Study guide
7.8 Tests for Means, Variances, and Proportions
ASQ Certified Six Sigma Black Belt CSSBB
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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
4.1 Team Types and Constraints 4.2 Team Roles, Responsibilities, and Member Selection 4.3 Team Success Factors and Ground Rules 4.4 Motivational Techniques and Overcoming Demotivators 4.5 Stages of Team Development and Conflict Resolution 4.6 Team Communication Plans 4.7 Team Leadership Models: Direct, Coach, Support, and Delegate 4.8 Group Behaviors, Conflict Resolution, and Meeting Management 4.9 Facilitation Techniques and Team Decision-Making Methods 4.10 Team Training: Needs Assessment, Delivery, and Evaluation
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
7.8 Tests for Means, Variances, and Proportions
Key Takeaways Explain This Topic AI
2-sample t-tests assess mean differences between independent groups; equal variance assumption uses pooled variance s_p^2, while unequal variance uses Welch's t-test with Satterthwaite df.
Paired t-tests reduce nuisance variability by analyzing paired differences d_i = x_1i - x_2i with df = n - 1 pairs, increasing statistical power for matched designs.
One-Way ANOVA decomposes total variance into between-group (SSB) and within-group (SSW) variance, evaluating F0 = MSB / MSW across k >= 3 factor levels.
Tukey's Honestly Significant Difference (HSD) post-hoc test controls the family-wise Type I error rate (alpha_FW) when conducting all pairwise comparisons following a significant ANOVA.
Chi-Square tests evaluate categorical data: Goodness-of-Fit tests categorical distribution fit (df = k - 1), while Independence tests evaluate contingency table associations (df = (r-1)(c-1)).
Last updated: August 2026
Comparative hypothesis testing evaluates process parameter differences across two or more experimental conditions, treatments, shifts, or operational lines. Choosing the correct parametric or non-parametric test depends on data type (continuous vs. discrete), sample independence (paired vs. independent), and variance homogeneity.
- Two-Sample Hypothesis Tests for Means and Proportions
- 1. Independent Two-Sample t t t -Test
Compares the means of two independent continuous populations ( μ 1 \mu_1 μ 1 vs. μ 2 \mu_2 μ 2 ).
Case A: Equal Variances ( σ 1 2 = σ 2 2 \sigma_1^2 = \sigma_2^2 σ 1 2 = σ 2 2 ) — Pooled t t t -Test
Pooled Variance ( s p 2 s_p^2 s p 2 ) : s p 2 = ( n 1 − 1 ) s 1 2 + ( n 2 − 1 ) s 2 2 n 1 + n 2 − 2 s_p^2 = \frac{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}{n_1 + n_2 - 2} s p 2 = n 1 + n 2 − 2 ( n 1 − 1 ) s 1 2 + ( n 2 − 1 ) s 2 2
Test Statistic : t 0 = x ˉ 1 − x ˉ 2 s p 2 ( 1 n 1 + 1 n 2 ) t_0 = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} t 0 = s p 2 ( n 1 1 + n 2 1 ) x ˉ 1 − x ˉ 2
Degrees of Freedom : df = n 1 + n 2 − 2 \text{df} = n_1 + n_2 - 2 df = n 1 + n 2 − 2 .
Case B: Unequal Variances ( σ 1 2 ≠ σ 2 2 \sigma_1^2 \neq \sigma_2^2 σ 1 2 = σ 2 2 ) — Welch's t t t -Test
Uses unpooled standard errors with Welch-Satterthwaite degrees of freedom approximation ( ν \nu ν ): ν = ( s 1 2 n 1 + s 2 2 n 2 ) 2 ( s 1 2 / n 1 ) 2 n 1 − 1 + ( s 2 2 / n 2 ) 2 n 2 − 1 \nu = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}} ν = n 1 − 1 ( s 1 2 / n 1 ) 2 + n 2 − 1 ( s 2 2 / n 2 ) 2 ( n 1 s 1 2 + n 2 s 2 2 ) 2
2. Paired Two-Sample t t t -Test
Used when observations are naturally paired or matched (e.g., before-and-after measurements on the exact same production parts, tools, or operators).
Difference Variable : d i = x 1 i − x 2 i d_i = x_{1i} - x_{2i} d i = x 1 i − x 2 i
Test Statistic : t 0 = d ˉ − μ d s d n t_0 = \frac{\bar{d} - \mu_d}{\frac{s_d}{\sqrt{n}}} t 0 = n s d d ˉ − μ d
Degrees of Freedom : df = n − 1 \text{df} = n - 1 df = n − 1 (where n n n is the number of matched pairs).
Operational Advantage : Eliminates unit-to-unit baseline variability, greatly reducing residual error variance and maximizing test sensitivity/power.
3. Two-Sample Proportion z z z -Test
Compares two independent binomial success proportions ( p 1 p_1 p 1 vs. p 2 p_2 p 2 ).
Pooled Proportion ( p ˉ \bar{p} p ˉ ) : p ˉ = x 1 + x 2 n 1 + n 2 \bar{p} = \frac{x_1 + x_2}{n_1 + n_2} p ˉ = n 1 + n 2 x 1 + x 2
Test Statistic : z 0 = p ^ 1 − p ^ 2 p ˉ ( 1 − p ˉ ) ( 1 n 1 + 1 n 2 ) z_0 = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\bar{p}(1-\bar{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} z 0 = p ˉ ( 1 − p ˉ ) ( n 1 1 + n 2 1 ) p ^ 1 − p ^ 2
4. F F F -Test for Equality of Two Variances
Evaluates whether two independent population variances are equal ( H 0 : σ 1 2 = σ 2 2 H_0: \sigma_1^2 = \sigma_2^2 H 0 : σ 1 2 = σ 2 2 ).
Test Statistic : F 0 = s 1 2 s 2 2 (placing larger sample variance in numerator, s 1 2 ≥ s 2 2 ) F_0 = \frac{s_1^2}{s_2^2} \quad \text{(placing larger sample variance in numerator, } s_1^2 \ge s_2^2 \text{)} F 0 = s 2 2 s 1 2 (placing larger sample variance in numerator, s 1 2 ≥ s 2 2 )
Degrees of Freedom : df 1 = n 1 − 1 \text{df}_1 = n_1 - 1 df 1 = n 1 − 1 , df 2 = n 2 − 1 \text{df}_2 = n_2 - 1 df 2 = n 2 − 1 .
Sensitivity Note : The F F F -test is highly sensitive to non-normality. Levene's test or Bartlett's test are preferred robust alternatives.
Non-Parametric Hypothesis Tests
When continuous data violates normality and transformation is unsuccessful, non-parametric tests based on data ranks must be deployed:
- Parametric Test Non-Parametric Equivalent Primary Metric Tested
- 1-Sample t t t -Test 1-Sample Wilcoxon Signed-Rank Median vs. Target
- 2-Sample t t t -Test (Independent) Mann-Whitney U U U Test (Wilcoxon Rank-Sum) Median Difference (2 Groups)
- Paired t t t -Test Paired Wilcoxon Signed-Rank Median Paired Difference
- One-Way ANOVA Kruskal-Wallis H H H Test Median Difference ( k ≥ 3 k \ge 3 k ≥ 3 Groups)
- Tests for Variances
- The Body of Knowledge requires tests for means, variances, and proportions . Variance tests are the
- ones teams most often skip, and they matter for two reasons: variation reduction is frequently the
- project's actual objective, and equality of variance is an assumption of the two-sample t-test and of
ANOVA.
One sample against a target variance χ 2 = ( n − 1 ) s 2 σ 0 2 , d f = n − 1 \chi^2 = \frac{(n-1)s^2}{\sigma_0^2}, \qquad df = n - 1 χ 2 = σ 0 2 ( n − 1 ) s 2 , df = n − 1
Compare against the chi-square distribution. This test is very sensitive to non-normality and should
not be used on visibly skewed data.
Two variances: the F test F = s 1 2 s 2 2 , d f 1 = n 1 − 1 , d f 2 = n 2 − 1 F = \frac{s_1^2}{s_2^2}, \qquad df_1 = n_1 - 1,\ df_2 = n_2 - 1 F = s 2 2 s 1 2 , d f 1 = n 1 − 1 , d f 2 = n 2 − 1
By convention the larger variance goes in the numerator so F ≥ 1 F \ge 1 F ≥ 1 , and the result is compared with F α / 2 , d f 1 , d f 2 F_{\alpha/2, df_1, df_2} F α /2 , d f 1 , d f 2 for a two-sided test.
Worked example. Line A: n 1 = 16 n_1 = 16 n 1 = 16 , s 1 = 2.4 s_1 = 2.4 s 1 = 2.4 . Line B: n 2 = 13 n_2 = 13 n 2 = 13 , s 2 = 1.5 s_2 = 1.5 s 2 = 1.5 . F = 2.4 2 1.5 2 = 5.76 2.25 = 2.56 F = \frac{2.4^2}{1.5^2} = \frac{5.76}{2.25} = 2.56 F = 1. 5 2 2. 4 2 = 2.25 5.76 = 2.56
with 15 and 12 degrees of freedom. The two-sided critical value F 0.025 , 15 , 12 ≈ 3.18 F_{0.025, 15, 12} \approx 3.18 F 0.025 , 15 , 12 ≈ 3.18 . Since 2.56 < 3.18 2.56 < 3.18 2.56 < 3.18 , do not reject: the evidence does not establish a difference in variability at the 5%
level, despite the sample standard deviations differing by 60%. Variance tests need substantial sample
sizes.
Like the chi-square test, the F test for variances is highly sensitive to departures from normality ,
much more so than the t-test is for means.
Several variances
- Test Assumption Use
- Bartlett's test Requires normality; very sensitive to violation Data confidently normal
Levene's test Uses absolute deviations from the group mean; robust to non-normality The general-purpose default
Brown-Forsythe Levene's using deviations from the median; still more robust Skewed data
Because Bartlett's test frequently rejects for non-normality rather than for unequal variance, Levene's test is the practical default when checking the equal-variance assumption before ANOVA.
When variances are unequal
Situation Response
Comparing two means with unequal variances Use Welch's t-test , which does not pool the variances and adjusts the degrees of freedom
Comparing several means with unequal variances Use Welch's ANOVA , or transform the response
Variance itself increases with the mean Apply a variance-stabilizing transformation: logarithm, square root, or Box-Cox
Data badly non-normal Use the non-parametric alternative: Mann-Whitney for two groups, Kruskal-Wallis for several
- Why variance tests matter to a Black Belt
- A project that reduces the mean of a cycle time while leaving the spread unchanged has improved the
- average experience and left the worst case where it was. Customers experience the tail, not the mean. In
- many projects the correct primary metric is the standard deviation or a high percentile, and the
corresponding hypothesis test is a variance test rather than a t-test.
The same logic drives the capability indices: C p C_p C p is entirely a statement about variance, and a project
targeting C p C_p C p rather than C p k C_{pk} C p k is targeting variation rather than centring.