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Statistical Process Control

统计过程控制（SPC）

Overview

概述

SPC uses control charts to monitor process stability over time. Upper and Lower Control Limits (UCL/LCL) are set at ±3σ from the process mean. Points within limits = common cause variation (stable). Points outside or showing patterns = special cause variation (investigate). Primary charts: X-bar/R, X-bar/S, I-MR, p-chart, c-chart.

SPC通过控制图随时间监控过程稳定性。上控制限（UCL）和下控制限（LCL）设置为偏离过程均值±3σ的位置。控制限内的点代表普通原因变异（过程稳定）。超出控制限或呈现特定模式的点代表特殊原因变异（需展开调查）。主要控制图类型包括：X-bar/R图、X-bar/S图、I-MR图、p图、c图。

When to Use

适用场景

Trigger conditions:

Monitoring production process for stability and detecting shifts
Setting statistically-based control limits for quality metrics
Distinguishing normal variation from assignable causes

When NOT to use:

For process capability assessment (use Cpk)
For root cause analysis of known problems (use fishbone/5-why)

触发条件：

监控生产过程稳定性并检测偏移
为质量指标设置基于统计方法的控制限
区分正常变异与可查明原因的变异

不适用场景：

用于过程能力评估（应使用Cpk）
用于已知问题的根本原因分析（应使用鱼骨图/5Why法）

Algorithm

算法

IRON LAW: Control Limits Are NOT Specification Limits
Control limits (±3σ) describe what the process IS doing.
Specification limits describe what the process SHOULD do.
A process can be in statistical control (stable) but still produce
out-of-spec products (incapable). Conversely, a capable process may
be out of control (drifting). Monitor control FIRST, then assess capability.

IRON LAW: Control Limits Are NOT Specification Limits
Control limits (±3σ) describe what the process IS doing.
Specification limits describe what the process SHOULD do.
A process can be in statistical control (stable) but still produce
out-of-spec products (incapable). Conversely, a capable process may
be out of control (drifting). Monitor control FIRST, then assess capability.

Phase 1: Input Validation

阶段1：输入验证

Collect: 25+ subgroups of measurements (5 per subgroup typical for X-bar/R). Verify: measurement system is adequate (gauge R&R < 10%), data collected in time order. Gate: Sufficient subgroups, time-ordered data, measurement system verified.

收集25个以上的测量子组（X-bar/R图通常每个子组包含5个样本）。验证：测量系统足够可靠（测量系统分析R&R < 10%），数据按时间顺序收集。 准入条件： 子组数量充足、数据按时间排序、测量系统已验证。

Phase 2: Core Algorithm

阶段2：核心算法

X-bar/R Chart (subgroup data):

Compute subgroup means (X̄) and ranges (R)
Compute grand mean (X̄̄) and average range (R̄)
UCL_X̄ = X̄̄ + A₂×R̄, LCL_X̄ = X̄̄ - A₂×R̄ (A₂ from statistical tables by subgroup size)
UCL_R = D₄×R̄, LCL_R = D₃×R̄
Plot points, apply Western Electric rules for out-of-control signals

X-bar/R图（子组数据）：

计算子组均值（X̄）和极差（R）
计算总均值（X̄̄）和平均极差（R̄）
UCL_X̄ = X̄̄ + A₂×R̄，LCL_X̄ = X̄̄ - A₂×R̄（A₂为根据子组大小从统计表格中获取的常数）
UCL_R = D₄×R̄，LCL_R = D₃×R̄
绘制数据点，应用Western Electric规则识别失控信号

Phase 3: Verification

阶段3：验证

Check for: points outside limits, runs (7+ consecutive on one side), trends (7+ consecutive increasing/decreasing), 2 of 3 beyond 2σ, 4 of 5 beyond 1σ. Gate: Chart constructed, out-of-control signals identified.

检查是否存在以下情况：点超出控制限、连续7个点位于均值单侧、连续7个点呈上升/下降趋势、3个点中有2个超出2σ范围、5个点中有4个超出1σ范围。 准入条件： 已完成图表构建、已识别失控信号。

Phase 4: Output

阶段4：输出

Return control chart data with signals and stability assessment.

返回包含失控信号和稳定性评估结果的控制图数据。

Output Format

输出格式

json

{
  "chart": {"type": "xbar_r", "center_line": 50.2, "ucl": 52.1, "lcl": 48.3},
  "signals": [{"subgroup": 18, "rule": "point_beyond_ucl", "value": 52.8}],
  "stability": "out_of_control",
  "metadata": {"subgroups": 30, "subgroup_size": 5}
}

json

{
  "chart": {"type": "xbar_r", "center_line": 50.2, "ucl": 52.1, "lcl": 48.3},
  "signals": [{"subgroup": 18, "rule": "point_beyond_ucl", "value": 52.8}],
  "stability": "out_of_control",
  "metadata": {"subgroups": 30, "subgroup_size": 5}
}

Examples

示例

Sample I/O

示例输入输出

Input: 25 subgroups of 5 measurements each, all within ±3σ, no patterns Expected: Process in control. No signals triggered.

输入： 25个各含5个测量值的子组，所有数据均在±3σ范围内，无异常模式 预期结果： 过程处于受控状态，未触发任何信号。

Edge Cases

边缘情况

Input	Expected	Why
One point just outside UCL	Signal, but may be false alarm	~0.27% chance per point even when in control
Gradual upward trend	Trend rule triggered	Process drifting, investigate
All points near center	Suspicious — check data	May indicate data manipulation or measurement issue

输入	预期结果	原因
一个点略微超出UCL	触发信号，但可能为误报	即使过程受控，每个点仍有约0.27%的概率超出控制限
缓慢上升趋势	触发趋势规则	过程正在偏移，需展开调查
所有点均接近中心线	存在异常——需检查数据	可能表明数据被篡改或测量系统存在问题

Gotchas

注意事项

Rational subgrouping: Subgroups must be collected under similar conditions (same shift, machine, operator). Poor subgrouping inflates within-group variation, making limits too wide.
Recalculating limits: Don't recalculate limits every time you add data. Establish limits from a stable baseline period and keep them fixed until a known process change.
Chart type selection: Variables data (measurements) → X-bar/R or I-MR. Attribute data (counts/proportions) → p-chart, np-chart, c-chart, u-chart. Wrong chart type = wrong limits.
Normality assumption: X-bar chart is robust to non-normality (central limit theorem). Individual charts (I-MR) require approximate normality — check with histogram.
Over-adjustment: Reacting to every small variation (tampering) INCREASES variability. Only investigate special cause signals, not common cause variation.

合理子组划分：子组必须在相似条件下收集（如同一班次、同一机器、同一操作员）。子组划分不当会扩大组内变异，导致控制限过宽。
控制限重算：不要每次添加数据都重算控制限。应基于稳定的基线周期建立控制限，除非已知过程发生变更，否则保持控制限固定。
图表类型选择：计量型数据（测量值）→ X-bar/R图或I-MR图。计数型数据（数量/比例）→ p图、np图、c图、u图。错误的图表类型会导致控制限错误。
正态性假设：X-bar图对非正态性具有鲁棒性（中心极限定理）。单值图（I-MR）要求数据近似正态分布——需通过直方图验证。
过度调整：对每一次微小变异都做出反应（干预）会增加变异程度。仅针对特殊原因信号展开调查，不要干预普通原因变异。

References

参考资料

For control chart constants tables, see
```
references/chart-constants.md
```
For Western Electric rules and pattern detection, see
```
references/we-rules.md
```

控制图常数表格，请参阅
```
references/chart-constants.md
```
Western Electric规则与模式检测，请参阅
```
references/we-rules.md
```