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):

1. Compute subgroup means (X̄) and ranges (R)
2. Compute grand mean (X̄̄) and average range (R̄)
3. UCL_X̄ = X̄̄ + A₂×R̄, LCL_X̄ = X̄̄ - A₂×R̄ (A₂ from statistical tables by subgroup size)
4. UCL_R = D₄×R̄, LCL_R = D₃×R̄
5. 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