One-Way ANOVA in SPSS: Example, Assumptions, Post Hoc Tests and APA Reporting

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What is a one-way ANOVA?

A one-way analysis of variance compares the average score of several groups that differ on one independent variable. "One-way" refers to that single factor; the factor can have three, four or more levels (conditions), but there is still only one of them. Typical psychology examples include comparing anxiety scores across three therapy types, reaction times across four levels of sleep deprivation, or exam scores across three teaching methods (Field, 2024).

The test works by comparing two sources of variability in the data:

• Between-groups variance: how far the group means are from the overall mean. If the conditions have an effect, this will be large.
• Within-groups variance: how much individual scores vary around their own group mean. This is the background noise, or error.

The F ratio is the between-groups variance divided by the within-groups variance. When the groups do not really differ, F is close to 1; the larger F is, the less likely it is that differences this big arose by chance. Because the F test looks at all the groups at once, it is called an omnibus test: a significant result tells you that the means are not all equal, but not which ones differ. That is the job of post hoc tests.

One-way ANOVA vs t-test vs two-way ANOVA

A common question is why you cannot simply run several t-tests. With three groups, you would need three t-tests (A vs B, A vs C, B vs C), and each one carries a 5% risk of a false positive. Across three tests, the chance of at least one false positive rises to about 14% (1 − .95³ = .14), and it keeps climbing as you add groups. The ANOVA tests all the groups in one analysis at α = .05, and post hoc tests then make the pairwise comparisons while controlling this inflated Type I error rate.

If your design has two independent variables, see our guide to main effects and interactions in a 2 × 2 factorial ANOVA. For a wider decision guide covering chi-square, correlation and regression, see which statistical test I should use.

One-way ANOVA assumptions (and what to do if they are violated)

Two recent papers are worth knowing. Blanca et al. (2017) simulated a wide range of non-normal distributions and found that the F test kept its false-positive rate under control in most conditions, so mild non-normality is rarely a reason to abandon ANOVA. Unequal variances are a bigger problem, particularly when group sizes also differ. Delacre et al. (2019) therefore recommend using Welch's F by default: it performs almost as well as the classic F when variances are equal and much better when they are not.

A dependent variable also needs a precise operational definition so readers know exactly what was measured and how it was scored.

Worked example: does a customer's clothing change how fast they are served?

This example comes from a classic textbook study (Smith & Davis, 2016). Salespeople were randomly assigned to see a customer who was dressy, sloppy or casually dressed, and the researchers timed how many seconds passed before the salesperson approached. The independent variable is clothing (three levels), the dependent variable is time to approach in seconds, and there are eight salespeople per group (N = 24). Higher times mean slower service.

Each mean is the sum divided by the number of scores: for the dressy group, 387 / 8 = 48.38 seconds. Looking at the means, salespeople seem to approach dressy and casual customers at about the same speed (48.38 and 48.88 seconds) but much more slowly when the customer is sloppily dressed (63.25 seconds). Whether those differences are bigger than chance is what the ANOVA tests.

How to run a one-way ANOVA in SPSS

Enter the data with one row per participant: one column for the group (coded 1 = dressy, 2 = sloppy, 3 = casual, with value labels) and one column for the time in seconds. Then:

1. Choose Analyze > Compare Means > One-Way ANOVA.
2. Move the outcome (Time to help) into the Dependent List and the grouping variable (Clothes) into Factor. SPSS calls the dependent variable the Dependent List and the independent variable the Factor.
3. Click Post Hoc, tick Tukey, and also tick Games-Howell so you have a post hoc test ready if the variances turn out to be unequal. Click Continue.
4. Click Options and tick Descriptive, Homogeneity of variance test, Welch and Means plot. Click Continue, then OK.

How to interpret one-way ANOVA output in SPSS

SPSS produces three tables you need and a plot. Read them in this order.

1. Descriptives

This table gives N, the mean, standard deviation, standard error and 95% confidence interval for each group and for everyone combined. Copy the means and standard deviations into your write-up; they show the direction of any difference the ANOVA finds. See our guide to descriptive statistics in SPSS for what each one means.

2. The ANOVA table

This is the omnibus test. Read across the Between Groups row:

• Sum of Squares splits the total variability (3688.00) into the part explained by the groups (1141.75) and the part within groups (2546.25).
• df gives the two degrees of freedom you report: between groups = number of groups − 1 = 2; within groups = N − number of groups = 21.
• Mean Square is each sum of squares divided by its df.
• F = 570.875 / 121.250 = 4.71.
• Sig. is the p-value. Here p = .020, which is below .05, so the group means are not all equal.

If you ticked Homogeneity of variance test, SPSS also prints Levene's test. Here it is not significant, F(2, 21) = 0.95, p = .404, so the equal-variances assumption is reasonable and Tukey is the right post hoc test.

3. Multiple Comparisons (post hoc tests)

Because the ANOVA was significant, the Tukey HSD table shows which pairs of groups differ. Each pair appears twice (A vs B and B vs A), so read each comparison once. Significant differences are marked with an asterisk in the Mean Difference column.

4. The means plot

The means plot shows the three group means. It makes the pattern obvious: the sloppy group sits well above the other two. Note that SPSS joins the points with lines even though the groups have no natural order; for a report, a dot plot with confidence intervals is clearer. Our guide to making figures for a scientific paper explains why.

Effect size for a one-way ANOVA: eta squared and omega squared

A p-value tells you whether the groups differ; an effect size tells you by how much. Report one with every ANOVA (American Psychological Association, 2020).

• Eta squared (η²) = between-groups sum of squares ÷ total sum of squares = 1141.75 ÷ 3688.00 = .31. Clothing accounts for about 31% of the variability in approach times in this sample.
• Omega squared (ω²) = (SS between − df between × MS within) ÷ (SS total + MS within) = .24. η² overestimates the population effect, especially in small samples, so ω² is the less biased choice (Albers & Lakens, 2018; Lakens, 2013).

Cohen's (1992) benchmarks for ANOVA are expressed as f: .10 small, .25 medium and .40 large, which correspond roughly to η² of .01, .06 and .14. Here f = .67, a large effect, although with only eight people per group the estimate is imprecise. Benchmarks are a rough guide; judge the size of an effect against what matters in your field. You can convert between η², ω², f and d with our effect size calculator.

How to report a one-way ANOVA in APA 7

An APA results paragraph for a one-way ANOVA states the test, the F ratio with both degrees of freedom, the exact p value, an effect size, the post hoc test used and the means and standard deviations of every group (American Psychological Association, 2020; Appelbaum et al., 2018). Italicise statistical symbols and give p without a leading zero; our guide to reporting statistics in APA 7 covers the details.

Report a non-significant result in the same format, for example F(2, 21) = 1.12, p = .345, η² = .10, and do not run or report post hoc tests after a non-significant omnibus test unless you planned specific comparisons in advance. If you used Welch's test, report its adjusted degrees of freedom, as in the next section.

When assumptions are violated: Welch's ANOVA and Kruskal-Wallis

Welch's ANOVA (unequal variances)

Welch's F adjusts the within-group degrees of freedom to allow for unequal variances, so the second df is usually a decimal. Delacre et al. (2019) argue it should be the default. For the example data, Welch's test gives F(2, 13.88) = 3.91, p = .045: the same conclusion as the classic F, with a slightly larger p-value. Pair Welch's F with Games-Howell post hoc tests, which do not assume equal variances.

Kruskal-Wallis test (ordinal or severely non-normal data)

The Kruskal-Wallis test compares groups using the ranks of the scores instead of the scores themselves (Kruskal & Wallis, 1952). Run it from Analyze > Nonparametric Tests > Independent Samples in SPSS. For the example, H(2) = 6.68, p = .035, with mean ranks of 9.50 (dressy), 17.75 (sloppy) and 10.25 (casual). Follow a significant result with Dunn's pairwise tests, which SPSS provides with a Bonferroni adjustment.

Common one-way ANOVA mistakes

• Running several t-tests instead of an ANOVA, which inflates the false-positive rate.
• Stopping at the F test. A significant F says the means differ somewhere; the post hoc table says where.
• Running post hoc tests after a non-significant F and reporting any difference that appears.
• Leaving out descriptive statistics. Every group's mean and standard deviation belongs in the write-up.
• Writing p = .000. SPSS rounds; report p < .001 instead.
• Using a one-way ANOVA for repeated measurements of the same people, or for a categorical outcome such as yes/no answers.
• Ignoring unequal variances when Levene's test is significant, and group sizes differ; use Welch's F and Games-Howell.

Getting help with ANOVA in your statistics course

If you would like a specialist to check your SPSS setup, help you interpret your output or give feedback on your APA results section, see our psychology research methods and statistics support. We explain each step so you can apply it confidently in your own work.

Frequently asked questions

Sources

1. Field A. Discovering Statistics Using IBM SPSS Statistics. 6th ed. London: Sage; 2024
2. Delacre M, Leys C, Mora YL, Lakens D. Taking parametric assumptions seriously: arguments for the use of Welch's F-test instead of the classical F-test in one-way ANOVA. Int Rev Soc Psychol 2019;32(1):13
3. American Psychological Association. Publication Manual of the American Psychological Association. 7th ed. Washington, DC: American Psychological Association; 2020
4. Albers C, Lakens D. When power analyses based on pilot data are biased: inaccurate effect size estimators and follow-up bias. J Exp Soc Psychol 2018;74:187-195
5. Appelbaum M, Cooper H, Kline RB, et al. Journal article reporting standards for quantitative research in psychology: the APA Publications and Communications Board task force report. Am Psychol 2018;73:3-25
6. Blanca MJ, Alarcón R, Arnau J, Bono R, Bendayan R. Non-normal data: is ANOVA still a valid option? Psicothema 2017;29(4):552-557
7. Lakens D. Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Front Psychol 2013;4:863
8. Cohen J. A power primer. Psychol Bull 1992;112(1):155-159
9. Kruskal WH, Wallis WA. Use of ranks in one-criterion variance analysis. J Am Stat Assoc 1952;47(260):583-621
10. Smith RA, Davis SF. The Psychologist as Detective: An Introduction to Conducting Research in Psychology. Updated ed. Boston, MA: Pearson; 2016

Dr. Keith DzirasaResearch Scientist, Psychology · PhD in Psychology, MSc in Research Methods
Keith has 17 years of experience in experimental psychology and cognitive and behavioral research. He brings an experimental research perspective to projects that examine cognition, perception, and behavior. Areas of expertise Experimental and behavioral studies Quantitative research projects and data analysis…