{"id":66999,"date":"2026-10-13T11:17:42","date_gmt":"2026-10-13T08:17:42","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66999"},"modified":"2026-10-13T11:17:42","modified_gmt":"2026-10-13T08:17:42","slug":"friedman-test-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/friedman-test-calculator\/","title":{"rendered":"Friedman Test Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Compare Three or More Related Conditions from Their Ranks<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66999\" data-favorite-title=\"Friedman Test\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/friedman-test-calculator\/\" data-favorite-excerpt=\"Enter one line per subject with a value for each condition and this Friedman test calculator returns the tie-corrected chi-square statistic, degrees of freedom, p-value, Kendall&#039;s W and a decision at your chosen significance level.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-wpcalc-bell-curve\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/statistics\/friedman-test-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66999\" data-post=\"66999\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Measurements (one subject per line)<\/label><textarea id=\"field-1\" name=\"measurements_one_subject_per_line\">7 6 5\n5 4 6\n8 7 6\n4 5 3\n6 6 4\n7 5 6\n5 3 4\n9 8 7<\/textarea><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Significance level<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"significance_level\"><option value=\"0.10\">0.10<\/option><option value=\"0.05\" selected>0.05<\/option><option value=\"0.025\">0.025<\/option><option value=\"0.01\">0.01<\/option><option value=\"0.001\">0.001<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66999\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Friedman Test Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Friedman statistic (\u03c7\u00b2, tie-corrected)<\/span><span class=\"result-term\">\u03c7\u00b2\u1d3f<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Degrees of freedom<\/span><span class=\"result-term\">df<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>p-value (\u03c7\u00b2 approximation)<\/span><span class=\"result-term\">p<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Kendall&#039;s W (effect size)<\/span><span class=\"result-term\">W<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Decision at \u03b1<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-6\"><span class=\"field-title\"><span>Rank sums by condition<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-7\"><span class=\"field-title\"><span>How the Friedman test works<\/span><\/span><span class=\"field-value\" id=\"result-7\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"66999\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Friedman Test<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\chi_F^2 = \\frac{12}{n\\,k\\,(k+1)} \\sum_{j=1}^{k} R_j^2 \\; - \\; 3n(k+1) \\qquad\\text{(tie-corrected: } \\chi^2 = \\chi_F^2 \/ C\\text{)} \\\\ C = 1 - \\frac{\\sum\\left(t^3 - t\\right)}{n\\left(k^3 - k\\right)} \\\\ W = \\frac{\\chi^2}{n(k-1)} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 number of subjects (blocks)<\/li>\n<li>\\(k\\) \u2014 number of related conditions (\\(k \\ge 3\\))<\/li>\n<li>\\(R_j\\) \u2014 sum of ranks of condition \\(j\\); each subject's values are ranked from 1 (smallest) to \\(k\\), with tied values receiving average ranks<\/li>\n<li>\\(t\\) \u2014 size of each group of tied values within a subject<\/li>\n<li>\\(W\\) \u2014 Kendall's coefficient of concordance, the effect size<\/li>\n<\/ul>\n<p>The statistic compares the observed rank sums with the value expected when all conditions are equal, \\(nk(k+1)\/4\\cdot 2\\); large departures produce large \\(\\chi^2\\). The tie correction inflates the statistic when ties are present, and the p-value is obtained from the \\(\\chi^2\\) distribution with \\(k-1\\) degrees of freedom.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Friedman Test \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> eight subjects each rate three products on a scale (the default data in the calculator). Within each subject the three scores are ranked 1\u20133; the rank sums come out as \\(R_1 = 21.5\\), \\(R_2 = 14.5\\), \\(R_3 = 12\\).<\/p>\n<p>\\[ \\chi_F^2 = \\frac{12}{8 \\cdot 3 \\cdot 4}\\left(21.5^2 + 14.5^2 + 12^2\\right) - 3 \\cdot 8 \\cdot 4 = 6.0625 \\]<\/p>\n<p>One subject gave two equal scores, so one tie group of size \\(t=2\\) contributes \\(t^3 - t = 6\\) to the correction:<\/p>\n<p>\\[ C = 1 - \\frac{6}{8(27 - 4 + -3)} = 1 - \\frac{6}{192} = 0.96875, \\qquad \\chi^2 = \\frac{6.0625}{0.96875} = 6.2581 \\]<\/p>\n<p>With \\(df = k - 1 = 2\\), the \\(\\chi^2\\) approximation gives \\(p = 0.0438\\), and Kendall's \\(W = 6.2581 \/ (8 \\cdot 2) = 0.3911\\).<\/p>\n<p><strong>Result:<\/strong> \\(\\chi^2 = 6.2581\\), \\(df = 2\\), \\(p = 0.0438\\). At \\(\\alpha = 0.05\\), reject \\(H_0\\): the three products differ significantly in location, with moderate agreement among subjects (\\(W \\approx 0.39\\)).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Paste or type your data into <strong>Measurements (one subject per line)<\/strong>. Each line must hold one subject's values, one per condition, separated by spaces or semicolons \u2014 for example <code>7 6 5<\/code> or <code>7; 6; 5<\/code>. Every line must contain the same number of values, at least three.<\/li>\n<li>If your numbers use decimal commas (e.g. <code>7,5<\/code>), you can enter them as-is; the calculator converts them automatically.<\/li>\n<li>Choose a <strong>Significance level<\/strong> (0.10, 0.05, 0.025, 0.01 or 0.001) \u2014 this is the threshold \\(\\alpha\\) used for the reject \/ fail-to-reject decision.<\/li>\n<li>Read the results: the tie-corrected Friedman statistic \\(\\chi^2\\), the degrees of freedom, the p-value, Kendall's \\(W\\), the decision at \\(\\alpha\\), a rank-sum table per condition, and a short written explanation.<\/li>\n<\/ol>\n<h2>What the Friedman Test Is<\/h2>\n<p>The Friedman test is a nonparametric alternative to the one-way repeated-measures ANOVA. It compares three or more <em>related<\/em> conditions \u2014 typically the same subjects measured under each condition, or a randomized block design in which each block receives every treatment \u2014 to test whether the conditions differ in location (whether scores tend to be systematically higher under some conditions than others).<\/p>\n<p>The test works purely on ranks. Each subject's values are ranked within that subject, which strips out all subject-to-subject differences: a subject who rates everything high contributes the same rank pattern as one who rates everything low. If the conditions are identical, each condition's ranks are a random permutation of \\(1, \\dots, k\\), so every condition's expected rank sum is \\(n \\cdot k(k+1)\/2\\). The Friedman statistic measures how far the observed rank sums deviate from this expectation: if one condition is consistently ranked highest, its rank sum is large, another's is small, and \\(\\chi^2\\) grows.<\/p>\n<p>The p-value comes from comparing the statistic with the \\(\\chi^2_{k-1}\\) distribution. If \\(p \\le \\alpha\\), you reject the null hypothesis and conclude that at least one condition tends to be ranked higher or lower than the others \u2014 but not which one. Kendall's \\(W = \\chi^2 \/ (n(k-1))\\) quantifies how consistent the subjects' orderings are: 0 means no agreement, 1 means every subject ranks the conditions in exactly the same order, and values around 0.3\u20130.5 are usually described as moderate agreement.<\/p>\n<p>A significant result is only the first step. To find <em>which<\/em> conditions differ, follow up with pairwise comparisons: the Nemenyi test (a critical difference of \\(q_\\alpha\\sqrt{k(k+1)\/(6n)}\\) on the mean ranks) or <a href=\"https:\/\/wpcalc.com\/en\/statistics\/wilcoxon-signed-rank-test-calculator\/\">Wilcoxon signed-rank tests<\/a> with a Bonferroni-adjusted significance level. If your groups are <em>independent<\/em> rather than matched, the Friedman test is the wrong tool \u2014 use the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/kruskal-wallis-test-calculator\/\">Kruskal\u2013Wallis test<\/a> instead.<\/p>\n<h2>Assumptions and Practical Tips<\/h2>\n<ul>\n<li><strong>Matched data:<\/strong> every subject (block) must have a value under every condition. Missing values cannot be handled \u2014 either drop the subject or impute the value before entering the data.<\/li>\n<li><strong>Independent subjects:<\/strong> subjects\/blocks must not influence each other, while the repeated measurements within a subject are of course related \u2014 that is the point of the test.<\/li>\n<li><strong>Ordinal scale or better:<\/strong> the values only need to be rankable within each subject, so Likert ratings, ranks and scores all qualify.<\/li>\n<li><strong>The p-value is an approximation.<\/strong> The \\(\\chi^2\\) reference distribution becomes accurate for larger \\(nk\\), but tends to be somewhat liberal (p-values too small) for small samples \u2014 for example fewer than about 13 subjects with \\(k = 3\\). In those cases, compare the statistic against exact Friedman critical values before trusting a borderline decision.<\/li>\n<li><strong>Ties are handled, but many ties lose power.<\/strong> The calculator applies the standard tie correction \\(C\\); if every subject gives identical values across conditions, the statistic is zero by construction and no difference can be detected.<\/li>\n<li><strong>Interpret W alongside p.<\/strong> With many subjects, even a tiny, practically irrelevant difference in rank sums can be statistically significant; Kendall's \\(W\\) tells you whether the agreement behind the result is actually substantial. For another view of rank agreement between raters, see the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/kendalls-tau-calculator\/\">Kendall's tau calculator<\/a> and the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/spearman-rank-correlation-calculator\/\">Spearman rank correlation calculator<\/a>.<\/li>\n<\/ul>\n<p>Direction of the effect is read from the rank-sum table: the condition with the largest mean rank tends to score highest, and the one with the smallest mean rank tends to score lowest. Comparing each condition's mean rank with the null expectation \\(k(k+1)\/2\\) shows which conditions pull the result.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter one line per subject with a value for each condition and this Friedman test calculator returns the tie-corrected chi-square statistic, degrees of freedom, p-value, Kendall's W and a decision at your chosen significance level.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[74],"tags":[79],"class_list":["post-66999","post","type-post","status-publish","format-standard","hentry","category-statistics","tag-hypothesis-testing"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66999","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=66999"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66999\/revisions"}],"predecessor-version":[{"id":67000,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66999\/revisions\/67000"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66999"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66999"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}