A signal while every part is good
A shaft diameter has a tolerance of 48.00 ± 0.10 mm. The hourly subgroup means have hovered around 48.004 for weeks. This morning the mean is 48.041. Every shaft in the subgroup is well inside the tolerance, so by the drawing nothing is wrong. Yet the control chart shows a point above the upper control limit. Should anyone act?
Yes: not because a part is bad, but because the process is no longer doing what it normally does. Finding out why now is far cheaper than finding out after the first reject.
What control limits are
Every process varies. Control limits mark the boundaries of the variation that is expected when only common, everyday causes are at work. They are typically placed three standard deviations either side of the centre line, so that in a stable process a point falls outside them by pure chance only very rarely.
- UCL, upper control limit
- LCL, lower control limit
- Centre line, usually the process average
The key word is calculated. Control limits are not chosen and not taken from the drawing. They come from data collected while the process was running normally.
Control limits are not tolerance limits
Tolerance limits (LSL and USL) come from the requirement and decide whether a part is accepted. Control limits come from the process and decide whether the process has changed. They answer different questions and are not interchangeable.
On an X̄ chart, the control limits are usually much narrower than the tolerance, because the chart plots subgroup means, which vary less than single parts. Drawing the tolerance limits on an X̄ chart is a classic mistake: it compares means with a requirement that applies to individual parts.
How the limits are calculated
For the common X̄-R chart, with subgroups of size n:
- X̄ chart: UCL = X̿ + A2 × R̄ and LCL = X̿ - A2 × R̄
- R chart: UCL = D4 × R̄ and LCL = D3 × R̄
X̿ is the grand mean, R̄ the average range, and A2, D3 and D4 are tabulated constants. For n = 5: A2 = 0.577, D3 = 0 and D4 = 2.114.
Worked example (illustrative values). Twenty-five subgroups of five shafts give X̿ = 48.004 mm and R̄ = 0.050 mm.
- A2 × R̄ = 0.577 × 0.050 = 0.029
- X̄ chart: UCL = 48.004 + 0.029 = 48.033 mm and LCL = 48.004 - 0.029 = 47.975 mm
- R chart: UCL = 2.114 × 0.050 = 0.106 mm and LCL = 0
This morning’s mean of 48.041 is above the UCL of 48.033, so it is a signal.
The A2 constant is not arbitrary: A2 × R̄ equals three times the estimated standard deviation of the subgroup means, 3 × (R̄ / d2) / √n. Other charts use other formulas. An individuals chart, for example, uses X̄ ± 2.66 × the average moving range.
Getting the limits right in practice
- Use enough data. The AIAG SPC manual recommends 25 or more subgroups containing at least 100 individual readings for initial limits.
- Use a stable period. If special causes were present while the data was collected, remove the affected subgroups when their cause is known, and recalculate.
- Check the R chart first. The X̄ limits are built on R̄, so they are only meaningful if the spread is stable.
- Recalculate only with reason. A new tool, process change or confirmed improvement justifies new limits. Recalculating every week lets the limits follow a drifting process and hides the drift.
Common misunderstandings
- “Outside the control limits means scrap.” No. It means investigate. The parts may all be good.
- “Inside the control limits means everything is fine.” Not necessarily. Patterns such as a run of rising values are also signals, which SPC rules look for.
- “Tighter limits are better.” Control limits reflect the process; you cannot improve the process by narrowing them. Reduce the variation and the limits will follow.
- “A stable process meets the requirement.” A process can be perfectly stable and still produce parts outside tolerance, if its spread is too wide or badly centred.
In short
Control limits are the process’s own voice: they show where normal variation ends. Use them to decide when to look for a cause, use tolerance limits to decide about parts, and keep the two apart on the chart and in conversations.
