Chapter 19: Nonparametric Methods
Student Resources
I use the 4 “P’s” framework to help you learn the material in this chapter: Prepare, Practice, Participate, and Perform. To increase the chances to succeed in this course, I strongly encourage you to complete all four “P’s” for each chapter.
1 Prepare
1.1 Chapter Overview
This chapter introduces nonparametric methods — statistical procedures that do not assume a specific distributional form and are appropriate when data are ordinal, categorical, severely non-normal in small samples, or subject to floor and ceiling effects. You will learn when and why to move beyond parametric tests (t-tests, ANOVA, Pearson r) and which nonparametric alternative to select for each research design. The chapter covers chi-square tests for categorical outcomes, Spearman rank-order correlation, Mann-Whitney U, Wilcoxon signed-rank, Kruskal-Wallis, and Friedman’s test — along with the effect sizes paired with each procedure and guidance on APA-style reporting.
1.2 Multimedia Resources
The following table provides access to video and slide resources for this chapter. Click the links to open them in an overlay for better viewing on all devices.
| Resource | Description | Link |
|---|---|---|
| Long Video Overview | A detailed video explaining nonparametric methods, when to use them, how ranks work, and how to interpret chi-square, Spearman, Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman’s tests in movement science research. | 🔗 Watch Video |
| 🆕 Interactive Self-Study Guide | NEW: An interactive, comprehensive module to be completed before attending class. Covers core concepts to prepare you for the lecture. | 🔗 View Guide |
| Slide Deck HTML | Interactive HTML slides for class. During class, the instructor controls the presentation; after class, review at your own pace. | 🔗 Open Slides |
| Slide Deck PDF | PDF version of the slide deck for download and offline viewing. | 🔗 Download PDF |
You can earn extra credit points by completing the Interactive Self-Study Guide before attending the lecture.
Instructions: 1. Click the link in the table above to start the guide. 2. Complete the activity (you may retake it as many times as you like). 3. Once satisfied with your result, take a screenshot of the Final Score Card. 4. Submit the screenshot to Canvas.
Important: Your screenshot must clearly show both your score and your full name (ensure you enter your name at the start of the activity). See Canvas for point details.
1.3 Read the Chapter
Read (Weir & Vincent, 2021, p. Ch.16) and (Furtado, 2026, p. Ch.19) to understand when nonparametric methods are appropriate and how to select, apply, and report the correct procedure for each research design.
To succeed in this course, you must read the textbook chapters assigned for each topic. This is the only way to learn the material in depth.
Once done, proceed to the next section to practice what you learned.
2 Practice
Practicing what you learned in the chapter is essential to mastering the material. Below are some resources to help you practice the material in this chapter.
2.1 Frequently Asked Questions
Nonparametric tests are most appropriate under four conditions:
Ordinal-level measurement. Pain ratings, Borg RPE scores, and Likert items are ordinal — the intervals between scale values are not necessarily equal. Rank-based tests respect the ordering without assuming equal intervals.
Severe non-normality in small samples. When n < 20 per group and there is strong evidence of skewness or outliers, the Central Limit Theorem cannot rescue the normality assumption. Rank-based methods are inherently insensitive to extreme scores.
Categorical outcome variables. When the outcome is a named category (injured/not injured, group membership), chi-square tests are the appropriate framework — no parametric alternative exists.
Floor or ceiling effects. Scores that cluster near the boundaries of a scale violate the symmetry assumption underlying many parametric tests.
Importantly, nonparametric tests are not the default safer choice. When parametric assumptions are reasonably met, parametric tests are more powerful. Use the nonparametric alternative only when there is a clear reason to do so.
Most nonparametric tests share one core mechanism: they convert raw scores to ranks and then perform arithmetic on those ranks instead of the original values. After ranking, the test depends only on the ordering of scores, not their exact magnitudes. This is what makes rank-based tests robust to outliers — an extreme score of 2 and an extreme score of −500 receive the same rank (rank 1) if both are the lowest value in the dataset.
When two or more participants share the same raw score (a tie), each receives the average of the ranks they would have occupied. Most statistical software handles ties automatically.
Both use the same chi-square formula, but they answer different questions:
Goodness-of-fit test: tests whether the observed frequency distribution of a single categorical variable matches a hypothesized distribution (e.g., is the sex split in our sample 50/50?). Each category is compared to its expected proportion.
Test of independence: tests whether two categorical variables are associated (e.g., is group assignment independent of sex?). Data are organized in a contingency table, and observed cell counts are compared to the counts expected if the variables were unrelated.
The key assumption for the test of independence is that all expected cell counts must be ≥ 5. When this is violated — typically in small samples or tables with many cells — Fisher’s exact test should be used instead (for 2×2 tables) or categories should be collapsed.
Fisher’s exact test is appropriate when a 2×2 contingency table has one or more expected cell counts below 5. The chi-square statistic is an approximation that requires reasonably large expected counts; when cells are small, the approximation breaks down and the p-value becomes unreliable. Fisher’s exact test calculates the exact probability of the observed distribution (and all more extreme distributions) without any large-sample approximation. SPSS reports Fisher’s exact p alongside the chi-square p in the crosstabs output — use Fisher’s when any expected cell count is below 5.
Both measure the association between two variables, but they differ in what kind of association they detect:
Pearson r quantifies the strength of the linear relationship between two continuous variables. It assumes both variables are normally distributed and that the relationship follows a straight line.
Spearman ρ quantifies the strength of the monotonic relationship — whether one variable tends to increase as the other increases (or decreases), regardless of whether the pattern is perfectly linear. It works by converting both variables to ranks and then correlating those ranks. Use Spearman ρ when one or both variables are ordinal, when the relationship is clearly nonlinear (e.g., diminishing returns), or when outliers substantially distort Pearson r.
The Mann-Whitney U test is the nonparametric alternative to the independent-samples t-test. It is used when you have two independent groups and either the outcome is ordinal, the data are severely non-normal in small samples, or both. Instead of comparing group means, it asks whether scores in one group tend to be systematically higher or lower than scores in the other group.
The test works by pooling all scores, ranking them together, and comparing rank sums between groups. A large discrepancy in rank sums indicates that one group consistently outranks the other. The appropriate effect size is rank-biserial r (ranging from −1 to +1), which has the same magnitude benchmarks as Pearson r.
The Wilcoxon signed-rank test is the nonparametric alternative to the paired-samples t-test. It is used when two related measurements are available for each participant (e.g., pre and post) and the difference scores are not normally distributed or are measured on an ordinal scale.
Unlike the sign test — which considers only whether each participant improved or worsened — the Wilcoxon test also accounts for the magnitude of each difference (by ranking the absolute differences and weighting positive vs. negative ranks). This makes it more powerful than the sign test whenever the magnitudes of the differences are interpretable. The sign test should be reserved for situations where only the direction of change is reliable.
The Kruskal-Wallis test is the nonparametric alternative to one-way between-subjects ANOVA. Use it when you have three or more independent groups and the outcome is ordinal or the normality assumption is untenable with small samples.
Like ANOVA, a significant Kruskal-Wallis result tells you only that at least one group differs — it does not identify which pairs. After a significant result, follow up with Dunn’s test (also called Dunn-Bonferroni pairwise comparisons) to pinpoint which groups differ while controlling the familywise Type I error rate.
Note: when you have only two groups, use the Mann-Whitney U test directly — Kruskal-Wallis with two groups is mathematically equivalent but less commonly reported.
Friedman’s test is the nonparametric alternative to one-way repeated-measures ANOVA. Use it when the same participants are measured under three or more conditions and the outcome is ordinal or severely non-normal. It works by ranking scores within each participant across conditions — each participant contributes their own set of ranks 1 through k. The test statistic is based on whether the rank sums across conditions are more unequal than chance alone would predict.
The appropriate effect size is Kendall’s W (concordance coefficient), which ranges from 0 (no consistency in ranking across participants) to 1 (all participants rank the conditions identically). After a significant Friedman result, follow up with pairwise Wilcoxon signed-rank tests with Bonferroni correction.
| Test | Effect size |
|---|---|
| Chi-square independence (2×2) | Phi (φ) |
| Chi-square independence (larger table) | Cramér’s V |
| Spearman ρ | ρ itself |
| Mann-Whitney U | Rank-biserial r |
| Wilcoxon signed-rank | Rank-biserial r |
| Kruskal-Wallis | η²_H |
| Friedman’s | Kendall’s W |
All effect sizes should be reported alongside the test statistic and p-value — a non-significant result does not mean the effect is zero, and an effect size gives readers the information needed to judge practical importance and evaluate statistical power.
Using nonparametric tests “just to be safe.” Rank-based methods discard information by converting to ranks, which reduces power when parametric assumptions are met. Choosing nonparametric tests by default — rather than because assumptions are violated — inflates Type II error.
Treating a non-significant result as confirming no difference. Failing to reject H₀ with a small sample may simply reflect lack of power. Always report the effect size alongside p.
Violating the expected cell count requirement for chi-square. Running a chi-square test when expected cell counts are < 5 can produce a misleading p-value. Check expected counts in SPSS output and switch to Fisher’s exact if needed.
Running multiple nonparametric follow-up tests without correction. Post-hoc Wilcoxon or Mann-Whitney comparisons after Friedman or Kruskal-Wallis inflate the familywise error rate just as multiple t-tests do. Apply Bonferroni or Holm correction.
Misinterpreting Kendall’s W as equivalent to eta-squared. Kendall’s W measures concordance in rank ordering across participants — it is not directly comparable to η² from parametric ANOVA.
2.2 Test your Knowledge
Take this low-stakes quiz to test your knowledge of the material in this chapter. This quiz is for practice only and will help you identify areas where you may need additional review.
3 Participate
This section includes activities and discussions that will be completed during class time. Your active participation is essential for deepening your understanding of the material.
During class, we will: - Identify which nonparametric test is appropriate for a given research design using the decision guide - Distinguish between chi-square goodness-of-fit and chi-square independence and verify the expected cell count assumption - Interpret Spearman ρ output from SPSS and compare it to a Pearson r result for the same data - Match Mann-Whitney U and Wilcoxon signed-rank results to their appropriate effect sizes - Interpret a Kruskal-Wallis result and determine whether post-hoc comparisons are needed - Interpret Friedman’s test output and Kendall’s W in the context of a repeated-measures design - Practice writing complete APA-style results paragraphs for each test
4 Perform
4.1 Apply Your Learning
Now that you’ve prepared, practiced, and participated, it’s time to demonstrate your mastery of the material through assignments and assessments.
I strongly encourage you to complete the previous “Ps” (Prepare, Practice, Participate) before attempting any assignments or assessments associated with this chapter.