things, and have different consequences when crossed.

A subgroup mean of five parts can be inside the specification while a part in the subgroup is outside it, and the X̄ limits are tighter than the specification by roughly √n for a capable process, so putting the two on one chart misleads in both directions at once. The AIAG & VDA manual is explicit that tolerance limits are not entered on a control chart, and it treats the process-related chart (limits from the data) and the tolerance-related chart (a different tool, for a different purpose) as separate concepts.^[3]

Why three sigma, and what 0.27 % means

If the plotted statistic were exactly normal and the process exactly stable, the chance that one point falls outside ±3σ is 0.27 %, about one in 370. That number is worth knowing, and it is not the reason for the limits. Shewhart chose three sigma because it worked across the distributions he met, and Wheeler's later argument is that for any homogeneous data, of whatever shape, three-sigma limits bracket nearly all of the points, which is what makes the chart usable without a normality check (Module 8).^[11] The 0.27 % is best read as an upper reference: the false-alarm rate of rule 1 alone on a well-behaved process is around one point in a few hundred. The section on average run length below turns that into a number for every rule set.

Building the charts by hand

Three charts cover most variables data. The X̄-R chart plots subgroup means and ranges for subgroup sizes of about 2 to 10; the X̄-S chart plots means and standard deviations and is preferred above about n = 10, where the range wastes information; the I-MR chart plots individual values and the moving range between successive values when subgroups are impossible or pointless (one batch a day, one temperature reading an hour, destructive tests).^[4][5] All three use the same logic: estimate the routine standard deviation from the within-subgroup spread, then set the limits at three of the plotted statistic's standard deviations. The constants absorb the arithmetic and are tabulated by subgroup size.^[4]

X̄-R chart: the bore diameter again

Constructed data, not a real production run: the Module 7 bore study, Ø12.000 ± 0.025 mm, 25 subgroups of 5 consecutive parts every 30 minutes, bore micrometer to 0.001 mm. The full table is on the Module 7 page; the subgroup means and ranges are what the chart needs.

x̄̄ = 12.0039 mm and R̄ = 0.01536 mm. A₂R̄ = 0.577 × 0.01536 = 0.00886 mm, so the X̄ limits are 11.9950 and 12.0128 mm. D₄R̄ = 2.114 × 0.01536 = 0.0325 mm is the upper limit of the R chart, and its lower limit is zero because D₃ = 0 for n ≤ 6. The routine standard deviation is σ̂ = 0.01536 / 2.326 = 0.00660 mm, and the standard deviation of a subgroup mean is 0.00660/√5 = 0.00295 mm; three of those is 0.00886 mm, which is A₂R̄ again. Compare that with the half-tolerance of 0.025 mm: the X̄ limits are 2.8 times tighter than the specification. No mean and no range crosses its limit, and, checking the rules in the next section, nothing else fires: the process was stable over the study. The calculator below redraws both charts from the Module 7 table.

Read the R chart first, always. The X̄ limits are computed from R̄, so if the ranges are out of control the X̄ limits are wrong before you have looked at them. A range chart that signals says the within-subgroup spread changed, which usually points to something different from a shift in the mean: a loose fixture, a worn tool making the size wander within minutes, a mixed lot inside a subgroup.

X̄-S chart: a larger subgroup

Constructed data, not a real production run. Setting: the seal width of a heat-sealed pouch in mm, measured with a calibrated loupe to 0.01 mm, ten consecutive pouches every hour for 20 hours.

x̄̄ = 4.9954 mm, S̄ = 0.05506 mm. A₃S̄ = 0.975 × 0.05506 = 0.05368, so the X̄ limits are 4.9417 and 5.0491 mm; the S chart runs from B₃S̄ = 0.01564 to B₄S̄ = 0.09448 mm. σ̂ = S̄/c₄ = 0.05661 mm, and three standard deviations of a mean of ten is 3 × 0.05661/√10 = 0.05370 mm, A₃S̄ again. No subgroup mean or standard deviation crosses a limit and no Western Electric rule fires. (Nelson's rule 7, fifteen points in a row within one sigma of the centre, does fire at subgroups 15 and 16 on the X̄ chart; with limits this well estimated and twenty subgroups that is a false alarm of the kind the next section prices, and a reason to use that rule only when stratification is a real suspicion.)

I-MR chart: one reading at a time

Constructed data, not a real production run. Setting: the zone-3 temperature of a braze furnace in °C, logged hourly from the controller's thermocouple to 0.1 °C, 50 consecutive hours.

x̄ = 610.342 °C, MR̄ = 1.3163 °C. σ̂ = 1.3163/1.128 = 1.1670 °C; 3σ̂ = 3.5009, so the I chart limits are 606.84 and 613.84 °C, and the MR chart's upper limit is 3.267 × 1.3163 = 4.300 °C. Reading 33, 615.7 °C, is 4.6 standard deviations above the mean, 1.9 °C above the upper limit, and the moving range from reading 33 back down to reading 34 (5.4 °C) is beyond the MR limit as well (the range into the spike, 3.8 °C, is just inside it). That is a special cause: one reading, not a shift. The log showed a door interlock fault that hour. The reading is real, the cause is known and fixed, and the right treatment is to record it, remove it from the limit calculation, and keep charting.

Recomputing the limits without the assigned cause. On the other 49 readings, x̄ = 610.233 °C, MR̄ = 1.1854 °C, σ̂ = 1.0509 °C and the limits are 607.08 to 613.39 °C, 0.70 °C narrower in total. A single spike inflated the moving-range average and with it every limit on the chart, which is the general lesson: limits computed from data that contain a special cause are too wide, and a chart with limits that are too wide misses the next special cause. (The inflated limits also produced a spurious Nelson rule 7 signal, fifteen readings within one sigma, at readings 25 and 26; it disappears with the corrected limits.) The AIAG & VDA manual's guidance on outliers is the same: an outlier with an identified cause is documented and excluded; one without a cause is data.^[3]

A published check: the NIST individuals chart

The NIST handbook's I-MR example, ten batch readings, gives x̄ = 50.81, MR̄ = 1.8778, and limits 45.8159 and 55.8041; the course's three routes reproduce all four figures (3 × 1.8778/1.128 = 4.9941 either side of the mean).^[5]

Calculator: control chart builder

Pre-loaded with the Module 7 bore data as an X̄-R chart. Switch the type, paste your own data, select which rules to flag, or freeze the limits from the first k subgroups (a baseline) and watch later subgroups against them, which is how a chart is used in production once its limits are established.

The runs rules

A point beyond three sigma is rule 1. It catches a large change quickly and a small one slowly: a shift of one standard deviation of the plotted statistic takes, on average, 44 points to produce a point beyond the limit (next section). The runs rules look for patterns that a stable process would rarely make, so that smaller changes are caught sooner. The four rules from the Western Electric Statistical Quality Control Handbook of 1956 divide each side of the chart into zones one sigma wide (C nearest the centre, then B, then A) and read:^[6][7]

1. One point beyond zone A (beyond 3σ).
2. Two out of three consecutive points in zone A or beyond (beyond 2σ), on the same side.
3. Four out of five consecutive points in zone B or beyond (beyond 1σ), on the same side.
4. Eight consecutive points on the same side of the centre line.

Nelson's 1984 note in the Journal of Quality Technology lists eight tests, which most software offers as a menu: (1) one point beyond zone A; (2) nine in a row on one side; (3) six in a row steadily increasing or decreasing; (4) fourteen in a row alternating up and down; (5) two of three beyond 2σ on one side; (6) four of five beyond 1σ on one side; (7) fifteen in a row within zone C; (8) eight in a row beyond zone C on both sides.^[8] Tests 1, 5, 6 and 2 are the Western Electric rules with the run length raised from eight to nine; tests 3 and 4 look for trends and systematic alternation; tests 7 and 8 look for stratification and mixtures, which are signs that the subgroups are not rational. Nelson's own advice was that the tests are for use on charts whose limits are established and that not all of them should be applied at once. The AIAG & VDA manual states its stability criteria in the same terms, violations of the limits, runs (usually seven or more on one side), trends (usually seven or more rising or falling), and an unexpected count in the middle third of the chart (for a 25-point chart, fewer than 11 or more than 23 at a 1 % error rate), and notes that the Western Electric and Nelson rule sets are ways of counting the same things.^[3]

Worked example: a small shift the 3σ rule does not see

Constructed data, not a real production run. Setting: the mass of adhesive dispensed per shot in mg, one shot weighed every 10 minutes on a balance reading to 0.01 mg, 50 shots; a new lot of adhesive was loaded before shot 26. The data were constructed with a mean of 20.0 mg and standard deviation 0.5 mg for the first 25 shots and a mean of 20.5 mg for the rest.

The I chart on all 50 values has x̄ = 20.183 mg, MR̄ = 0.5418 mg, σ̂ = 0.4804 mg and limits 18.742 to 21.624 mg. (In production the limits would come from a baseline period; here the whole record is used, which pulls the centre line up toward the shifted values and makes the exercise slightly harder than real life.) The first 25 shots average 19.845 mg and the last 25 20.521 mg: a shift of 0.676 mg, about one routine standard deviation. What the rules see:

• Rule 1 never fires. No shot in 25 after the change reaches the upper limit; a one-sigma shift moves the mean to 2σ below the limit, and a point beyond it is still a one-in-40 event.
• Rule 4 (eight on one side) fires first, at shot 40, 15 shots after the change, and again at 49 and 50. Nelson's test 3 (six rising) fires at 40 and 41 as the run builds.
• Rules 2 and 3 fire at shot 50, 25 shots after the change, when two large values arrive together.
• Nelson's test 4 (fourteen alternating) fires at shots 14 to 16, before the change. That is a false alarm: with all eight tests switched on, a stable process produces one every few dozen points, and this one costs an investigation of a lot that was fine.

The MR chart is quiet throughout: the shot-to-shot spread did not change, only the level, which is how a chart distinguishes a shift in the mean from a change in the noise. The lesson runs both ways. Without the runs rules, a one-sigma shift in a dispensed mass would have run for a long time; with all eight Nelson tests, a stable period produced a false alarm in 25 points. The next section puts numbers on that trade.

False alarms and average run length

Every rule you add catches real changes sooner and raises false alarms more often. The way to compare rule sets is the average run length (ARL): the mean number of points plotted until the chart signals. In control, you want the ARL long (few false alarms); after a shift you want it short. For rule 1 alone on a normal statistic the in-control ARL is 1/p where p is the probability beyond ±3σ: 1/0.0027 = 370.4. With runs rules the arithmetic needs a Markov chain, which Champ and Woodall set up in 1987 and used to show exactly how much each supplementary rule shortens the in-control run.^[10] Table 5 was computed by the course's own implementation of that chain and checked by simulation; the values are not copied from the paper, and the rule-1 column is the closed form.

Read the first column. Rule 1 alone gives a false alarm every 370 points on average; adding rule 2 brings that to 225, rule 3 to 166, rule 4 to 153, and all four together to 92: four times as many false alarms as the plain chart. Now read the one-sigma column. The plain chart takes 43.9 points on average to see a shift of one standard deviation of the plotted statistic; all four rules together take 9.2, 4.8 times fewer. For a shift of three sigma the plain chart needs 2.0 points and the rules barely help. That is the trade in numbers: the runs rules buy sensitivity to small shifts at the price of four times the false alarms, and do nothing for large ones.

Two things to add. The shift in Table 5 is in units of the plotted statistic. On an X̄ chart with n = 5, a shift of one standard deviation of the individual parts is a shift of √5 = 2.24 standard deviations of the mean, so the plain X̄ chart of the bore would see it in between 6.3 and 2.0 subgroups, a couple of hours at one subgroup per half hour; subgrouping is how a Shewhart chart gets its sensitivity. And the in-control figures are exact only for a normal statistic on a perfectly stable process with limits known exactly; with limits estimated from 25 subgroups the true false-alarm rate is somewhat higher, which is one more reason not to switch on every rule.^[9] The practical guidance this course follows, which is also Nelson's and Wheeler's, is to use rule 1 always, add rules 2 to 4 when small sustained shifts matter and you can afford the investigations, and keep the trend and alternation tests for diagnosis rather than routine monitoring.^[8][11]

Simulator: watch the rules catch a shift

The simulator draws subgroups from a parent distribution, applies a shift at a chosen subgroup, and counts how many subgroups each chart and rule set needs to signal, over many repetitions. Pre-loaded with a one-sigma shift at subgroup 25 and subgroups of five; try n = 1 to see the I chart struggle, and a non-normal parent to see how little it matters at n = 5.

How often to sample, and rational subgroups again

The chart answers the question the subgrouping asks. Wheeler's rule for a rational subgroup is that it should come from a small region of time, space or product, so that the variation inside it is routine variation only; the chart then asks whether the variation between subgroups is more than routine.^[12] Five consecutive bores every 30 minutes asks "does the reaming process drift or jump between half-hour checks?" Five bores picked at random across a shift asks nothing useful, because a shift-to-shift change is now inside the subgroup and inflates R̄, and the limits widen until they miss everything. The AIAG & VDA manual makes the same two points about frequency: sample at a rate matched to how fast the process can change, since samples taken too far apart miss signals, and not so close together that consecutive values are autocorrelated, which makes the moving-range estimate of σ too small and the limits too tight.^[3]

A working procedure: list the ways the process can change (tool wear over hours, a lot change at each material delivery, a warm-up drift each morning, an operator change each shift), decide which of them the chart must catch and how soon, and set the subgroup so that none of those changes falls inside it and the sampling interval so that the fastest one of interest produces several subgroups before it matters. Then choose the chart: subgroups of 2 to 10 give X̄-R, larger ones X̄-S, and if the process changes so slowly that consecutive parts are effectively identical, or so rarely that one measurement per event is all there is, the I-MR chart with a longer interval. The subgroup size trades sensitivity against cost, and Table 5 with the √n scaling tells you what you are buying.

Common mistakes

1. Specification limits on the chart. Consequence: operators adjust a stable process toward nominal and its variation grows (the funnel experiment). Fix: control limits from the data on the chart; the specification on the histogram.
2. Limits from the specification, or from a target. Consequence: the chart has nothing to do with the process. Fix: x̄̄ ± A₂R̄ from at least 20 to 25 subgroups of real data.
3. Reading the X̄ chart before the R chart. Consequence: X̄ limits computed from an out-of-control range are wrong. Fix: range or S chart first, every time.
4. Limits recomputed from data that include a known special cause. Consequence: limits too wide, the next cause missed. Fix: document the cause, exclude the point, recompute; keep undocumented points in.
5. Recomputing limits every week. Consequence: a slowly drifting process keeps redrawing its limits around itself and never signals. Fix: establish limits from a baseline, freeze them, and change them only when a deliberate process change has been made and verified.
6. All eight tests switched on. Consequence: a false alarm every few dozen points, investigations that find nothing, and operators who stop looking. Fix: rule 1 always, rules 2 to 4 by decision, the rest for diagnosis.
7. Subgroups that span a change. Consequence: R̄ inflated, limits too wide, the chart blind. Fix: consecutive parts, one machine, one lot, one operator inside each subgroup.
8. Charting a characteristic nobody acts on. Consequence: a wall of charts and no reaction plan. Fix: every chart has an owner and a written reaction to each signal (Module 18).

Exercises

Exercise 1: an X̄-R chart from scratch

Constructed data, not a real production run. Setting: the length of a turned pin in mm on a height gauge reading to 0.001 mm, four consecutive pins every 15 minutes, 20 samples.

Tasks. (a) Compute x̄̄ and R̄. (b) With A₂ = 0.729, D₃ = 0, D₄ = 2.282 and d₂ = 2.059 for n = 4, compute the limits of both charts. (c) Is the process stable? Check rule 1 and the Western Electric rules. (d) Estimate σ̂ and the standard deviation of a subgroup mean. (e) The next subgroup after the study has a mean of 30.020 mm and a range of 0.030 mm. What do you do?

Exercise 2: what a trend looks like

Constructed data, not a real production run. Setting: the concentration of a machining coolant in % measured by refractometer to 0.01 %, one reading per shift for 30 shifts; the sump lost water by evaporation and was not topped up. The data were constructed with a steady downward drift.

Tasks. (a) Build the I-MR chart on all 30 readings. (b) Which rules fire, in what order, and which chart (I or MR) sees the problem? (c) Compare σ̂ from the moving range with the sample standard deviation of the 30 readings and explain the difference. (d) What is the right response, and what should the limits be based on afterwards?

Quiz

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

Key takeaways

• A control chart separates routine variation from special causes with an operational rule: limits at three standard deviations of the plotted statistic, estimated from within rational subgroups.
• Control limits come from the data and describe the process; specification limits come from the drawing and describe the customer. Never on the same chart, for the second of three times in this course.
• X̄-R for subgroups of 2 to 10, X̄-S above that, I-MR for individual readings. The constants (A₂, D₃, D₄, d₂; A₃, B₃, B₄, c₄; 1.128, 3.267) turn R̄, S̄ or MR̄ into limits. Read the spread chart first.
• The Western Electric rules (beyond 3σ; 2 of 3 beyond 2σ; 4 of 5 beyond 1σ; 8 on one side) and Nelson's eight tests catch small shifts sooner at the cost of more false alarms; the AIAG & VDA stability criteria say the same things in a different counting.
• The in-control ARL is about 370 for rule 1 alone and about 92 with all four Western Electric rules; a one-sigma shift takes about 44 points to detect with rule 1 and about 9 with all four. Choose rules deliberately.
• A documented special cause is excluded from the limits; an undocumented outlier is data. Limits are established from a baseline, frozen, and changed only after a deliberate, verified process change.
• The subgroup and the sampling interval decide what the chart can see. Put only routine variation inside a subgroup and sample fast enough for the changes you need to catch.

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. Best, M., and Neuhauser, D. "Walter A Shewhart, 1924, and the Hawthorne factory." Quality and Safety in Health Care 15(2):142 to 143, 2006. https://pmc.ncbi.nlm.nih.gov/articles/PMC2464836/
2. Shewhart, W. A. Economic Control of Quality of Manufactured Product. Van Nostrand, 1931 (reissued ASQ, 1980). https://archive.org/details/in.ernet.dli.2015.150272 (catalogue; the economic justification of three-sigma limits)
3. 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). Section 7.5 (pp. 41 to 42, stability criteria and the rule sets; operating characteristic and ARL), Section 7.6 (outliers), Section 10.1 (process- versus tolerance-related control), Section 10.2.2 (p. 80, stability criteria), Section 10.2.4.2 (p. 83, average run length), Section 10.4 (p. 128, rational sampling and subgrouping, sampling frequency). https://www.aiag.org/training-and-resources/manuals/details/SPCAV-1 (licensed copy read by the author)
4. NIST/SEMATECH. "6.3.2.1. Shewhart X-bar and R and S Control Charts." e-Handbook of Statistical Methods. Formulas and the A₂, D₃, D₄ (and A₃, B₃, B₄) constants; the range method up to about n = 10. https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm
5. NIST/SEMATECH. "6.3.2.2. Individuals Control Charts." e-Handbook of Statistical Methods. The I-MR chart and the worked example reproduced by this course. https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc322.htm
6. Western Electric Co. Statistical Quality Control Handbook, 1st ed. Indianapolis, 1956. (secondary: out of print; the four zone rules are cited as reproduced in [7])
7. Wikipedia. "Western Electric rules." https://en.wikipedia.org/wiki/Western_Electric_rules (secondary for the 1956 handbook; the rule statements and zones)
8. Nelson, L. S. "The Shewhart Control Chart: Tests for Special Causes." Journal of Quality Technology 16(4):237 to 239, 1984. https://www.leansixsigmadefinition.com/.../The-Shewhart-Control-Chart-Tests-for-Special-Causes-Lloyd-Nelson-Journal-of-Quality-Technology.pdf (the eight tests and Nelson's guidance on their use; read in full from this reproduction)
9. Woodall, W. H. "Controversies and Contradictions in Statistical Process Control (with discussion)." Journal of Quality Technology 32(4):341 to 378, 2000. https://www.tandfonline.com/doi/abs/10.1080/00224065.2000.11980013 (abstract read)
10. Champ, C. W., and Woodall, W. H. "Exact Results for Shewhart Control Charts with Supplementary Runs Rules." Technometrics 29(4):393 to 399, 1987. https://www.tandfonline.com/doi/abs/10.1080/00401706.1987.10488266 (abstract read; the Markov-chain method. Table 5 was computed by the course's own implementation and checked by simulation, not copied)
11. Wheeler, D. J., and Chambers, D. S. Understanding Statistical Process Control, 3rd ed. SPC Press, 2010. https://www.spcpress.com/book_understanding_statistical_process_control.php (catalogue; three-sigma limits without a normality requirement, and the use of few rules; see also Wheeler's 2011 columns cited in Module 8)
12. Wheeler, D. J. "Rational Subgrouping: The Conceptual Foundation of Process Behavior Charts." Quality Digest, 1 June 2015 (SPC Press manuscript 282); and "Rational Sampling," 1 July 2015 (manuscript 283). https://spcpress.com/pdf/DJW282.pdf
13. ISO 7870-2:2023. Control charts. Part 2: Shewhart control charts, 2nd ed. International Organization for Standardization, 2023. https://www.evs.ee/en/iso-7870-2-2023 (catalogue: current edition and scope, including warning limits and pattern tests; the international standard for the charts of this module, read at scope level only, so no single claim here rests on it)

Further reading

• Montgomery, D. C. Introduction to Statistical Quality Control, 8th ed. Wiley, 2019. The standard textbook treatment of the charts of this module, with OC curves and ARL tables.
• Module 17 for attribute charts, EWMA and CUSUM (which detect small shifts without runs rules), and what to do when a chart signals; Module 18 for control plans and reaction plans.