14.5 SPC Charts and Variation Types
Key Takeaways
- Common-cause variation is inherent, random, and built into the system, so only management changing the process removes it; assignable (special) cause variation is identifiable and external to the normal process, so local action finds and removes it
- A process with only common-cause variation is in statistical control, which is stable and predictable but says nothing about capability — in control does not mean capable, and that distinction is a heavily tested trap
- Control limits (typically the centre line ±3 sigma) are calculated from the process's own data, while specification limits come from the customer or designer; specification limits never belong on an X-bar chart
- X-bar tracks the subgroup average and R tracks the within-subgroup range; read them together and check R first, because the X-bar limits are computed as X-double-bar ± A2 × R-bar
- A p chart plots the proportion of defective units while a c chart plots the count of defects per unit or constant area, so one panel with six scratches adds 6 to a c chart but only 1 to a p chart
Statistical process control (SPC) answers one question before you act: is this variation a signal or is it noise? Adjust a stable process because of a random wiggle and you make it worse — Deming called that tampering, ASCM materials call it over-adjustment, and either way you have added the variation you meant to remove. Ignore a genuine shift and you ship defects. Everything below is machinery for telling those two cases apart.
Two Kinds of Variation
| Common cause | Assignable (special) cause | |
|---|---|---|
| Source | Inherent in the process design; many small random influences | A specific, identifiable event outside the normal process |
| Pattern | Random and stable; predictable within limits | A spike, shift, trend, or new pattern |
| Examples | Normal tooling clearance, routine lot-to-lot material spread, ordinary operator variation | A worn insert, a new material lot, an untrained operator, a mis-set machine |
| Who must act | Management — only a change to the system changes it | Local action — operator, supervisor, or technician finds and removes it |
| Shorthand | The system the workers work in | Something that happened to the workers |
Hunting a culprit when the variation is common cause is the most common waste in a quality meeting: the process is delivering exactly what it was designed to deliver, and only redesign will change it.
A process showing only common-cause variation is in statistical control — stable and predictable. Now the trap the exam sets repeatedly: in control does not mean capable. A perfectly stable process can be stably, predictably producing scrap if its natural spread is wider than the tolerance or if it sits off-centre. Control describes the voice of the process; capability (Cp and Cpk, section 14.1) describes whether that voice fits inside the customer's specification.
Control Chart Anatomy
Every control chart carries three horizontal references: the centre line (CL), the average of the plotted statistic; and the upper control limit (UCL) and lower control limit (LCL), conventionally the centre line ±3 standard deviations of that statistic.
The most-tested fact in this area: control limits are calculated from the process's own data; specification limits come from the customer or the design engineer. Different origins, and no arithmetic connects them. Specification limits therefore never belong on an X-bar chart — comparing a subgroup average against a limit written for individual units is a guaranteed wrong answer. A process can sit inside its control limits and outside spec (stable but incapable), or inside spec and out of control (currently lucky).
The Four Charts the ECM Names
Split on data type first. Variable data is measured on a continuous scale — millimetres, grams, minutes. Attribute data is counted or classified — pass/fail, number of blemishes.
- X-bar chart — plots the average of each subgroup of a measured variable, tracking where the process is centred.
- R chart — plots the range (largest minus smallest) within each subgroup, tracking spread. Read the pair together, because averaging conceals a widening spread: the subgroup mean holds steady while individual parts scatter further apart. Read R first — the X-bar limits are computed from the average range (X-double-bar ± A2 × R-bar), so an unstable range puts the X-bar limits on a moving foundation.
- p chart — plots the proportion (fraction) defective in each sample. Attribute data; sample size may vary, in which case the limits move with n. (The np chart plots the count of defectives when n is constant.)
- c chart — plots the count of defects per unit or per constant inspection area. Attribute data with a fixed opportunity size. (The u chart handles defects per unit when the area varies.)
The classic distinction: a p chart counts defective units; a c chart counts defects. One painted panel with six scratches contributes 6 to a c chart but 1 to a p chart — and a panel with a single scratch also contributes 1. "Flaws per panel" wants a c chart; "percentage of panels rejected" wants a p chart.
| Data you have | Question you are asking | Correct chart |
|---|---|---|
| Measurements, taken in subgroups | Has the process centre moved? | X-bar chart |
| Measurements, taken in subgroups | Has the process spread changed? | R chart |
| Units classified good or bad | What fraction is defective? | p chart |
| Defects counted per unit or constant area | How many defects per unit? | c chart |
Worked X-bar and R example
Five subgroups of n = 4 fill weights in grams, tolerance 248–252 g:
| Subgroup | Readings | Average | Range |
|---|---|---|---|
| 1 | 250.2, 249.6, 250.4, 249.8 | 250.0 | 0.8 |
| 2 | 249.9, 250.5, 250.1, 249.5 | 250.0 | 1.0 |
| 3 | 250.6, 250.2, 249.8, 250.2 | 250.2 | 0.8 |
| 4 | 249.6, 250.0, 250.2, 249.8 | 249.9 | 0.6 |
| 5 | 251.8, 251.2, 251.6, 251.4 | 251.5 | 0.6 |
Subgroup 1 in full: (250.2 + 249.6 + 250.4 + 249.8) ÷ 4 = 1000.0 ÷ 4 = 250.0; range = 250.4 − 249.6 = 0.8.
Grand mean (X-double-bar) = (250.0 + 250.0 + 250.2 + 249.9 + 251.5) ÷ 5 = 1251.6 ÷ 5 = 250.32. Average range (R-bar) = (0.8 + 1.0 + 0.8 + 0.6 + 0.6) ÷ 5 = 3.8 ÷ 5 = 0.76. With the standard factors for n = 4 (A2 = 0.729, D4 = 2.282, D3 = 0):
- X-bar chart: UCL = 250.32 + (0.729 × 0.76) = 250.32 + 0.55 = 250.87; LCL = 250.32 − 0.55 = 249.77.
- R chart: UCL = 2.282 × 0.76 = 1.73; LCL = 0 × 0.76 = 0.
Every range sits between 0.6 and 1.0, well inside 1.73, so the R chart is in control and the spread is stable. But subgroup 5's average of 251.5 exceeds the X-bar UCL of 250.87 — a special-cause signal. Something moved the process centre without changing its spread, so look for a discrete event between subgroups 4 and 5: a new material lot, a changed setting, a refilled hopper.
Notice what is not true here: every individual reading, 251.8 included, is inside the 248–252 g tolerance. The parts all pass, and the process is still out of control. That is the economic argument for SPC — the chart catches the shift while the parts are still good. One step further: estimated process sigma = R-bar ÷ d2 = 0.76 ÷ 2.059 = 0.369 g, so the natural spread is ±3 × 0.369 = ±1.11 g. Centred at 251.5 the upper tail reaches 252.6 g, past the 252 g upper specification. Left alone, this shift starts making rejects.
Signals beyond a single point outside the limits
- A run — seven or more consecutive points on the same side of the centre line (some references use eight). The process has shifted even though no point escaped.
- A trend — seven or more consecutive points steadily rising or falling: tool wear, a heating bath, a fouling filter.
- Other non-random patterns — cycles, points hugging the centre line, or too many points crowding the limits, which usually means mixed streams or mis-calculated limits.
The Planner's Stake
An out-of-control process is by definition unpredictable, and every planning parameter you own is a prediction. Unstable yield makes the scrap factor in material requirements planning (MRP) a guess; unstable cycle time makes the routing's run and queue times a guess; together they make the promise date a guess. The organization does not absorb that uncertainty for free — it buys protection with safety stock and safety lead time, and both are inventory and cash. When quality reports a critical process out of control, raise buffers temporarily, keep load off that resource in the master schedule, and lower the buffers only after the chart shows the gain held. Quality instability is paid for in inventory.
Exam traps
- Control limits are not specification limits, and specification limits never go on an X-bar chart.
- In control, capable, and in specification are three separate ideas; a stable process can still be incapable.
- Common cause is a management and system responsibility; only a special cause justifies a local hunt.
- p charts count defective units; c charts count defects, so one unit can add several points to a c chart.
- Adjusting a stable process because of one unremarkable point is tampering, and it increases variation.
Subgroups of n = 4 measurements produce a grand mean of 12.40 mm and an average range of 0.20 mm. Using the standard factor A2 = 0.729 for n = 4, what is the upper control limit of the X-bar chart?
A control chart for a machining operation shows no points outside the limits, no runs, and no trends, yet parts scatter across most of the tolerance band. What is the correct conclusion?
Each finished laminate panel is inspected over the same fixed area and the number of surface blemishes on it is recorded; some panels show none and some show four. Which control chart applies?
A process has been in statistical control for six months with no special-cause signals, yet about 3% of units still fall outside the customer's specification limits. What does this indicate?