{"id":1031,"date":"2026-10-14T05:16:00","date_gmt":"2026-10-14T05:16:00","guid":{"rendered":"https:\/\/freestatisticaltools.com\/blog\/?p=1031"},"modified":"2026-10-14T05:16:00","modified_gmt":"2026-10-14T05:16:00","slug":"one-way-anova","status":"publish","type":"post","link":"https:\/\/freestatisticaltools.com\/blog\/one-way-anova\/","title":{"rendered":"One-Way ANOVA Explained: Assumptions, Post Hoc Tests and APA Reporting"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">A <strong>one-way ANOVA<\/strong> (analysis of variance) tests whether the means of three or more independent groups differ. It answers one question, &quot;is there <em>any<\/em> difference among these group means?&quot;, without inflating the false-positive rate the way a long series of t-tests would.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This guide explains the logic of ANOVA, its assumptions, a full worked example, how to follow up a significant result with post hoc tests, and how to report it in APA 7 style.<\/p>\n\n\n<div class=\"fst-tool\"><span class=\"ico\"><svg viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.8\" stroke-linecap=\"round\" stroke-linejoin=\"round\" aria-hidden=\"true\"><path d=\"M4 20V10M10 20V4M16 20v-7M22 20H2\"\/><\/svg><\/span><span><span class=\"t-eyebrow\">Free tool \u00b7 runs in your browser<\/span><a class=\"t-name\" href=\"https:\/\/freestatisticaltools.com\/one-way-anova-calculator\/\" style=\"text-decoration:none;color:inherit\">One-Way ANOVA Calculator<\/a><span class=\"t-desc\">Free one-way ANOVA calculator with Levene\u2019s test, Welch\u2019s F, eta squared, omega squared and Tukey HSD post-hoc comparisons. APA tables in seconds.<\/span><\/span><a class=\"btn accent\" href=\"https:\/\/freestatisticaltools.com\/one-way-anova-calculator\/\" data-tool=\"one-way-anova-calculator\">Run a one-way ANOVA \u2192<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Why not just run several t-tests?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">With three groups you would need three t-tests (A vs B, A vs C, B vs C). If each uses \u03b1 = .05, the chance of at least one false positive across the three is about 1 \u2212 0.95\u00b3 \u2248 14%. With five groups (10 comparisons) it is about 40%. ANOVA tests all groups at once with a single F-test, keeping the overall error rate at 5%. Post hoc tests then find <em>which<\/em> groups differ, with corrections built in.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">When to use a one-way ANOVA<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>One continuous outcome<\/strong> (exam score, satisfaction scale, reaction time).<\/li>\n\n<li><strong>One categorical factor with three or more independent groups<\/strong> (three teaching methods, four age bands, five regions).<\/li>\n\n<li><strong>Different participants in each group.<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">If the same people are measured several times, use a <a href=\"\/repeated-measures-anova-calculator\/\">repeated measures ANOVA<\/a>. If you have two factors (say, method and gender), use a <a href=\"\/two-way-anova-calculator\/\">two-way ANOVA<\/a>. If you want to adjust for a covariate such as pre-test score, use an <a href=\"\/ancova-calculator\/\">ANCOVA<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">How ANOVA works (in one paragraph)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">ANOVA splits the total variability in the outcome into two parts: variability <strong>between<\/strong> groups (how far the group means are from the grand mean) and variability <strong>within<\/strong> groups (how far individuals are from their own group mean). The F-ratio compares them:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><code>F = mean square between \u00f7 mean square within<\/code><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If the groups really have the same population mean, F should be close to 1. A large F means the group means are further apart than within-group noise would explain.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Assumptions<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Independent observations<\/strong>, guaranteed by the design.<\/li>\n\n<li><strong>Normally distributed residuals<\/strong> in each group. ANOVA is robust to moderate non-normality when groups are of similar size and have 20 or more participants each. Check with a Q-Q plot or the <a href=\"\/shapiro-wilk-test-calculator\/\">Shapiro-Wilk test<\/a>.<\/li>\n\n<li><strong>Homogeneity of variance.<\/strong> Check with Levene&#39;s test. If it is significant, report <strong>Welch&#39;s F<\/strong> instead of the classic F.<\/li>\n\n<li><strong>No extreme outliers<\/strong> within groups.<\/li>\n<\/ol>\n\n\n\n<div class=\"fst-note tip\"><b>Tip<\/b><p>If your data are ordinal or badly skewed with small groups, use the <a href=\"\/kruskal-wallis-test-calculator\/\">Kruskal-Wallis test<\/a>, the nonparametric alternative to one-way ANOVA. It comes with Dunn&#39;s post hoc tests.<\/p><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Worked example<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A school compares end-of-term maths scores for three teaching methods, with 20 students in each group.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Method<\/th><th>n<\/th><th>Mean<\/th><th>SD<\/th><\/tr><\/thead><tbody><tr><td>Lecture<\/td><td>20<\/td><td>70.2<\/td><td>8.4<\/td><\/tr><tr><td>Flipped classroom<\/td><td>20<\/td><td>75.8<\/td><td>9.1<\/td><\/tr><tr><td>Problem-based<\/td><td>20<\/td><td>78.1<\/td><td>8.7<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Check equal variances<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The SDs are similar (8.4 to 9.1), and Levene&#39;s test is not significant, so the classic F-test is appropriate. (Welch&#39;s F gives the same conclusion here: F(2, 37.96) = 4.46, p = .018.)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Read the ANOVA table<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Source<\/th><th>SS<\/th><th>df<\/th><th>MS<\/th><th>F<\/th><th>p<\/th><\/tr><\/thead><tbody><tr><td>Between groups<\/td><td>660.4<\/td><td>2<\/td><td>330.2<\/td><td>4.32<\/td><td>.018<\/td><\/tr><tr><td>Within groups<\/td><td>4352.1<\/td><td>57<\/td><td>76.4<\/td><td><\/td><td><\/td><\/tr><tr><td>Total<\/td><td>5012.5<\/td><td>59<\/td><td><\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">F(2, 57) = 4.32, p = .018. Because p &lt; .05, at least one group mean differs from the others.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Effect size<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Eta squared (\u03b7\u00b2)<\/strong> = SS between \u00f7 SS total = 660.4 \u00f7 5012.5 = <strong>.13<\/strong>. Teaching method accounts for about 13% of the variance in scores in this sample.<\/li>\n\n<li><strong>Omega squared (\u03c9\u00b2)<\/strong> = <strong>.10<\/strong>. It is less biased than \u03b7\u00b2, especially in small samples, and many methodologists prefer it.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Common rough benchmarks for \u03b7\u00b2 are .01 (small), .06 (medium) and .14 (large).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Post hoc tests<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The F-test says <em>something<\/em> differs, not <em>what<\/em>. The Tukey HSD test compares every pair of groups while controlling the familywise error rate:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Comparison<\/th><th>Mean difference<\/th><th>p (Tukey)<\/th><\/tr><\/thead><tbody><tr><td>Flipped \u2212 Lecture<\/td><td>5.6<\/td><td>.115<\/td><\/tr><tr><td>Problem-based \u2212 Lecture<\/td><td>7.9<\/td><td>.016<\/td><\/tr><tr><td>Problem-based \u2212 Flipped<\/td><td>2.3<\/td><td>.685<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Only the problem-based vs lecture difference is significant. The flipped classroom sits in between and does not differ significantly from either.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Which post hoc test should I use?<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Situation<\/th><th>Recommended test<\/th><\/tr><\/thead><tbody><tr><td>Equal variances, all pairwise comparisons<\/td><td>Tukey HSD<\/td><\/tr><tr><td>Unequal variances or very unequal group sizes<\/td><td>Games-Howell<\/td><\/tr><tr><td>A few planned comparisons only<\/td><td>Bonferroni or Holm-corrected t-tests<\/td><\/tr><tr><td>Every group vs one control group<\/td><td>Dunnett&#39;s test<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The free <a href=\"\/one-way-anova-calculator\/\">one-way ANOVA calculator<\/a> gives Tukey HSD comparisons automatically, along with Levene&#39;s test, Welch&#39;s F, \u03b7\u00b2 and \u03c9\u00b2.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">How to report a one-way ANOVA in APA 7<\/h2>\n\n\n\n<div class=\"fst-report\"><b>How to report it (APA 7)<button type=\"button\" data-copy-report>Copy<\/button><\/b><div><p>A one-way ANOVA showed that maths scores differed by teaching method, <i>F<\/i>(2, 57) = 4.32, <i>p<\/i> = .018, \u03b7\u00b2 = .13. Tukey post hoc tests indicated that students taught with problem-based learning (<i>M<\/i> = 78.1, <i>SD<\/i> = 8.7) scored significantly higher than students taught by lecture (<i>M<\/i> = 70.2, <i>SD<\/i> = 8.4), <i>p<\/i> = .016. The flipped-classroom group (<i>M<\/i> = 75.8, <i>SD<\/i> = 9.1) did not differ significantly from either group.<\/p><\/div><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Checklist for your write-up:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Means and SDs for every group (in text or a table).<\/li>\n\n<li>F with both degrees of freedom, the exact p-value and an effect size.<\/li>\n\n<li>Which post hoc test you used, and its results.<\/li>\n\n<li>If Levene&#39;s test was significant, say you used Welch&#39;s F and Games-Howell comparisons.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Common mistakes<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Running post hoc tests after a non-significant F<\/strong> and reporting the one comparison that &quot;worked&quot;. Unless the comparisons were planned in advance, stop at the non-significant F.<\/li>\n\n<li><strong>Ignoring unequal variances<\/strong> when group sizes differ. That combination makes the classic F-test unreliable.<\/li>\n\n<li><strong>Reporting only p-values.<\/strong> Always add \u03b7\u00b2 or \u03c9\u00b2.<\/li>\n\n<li><strong>Using ANOVA for repeated measurements<\/strong> of the same people. That needs a repeated measures ANOVA.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Frequently asked questions<\/h2>\n\n\n\n<div class=\"fst-faq\"><details><summary>What does the F value mean in ANOVA?<\/summary><div class=\"a\"><p>F is the ratio of the variance between group means to the variance within groups. A value near 1 suggests no difference between groups; larger values suggest the group means differ more than chance would explain.<\/p><\/div><\/details><details><summary>What is a good eta squared value?<\/summary><div class=\"a\"><p>As rough benchmarks, \u03b7\u00b2 of .01 is small, .06 is medium and .14 is large. Interpret it in the context of your field and outcome rather than relying only on these labels.<\/p><\/div><\/details><details><summary>Can I run ANOVA with unequal group sizes?<\/summary><div class=\"a\"><p>Yes. Unequal group sizes are fine, but combined with unequal variances they make the classic F-test unreliable. Check Levene&#39;s test and use Welch&#39;s F with Games-Howell comparisons if needed.<\/p><\/div><\/details><details><summary>What should I do if Levene&#039;s test is significant?<\/summary><div class=\"a\"><p>Report Welch&#39;s F instead of the standard F, and use the Games-Howell post hoc test, which does not assume equal variances.<\/p><\/div><\/details><details><summary>What is the nonparametric alternative to one-way ANOVA?<\/summary><div class=\"a\"><p>The Kruskal-Wallis H test. It compares the rank distributions of three or more independent groups and is followed by Dunn&#39;s post hoc tests.<\/p><\/div><\/details><\/div>\n\n\n<div class=\"fst-report\"><b>Cite the tool (APA 7)<button type=\"button\" data-copy-report>Copy<\/button><\/b><div>Xavier, B. (2026). <i>Free Statistical Tools<\/i> [Computer software]. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.23372149 (web app: https:\/\/freestatisticaltools.com\/one-way-anova-calculator\/)<\/div><small>Name the tool in your methods: \u201cThe analysis was carried out with the One-Way ANOVA Calculator in Free Statistical Tools.\u201d<\/small><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Compare three or more group means the right way, with a worked example covering the F-test, Welch&#8217;s F, Tukey post hoc comparisons, effect sizes and APA reporting.<\/p>\n","protected":false},"author":1,"featured_media":1032,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_fst_seo_title":"One-Way ANOVA: Step-by-Step Guide, Post Hoc Tests and APA","_fst_meta_desc":"Learn one-way ANOVA step by step: assumptions, Levene's test, Welch's F, Tukey post hoc tests, eta squared and an APA write-up, with a free calculator.","_fst_primary_tool":"one-way-anova-calculator","_fst_canonical":"","_fst_noindex":false,"_fst_no_toc":false,"_fst_no_autolink":false,"footnotes":""},"categories":[3],"tags":[20,19,18,21],"class_list":["post-1031","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-compare-means","tag-anova","tag-apa-style","tag-effect-size","tag-post-hoc-tests"],"_links":{"self":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts\/1031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/comments?post=1031"}],"version-history":[{"count":0,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts\/1031\/revisions"}],"wp:attachment":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/media?parent=1031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/categories?post=1031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/tags?post=1031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}