One-way Analysis of Variance
Also known as: one-factor ANOVA, single-factor ANOVA, analysis of variance, tek yönlü ANOVA
One-way ANOVA is a parametric hypothesis test that compares the means of three or more independent groups on a single continuous outcome to decide whether at least one group mean differs. It rests on the variance-partitioning framework introduced by Ronald A. Fisher in 1925.
Key highlights
- Compares many groups in one test while controlling the overall false-positive rate.
- Cleanly partitions total variation into between-group and within-group components.
- Founded on Fisher's classic theory and supported by an interpretable effect size, eta-squared.
Intuition
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How it works
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When to use it
Use it to compare three or more independent groups on a single continuous outcome. Three assumptions should hold: the outcome is approximately normally distributed within each group (check with Shapiro-Wilk or K-S), the group variances are homogeneous (check with Levene's test), and observations are independent. A reasonable sample size matters; with very small groups or clear non-normality the F test becomes unreliable and the nonparametric Kruskal-Wallis test is a better choice.
Strengths & limitations
- Compares many groups in one test while controlling the overall false-positive rate.
- Cleanly partitions total variation into between-group and within-group components.
- Founded on Fisher's classic theory and supported by an interpretable effect size, eta-squared.
- Sensitive to non-normality and outliers, which inflate group variances and undermine post-hoc reliability.
- Assumes homogeneous variances; when this fails, the F statistic is biased.
- An omnibus result only signals that some difference exists — it does not identify which groups differ.
- Needs an adequate sample; with fewer than about ten observations per group it is unreliable.
Common pitfalls
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Applications
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Frequently asked
Why not just run several t-tests instead?
Running a separate t-test for every pair of groups inflates the overall false-positive rate, because each test carries its own chance of a Type I error. One-way ANOVA makes a single omnibus comparison that keeps the overall error rate under control, then post-hoc tests pinpoint the specific differences.
The F test is significant — now what?
A significant F only tells you that at least one group mean differs, not which ones. Follow it with post-hoc comparisons such as Tukey HSD for equal group sizes, Bonferroni for a small number of comparisons, or Scheffé when group sizes differ, and report eta-squared as the effect size.
What if the variances are not equal?
If Levene's test is significant, the equal-variance assumption is violated and the classic F statistic is biased. Use Welch ANOVA, which does not assume homogeneous variances, and pair it with the Games-Howell post-hoc test.
When should I use Kruskal-Wallis instead?
When the outcome is clearly non-normal, contains influential outliers, or the groups are very small, the parametric F test becomes unreliable. The Kruskal-Wallis test is the rank-based nonparametric alternative for comparing three or more independent groups.
Sources
- 1.Fisher, R. A. (1925). Statistical Methods for Research Workers. Edinburgh: Oliver and Boyd.
- 2.Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). SAGE.ISBN 978-1446249185
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Cite this page
ScholarGate. (2026, June 1). One-way ANOVA. ScholarGate. https://scholargate.app/statistics/one-way-anova