One-Way ANOVA (3+ groups)

Verified

ANOVA

Independently verified. Every statistic this test reports has been re-derived against an independent reference — never the library the pipeline itself calls — the rendered output was read back in a browser, and the result is locked with a committed regression suite.

Run this test straight away on a free built-in teaching dataset — no data of your own needed — or bring your own. Either opens the guided workspace: variable setup, assumption diagnostics, results with effect sizes and confidence intervals, figures, and APA-ready reporting.

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Determines whether there are statistically significant differences in the means of a continuous dependent variable across three or more INDEPENDENT groups.

Computes the F statistic = MSB / MSW (between-groups mean square over within-groups mean square) with df = (k−1, N−k) and reports the omnibus p, η² / ω² effect sizes, and post-hoc pairwise comparisons (Tukey HSD by default, or Games-Howell when Welch's correction is preferred). Engine pre-checks homogeneity of variance (Levene), within-group normality (Shapiro-Wilk), and outliers; auto-routes to Welch's ANOVA or Kruskal-Wallis when assumptions fail.

Worked example

Do three teaching methods produce different exam scores?

90 students, 30 per method (lecture, flipped, online); a one-way ANOVA on exam scores, followed by Tukey post-hoc tests.

Result

Method significantly affected scores, F(2, 87) = 5.75, p = .005, η² = .12; Tukey post-hoc showed the flipped classroom scored highest.

How you'd report it (APA)

A one-way ANOVA showed a significant effect of teaching method on exam scores, F(2, 87) = 5.75, p = .005, η² = .12.

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

  • Experimental design (3+ treatment groups)
    A researcher tests three teaching methods (traditional, flipped classroom, online) on exam scores.
  • Quasi-experimental (3+ naturally occurring groups)
    A sociologist examines whether income differs across education levels: high school, bachelor's, master's, PhD.

When NOT to — use instead

Hypotheses

H₀: μ₁ = μ₂ = … = μₖ — all k group means are equal in the population.
Hₐ: at least one μᵢ ≠ μⱼ — the population means are NOT all equal (at least one pair of group means differs).

Parameter tested: set of population group means {μ₁, …, μₖ}

Assumptions (and what to do if they fail)

No significant outliers in any groupmedium

Check: Inspect the per-group boxplot; flag points beyond the whiskers.

If violated: Outliers can inflate within-group variance and mask between-group differences (or fabricate them).

Dependent variable is approximately normally distributed within each groupmedium

Check: Run Shapiro-Wilk per group; inspect per-group Q-Q plots.

If violated: Severely non-normal data with small per-group n threatens the F-test's nominal Type I error rate.

Homogeneity of variances across groups (Levene's test)medium

Check: Run Levene's test (centred at the median, not the mean — more robust).

If violated: Unequal variances inflate the Type I error rate of standard ANOVA, especially when group sizes are unbalanced.

Ready to run a One-Way ANOVA (3+ groups) on your own data?

Guided setup, automatic assumption checks, effect sizes, figures and an APA write-up.

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