One-Way ANOVA: Assumptions, Example, and APA Reporting

A one-way ANOVA (analysis of variance) tests whether the means of three or more independent groups differ on a continuous outcome. It has one categorical independent variable (the factor, such as teaching method) and one continuous dependent variable (such as exam score). A significant F test tells you that at least one group mean differs; post hoc tests tell you which ones.

This guide walks through the assumptions, computes an ANOVA by hand on a small dataset, adds effect sizes and post hoc tests, and ends with an APA 7 reporting template.

When to use a one-way ANOVA

Use it when:

• You have one grouping variable with three or more levels (with two groups, an independent-samples t test gives the same p value, since F = t²).
• Groups are independent (different participants in each group).
• The outcome is continuous (interval or ratio).

Why not just run several t tests? With three groups you would need three t tests, and each one carries its own 5% chance of a false positive. The familywise error rate grows with every extra test. ANOVA tests all groups in one omnibus test, and post hoc procedures then control error across the pairwise comparisons.

If you have two factors (for example, teaching method and gender), you need a two-way ANOVA instead. If the same participants are measured repeatedly, you need a repeated-measures ANOVA.

Hypotheses

• H0: μ1 = μ2 = μ3 (all population means are equal).
• H1: At least one population mean differs from the others.

Note that H1 does not say all means differ, and the omnibus test does not tell you which ones do.

Assumptions

1. Independence of observations. Each participant contributes one score and belongs to one group. This is a design issue; no test can fix a violation after the fact.
2. Normality of residuals. The outcome should be approximately normal within each group. Check with Q-Q plots and, optionally, Shapiro–Wilk tests. ANOVA is fairly robust to moderate non-normality when group sizes are similar and not tiny.
3. Homogeneity of variance. Group variances should be similar. Check with Levene's test. If it is violated, especially with unequal group sizes, use Welch's ANOVA and the Games-Howell post hoc test.
4. No extreme outliers within groups, since they inflate variance and can distort means.

If normality is badly violated and samples are small, the Kruskal–Wallis test is the usual nonparametric alternative.

Worked example: computing F by hand

A researcher compares exam scores under three teaching methods, with 5 students per method (N = 15).

The grand mean (mean of all 15 scores) is 1105 / 15 = 73.67.

Step 1: Between-groups sum of squares (SSB)

SSB measures how far group means are from the grand mean, weighted by group size:

SSB = Σ n × (group mean − grand mean)²

• Lecture: 5 × (70.0 − 73.667)² = 5 × 13.444 = 67.22
• Flipped: 5 × (78.0 − 73.667)² = 5 × 18.778 = 93.89
• Online: 5 × (73.0 − 73.667)² = 5 × 0.444 = 2.22

SSB = 67.22 + 93.89 + 2.22 = 163.33

Step 2: Within-groups sum of squares (SSW)

SSW adds up squared deviations of each score from its own group mean:

• Lecture: 0² + 2² + (−2)² + 5² + (−5)² = 58
• Flipped: 0² + 2² + (−4)² + 4² + (−2)² = 40
• Online: (−1)² + 2² + (−3)² + 5² + (−3)² = 48

SSW = 58 + 40 + 48 = 146.00

Check: the total sum of squares (each score minus the grand mean, squared and summed) is 309.33, and SSB + SSW = 163.33 + 146.00 = 309.33.

Step 3: Degrees of freedom and mean squares

• df between = k − 1 = 3 − 1 = 2
• df within = N − k = 15 − 3 = 12
• MSB = SSB / df between = 163.33 / 2 = 81.67
• MSW = SSW / df within = 146.00 / 12 = 12.17

Step 4: The F ratio

F = MSB / MSW = 81.67 / 12.17 = 6.71

With df = (2, 12), the p value is .011. (The critical F at α = .05 is 3.89, and 6.71 exceeds it.) We reject H0: mean exam scores differ across teaching methods.

The ANOVA table

You can reproduce these numbers by entering the three groups into the ANOVA calculator.

Checking the assumptions in this example

Levene's test (mean-centered, as in SPSS) gives F(2, 12) = 0.08, p = .92, so there is no evidence of unequal variances. Shapiro–Wilk p values for the three groups are all above .30. With only 5 observations per group these tests have little power, so in a real study you would also inspect plots and rely on design considerations.

Effect size: eta squared

A p value does not tell you how large the differences are. The usual effect size for one-way ANOVA is eta squared:

η² = SSB / SST = 163.33 / 309.33 = .53

So 53% of the variance in exam scores is associated with teaching method. In a one-way design, eta squared and partial eta squared are identical.

Eta squared is positively biased in small samples. Omega squared is a less biased alternative:

ω² = (SSB − (k − 1) × MSW) / (SST + MSW) = (163.33 − 2 × 12.17) / (309.33 + 12.17) = .43

Cohen's (1988) rough benchmarks for η² are .01 (small), .06 (medium), and .14 (large), but interpret effects in the context of your field. The effect size calculator can convert between common effect size measures.

Post hoc tests: which groups differ?

Tukey's HSD (equal variances)

Tukey's honestly significant difference compares every pair of means while controlling the familywise error rate. It assumes equal variances and works best with equal or similar group sizes.

For this example, the critical difference is HSD = q × √(MSW / n) = 3.773 × √(12.17 / 5) = 5.89, where q is the studentized range value for 3 groups and 12 df at α = .05. Any pair of means differing by more than 5.89 points is significant.

Only the Flipped vs. Lecture difference is significant. The Flipped vs. Online difference (5.0 points) is in the expected direction but does not reach significance with this small sample.

Games-Howell (unequal variances)

When Levene's test or inspection of SDs suggests unequal variances, use Games-Howell. It does not assume equal variances or equal sample sizes, and it pairs naturally with Welch's ANOVA.

Planned contrasts

If you had specific hypotheses before seeing the data (for example, "flipped classroom beats the average of the other two"), planned contrasts are more powerful than testing every pair. Specify them in advance and report them as such.

Welch's ANOVA

The classic F test assumes equal variances. Welch's ANOVA adjusts the denominator degrees of freedom and weights groups by their precision, so it remains accurate when variances differ. Many methodologists recommend reporting Welch's ANOVA by default, because it loses little power when variances are equal.

For our data, Welch's ANOVA gives F(2, 7.95) = 6.41, p = .022. The conclusion is the same as the classic F test, which is expected because the variances are similar. Note the non-integer denominator df, which you should report with two decimals.

How to report a one-way ANOVA in APA 7

Report descriptive statistics, the F test with both degrees of freedom, the exact p value, an effect size, and the post hoc results.

APA formatting reminders:

• Italicize statistical symbols (F, p, M, SD).
• Report exact p values to two or three decimals; use p < .001 only for very small values.
• No leading zero for statistics that cannot exceed 1 (p, η²).
• Include a table of means and SDs if there are more than a few groups.

If variances were unequal, the sentence would instead read: "Welch's ANOVA indicated a significant effect, F(2, 7.95) = 6.41, p = .022, and Games-Howell post hoc tests showed..."

Common mistakes

• Running multiple t tests instead of ANOVA plus a post hoc procedure.
• Interpreting a significant F as "all groups differ." It only says at least one differs.
• Running post hoc tests after a non-significant F without a planned rationale, and then reporting only the significant pair.
• Ignoring unequal variances with unequal group sizes. This combination is where the classic F test is least trustworthy.
• Reporting only p values. Include η² or ω² and confidence intervals for pairwise differences.
• Using one-way ANOVA for repeated measures. Scores from the same people are not independent.

FAQ

What is the difference between a one-way ANOVA and a t test?

A t test compares two means. A one-way ANOVA compares three or more. With exactly two groups, both give the same p value (F = t²). For two groups, the t-test calculator is the simpler choice.

What if Levene's test is significant?

Report Welch's ANOVA instead of (or alongside) the classic F test, and use Games-Howell for post hoc comparisons.

How many participants do I need for a one-way ANOVA?

It depends on the effect size you expect, α, and the power you want. Run an a priori power analysis before data collection rather than relying on a rule of thumb.

Should I report eta squared or partial eta squared?

In a one-way ANOVA they are the same number. In factorial designs they differ, and partial eta squared is the more common choice, as explained in the two-way ANOVA guide.

What is the nonparametric alternative to one-way ANOVA?

The Kruskal–Wallis test, followed by Dunn's test with a correction for multiple comparisons.

Turning results into a paper

For quick checks, the ANOVA calculator handles a single comparison. If you have a full dataset and need the whole paper, Data2Paper's Generate Paper produces a draft with methods, results (ANOVA tables, post hoc tests, effect sizes), figures, and references. You can preview the first half for free and unlock the rest for $19.90. Check all statistics and interpretations before using the draft.