Overview
Analyze is where suspected causes become verified causes. Measuring and modeling relationships between variables covers correlation, simple and multiple linear regression, residual analysis, multi-vari studies and multivariate tools such as principal components and discriminant analysis. Hypothesis testing covers the logic of null and alternative hypotheses, Type I and Type II error, power, sample size, point and interval estimates, and the tests themselves: z and t tests for means, paired t, F and chi-square tests for variances, proportion tests, one-way and two-way ANOVA, chi-square goodness of fit and contingency tables, and nonparametric alternatives. Failure mode and effects analysis (FMEA) covers design and process FMEA and risk priority numbers. Additional analysis methods covers gap analysis, root cause analysis tools and waste analysis.
Candidates must select the correct test for a data situation, interpret p-values and regression output, and prioritize risks. Analyze depends on Domain 5's validated data and feeds Domain 7's experiments and solutions.
How it's tested
Expect test-selection items (which test fits this data), interpretation items (read a p-value, confidence interval, ANOVA table or regression output), and short calculations (t statistic, R², RPN, sample size). Scenarios often hide the deciding detail, such as paired measurements or attribute data, in one phrase.
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Choose the test
A comparison is described by number of groups, data type, pairing and normality, and you select the matching parametric or nonparametric test.
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Read the output
An ANOVA table, regression output or confidence interval is shown and you draw the correct conclusion about significance, effect size or model fit.
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Prioritize risk or cause
FMEA ratings or root-cause data are given and you decide which failure mode or cause to address first.
A team compares defect proportions across three shifts, with counts of good and defective units for each. Options include one-way ANOVA, a two-sample t-test and a chi-square test. The response is categorical (good or defective) across three groups, so means-based tests do not fit. A chi-square test of independence on the 3 × 2 contingency table is the correct choice.
Subsections 4
Official subsections from the ASQ outline, in outline order.
6.1 Measuring and modeling relationships between variablesCorrelation, regression, residual analysis, multi-vari studies and multivariate tools.
- What it means
- This covers measuring the strength of linear association with the correlation coefficient r, modeling a response with simple and multiple linear regression, checking assumptions with residual plots, using multi-vari charts to separate positional, cyclical and temporal variation, and multivariate tools such as principal components, factor analysis, discriminant analysis and MANOVA.
- What to understand
- Correlation measures association, not causation, and only linear association. R² is the proportion of response variation explained by the model; adjusted R² penalizes extra predictors. Residuals should be random, normal and constant in spread; patterns signal a missing term or non-constant variance. In multiple regression, correlated predictors (multicollinearity, often flagged by variance inflation factors) make coefficients unstable.
- How it shows up
- Regression output is shown and you interpret a coefficient, R² or p-value, or a scatter or residual plot is described and you identify the problem.
- Common confusion
- Concluding that X causes Y because r is strong, when a lurking variable or reverse causation could explain the association.
6.2 Hypothesis testingTest logic, errors, power, sample size, interval estimates and tests for means, variances and proportions.
- What it means
- This covers stating null and alternative hypotheses, significance level α (Type I risk), β (Type II risk) and power (1 − β), statistical versus practical significance, sample size, confidence intervals, and tests including z, one-sample and two-sample t, paired t, F and chi-square for variances, one and two proportions, ANOVA, chi-square goodness of fit and contingency tables, and nonparametric tests such as Mann-Whitney, Kruskal-Wallis, Mood's median and Wilcoxon.
- What to understand
- Reject the null hypothesis when p < α; otherwise fail to reject, which is not proof the null is true. Larger samples raise power and can make tiny, unimportant differences statistically significant. Pick the test by data type, number of groups, pairing and assumptions such as normality and equal variance.
- How it shows up
- A comparison is described and you choose the test, or a p-value or interval is shown and you state the conclusion and error risk.
- Common confusion
- Saying the null hypothesis is accepted or proven when the test fails to reject it.
6.3 Failure mode and effects analysis (FMEA)Design and process FMEA, severity, occurrence, detection and risk priority.
- What it means
- FMEA systematically lists how a design or process can fail, the effects and causes of each failure mode, current controls, and ratings for severity, occurrence and detection, usually on 1 to 10 scales, to prioritize risk reduction. Design FMEA (DFMEA) addresses the product; process FMEA (PFMEA) addresses how it is made or delivered.
- What to understand
- Risk priority number RPN = severity × occurrence × detection, where a higher detection rating means the failure is harder to detect. High-severity modes (9 or 10, safety or regulatory) need action regardless of RPN. Actions should reduce occurrence or improve detection, then the team re-rates and recalculates.
- How it shows up
- An FMEA table is given and you choose which failure mode to address first or how a proposed action changes the ratings.
- Common confusion
- Ranking strictly by RPN, so a severity-10 mode with low RPN is ignored in favor of a nuisance mode with high occurrence.
6.4 Additional analysis methodsGap analysis, root cause analysis and waste analysis.
- What it means
- This covers gap analysis (current versus desired state), root cause analysis tools (5 whys, cause-and-effect or fishbone diagrams, fault tree analysis, Pareto charts, relational matrices) and waste analysis that identifies the lean wastes and their causes.
- What to understand
- Root cause analysis keeps asking why until it reaches a cause that, if removed, prevents recurrence, then verifies that cause with data. Fishbone diagrams organize hypotheses but prove nothing. Fault tree analysis works top-down from an undesired event using AND and OR gates; Pareto focuses effort on the vital few categories.
- How it shows up
- A problem investigation is described and you choose the next analysis step, or decide whether a proposed root cause has been verified.
- Common confusion
- Treating the fishbone diagram's most-voted cause as the verified root cause without data.
High-yield concepts
The terms, rules and formulas this domain turns into questions most often.
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Type I and Type II errors, power
Type I error (α) rejects a true null hypothesis, a false alarm; Type II error (β) fails to reject a false null, a missed effect; power = 1 − β is the probability of detecting a real effect.
- Remember
- For a fixed sample size, lowering α raises β; increasing sample size raises power.
- Tested as
- Definition and consequence items, such as which error occurs when a good batch is rejected.
- Trap
- Mixing up α as producer's risk and β as consumer's risk in acceptance contexts.
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p-value decision rule
The p-value is the probability of observing a result at least as extreme as the sample if the null hypothesis were true; reject the null when p < α.
- Remember
- "If p is low, the null must go"; failing to reject is not acceptance.
- Tested as
- Every output-interpretation question depends on it.
- Trap
- Reading p as the probability that the null hypothesis is true.
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Test statistics for means
z = (x̄ − μ₀) / (σ/√n) when σ is known; t = (x̄ − μ₀) / (s/√n) with n − 1 degrees of freedom when σ is estimated; paired t uses the differences d̄ / (s_d/√n).
- Remember
- Paired data (same unit measured twice) need the paired t-test, which removes unit-to-unit variation.
- Tested as
- Test-selection and small calculation items.
- Trap
- Using a two-sample t-test on before-and-after measurements of the same units.
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Confidence interval for a mean
x̄ ± t(α/2, n − 1) × s/√n, or x̄ ± z(α/2) × σ/√n when σ is known; for 95% confidence z = 1.96.
- Remember
- If the hypothesized value lies outside the interval, the corresponding two-sided test rejects the null.
- Tested as
- Interval calculations and interpretation of interval output.
- Trap
- Saying there is a 95% probability the true mean is in this specific interval rather than describing the method's long-run coverage.
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Sample size for a mean
n = (z(α/2) × σ / E)², where E is the desired margin of error; round up.
- Remember
- Halving the margin of error requires four times the sample.
- Tested as
- Calculation items in study planning.
- Trap
- Rounding the result down.
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Tests for variances
The F-test compares two variances, F = s₁² / s₂²; chi-square tests one variance against a standard, χ² = (n − 1)s² / σ₀²; Bartlett's (normal data) and Levene's (robust to non-normality) compare several variances.
- Remember
- Check variance equality before choosing a pooled two-sample t or ANOVA.
- Tested as
- Test-selection items about consistency or variation.
- Trap
- Using Bartlett's test on clearly non-normal data instead of Levene's.
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One-way ANOVA
ANOVA tests whether three or more group means are equal using F = MS_between / MS_within, where MS = SS / df.
- Remember
- A significant F says at least one mean differs, not which; follow with multiple comparisons such as Tukey.
- Tested as
- Reading an ANOVA table and completing missing values.
- Trap
- Running multiple t-tests across groups, which inflates the overall Type I error.
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Chi-square tests
Goodness of fit and contingency-table tests use χ² = Σ (O − E)² / E; for a contingency table, E = row total × column total / grand total and df = (r − 1)(c − 1).
- Remember
- Expected counts should generally be at least 5 per cell.
- Tested as
- Test selection for categorical data and small calculations.
- Trap
- Using ANOVA or t-tests on counts of categories.
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Correlation and R²
r ranges from −1 to +1 and measures linear association; in simple regression R² = r², the share of response variation explained.
- Remember
- r = 0.8 explains 64% of variation, not 80%.
- Tested as
- Interpretation items for scatter plots and regression output.
- Trap
- Inferring causation or nonlinearity absence from r.
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Regression residual checks
Residuals (observed − fitted) should be independent, normally distributed and of constant variance; funnel shapes show heteroscedasticity, curves show a missing term.
- Remember
- A high R² does not validate a model whose residuals show patterns.
- Tested as
- Asked with residual plot descriptions.
- Trap
- Judging model adequacy by R² alone.
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Nonparametric alternatives
Mann-Whitney replaces two-sample t, Wilcoxon signed rank replaces one-sample or paired t, Kruskal-Wallis and Mood's median replace one-way ANOVA; they compare medians or ranks.
- Remember
- Use them when data are clearly non-normal with small samples or ordinal; Mood's median is more robust to outliers than Kruskal-Wallis.
- Tested as
- Test-selection items where normality fails.
- Trap
- Choosing a parametric test for heavily skewed small samples.
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Risk priority number (RPN)
RPN = severity × occurrence × detection, each typically rated 1 to 10, ranging from 1 to 1,000.
- Remember
- Address high severity first; detection is rated high when the failure is hard to detect.
- Tested as
- FMEA prioritization and re-rating calculations.
- Trap
- Thinking an action that improves detection lowers severity; severity changes only through design changes that alter the effect.
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Statistical vs. practical significance
A statistically significant result is unlikely under the null; practical significance asks whether the difference matters to customers or cost.
- Remember
- Large samples can make trivial differences significant; small samples can miss important ones.
- Tested as
- Interpretation items where p is tiny but the effect is negligible.
- Trap
- Recommending an expensive change because p < 0.05 when the effect size is too small to matter.
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Multi-vari analysis
A multi-vari chart separates variation into positional (within part), cyclical (part to part) and temporal (time to time) families.
- Remember
- It narrows the search to the dominant family of variation before detailed experiments.
- Tested as
- Asked when a team must find where variation comes from in a production process.
- Trap
- Confusing it with a multivariate statistical technique such as MANOVA.
Side by side
Concepts candidates mix up, compared on the point that decides the answer.
Which test? Means, variances, proportions and counts
| Concept | Meaning | Key difference | Common trap |
|---|---|---|---|
| Two-sample t vs. paired t | Compare means of two independent groups vs. two measurements on the same units | Pairing removes unit-to-unit variation | Using two-sample t on before-and-after data |
| One-way ANOVA | Compare three or more means | One F-test controls overall α | Running several t-tests instead |
| F-test, Bartlett, Levene | Compare variances | F for two normal groups, Bartlett for several normal, Levene when non-normal | Bartlett on skewed data |
| Chi-square contingency | Test association between categorical variables or compare proportions across groups | Works on counts, not measurements | ANOVA on pass/fail counts |
| Mann-Whitney, Kruskal-Wallis, Mood's median | Nonparametric comparisons of medians or ranks | No normality assumption | Assuming they compare means |
Type I vs. Type II error
| Concept | Meaning | Key difference | Common trap |
|---|---|---|---|
| Type I (α) | Rejecting a true null hypothesis | False alarm; producer's risk in acceptance sampling | Thinking a smaller α has no cost |
| Type II (β) | Failing to reject a false null hypothesis | Missed effect; consumer's risk in acceptance sampling | Ignoring it when the test is not significant |
| Power (1 − β) | Probability of detecting a real effect | Increases with sample size and effect size | Concluding no effect from an underpowered test |
Correlation vs. regression vs. causation
| Concept | Meaning | Key difference | Common trap |
|---|---|---|---|
| Correlation | Strength and direction of linear association | Symmetric; no dependent variable | Read as cause and effect |
| Regression | Equation predicting Y from X | Gives slope, intercept and prediction | Extrapolating beyond the data range |
| Causation | Changing X changes Y | Established by designed experiments or strong mechanism | Inferred from observational data alone |
Exam traps
Each trap is an answer that looks right. Learn the clue that should change your mind.
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Independent instead of paired
- What candidates pick
- Two-sample t-test for the same 20 parts measured on two gages.
- Why it looks right
- There are two sets of numbers to compare.
- Why it's wrong
- Each part appears in both sets, so the data are paired; the paired t-test removes part-to-part variation and has more power.
Clue to watch for The same units, people or parts are measured under both conditions.
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Many t-tests
- What candidates pick
- Run pairwise t-tests among four suppliers.
- Why it looks right
- Each test is familiar and simple.
- Why it's wrong
- Six separate tests at α = 0.05 inflate the chance of at least one false alarm; one-way ANOVA tests all means together.
Clue to watch for Three or more groups are compared on a continuous response.
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Significant means important
- What candidates pick
- Implement a costly change because p = 0.001.
- Why it looks right
- A very small p-value looks decisive.
- Why it's wrong
- With large samples tiny differences become significant; the effect size must be practically meaningful.
Clue to watch for A huge sample and a very small difference in means.
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RPN over severity
- What candidates pick
- Address the failure mode with RPN 280 (S 4, O 10, D 7) before one with RPN 90 (S 10, O 3, D 3).
- Why it looks right
- RPN appears to rank total risk.
- Why it's wrong
- A severity of 10 indicates a safety or regulatory effect that needs action regardless of a lower RPN.
Clue to watch for One failure mode has severity 9 or 10.
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Correlation proves cause
- What candidates pick
- Conclude oven temperature causes defects because r = 0.85.
- Why it looks right
- The relationship is strong and plausible.
- Why it's wrong
- Observational correlation can come from a lurking variable; causation needs a designed experiment or verification.
Clue to watch for Data are historical or observational, not from a controlled experiment.
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Fishbone vote as root cause
- What candidates pick
- Implement a fix for the cause most team members voted for on the fishbone.
- Why it looks right
- It reflects team consensus and expertise.
- Why it's wrong
- A fishbone organizes hypotheses; root causes must be verified with data before Improve.
Clue to watch for No data have yet been collected to test the suspected cause.
Answer framework
For any Analyze-phase question, use these steps to choose and interpret the analysis.
- Identify the response and input data types, continuous or categorical, and whether observations are independent or paired on the same units.
- Count the groups and check assumptions such as normality and equal variance to choose between parametric and nonparametric tests.
- State the null and alternative hypotheses, then compare the p-value with α, and phrase the conclusion as reject or fail to reject.
- Judge practical significance and model adequacy, including effect size, residual patterns, and whether the evidence shows correlation or causation.
- For risk and root cause, prioritize high severity before RPN, and verify any suspected cause with data before moving to Improve.
Practice question
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Checklist
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Study strategy
Build a test-selection flowchart first; it is the single most useful study aid for this section. Branch on data type, number of groups, pairing, and normality, and end each branch at a named test. Practice by reading short scenarios and naming the test in under thirty seconds.
Next, master interpretation. Practice reading ANOVA tables (fill in missing SS, df, MS and F), regression outputs (coefficients, p-values, R², residual plots) and confidence intervals. Always phrase conclusions as reject or fail to reject, and ask whether the effect is practically meaningful.
Then cover calculations: t statistics, confidence intervals, sample size, chi-square expected counts and degrees of freedom. Keep these on a formula page with t, chi-square and F tables tabbed in your open-book reference.
The commonly underestimated areas are FMEA and nonparametric tests. FMEA looks simple, but questions turn on severity overriding RPN and on which rating an action can change. Nonparametric tests appear as distractors and as correct answers when normality fails.
Afterwards, practice mixed Analyze sets under time pressure, then move to Improve, where verified causes become designed experiments and lean solutions.
FAQ
How many questions come from Analyze?
ASQ assigns 22 of the 150 scored questions to Analyze, about 14.7% of the exam, the second-largest section.
How do I choose between a t-test and ANOVA?
Use a t-test to compare two means and one-way ANOVA for three or more. If the same units appear in both conditions, use a paired t-test.
What does failing to reject the null hypothesis mean?
The data do not provide enough evidence of a difference at the chosen α. It does not prove the null is true, especially if the test had low power.
Why does severity override RPN in FMEA?
RPN multiplies three ratings, so a catastrophic but rare failure can score lower than a frequent nuisance. Safety and regulatory effects rated 9 or 10 need action regardless of RPN.
When should I use a nonparametric test?
When the data are ordinal or clearly non-normal and the sample is too small for the central limit theorem to help. They compare medians or ranks rather than means.
Does a high R² mean the model is good?
Not by itself. Check residuals for patterns, check for multicollinearity, and confirm the relationship makes engineering sense before using the model.
How does Analyze connect to Improve?
Analyze verifies which inputs drive the output. Improve then uses designed experiments to optimize those inputs and lean methods to implement the solution.
Other Quality Management certifications
Where the same subject shows up in other exams. Cards marked "Same subject" open the matching domain.
Test your Domain 6 knowledge
You have covered regression and multivariate tools, hypothesis testing, FMEA and root cause analysis. Test whether you can choose and interpret the right analysis under exam conditions.