Two Types of Variation
Walter Shewhart, who invented the control chart at Bell Labs in 1924, observed that all processes have variation — but not all variation has the same source. He divided variation into two categories that every quality practitioner must understand.
Common cause variation (偶然原因) is the random background noise of a stable process. It comes from dozens of small, interacting factors — slight differences in raw material, minor temperature fluctuations, vibration, tool wear within its designed life. This variation is predictable within limits. You cannot and should not try to eliminate it by reacting to individual high or low readings. Doing so — called tampering — actually makes the process worse.
Special cause variation (異常原因) comes from a specific, identifiable change: a different raw material lot, a worn tool beyond its life, a new operator, a machine that shifted after a minor collision. Special cause variation is a signal. It should always be investigated, the root cause identified, and the cause either corrected (if harmful) or standardised (if beneficial).
The entire purpose of a control chart is to separate these two types of variation in real time, so the operator knows whether to react or leave the process alone.
Anatomy of an X̄/R Chart
The most common control chart for manufacturing is the X̄/R chart — the average-and-range chart. It comes in pairs: the top chart tracks the subgroup average (X̄), the bottom chart tracks the subgroup range (R). Both must be in control for the process to be considered stable.
UCL (Upper Control Limit) = X̄̄ + A₂ × R̄
LCL (Lower Control Limit) = X̄̄ − A₂ × R̄
A₂ is a constant that depends on subgroup size n (for n=5, A₂ = 0.577)
The constants A₂, D₃, D₄ are tabulated in every SPC textbook and on most control chart forms. You do not need to calculate them from scratch. The key is understanding what they represent: control limits are set at ±3 standard deviations of the subgroup average, derived from the actual process variation measured in the R chart. They are not pulled from the specification or from gut feel.
Control Limits Are Not Specification Limits
This distinction is the most important concept in SPC, and the most commonly confused. It deserves its own section.
Specification limits come from the drawing. They define what the customer requires. They are fixed.
Control limits come from the process itself. They define what the process naturally produces when stable. They are calculated from data and will change if the process fundamentally changes.
A process can be statistically in control (all points within control limits, no patterns) and still produce defects — if the process variation is wider than the specification. A process can also have a point outside control limits and still produce zero defects — if the specification is much wider than the process variation. These are separate problems requiring separate solutions.
The ratio of specification width to process variation is quantified by the process capability indices Cp and Cpk — a subject for a separate article. For now, remember: control limits tell you about process stability; specification limits tell you about product conformance. Never draw spec limits on a control chart.
The Eight Nelson Rules for Special Causes
A single point outside the control limits is the most obvious signal. But special causes also produce patterns that occur entirely within the control limits. Lloyd Nelson codified eight detection rules:
| Rule | Pattern | Common cause |
|---|---|---|
| 1 | 1 point outside ±3σ | Sudden shift, measurement error, calculation error |
| 2 | 9 consecutive points same side of centreline | Process mean has shifted (new material lot, tool change) |
| 3 | 6 points in a row trending up or down | Tool wear, gradual temperature drift |
| 4 | 14 points alternating up-down-up-down | Two alternating machines, operators, or material sources mixed in one chart |
| 5 | 2 of 3 points beyond ±2σ same side | Process mean shift of moderate size |
| 6 | 4 of 5 points beyond ±1σ same side | Gradual process mean shift |
| 7 | 15 points in a row within ±1σ | Data stratification problem — two different populations mixed |
| 8 | 8 points in a row beyond ±1σ both sides | Bimodal distribution — two different processes mixed |
In daily shop floor practice, rules 1, 2, and 3 catch the vast majority of real problems. Teaching operators all eight rules at once is overwhelming. Start with rule 1 (out of limits) and rule 2 (run of 9). Add rule 3 (trend) once the first two are habitual.
Setting Up a Chart from Scratch — Practical Steps
Starting a new control chart requires a baseline data collection period. Here is the practical sequence I have used dozens of times:
Step 1: Define subgroup size (typically n=4 or n=5) and subgroup frequency (every hour, every 50 parts, etc.). Subgroups should be rational — parts made close together in time, on the same machine, same tool. Do not mix parts from different machines in one subgroup.
Step 2: Collect 20–25 subgroups under normal operating conditions. Do not introduce any changes during this baseline period. Do not discard data because it looks bad.
Step 3: Calculate X̄̄ and R̄ from the baseline data. Calculate control limits using the appropriate constants for your subgroup size.
Step 4: Review the baseline data against the calculated limits. If any points are outside limits or obvious patterns exist, investigate those periods, identify the special cause, remove those subgroups from the baseline calculation, and recalculate. Repeat until the baseline data is clean.
Step 5: Deploy the chart on the shop floor. Operators plot new subgroups in real time. When a signal occurs, they mark it on the chart and follow the reaction plan.
The Reaction Plan — What Operators Actually Do
A control chart without a reaction plan is decoration. The reaction plan defines specifically what an operator must do when each type of signal occurs. It typically lives on the back of the chart form or is posted at the machine.
At minimum, a reaction plan covers: stop the machine or continue? Who to notify? What to check first? What to measure to distinguish between the most likely special causes? How to document the investigation? The reaction plan should be written by the engineer who set up the chart, in language the operator can follow without calling a supervisor.
統計的手法を用いて製造工程を監視・管理する手法。中心は管理図（コントロールチャート）であり、工程の変動を「偶然原因による変動」と「異常原因による変動」に区別する。W.A. シューハートが1924年に考案し、デミング博士が日本に広めた。規格限界と管理限界は別物であることが最も重要な理解ポイント。
The One Question That Reveals Maturity
When I visit a factory using control charts, I ask the operator one question: "What does it mean when a point is inside the control limits?" Many operators answer "the part is good." That is the wrong answer. Inside the control limits means the process has not changed unexpectedly — it says nothing directly about conformance to specification.
When an operator can answer that question correctly, the chart on the wall is doing its job. When they cannot, the chart is there for auditors, not for quality. The difference between those two situations is not the chart — it is whether someone took the time to explain what it means.