8.4 One-Factor Experiments: Completely Randomized, Randomized Block, and Latin Square Designs

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1 Chapter 1: Introduction & CSSBB Exam Strategy

1.1 CSSBB Exam Format, Scoring, and Testing Windows 1.2 Eligibility, Project Affidavits, Application, and Recertification 1.3 Open-Book Strategy, Reference Materials, and a 16-Week Study Plan

2 Chapter 2: Organization-Wide Planning and Deployment

2.1 Fundamentals of Six Sigma and Lean Methodologies 2.2 Selecting the Right Methodology: DMAIC, DFSS, and Lean 2.3 Business Systems, Processes, and Stakeholder Relationships 2.4 Strategic Planning and Deployment of Six Sigma Initiatives 2.5 Six Sigma Roles and Responsibilities 2.6 Organizational Barriers and Common Causes of Six Sigma Failure 2.7 Change Management: Stakeholder Analysis, Readiness, and Communication

3 Chapter 3: Organizational Process Management and Measures

3.1 Impact of Six Sigma Projects on Customers, Suppliers, and Stakeholders 3.2 Benchmarking Types and Performance Goal Setting 3.3 Performance Measures: Balanced Scorecard, KPIs, KBIs, OKRs, and Line of Sight 3.4 Leading and Lagging Indicators and Performance Dashboards 3.5 Financial Measures: Revenue Growth, Margin, NPV, ROI, and Cost-Benefit Analysis 3.6 Cost of Poor Quality and Hard versus Soft Savings

4 Chapter 4: Team Management

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5 Chapter 5: Define Phase

5.1 Customer Identification and Segmentation 5.2 Customer Data Collection: Methods, Validity, and Reliability 5.3 Customer Requirements: CTX, CTQ Trees, and Operational Definitions 5.4 The Kano Model and Quality Function Deployment 5.5 SIPOC and Process Mapping 5.6 Business Case Justification 5.7 The Project Charter and the Problem Statement 5.8 Project Scope and Boundaries 5.9 SMART Goals and Project Performance Measurements 5.10 Project Charter Reviews and Toll-Gate Governance 5.11 Project Management Tools: Gantt Charts, Toll-Gate Reviews, WBS, and RACI 5.12 Analytical Tools Used Throughout the DMAIC Cycle

6 Chapter 6: Measure Phase

6.1 Process Flow Metrics and the Hidden Factory 6.2 Process Analysis Tools and Value Stream Mapping 6.3 Types of Data and Measurement Scales 6.4 Sampling Concepts, Methods, and Sample Size for Estimation 6.5 Data Collection Plans and Common Pitfalls 6.6 Gage R&R for Variable Measurement Systems 6.7 Attribute Agreement Analysis and Audit MSA 6.8 Measurement Systems Across the Organization, and Metrology 6.9 Population Parameters, Sample Statistics, and Valid Statistical Conclusions 6.10 Descriptive Statistics: Central Tendency, Dispersion, and Shape 6.11 Graphical Methods for Exploratory Data Analysis 6.12 Probability Concepts, Common Distributions, and the Central Limit Theorem 6.13 Additional Distributions: Hypergeometric, Exponential, Lognormal, Weibull, and Bivariate 6.14 Process Capability and Performance Indices 6.15 Capability Studies, Attributes Capability, Yield Metrics, and Long-Term Capability

7 Chapter 7: Analyze Phase

7.1 The Correlation Coefficient, Its Confidence Interval, and Causation 7.2 Linear Regression and Residual Analysis 7.3 Multivariate Tools: Factor Analysis, Discriminant Analysis, and MANOVA 7.4 Hypothesis Testing Terminology, Errors, and Single-Sample Tests 7.5 Statistical Versus Practical Significance 7.6 Sample Size for Hypothesis Tests 7.7 Point Estimates, Confidence, Prediction, and Tolerance Intervals 7.8 Tests for Means, Variances, and Proportions 7.9 Analysis of Variance 7.10 Chi-Square Goodness-of-Fit and Tests of Independence 7.11 Types of Risk and Failure Mode and Effects Analysis 7.12 Gap Analysis and Root Cause Analysis 7.13 Waste Analysis: The Seven Classic Wastes and Resource Under-Utilization

8 Chapter 8: Improve Phase

8.1 Design of Experiments: Objectives, Terminology, and Effects 8.2 DOE Design Principles: Randomization, Blocking, Replication, and Resolution 8.3 Planning and Evaluating Designed Experiments 8.4 One-Factor Experiments: Completely Randomized, Randomized Block, and Latin Square Designs 8.5 Full Factorial Experiments and Design Selection 8.6 Two-Level Fractional Factorial Experiments and Confounding 8.7 Response Surface Methodology and Optimization 8.8 Waste Elimination: Pull Systems, Kanban, 5S, and Poka-Yoke 8.9 Cycle-Time Reduction: Continuous Flow, SMED, and Heijunka 8.10 Kaizen, Kaizen Blitz, Theory of Constraints, and OEE 8.11 Implementation Planning, Piloting, and Solution Selection

9 Chapter 9: Control Phase

9.1 SPC Objectives, Variable Selection, and Rational Subgrouping 9.2 Control Charts for Variables Data 9.3 Control Charts for Attributes Data 9.4 Short-Run SPC and Moving Average Charts 9.5 Control Chart Analysis: Common Cause, Special Cause, and Run Rules 9.6 Total Productive Maintenance and Visual Controls 9.7 Measurement System Reanalysis 9.8 Control Plan Development and Process Owner Handoff 9.9 Lessons Learned, Documentation, and Training for Sustainment 9.10 Ongoing Evaluation of the Improved Process

10 Chapter 10: Design for Six Sigma (DFSS)

10.1 DFSS Frameworks: DMADV, DMADOV, and Design Transfer 10.2 Design for X (DFX) 10.3 Robust Design, Parameter Design, and Verification 10.4 Tolerance Design and Statistical Tolerancing

8.4 One-Factor Experiments: Completely Randomized, Randomized Block, and Latin Square Designs

Key Takeaways Explain This Topic AI

A completely randomized design assigns treatments entirely at random and is correct when no nuisance variable is known.

A randomized block design controls exactly one known nuisance variable by running a complete set of treatments within each block.

A Latin square controls two nuisance variables simultaneously using k squared runs instead of k cubed.

Blocking removes nuisance variation from the error term, which increases the sensitivity of the test for the treatment effect.

A Latin square assumes no interaction among the treatment and the two blocking factors.

Last updated: August 2026

Choosing by the number of nuisance variables

• Nuisance variables to control Design Runs for k k k treatments Error degrees of freedom
• 0 Completely randomized (CRD) k × n k \times n k × n replicates N − k N - k N − k

1 Randomized complete block (RCBD) k × b k \times b k × b blocks ( k − 1 ) ( b − 1 ) (k-1)(b-1) ( k − 1 ) ( b − 1 )

• 2 Latin square k 2 k^2 k 2 ( k − 1 ) ( k − 2 ) (k-1)(k-2) ( k − 1 ) ( k − 2 )
• 3 Graeco-Latin square k 2 k^2 k 2 ( k − 1 ) ( k − 3 ) (k-1)(k-3) ( k − 1 ) ( k − 3 )
• The progression is the point: each additional blocking factor removes another known source of variation

from the error term, at the cost of degrees of freedom and additional assumptions.

• Completely randomized design
• Every experimental unit has an equal chance of receiving any treatment, and run order is fully
• randomized. This is the correct design when experimental units are homogeneous and no nuisance variable

is known.

Model: y i j = μ + τ i + ε i j y_{ij} = \mu + \tau_i + \varepsilon_{ij} y ij ​ = μ + τ i ​ + ε ij ​ , where τ i \tau_i τ i ​ is the effect of treatment i i i .

Analysis is a one-way ANOVA:

• Source df
• Treatment k − 1 k - 1 k − 1
• Error N − k N - k N − k
• Total N − 1 N - 1 N − 1
• Strengths: simplest to run and analyze, tolerates unequal group sizes, and maximizes error degrees of
• freedom. Weakness: any nuisance variation stays in the error term, so a real treatment effect can be

masked.

• Randomized complete block design
• A block is a group of units expected to be more homogeneous within than between: one raw material
• lot, one day, one operator, one machine. Every treatment appears exactly once in every block , and

run order is randomized within each block.

Model: y i j = μ + τ i + β j + ε i j y_{ij} = \mu + \tau_i + \beta_j + \varepsilon_{ij} y ij ​ = μ + τ i ​ + β j ​ + ε ij ​ .

• Source df
• Treatment k − 1 k - 1 k − 1
• Block b − 1 b - 1 b − 1
• Error ( k − 1 ) ( b − 1 ) (k-1)(b-1) ( k − 1 ) ( b − 1 )
• Total k b − 1 kb - 1 kb − 1

Worked example. Four cutting-tool coatings ( k = 4 k = 4 k = 4 ) are compared, and tool life is known to depend

strongly on the bar-stock lot. Using five lots as blocks ( b = 5 b = 5 b = 5 ) requires 20 runs, with each coating

run once per lot.

• Lot 1 Lot 2 Lot 3 Lot 4 Lot 5
• Order within block Randomized Randomized Randomized Randomized Randomized
• Coatings run A, B, C, D A, B, C, D A, B, C, D A, B, C, D A, B, C, D
• Because lot-to-lot variation is estimated as its own term, it leaves the error, and a coating difference
• that a CRD would have missed becomes detectable. Error degrees of freedom drop from 16 in a CRD with the

same 20 runs to ( 4 − 1 ) ( 5 − 1 ) = 12 (4-1)(5-1) = 12 ( 4 − 1 ) ( 5 − 1 ) = 12 ; the trade is almost always worth it when the blocking factor genuinely

matters.

Assumption : no treatment-by-block interaction. If one coating is best in some lots and worst in

others, the additive model is wrong and a factorial treating lot as a factor is required.

• Complete means every treatment appears in every block. When blocks are too small to hold all
• treatments -- a furnace that holds three parts but four treatments are being compared -- a balanced

incomplete block design is used instead.

• Latin square
• Controls two nuisance variables at once. Treatments are arranged in a k × k k \times k k × k grid so each

treatment appears exactly once in every row and exactly once in every column.

A 4 × 4 4 \times 4 4 × 4 Latin square with operators as rows and days as columns:

• Day 1 Day 2 Day 3 Day 4
• Operator 1 A B C D
• Operator 2 B C D A
• Operator 3 C D A B
• Operator 4 D A B C
• Sixteen runs control both operator and day effects. A full factorial of treatment by operator by day

would need 64.

• Source df
• Treatment k − 1 k - 1 k − 1
• Rows k − 1 k - 1 k − 1
• Columns k − 1 k - 1 k − 1
• Error ( k − 1 ) ( k − 2 ) (k-1)(k-2) ( k − 1 ) ( k − 2 )
• Total k 2 − 1 k^2 - 1 k 2 − 1

Constraints and assumptions:

The number of treatments must equal the number of rows and columns.

No interaction is permitted among treatment, row, and column factors; the model is strictly

additive.

Error degrees of freedom are small for small k k k . A 3 × 3 3 \times 3 3 × 3 square leaves only 2, which is rarely

enough; k ≥ 4 k \ge 4 k ≥ 4 is preferred, or replicate the square.

The particular square used should be selected at random from the available squares of that size.

A Graeco-Latin square extends the idea to three nuisance variables by superimposing a second set of

treatment labels, still in k 2 k^2 k 2 runs, but the error degrees of freedom shrink further to ( k − 1 ) ( k − 3 ) (k-1)(k-3) ( k − 1 ) ( k − 3 ) .

Choosing in practice

• Situation Design
• Comparing four solder pastes on a single stable line, one shift, one day CRD
• Comparing four solder pastes where board lot strongly affects the result RCBD, blocking on lot
• Comparing four solder pastes where both operator and day matter Latin square

Comparing four pastes where paste behaves differently by operator Factorial with operator as a factor -- the additive assumption fails

• The last row is the decision that matters. Blocking assumes the nuisance factor shifts the response by
• a constant amount regardless of treatment. If the nuisance factor genuinely interacts with the

treatment, it is not a nuisance factor -- it is a factor, and the design must treat it as one.