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Stat Dojo / Quality Statistics

Control Chart Theory

Why 3-sigma limits work, how the chart constants are derived, and the false-alarm and detection rates behind every control chart.

Question it answers
Is this process stable, or has something changed?
Data needed
Time-ordered measurements, individually or in subgroups
Key output
A center line, control limits, and signals
Core idea
Limits = center ± 3 standard errors, estimated from within-subgroup variation
Assumptions
Rational subgroups; independent observations
Excel
Formulas with the constants table; Line chart
Minitab
Stat > Control Charts > Variables Charts
Why it matters
Tells you when to act and when to leave the process alone

The Idea in Plain Language

A control chart separates two kinds of variation. Common-cause variation is the steady background noise built into the process. Special-cause variation comes from something specific that changed: a worn tool, a new batch, a different operator. The chart draws limits that describe what common-cause variation looks like, so a point outside them says “this is unlikely to be just noise.”

The limits are not specification limits. They come from the process itself: the center line is the average, and the control limits sit three standard errors either side. The theory below explains where that number comes from and how the constants in the chart tables are made.

Why it matters. Acting on noise (tampering) makes a process worse, and ignoring a real signal lets a problem run. The chart is a decision rule for when to act, and its false-alarm and detection rates are exactly what the theory describes.

Why Three Sigma?

If a process is stable and roughly normal, a plotted statistic falls beyond the 3-sigma limits with probability 2 × P(Z > 3) = 0.0027, about 1 chance in 370. So on average a stable process gives one false alarm every 370 points. This is the average run length when nothing has changed (ARL0).

Limit widthFalse alarm per pointAverage points between false alarmsEffect
± 2 sigma0.045522Frequent false alarms, fast detection
± 2.5 sigma0.012481A compromise sometimes used
± 3 sigma0.0027370Rare false alarms; the standard

Walter Shewhart chose three sigma as a practical balance: wide enough that you rarely chase noise, narrow enough to catch real shifts. It does not depend on the data being exactly normal, because even for skewed data nearly all values fall within three standard deviations.

The flip side is the speed of detection. A chart can take many points to notice a small shift:

155.2 33.4 0.5 sigma shift 43.9 4.5 1 sigma shift 15.0 1.6 1.5 sigma shift 6.3 1.1 2 sigma shift n = 1 n = 5 Average points until a signal
A 1-sigma shift takes about 44 individual points to catch on average, but a subgroup of 5 turns it into a 2.2-sigma shift in the average and catches it in about 4.5 subgroups.
Subgroups are power. Averaging n values shrinks the standard error by √n, so the same shift looks √n times larger on an X-bar chart. That is also why rational subgrouping matters: subgroups should capture only common-cause variation within them.

Where the Constants Come From

The estimate of the process standard deviation from a subgroup range is σ̂ = R̄ / d2, where d2 is the expected range of n values from a standard normal distribution. For n = 2 the expected gap between two standard normal values is 1.128, which is where the individuals chart constant comes from. The standard deviation of that range is d3, which gives the limits for the range chart.

nd2d3A2 = 3/(d2√n)D3 = 1 − 3d3/d2D4 = 1 + 3d3/d2
21.1280.8521.8800.0003.266
31.6930.8881.0230.0002.575
42.0590.8800.7290.0002.282
52.3260.8640.5770.0002.114
62.5340.8480.4830.0002.004
72.7040.8330.4190.0761.924
82.8470.8200.3730.1361.864
92.9700.8080.3370.1841.816
103.0780.7970.3080.2231.777
  • X-bar chart: X̄̄ ± A2 R̄ (which equals X̄̄ ± 3σ̂/√n).
  • R chart: center R̄, limits D3 R̄ and D4 R̄. The lower limit is zero for subgroups under 7 because a range cannot be negative.
  • Individuals chart: X̄ ± 2.660 M̄R (since 3/1.128 = 2.660), and the moving range chart has upper limit 3.267 M̄R.
  • Why range, not standard deviation? For small subgroups the range is almost as efficient and far easier to calculate by hand. For n above about 10, an S chart is better.

These values were computed numerically from the normal distribution and match the standard published tables.

Signals Beyond One Point

A single point beyond the limits is the main signal, but patterns inside the limits also signal change. The zone lines in the figure below divide the space into one-, two-, and three-sigma bands.

RuleSignalWhat it often means
1One point beyond 3 sigmaA sudden special cause: a mistake, a failure
2Eight points in a row on one side of the center lineA small, sustained shift in the average
3Six points in a row steadily rising or fallingA trend, such as tool wear
4Two of three points beyond 2 sigma on the same sideA shift beginning
5Fifteen points in a row within 1 sigmaStratification: data mixed from different sources, or limits too wide

Each additional rule catches more changes but also raises the false-alarm rate, so use only the rules that match the failure modes you care about.

Worked Example 1: An Individuals (I-MR) Chart

Twenty-four consecutive pH readings from a bath, one per hour: 7.02, 6.96, 7.03, 7.05, 7.02, 7.00, 7.02, 7.08, 6.91, 7.10, 6.97, 6.98, 6.97, 6.96, 6.99, 7.03, 7.34, 7.02, 6.99, 7.05, 6.94, 7.03, 7.02, 7.05.

  1. Center line = mean of the readings = 7.022.
  2. Moving ranges (absolute differences between successive readings) average M̄R = 0.0796.
  3. Sigma estimate = M̄R / 1.128 = 0.0705.
  4. Limits = 7.022 ± 3 × 0.0705 = 6.810 to 7.234 (equivalently ± 2.66 × M̄R).
  5. Moving range limit = 3.267 × 0.0796 = 0.260.
6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 UCL = 7.23 Mean = 7.02 LCL = 6.81 Reading 17: special cause Reading number pH
Reading 17 is above the upper control limit. The dotted lines mark the one- and two-sigma zones.

Reading 17 (7.34) is a signal. Investigation found a wrong reagent was added that hour, a clear special cause. With that reading removed the center line becomes 7.008 and the limits tighten to 6.862 to 7.155, because the extreme reading had inflated the moving range.

Conclusion. The process is stable apart from the one assignable cause at reading 17. After removing it, use the revised limits (6.862 to 7.155) to monitor future readings. Do not remove points just to tighten limits: only remove those with a known and corrected cause.

Worked Example 2: An X-bar and R Chart

Five bottles were measured every hour for 20 hours (fill level, mm). The first three subgroups: (24.4, 24.6, 24.2, 25.4, 24.7) with mean 24.66 and range 1.2; (25.7, 25.8, 24.6, 26.2, 24.1) with mean 25.28 and range 2.1; (23.8, 24.2, 24.6, 25.5, 26.6) with mean 24.94 and range 2.8.

  1. Grand mean X̄̄ = 25.094; average range R̄ = 1.680.
  2. Constants for n = 5: A2 = 0.577, D3 = 0, D4 = 2.114.
  3. X-bar limits = 25.094 ± 0.577 × 1.680 = 24.125 to 26.063.
  4. R chart limits: lower = 0; upper = 2.114 × 1.680 = 3.552.
  5. Within-process sigma = R̄ / d2 = 1.680 / 2.326 = 0.722.
23.8 24.2 24.6 25.0 25.4 25.8 26.2 UCL = 26.1 Mean = 25.1 LCL = 24.1 Subgroup number Subgroup mean
All 20 subgroup averages are inside the limits and the largest range is 2.8, below the R chart limit of 3.55. The process looks stable.
Conclusion. With the R chart in control and the X-bar chart in control, the process is stable, and the within-subgroup sigma of 0.72 mm can be used for capability. Always check the R chart first: if the variation is out of control, the X-bar limits are not meaningful.

Run It in Excel and Minitab

ExcelStep by step

  1. Put the readings in column A. In B3 enter =ABS(A3-A2) and fill down for the moving ranges.
  2. Center: =AVERAGE(A2:A25) (7.022). Average moving range: =AVERAGE(B3:B25) (0.0796).
  3. UCL: =center+2.66*MRbar (7.234). LCL: =center-2.66*MRbar (6.810).
  4. For X-bar and R, compute each subgroup mean with =AVERAGE(row) and range with =MAX(row)-MIN(row), then use the constants table above.
  5. Chart: select the readings and the three limit columns, then Insert > Line with Markers. Excel has no built-in control chart or run rule tests.

MinitabStep by step

  1. Individuals: Stat > Control Charts > Variables Charts for Individuals > I-MR. Enter the column as the Variable.
  2. Subgrouped data: Stat > Control Charts > Variables Charts for Subgroups > Xbar-R. Choose either one column with a subgroup size, or several columns across.
  3. Click I-MR Options > Tests to select the signal rules (such as one point beyond 3 sigma, eight in a row on one side). Click Estimate to omit known special causes when computing the limits.
  4. Quicker: Assistant > Control Charts chooses the chart type and checks the rules for you.
  5. Minitab flags failed points in red and lists them in the session window.
Minitab session window: I-MR chart (typed excerpt, simplified)
I-MR Chart of pH

Test Results for I Chart of pH

TEST 1. One point more than 3.00 standard deviations from center line.
Test Failed at points: 17

Test Results for MR Chart of pH

TEST 1. One point more than 3.00 standard deviations from center line.
Test Failed at points: 17, 18

Estimates

Mean = 7.022   UCL = 7.234   LCL = 6.810   Average moving range = 0.0796   MR UCL = 0.2599

Reading and Reporting

  1. Check the range or moving range chart first. If the variation is not stable, the averages chart is not interpretable.
  2. Investigate each signal and record the cause. Remove only points with a known, corrected cause when recalculating limits.
  3. Collect enough data before fixing limits: at least 20 to 25 subgroups.
  4. Never put specification limits on a control chart for individual values: they answer a different question.
  5. Say what you did about each signal, not only that it occurred.

Common Mistakes

MistakeWhy it misleadsBetter
Using specification limits as control limitsSpecifications are what the customer wants; control limits are what the process doesCalculate limits from the data
Using the overall standard deviation of all the data for the limitsIncludes between-subgroup change, so the limits are too wide and hide signalsUse R̄/d2 or M̄R/1.128
Recalculating limits every time a point goes outChases noise and hides real changeFix limits after a stable baseline
Adjusting the process after every pointTampering increases variationAct only on signals
Subgroups that mix sources (two machines)Inflates the range and the limitsForm rational subgroups
Too few points (under 20) to set limitsLimits are unreliableCollect more before concluding

Try It Yourself

A process is sampled in subgroups of 4. After 25 subgroups, X̄̄ = 100.0 and R̄ = 6.0.

  • Find the X-bar chart limits.
  • Find the upper limit of the R chart.
  • Estimate the process standard deviation.
Show the answer

For n = 4: A2 = 0.729, D4 = 2.282, d2 = 2.059.

X-bar limits: 100 ± 0.729 × 6 = 95.63 to 104.37. R chart upper limit: 2.282 × 6 = 13.69 (the lower limit is 0). Process standard deviation = 6 / 2.059 = 2.91.

Control Chart Theory: Frequently Asked Questions

Why are control limits at three standard deviations?

It balances false alarms against detection. At three sigma a stable, normal process gives about 1 false alarm per 370 points, which is rare enough to investigate every signal, while real shifts still show up quickly.

What is the difference between control limits and specification limits?

Control limits describe what the process actually does, calculated from its own data. Specification limits describe what the customer or design requires. A process can be in control and out of spec, or out of control and in spec.

How many subgroups do I need to set control limits?

At least 20 to 25 subgroups of size 4 or 5, or 25 to 30 individual readings. Fewer gives unreliable limits.

Why do the individuals chart limits use 2.66?

Because the sigma estimate is the average moving range divided by 1.128, and 3 divided by 1.128 is 2.66.

When should I use an S chart instead of an R chart?

When subgroups are larger than about 8 to 10, the range loses efficiency and the standard deviation of each subgroup gives a better estimate.

Can I use a control chart on non-normal data?

Usually yes for X-bar charts because averages are close to normal. For individuals charts with strongly skewed data, consider a transformation or limits based on the actual distribution.

Sources and Further Reading

  • Walter A. Shewhart, Economic Control of Quality of Manufactured Product, Van Nostrand, 1931.
  • Douglas C. Montgomery, Introduction to Statistical Quality Control, Wiley (Chapters on control chart design and ARL).
  • NIST/SEMATECH, e-Handbook of Statistical Methods, Process or Product Monitoring and Control (itl.nist.gov/div898/handbook).
  • Minitab Support, “Methods and formulas for I-MR Chart and Xbar-R Chart” (support.minitab.com).

This content is educational. Worked examples use made-up data. Menu names for Minitab follow recent versions of Minitab Statistical Software and can differ slightly in older releases; Excel steps use Microsoft 365 and the Analysis ToolPak.