You ran a one-way ANOVA, got a significant p-value, and now your committee is asking whether your assumptions were met. If you're staring at your output wondering what Levene's test actually tells you — or whether that Shapiro-Wilk p-value of .03 invalidates your whole analysis — you're in the right place. This guide walks through the one-way ANOVA assumptions, how to check them, and what to do when they fail.
The Four One-Way ANOVA Assumptions
A one-way ANOVA tests whether means differ across three or more independent groups. For the F-statistic to be trustworthy, four assumptions need to hold:
- Independence of observations — each participant contributes one score, and groups don't influence each other.
- Normality — the dependent variable is approximately normally distributed within each group.
- Homogeneity of variance — group variances are roughly equal.
- Continuous dependent variable — the outcome is measured on an interval or ratio scale.
Independence and measurement level are design issues — you handle them when you plan the study, not after. The other two are diagnostic, and that's where most students get stuck.
How to Check the Normality Assumption
Normality applies to the dependent variable within each group, not the full dataset. Run these checks for each level of your independent variable:
Visual checks (do these first)
- Q-Q plots: Points should fall along the diagonal line. Small wobbles at the tails are fine.
- Histograms: Look for roughly bell-shaped distributions, not skewed cliffs.
- Boxplots: Symmetric boxes with similar whiskers across groups.
Formal tests
- Shapiro-Wilk test for samples under 50 per group. A non-significant result (p > .05) supports normality.
- Kolmogorov-Smirnov test for larger samples, though Shapiro-Wilk is generally preferred.
A key point most textbooks bury: ANOVA is robust to mild normality violations when group sizes are equal and n per group is above ~25. Don't panic if Shapiro-Wilk gives you p = .04 — check skewness (should be between -2 and 2) and kurtosis (between -7 and 7) before abandoning the test.
In StatRyx, normality checks run automatically when you select ANOVA — you get Q-Q plots, Shapiro-Wilk results, and skewness values for every group in one click.
How to Check Homogeneity of Variance
This is the assumption that trips up the most theses. Equal variance across groups means the spread of scores should look similar regardless of group membership.
Levene's Test
The standard check. The null hypothesis is that variances are equal, so you want a non-significant result.
- p > .05 → variances are equal, proceed with standard ANOVA
- p < .05 → variances differ, use Welch's ANOVA instead
Brown-Forsythe Test
A more robust variant of Levene's that uses medians instead of means. Better when data are skewed.
The rule-of-thumb check
If your largest group variance is less than 4× the smallest, ANOVA is generally robust enough — especially with equal sample sizes.
Worked Example: Checking Assumptions in a Real Study
Suppose you're comparing exam scores across three teaching methods (lecture, flipped, hybrid) with 20 students per group.
Normality check (Shapiro-Wilk):
- Lecture: W = .96, p = .54
- Flipped: W = .94, p = .28
- Hybrid: W = .92, p = .09
All three groups pass. Normality assumption met.
Homogeneity check (Levene's test):
F(2, 57) = 1.84, p = .168
Variances are equal. Homogeneity assumption met.
ANOVA result:
F(2, 57) = 5.42, p = .007, η² = .16
You can now report: "A one-way ANOVA revealed a significant effect of teaching method on exam scores, F(2, 57) = 5.42, p = .007, η² = .16. Assumptions of normality (all Shapiro-Wilk ps > .05) and homogeneity of variance (Levene's p = .168) were satisfied."
The η² of .16 means teaching method explains 16% of the variance in scores — a large effect by Cohen's conventions.
When Assumptions Are Violated: Choose the Right Alternative
| Violation | Use Instead | Why |
|---|---|---|
| Normality violated, equal variances | Kruskal-Wallis test | Non-parametric, ranks-based, no normality required |
| Equal variances violated, normal data | Welch's ANOVA | Adjusts degrees of freedom for unequal variances |
| Both violated | Kruskal-Wallis test | Most robust option for small, messy samples |
| Severe outliers driving the violation | Trimmed-means ANOVA | Reduces outlier influence |
| Independence violated (repeated measures) | Repeated-measures ANOVA | Models within-subject correlation |
Standard ANOVA vs Welch's ANOVA: Key Differences
| Feature | Standard ANOVA | Welch's ANOVA |
|---|---|---|
| Variance assumption | Equal variances required | Handles unequal variances |
| Degr |
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