Chapter 14: Analysis of Variance
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 Analysis of Variance (ANOVA)—the essential technique for comparing means across three or more groups in Movement Science research. You’ll learn why running multiple t-tests is problematic, how ANOVA partitions total variability into meaningful components, how to interpret the F-ratio and ANOVA source table, how to apply post hoc tests to identify which specific groups differ, and how to quantify the magnitude of group differences using effect sizes (η² and ω²).
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 one-way ANOVA, the F-ratio, assumptions, post hoc tests, effect sizes, and interpretation 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.11) and (Furtado, 2026, p. Ch.14) to understand the theoretical foundations and practical application of one-way ANOVA for comparing three or more group means.
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
Running multiple pairwise t-tests instead of ANOVA inflates the familywise error rate—the probability of making at least one Type I error across all comparisons. At α = .05, each individual test carries a 5% false-positive risk. With three groups and three pairwise comparisons, the familywise error rate rises to approximately .14—nearly three times the intended level. With five groups requiring ten comparisons, it reaches approximately .40. ANOVA solves this problem by testing a single omnibus null hypothesis (all group means are equal) in one analysis that maintains the stated α level. Post hoc procedures, applied only after a significant omnibus F, then control the familywise error rate when making specific pairwise comparisons.
ANOVA decomposes total variability into two components: between-groups variance and within-groups variance. Between-groups variance (\(SS_{\text{between}}\)) reflects how much group means differ from the grand mean—it captures the effect of the independent variable. Within-groups variance (\(SS_{\text{within}}\)) reflects how much individual scores vary around their own group mean—it captures random error and individual differences. Total variance is simply their sum: \(SS_{\text{total}} = SS_{\text{between}} + SS_{\text{within}}\). This partitioning is why ANOVA is so powerful: it quantifies “signal” (between-groups differences) relative to “noise” (within-groups variability) in a single, coherent framework.
The F-ratio is the heart of ANOVA: \(F = MS_{\text{between}} / MS_{\text{within}}\). Think of it as a signal-to-noise ratio. When the null hypothesis is true and all group means are equal, both mean squares estimate the same population variance, so F ≈ 1. When the alternative hypothesis is true and at least one group mean differs, \(MS_{\text{between}}\) is inflated by true group differences, producing F > 1. The larger the F, the stronger the evidence against H₀. To evaluate significance, compare the obtained F against a critical value from the F-distribution with \(df_1 = k - 1\) (numerator) and \(df_2 = N - k\) (denominator) degrees of freedom, where \(k\) = number of groups and \(N\) = total sample size.
One-way ANOVA rests on three key assumptions:
- Independence of observations: Scores from different participants must be independent. This is ensured by research design (e.g., random assignment, no repeated measures). Violations produce serious errors that cannot be corrected statistically.
- Normality: The dependent variable should be approximately normally distributed within each group. ANOVA is robust to moderate departures from normality when group sizes are approximately equal and \(n \geq 30\) per group. Use histograms, Q-Q plots, and the Shapiro-Wilk test to evaluate this assumption within each group separately.
- Homogeneity of variance: Population variances should be approximately equal across all groups. Evaluate with Levene’s test: a non-significant result (\(p > .05\)) supports the assumption. If violated, use Welch’s ANOVA or Brown-Forsythe ANOVA instead of the standard F-test.
The choice depends on whether the homogeneity of variance assumption is met:
- Tukey HSD: Use when Levene’s test is non-significant (equal variances assumed) and group sizes are equal or approximately equal. It provides the best balance between Type I and Type II error control and is the most widely used post hoc test in Movement Science.
- Games-Howell: Use when Levene’s test is significant (variances are unequal). It does not assume equal variances and performs well with unequal group sizes. Always switch to Games-Howell when heteroscedasticity is detected.
Run post hoc tests only after obtaining a significant omnibus F. Applying them to a non-significant F inflates Type I error and produces unreliable conclusions.
Both quantify the proportion of total variance in the dependent variable explained by the independent variable, but they differ in how they estimate the population effect:
- Eta-squared (η²) = \(SS_{\text{between}} / SS_{\text{total}}\). Simple to compute and interpret, but it slightly overestimates the population effect size, especially in small samples.
- Omega-squared (ω²) corrects for this upward bias and provides a less biased estimate of the true population effect. It is the preferred metric when reporting results in a manuscript, particularly with small-to-moderate sample sizes.
Cohen’s (1988) benchmarks apply to both: small = .01, medium = .06, large = .14.
A significant F-test tells you that at least one group mean differs from the others—nothing more. It does not tell you which specific pairs of groups are different. Think of it as a signal alerting you that something is going on among the groups. To determine exactly which group pairs are responsible, you must follow up with post hoc tests (e.g., Tukey HSD or Games-Howell). This two-step process—omnibus test first, then post hoc comparisons—is the methodologically correct approach and is required for APA-style reporting of ANOVA results.
A complete APA-style results statement includes the F-statistic, degrees of freedom, p-value, and effect size. For example:
A one-way ANOVA revealed a significant effect of training group on VO₂max, F(2, 57) = 14.87, p < .001, η² = .34. Tukey HSD post hoc comparisons indicated that the endurance training group (M = 52.3, SD = 6.1) had significantly higher VO₂max than both the resistance training group (M = 45.1, SD = 5.8; p = .003) and the control group (M = 40.5, SD = 5.9; p < .001). The resistance training group did not significantly differ from the control group (p = .082).
Always include means and standard deviations for each group, the specific post hoc comparisons with adjusted p-values, and an effect size to convey practical significance.
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 research scenarios that require one-way ANOVA versus a t-test - Check ANOVA assumptions (normality, homogeneity of variance) using SPSS - Run a one-way ANOVA in SPSS and interpret the source table output - Apply Tukey HSD and Games-Howell post hoc tests and compare their results - Compute and interpret effect sizes (η² and ω²) from SPSS output - Practice writing APA-style results statements for one-way ANOVA
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.