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Control charts (SPC)

Paste process measurements in production order to see whether the process is in statistical control, which points indicate a special cause and which sigma to use for capability.

Pre-sizing and estimating tool: results do not replace design, calculations or reports signed by a licensed professional, nor verification against the applicable codes and standards.

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With several values per line, each line is a subgroup. With a single column, values are grouped in order by the size above. For I-MR, all values are used in order.

Specification (for Cpk)

In the same unit as the measurements. Leave USL ≤ LSL to skip Cpk.

—Points signaling a special cause
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Average, individuals or fraction defective chart

Center line, 3σ limits and 1σ and 2σ bands. Red points trigger a rule.

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Dispersion chart (R, S or MR)

Range, standard deviation or moving range of each subgroup, with control limits.

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Signals found

Each flagged point and the rules triggered. Find the cause before recomputing limits without those points.

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Method, assumptions and references

Equations used

X̄-R: UCL/LCL = X̿ ± A₂ R̄; R: UCL = D₄ R̄, LCL = D₃ R̄; σ = R̄ ÷ d₂
X̄-S: UCL/LCL = X̿ ± A₃ S̄; S: UCL = B₄ S̄, LCL = B₃ S̄; σ = S̄ ÷ c₄
I-MR: UCL/LCL = X̄ ± 2.66 MR̄; MR: UCL = 3.267 MR̄; σ = MR̄ ÷ 1.128
p: p̄ = Σd ÷ Σn; UCL/LCL = p̄ ± 3 √[p̄ (1 − p̄) ÷ nᵢ] (LCL ≥ 0)
Rules on z = (point − center line) ÷ σ of the point; Cpk = min(USL − X̿, X̿ − LSL) ÷ 3σ

Assumptions and limits

  • Phase I limits computed from the data itself; for monitoring (phase II), freeze limits from a stable period. Rational subgroups of constant size, 2 to 25.
  • The rules assume an approximately normal distribution and independent points. Autocorrelated data (temperature, level) trigger false signals.
  • Constants A₂, A₃, B₃, B₄, D₃, D₄, d₂ and c₄ tabulated for n from 2 to 25, checked by numerical integration of the normal range distribution.

Standards and technical documents

  • ISO 7870-2:2023: Control charts — Part 2: Shewhart control charts
  • ISO 7870-1:2019: Control charts — Part 1: General guidelines
  • AIAG. Statistical Process Control (SPC) Reference Manual, 2nd ed., 2005
  • Western Electric Co. Statistical Quality Control Handbook, 1956
  • Nelson, L. S. The Shewhart control chart — tests for special causes. Journal of Quality Technology, 16(4), 237–239, 1984
  • Montgomery, D. C. Introduction to Statistical Quality Control, 8th ed. Wiley, 2019

U.S. regulations

  • NIST Handbook 133: Checking the Net Contents of Packaged Goods (statistical lot acceptance)
  • 21 CFR Part 820 (QMSR, incorporating ISO 13485:2016): statistical techniques for process monitoring of medical devices

Frequently asked questions

Which control chart should I use?

For measurements in subgroups of 2 to 10, X̄-R; above 10, X̄-S. With one reading at a time (batches, lab tests), I-MR. For items counted as good or bad, a p chart.

How are X̄-R limits computed?

UCL = X̿ + A₂ R̄ and LCL = X̿ − A₂ R̄. With subgroups of 5, A₂ = 0.577; if X̿ = 0.5000 in and R̄ = 0.0035 in, limits are 0.4980 and 0.5020 in.

Is a control limit the same as a specification?

No. Control limits come from the process and show its natural variation. Specifications come from the customer. A process can be in control and out of spec.

What do I do when a point falls outside?

Record what happened, investigate the special cause and fix it. Only then recompute limits without the explained points.