ProcessBehavior follows Donald Wheeler’s terminology and methodology as presented in his books, particularly Understanding Statistical Process Control and Advanced Topics in Statistical Process Control. This glossary maps Wheeler’s terms to common alternatives.
Core Terminology¶
Process Behavior Chart¶
Wheeler’s Term: Process Behavior Chart
Common Alternatives: Control Chart, SPC Chart, Shewhart Chart
Definition: A graphical tool that plots data over time against statistically-derived control limits. Used to distinguish between common cause and special cause variation.
Wheeler’s Insight: “Process behavior charts are about understanding variation, not about control.”
Common Cause Variation¶
Wheeler’s Term: Common Cause Variation (also: Routine Variation)
Common Alternatives: Random Variation, Inherent Variation, Natural Variation, Noise
Definition: Variation that is inherent to the process and always present. Characterized by statistical stability.
Wheeler’s Insight: “Common causes are like the voice of the process itself.”
Special Cause Variation¶
Wheeler’s Term: Special Cause Variation (also: Exceptional Variation)
Common Alternatives: Assignable Cause, Non-random Variation, Signal
Definition: Variation from sources outside the usual process. Indicates something different happened.
Wheeler’s Insight: “Special causes speak loudly enough to be heard above the noise of common causes.”
Rational Subgroup¶
Wheeler’s Term: Rational Subgroup
Common Alternatives: Sample Group, Factor Level, Category
Definition: A subgroup where items are selected to maximize similarity within and highlight differences between subgroups.
In ProcessBehavior: Specified via the factors parameter in formulate().
Natural Process Limits¶
Wheeler’s Term: Natural Process Limits
Common Alternatives: Control Limits, 3-Sigma Limits
Definition: Limits calculated from the data that describe the natural range of common cause variation. NOT specification limits.
Wheeler’s Insight: “Natural Process Limits are the Voice of the Process. Specification Limits are the Voice of the Customer. These are different voices.”
Design States (DS)¶
Wheeler identifies six design states that determine valid analysis approaches.
DS 1: Full Replication¶
Definition: Every (factor × time) cell contains 2+ observations.
ProcessBehavior Detection: All cell counts >= 2
Capabilities: Full VAS analysis, exact variance estimation
DS 2: No Replication¶
Definition: Every cell contains exactly 1 observation.
ProcessBehavior Detection: All cell counts == 1
Capabilities: MR-based variance estimation, approximate VAS
DS 3: Partial Replication¶
Definition: Mix of replicated and unreplicated cells.
ProcessBehavior Detection: Some cells with n=1, others with n>=2
Capabilities: Hybrid variance estimation
DS 4: Incomplete, No Singletons¶
Definition: Incomplete grid — empty cells present, all observed cells have N_kt >= 2.
After cleansing: Collapses to ADS 1 (Full Replication)
DS 5: Incomplete, No Replication¶
Definition: Incomplete grid — empty cells present, all observed cells have N_kt = 1.
After cleansing: Collapses to ADS 2 (No Replication)
DS 6: Incomplete, With Singletons¶
Definition: Incomplete grid — empty cells present, observed cells have mixed N_kt.
After cleansing: Collapses to ADS 3 (Partial Replication)
Variance Analysis System (VAS)¶
Dr. Thomas A. Bishop’s framework for decomposing variation into meaningful components. VAS extends Wheeler’s process behavior chart methodology with a hierarchical residual decomposition (R1-R5) that isolates within-cell, interaction, time, and factor effects.
R1: Total Deviation¶
Formula: R1 = Y - Y̅
Meaning: How far each observation is from the grand mean.
R2: Within-Cell Residual¶
Formula (DS 1): R2 = Y - Y̅kt
Meaning: Variation within subgroups, the “unexplained” portion.
Wheeler’s Insight: This represents measurement error and short-term variation.
R3: Interaction Residual¶
Formula: R3 = Y - Y̅k - Y̅t + Y̅
Meaning: How factor effects change over time.
Wheeler’s Insight: Significant R3 signals mean factor behavior is inconsistent.
R4: Time Effect + Unexplained¶
Formula: R4 = Y̅t - Y̅ + R2
Meaning: Time-related patterns combined with within-cell variation.
Wheeler’s Insight: Chart R4 to detect trends, shifts, and cycles.
R5: Factor Effect + Unexplained¶
Formula: R5 = Y̅k - Y̅ + R2
Meaning: Factor differences combined with within-cell variation.
Wheeler’s Insight: Chart R5 to identify true factor differences.
Chart Types¶
X Chart (Individual)¶
Wheeler’s Term: X Chart, Individual Chart
Common Alternatives: I Chart, Individuals Chart
Definition: Plots individual observations. When paired with its companion mR chart, forms the classic XmR pair.
In ProcessBehavior: study.charts.X
mR Chart (Moving Range)¶
Wheeler’s Term: mR Chart
Common Alternatives: MR Chart, Moving Range Chart
Definition: Moving range chart, the companion to the X chart. Plots |Xi - Xi-1|, the absolute difference between consecutive points.
In ProcessBehavior: study.charts.mR
Average Chart¶
Wheeler’s Term: Average Chart, X̄ Chart
Common Alternatives: Xbar Chart, X-bar Chart
Definition: Plots subgroup averages.
In ProcessBehavior: study.charts.Xbar
s Chart¶
Wheeler’s Term: s Chart
Common Alternatives: S Chart, Sigma Chart
Definition: Plots subgroup standard deviations.
In ProcessBehavior: study.charts.S
Control Limit Constants¶
Wheeler uses the standard SPC constants from Shewhart’s work:
| Constant | Purpose | Formula Source |
|---|---|---|
| c₄ | Unbiasing s | Related to gamma function |
| A₃ | Xbar limits from s | 3 / (c₄√n) |
| B₃, B₄ | S chart limits | Functions of c₄ and n |
| d₂ | Unbiasing range | Tabulated |
| D₃, D₄ | mR chart limits | Functions of d₂ |
Key Principles¶
The 3-Sigma Rule¶
Wheeler’s Principle: Use 3-sigma limits, not 2-sigma or other values.
Rationale: Balances sensitivity with false alarm rate. Provides approximately 99.73% coverage for normally distributed data, but works for most distributions.
Shewhart’s Empirical Rule¶
Wheeler’s Statement: “The power of a process behavior chart does not come from statistical theory, but from the ability to detect economically important shifts in the process.”
Implication: Focus on practical significance, not just statistical significance.
Limits ≠ Specifications¶
Wheeler’s Principle: Natural Process Limits (calculated from data) are fundamentally different from Specification Limits (set by requirements).
Common Mistake: Comparing control limits to specifications or treating them as equivalent.
First Analyze, Then Improve¶
Wheeler’s Principle: Understand your process variation before attempting improvement.
Implication: Establish a baseline, then make changes and measure effects.
Wheeler’s Books¶
Essential references:
Understanding Statistical Process Control (with David Chambers)
Foundation text for SPC
Introduces process behavior charts
Advanced Topics in Statistical Process Control
VAS residual analysis
Design States
Complex designs
Making Sense of Data
Data analysis philosophy
Interpretation guidelines
The Six Sigma Practitioner’s Guide to Data Analysis
Practical applications
Case studies
ProcessBehavior Mapping¶
| Wheeler Term | ProcessBehavior API |
|---|---|
| Process Behavior Chart | result.plot() |
| Rational Subgroup | factors parameter |
| Time Sequence | time parameter |
| X Chart | study.charts.X |
| mR Chart | study.charts.mR |
| Average Chart | study.charts.Xbar |
| s Chart | study.charts.S |
| R1-R5 Residuals | result.residuals |
| DS Detection | study.observed_design_state / study.analytical_design_state |
| Natural Process Limits | result.get_statistics() |
Further Reading¶
Wheeler, D.J. & Chambers, D.S. (1992). Understanding Statistical Process Control, 2nd ed. SPC Press.
Wheeler, D.J. (1995). Advanced Topics in Statistical Process Control. SPC Press.
Wheeler, D.J. (2000). Understanding Variation: The Key to Managing Chaos, 2nd ed. SPC Press.
Shewhart, W.A. (1931). Economic Control of Quality of Manufactured Product. Van Nostrand.