Practical guide · Quality management
Control chart (SPC chart): how to build one, calculate the limits and read the signals
Short answer
A control chart plots a process measurement over time against a centre line and upper and lower control limits calculated from the process's own data. A point outside the limits, or a long pattern on one side, signals a special cause worth finding. Points inside are normal variation.
A control chart, also called an SPC chart or Shewhart chart, plots samples taken in time order against a centre line and two control limits worked out from the process's own data. It separates the normal background variation of a process (common cause) from changes that have a specific, findable reason (special cause). Pick the chart by data type, set limits from at least 20 to 25 stable subgroups, and react to signals, not to every up and down.
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What a control chart tells you
Every process varies. A control chart asks whether the variation you see is the ordinary noise of the process or a signal that something changed. The ordinary noise is called common cause variation: it is always there, it comes from the process as designed, and it can only be reduced by changing the process. A special cause is a specific event, such as a worn tool, a new material lot or a changed setting, and it can be found and removed or kept.
The chart has three lines. The centre line is the average. The upper and lower control limits sit three standard deviations either side of it, calculated from the data, usually with constants for the subgroup size. A process whose points stay inside the limits with no pattern is said to be in statistical control: stable and predictable. That is not the same as good. A stable process can still make parts out of specification, and an unstable one can make good parts today. For how the chart sits among the other tools, see the statistical process control glossary entry.
| Control limits | Specification limits | |
|---|---|---|
| Where they come from | The process's own data | The customer or the drawing |
| What they describe | What the process does | What the part must do |
| Applied to | Averages or counts of samples | Individual parts |
| A point outside means | Something changed in the process | A part is out of tolerance |
Which control chart to use
The type is set by the kind of data you collect, not by preference.
| Data | Sample | Chart | Template |
|---|---|---|---|
| A measurement (length, weight, temperature) | Subgroups of 2 to 10 | X-bar and R | X-bar and R template |
| A measurement | One reading at a time | Individuals and moving range (I-MR) | I-MR template |
| Defective or not, sample size varies | Share defective | p chart | Attribute chart template |
| Defective or not, sample size fixed | Number defective | np chart | Attribute chart template |
| Defects per unit, fixed area of opportunity | Count of defects | c chart | Attribute chart template |
| Defects per unit, area varies | Defects per unit | u chart | Attribute chart template |
A defective is a whole unit that fails. A defect is a single flaw, and one unit can have several.
How to build a control chart in six steps
- 1
Choose the characteristic
One that matters to the customer and that you can measure well. Check the gauge first with a gage R&R study, or the chart will show gauge noise.
- 2
Choose the chart and the sampling plan
Use the table above. For X-bar and R, take a small subgroup of 4 or 5 consecutive parts at each sampling time, and sample often enough to catch a shift between samples.
- 3
Collect at least 20 to 25 subgroups
Collect them in time order while the process runs normally, and write down anything unusual that happens, so a later signal can be matched to an event.
- 4
Calculate the centre line and limits
From the data, with the constants for your subgroup size (see the worked example).
- 5
Plot the points and read them
Look for points outside the limits and for patterns, using the rules below. If the limits came from a period that included a known special cause, find it, remove those points and recalculate.
- 6
Keep plotting, and react to signals only
Extend the limits forward. When a signal appears, investigate, record the cause and the action, and leave the process alone otherwise.
Worked example: X-bar and R chart, step by step (illustrative)
This example is made up and uses 10 subgroups to keep the arithmetic short. In practice you would use 20 to 25. A shaft diameter is measured on 5 consecutive parts every hour. The table gives each subgroup's mean as a difference from 25.000 mm in thousandths of a millimetre, and its range in thousandths.
| Subgroup | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mean − 25.000 (µm) | +2 | −2 | +4 | −3 | +1 | +6 | −1 | +3 | −3 | 0 | +7 |
| Range R (µm) | 8 | 6 | 10 | 7 | 9 | 12 | 6 | 8 | 7 | 7 | 80 |
Means: 2 − 2 + 4 − 3 + 1 + 6 − 1 + 3 − 3 + 0 = 7. Ranges: 8 + 6 + 10 + 7 + 9 + 12 + 6 + 8 + 7 + 7 = 80.
| Quantity | Working | Result |
|---|---|---|
| Grand mean (centre line of X-bar) | 25.000 + 7 ÷ 10 thousandths | 25.0007 mm |
| Average range, R-bar | 80 ÷ 10 thousandths | 0.008 mm |
| X-bar upper limit | 25.0007 + 0.577 × 0.008 (A2 = 0.577 for n = 5) | 25.0053 mm |
| X-bar lower limit | 25.0007 − 0.577 × 0.008 | 24.9961 mm |
| R upper limit | 2.114 × 0.008 (D4 = 2.114 for n = 5) | 0.0169 mm |
| R lower limit | 0 × 0.008 (D3 = 0 for n = 5) | 0 mm |
Now read it. Subgroup 6 has a mean of 25.006, which is above the upper limit of 25.0053, so it is a signal. Its range of 12 thousandths is inside the R limit of 16.9, and every other mean sits between 24.997 and 25.004, inside the limits. Something happened in hour 6. The team checks the log and finds a tool insert was changed at the start of that hour, and the setting was corrected in hour 7. That is an identified special cause that has been dealt with.
Because the first limits were calculated with that point in them, the team removes subgroup 6 and recalculates from the other nine. Means total 7 − 6 = 1, so the grand mean is 25.000 + 1 ÷ 9 thousandths = 25.0001. Ranges total 80 − 12 = 68, so R-bar is 68 ÷ 9 = 7.56 thousandths = 0.00756 mm. The X-bar limits become 25.0001 ± 0.577 × 0.00756, which is 25.0001 ± 0.00436: 24.9958 to 25.0045. The R upper limit is 2.114 × 0.00756 = 0.0160. All nine remaining means (between 24.997 and 25.004) and ranges (6 to 10 thousandths) are inside the new limits, so these are the limits to carry forward.
The constants A2, D3 and D4 for subgroups of 2 to 10 are on the X-bar and R template, which calculates all of this. For individual readings, the I-MR chart uses 2.66 times the average moving range for the limits on the individuals chart and 3.267 times it for the moving range chart.
How to read the signals
| Signal | What to look for | What it suggests |
|---|---|---|
| A point beyond a limit | One point above the upper or below the lower limit | A sudden change, such as a tool, material or setting |
| A run on one side | A set run of points in a row on one side of the centre line (rule sets use 7, 8 or 9) | A shift in the average |
| A trend | Six or so points steadily rising or falling | Drift, such as tool wear or warm-up |
| Alternating up and down | Fourteen or so points in a row | Two alternating sources, such as two heads, or over-adjustment |
| Points hugging a limit | Several points near the same limit | A smaller shift that is building |
| Points hugging the centre line | Almost no spread in a long run | Mixed streams or limits set too wide |
The exact counts differ between rule sets (Western Electric, Nelson, AIAG) and software packages. Pick one, write it down and apply it the same way every time. The first rule on its own is the one nobody argues about.
On an R chart, a point above the upper limit means the spread within a subgroup grew: parts made close together differ more than before. Read the R chart first. If the range is out of control, the X-bar limits, which are built from it, are not reliable.
Mistakes that cause over-reaction or blindness
| Mistake | What goes wrong | Do this instead |
|---|---|---|
| Using spec limits as control limits | Real shifts are missed until parts are scrap | Calculate control limits from the process data |
| Adjusting after every point that is off target | Tampering adds variation to a stable process | Act only on a signal |
| Too few subgroups for the limits | Limits that move every week | At least 20 to 25 subgroups before trusting them |
| Mixed streams in one subgroup | The range is inflated and the limits are too wide | A subgroup is parts made close together under the same conditions |
| Recalculating limits after every signal | The limits chase the process and hide a real change | Recalculate only after a confirmed cause is removed or a deliberate improvement |
| Wrong chart type for the data | Meaningless limits | Choose by data type (table above) |
| Charting without a reaction plan | Signals are seen and nothing happens | Write what to check and who to call for each signal; the template has a reaction log |
Control, then capability
A control chart answers whether the process is stable. A capability study answers whether a stable process fits the specification. Do them in that order, because Cp and Cpk only mean something when the process is in control. A histogram of the same data shows the shape, and the process capability calculator and Cpk study sheet work out the indices. For a plant-wide view of quality, see how the signals feed corrective action in the quality improvement page.
Definitions and notes
- The control chart was developed by Walter Shewhart at Bell Laboratories in the 1920s; it is one of the seven basic quality tools (ASQ, "Control chart").
- The constants for subgroups of 5 (A2 = 0.577, D3 = 0, D4 = 2.114) and for the I-MR chart (2.66 and 3.267) are the standard published values, as on the X-bar and R template and the I-MR template.
- The shaft measurements, ranges, the tool insert and the limits are made up for illustration. They are not from a real plant. All arithmetic in the worked example is checked in the tables.
- Signal rule counts differ between rule sets. The ones in the table are common ones, not a single standard.
Go deeper
This page is the short version. These guides cover each part in full.
- Control Charts: Reading and Interpreting Manufacturing DataThe full blog guide to reading control charts, with more on interpreting patterns.
- Statistical Process Control Basics in ManufacturingWhere control charts fit in an SPC program and how to start one.
- Process Capability (Cp and Cpk) ExplainedWhat to do once the chart shows the process is stable.
- Gauge R&R: Repeatability and Reproducibility StudiesHow to check that the gauge is good enough before you chart its readings.
All our articles on this subject are in the quality management topic.
Free templates and tools
- X-bar and R control chart template25 subgroups with means and ranges, control limits from A2, D3 and D4, signal flags, both charts and a reaction log.
- I-MR chart template (individuals and moving range)Up to 100 single readings with moving ranges, limits from a baseline, five run-rule flags, both charts, a second phase and a reaction log.
- Attribute control chart template (p, np, c and u charts)p, np, c and u charts in one workbook: a chart selector, a sample size check, limits for each subgroup, signal flags and four worked examples.
- Histogram templateReadings, bins set by the square root rule, a tally and counts, the spec limits drawn and the share out of spec, with the common shapes explained.
- Process capability (Cpk) study sheet25 subgroups of readings with Cp, Cpk, Pp and Ppk worked out, control limits to check stability, and a histogram.
- Gage R&R study sheet10 parts, 2 or 3 appraisers, 2 or 3 trials: EV, AV, GRR, PV, %GRR and ndc by the average and range method.
- Process capability (Cp, Cpk) calculatorCp and Cpk against your specification limits, plus Pp and Ppk from pasted measurements, with a distribution chart.
- First pass yield (FPY) calculatorFirst pass yield at each step and rolled throughput yield (RTY) across the whole process.
- DPMO and sigma level calculatorDPMO, defects per unit, yield and sigma level from the defects found, units checked and opportunities per unit.
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FAQ
Questions about control chart
It is a line chart of a process measurement taken in time order, with a centre line and upper and lower control limits calculated from the process's own data. It shows whether the variation is normal background noise or a signal of a change that has a specific cause. It is the main tool of statistical process control.
Control limits come from the process's own data and show what the process does. Specification limits come from the customer or drawing and show what the part must do. A process can be inside its control limits and still make parts outside the specification, and the reverse. Do not use spec limits on a control chart.
For an X-bar and R chart, take the grand mean of the subgroup means and the average range R-bar. The X-bar limits are the grand mean plus and minus A2 times R-bar. The R limits are D4 times R-bar and D3 times R-bar. For a subgroup of 5, A2 is 0.577, D4 is 2.114 and D3 is 0.
A common rule of thumb is at least 20 to 25 subgroups collected while the process runs normally, before you trust the limits. Fewer points give limits that move every time you add one. Check the gauge first, because measurement noise ends up in the limits.
It means a point or a pattern is unlikely to be normal variation, such as a point beyond a limit or a long run on one side of the centre line. It points to a special cause to investigate. It does not mean the parts are bad. A process can be out of control and still within specification.
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