{"id":1029,"date":"2026-10-14T04:16:00","date_gmt":"2026-10-14T04:16:00","guid":{"rendered":"https:\/\/freestatisticaltools.com\/blog\/?p=1029"},"modified":"2026-10-14T04:16:00","modified_gmt":"2026-10-14T04:16:00","slug":"mann-whitney-u-test","status":"publish","type":"post","link":"https:\/\/freestatisticaltools.com\/blog\/mann-whitney-u-test\/","title":{"rendered":"Mann-Whitney U Test: When to Use It Instead of a T-Test (With Examples)"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">The <strong>Mann-Whitney U test<\/strong> (also called the Wilcoxon rank-sum test) compares two independent groups when you cannot, or should not, compare their means with a t-test. Instead of means, it works with the <strong>ranks<\/strong> of the scores, which makes it a good choice for ordinal data, small skewed samples and data with outliers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This guide explains when to choose it, what it actually tests, how to read the output and how to report it.<\/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=\"M3 17h4l3-10 4 14 3-8h4\"\/><\/svg><\/span><span><span class=\"t-eyebrow\">Free tool \u00b7 runs in your browser<\/span><a class=\"t-name\" href=\"https:\/\/freestatisticaltools.com\/mann-whitney-u-test-calculator\/\" style=\"text-decoration:none;color:inherit\">Mann-Whitney U Test Calculator<\/a><span class=\"t-desc\">Free Mann-Whitney U test (Wilcoxon rank-sum) calculator with mean ranks, z, exact or asymptotic p-value and effect size r. For ordinal or skewed data.<\/span><\/span><a class=\"btn accent\" href=\"https:\/\/freestatisticaltools.com\/mann-whitney-u-test-calculator\/\" data-tool=\"mann-whitney-u-test-calculator\">Run a Mann-Whitney U test \u2192<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">When should you use the Mann-Whitney U test?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Choose it over the <a href=\"\/blog\/independent-samples-t-test\/\">independent samples t-test<\/a> when:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>your outcome is <strong>ordinal<\/strong>, for example a single 5- or 7-point Likert item, a pain rating or a ranking;<\/li>\n\n<li>your sample is <strong>small<\/strong> (say, fewer than 20\u201330 per group) <strong>and clearly skewed<\/strong>;<\/li>\n\n<li>there are <strong>outliers<\/strong> you cannot justify removing;<\/li>\n\n<li>you are comparing things like waiting times, incomes or counts, which are often heavily right-skewed.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The design requirements are the same as for the independent t-test: two groups, and different people in each group. For two related measurements (the same people before and after), use the <a href=\"\/wilcoxon-signed-rank-test-calculator\/\">Wilcoxon signed-rank test<\/a>. For three or more groups, use the <a href=\"\/kruskal-wallis-test-calculator\/\">Kruskal-Wallis test<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">What does the Mann-Whitney U test actually test?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The test pools both groups, ranks all the scores from lowest to highest, and checks whether one group&#39;s ranks tend to be higher than the other&#39;s. Formally, it tests whether a randomly chosen score from group A is equally likely to be higher or lower than a randomly chosen score from group B.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It is often described as a &quot;test of medians&quot;, but that is only true if the two distributions have the <strong>same shape<\/strong> (one is just shifted relative to the other). If the shapes differ, report it as a difference in rank distributions or mean ranks, not medians.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Assumptions<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Two independent groups.<\/li>\n\n<li>An outcome that is at least ordinal.<\/li>\n\n<li>Independent observations.<\/li>\n\n<li>For a median interpretation: similar distribution shapes in both groups.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Normality is <strong>not<\/strong> required.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">How the U statistic is calculated<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Rank all N = n\u2081 + n\u2082 scores together. Tied scores get the average of the ranks they span.<\/li>\n\n<li>Add up the ranks in each group (R\u2081 and R\u2082).<\/li>\n\n<li>Compute U\u2081 = n\u2081n\u2082 + n\u2081(n\u2081 + 1)\/2 \u2212 R\u2081, and U\u2082 = n\u2081n\u2082 \u2212 U\u2081.<\/li>\n\n<li>The test statistic U is the smaller of U\u2081 and U\u2082.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">For larger samples, U is converted to a z-score (with a correction for ties), which gives the p-value. For small samples without many ties, an exact p-value is better. The free <a href=\"\/mann-whitney-u-test-calculator\/\">calculator<\/a> chooses between exact and asymptotic p-values for you.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Worked example<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A company asks customers to rate their support experience from 1 (very poor) to 7 (excellent). 18 customers used in-app chat and 20 used email. Because the outcome is a single rating item, the Mann-Whitney U test is appropriate.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Group<\/th><th>n<\/th><th>Median<\/th><th>Mean rank<\/th><\/tr><\/thead><tbody><tr><td>In-app chat<\/td><td>18<\/td><td>5.5<\/td><td>23.50<\/td><\/tr><tr><td>Email<\/td><td>20<\/td><td>4.5<\/td><td>15.90<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Result:<\/strong> U = 108, z = 2.15, p = .032.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Interpretation: customers who used in-app chat gave significantly higher ratings than customers who used email. Their mean rank was 23.50 compared with 15.90.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Effect size r<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The usual effect size is <strong>r = z \/ \u221aN<\/strong>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">r = 2.15 \/ \u221a38 = <strong>.35<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Cohen&#39;s rough benchmarks for r are .10 (small), .30 (medium) and .50 (large), so this is a medium effect. Some researchers report the <strong>rank-biserial correlation<\/strong> instead; both describe how strongly group membership relates to the ranks.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">How to report the Mann-Whitney U test in APA 7<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Report the medians (or mean ranks), U, z, the exact p-value and r.<\/p>\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 Mann-Whitney U test showed that customers who used in-app chat rated their support experience higher (<i>Mdn<\/i> = 5.5) than customers who used email (<i>Mdn<\/i> = 4.5), <i>U<\/i> = 108, <i>z<\/i> = 2.15, <i>p<\/i> = .032, <i>r<\/i> = .35.<\/p><\/div><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">If the distributions have different shapes, report mean ranks instead of medians:<\/p>\n\n\n\n<div class=\"fst-report\"><b>Mean-rank version<button type=\"button\" data-copy-report>Copy<\/button><\/b><div><p>Ratings were higher in the chat group (mean rank = 23.50) than in the email group (mean rank = 15.90), <i>U<\/i> = 108, <i>z<\/i> = 2.15, <i>p<\/i> = .032, <i>r<\/i> = .35.<\/p><\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Mann-Whitney U vs t-test: which has more power?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A common worry is that nonparametric tests &quot;lose power&quot;. When data really are normal, the Mann-Whitney U test is about 95% as efficient as the t-test, so you lose very little. When data are heavily skewed or have outliers, it can be considerably <em>more<\/em> powerful than the t-test. In other words, it is a safe choice when you are unsure.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th><\/th><th>Independent t-test<\/th><th>Mann-Whitney U test<\/th><\/tr><\/thead><tbody><tr><td>Compares<\/td><td>Means<\/td><td>Rank distributions<\/td><\/tr><tr><td>Outcome<\/td><td>Continuous<\/td><td>Ordinal or continuous<\/td><\/tr><tr><td>Normality needed<\/td><td>Approximately (or large n)<\/td><td>No<\/td><\/tr><tr><td>Sensitive to outliers<\/td><td>Yes<\/td><td>Much less<\/td><\/tr><tr><td>Effect size<\/td><td>Cohen&#39;s d<\/td><td>r or rank-biserial correlation<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Common mistakes<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Calling it a test of medians<\/strong> when the group distributions have clearly different shapes.<\/li>\n\n<li><strong>Using it for paired data.<\/strong> Before-and-after data from the same people need the Wilcoxon signed-rank test.<\/li>\n\n<li><strong>Leaving out the effect size.<\/strong> A p-value alone does not tell readers how big the difference is.<\/li>\n\n<li><strong>Running it on summed scale scores with large samples<\/strong> just because a normality test was significant. With large samples, the t-test is usually fine; look at the plots before deciding.<\/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>Is the Mann-Whitney U test the same as the Wilcoxon rank-sum test?<\/summary><div class=\"a\"><p>Yes. They are two versions of the same test and give the same p-value. Note that the Wilcoxon signed-rank test is a different test, used for paired data.<\/p><\/div><\/details><details><summary>When should I use Mann-Whitney instead of a t-test?<\/summary><div class=\"a\"><p>Use it when the outcome is ordinal (such as a single Likert item), when samples are small and skewed, or when outliers would distort the means.<\/p><\/div><\/details><details><summary>Does the Mann-Whitney U test compare medians?<\/summary><div class=\"a\"><p>Only if both groups have distributions of similar shape. Otherwise it compares the overall rank distributions, and you should report mean ranks.<\/p><\/div><\/details><details><summary>How do I calculate the effect size for a Mann-Whitney U test?<\/summary><div class=\"a\"><p>Divide the z statistic by the square root of the total sample size: r = z \/ \u221aN. Values around .10, .30 and .50 are commonly read as small, medium and large.<\/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\/mann-whitney-u-test-calculator\/)<\/div><small>Name the tool in your methods: \u201cThe analysis was carried out with the Mann-Whitney U Test Calculator in Free Statistical Tools.\u201d<\/small><\/div>\n","protected":false},"excerpt":{"rendered":"<p>The nonparametric alternative to the independent t-test, explained with a worked example, effect size r and an APA-ready write-up.<\/p>\n","protected":false},"author":1,"featured_media":1030,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_fst_seo_title":"Mann-Whitney U Test: When to Use It, Interpretation and APA","_fst_meta_desc":"When to use the Mann-Whitney U test instead of a t-test, how it works with ranks, how to interpret U, z and effect size r, and how to report it in APA style.","_fst_primary_tool":"mann-whitney-u-test-calculator","_fst_canonical":"","_fst_noindex":false,"_fst_no_toc":false,"_fst_no_autolink":false,"footnotes":""},"categories":[4],"tags":[19,18,23,22],"class_list":["post-1029","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-nonparametric-tests","tag-apa-style","tag-effect-size","tag-likert-data","tag-nonparametric"],"_links":{"self":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts\/1029","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=1029"}],"version-history":[{"count":0,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts\/1029\/revisions"}],"wp:attachment":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/media?parent=1029"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/categories?post=1029"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/tags?post=1029"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}