Statistical Process Control
Overview
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.
When to Use
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)
Algorithm
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
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.
Phase 2: Core Algorithm
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
Phase 3: Verification
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.
Phase 4: Output
Return control chart data with signals and stability assessment.
Output Format
{
"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
Sample I/O
Input: 25 subgroups of 5 measurements each, all within ±3σ, no patterns Expected: Process in control. No signals triggered.
Edge Cases
| 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 |
Gotchas
- 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.
References
- For control chart constants tables, see
references/chart-constants.md - For Western Electric rules and pattern detection, see
references/we-rules.md