{"id":66987,"date":"2026-10-13T11:01:17","date_gmt":"2026-10-13T08:01:17","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66987"},"modified":"2026-10-13T12:06:35","modified_gmt":"2026-10-13T09:06:35","slug":"wilcoxon-signed-rank-test-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/wilcoxon-signed-rank-test-calculator\/","title":{"rendered":"Wilcoxon Signed-Rank 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 Paired Measurements by Signed Ranks<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66987\" data-favorite-title=\"Wilcoxon Signed-Rank Test\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/wilcoxon-signed-rank-test-calculator\/\" data-favorite-excerpt=\"Enter two paired samples (or ready-made differences) and this Wilcoxon signed-rank test calculator computes W\u207a, W\u207b, the z statistic with tie correction and continuity correction, the p-value, and the 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\/wilcoxon-signed-rank-test-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66987\" data-post=\"66987\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Data format<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"data_format\"><option value=\"two\" selected>Two paired samples<\/option><option value=\"diffs\">Differences only<\/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><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Sample 1 (first \/ before)<\/label><textarea id=\"field-2\" name=\"sample_1_first_before\">85; 92; 78; 88; 95; 82; 90; 76; 89; 84<\/textarea><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Sample 2 (second \/ after)<\/label><textarea id=\"field-3\" name=\"sample_2_second_after\">91; 90; 85; 95; 99; 86; 88; 79; 92; 82<\/textarea><\/div><div class=\"field\" id=\"input-4\" hidden><label for=\"field-4\">Paired differences (second \u2212 first)<\/label><textarea id=\"field-4\" name=\"paired_differences_second_first\">6; -2; 7; 7; 4; 4; -2; 3; 3; -2<\/textarea><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">Alternative hypothesis<\/label><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"alternative_hypothesis\"><option value=\"two\" selected>Median difference \u2260 0 (two-sided)<\/option><option value=\"pos\">Median difference &gt; 0 (one-sided)<\/option><option value=\"neg\">Median difference &lt; 0 (one-sided)<\/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><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">Significance level<\/label><div class=\"term\">\u03b1<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"significance_level\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 0.05\" min=\"0.0001\" max=\"0.5\" value=\"0.05\"\/><small id=\"calc-info-6\">Type I error rate \u03b1, the threshold below which the p-value counts as statistically significant. 0.05 is the conventional choice.<\/small><\/div><div class=\"field is-checkbox is-single\" id=\"input-7\"><input type=\"checkbox\" id=\"field-7\" value=\"1\" checked=\"checked\"\/><label for=\"field-7\">Apply continuity correction<\/label><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66987\">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> Wilcoxon Signed-Rank Test Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Non-zero differences used<\/span><span class=\"result-term\">n<\/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>Sum of positive ranks<\/span><span class=\"result-term\">W\u207a<\/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>Sum of negative ranks<\/span><span class=\"result-term\">W\u207b<\/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>Test statistic (smaller of W\u207a, W\u207b)<\/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>z statistic (normal approximation)<\/span><span class=\"result-term\">z<\/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\" id=\"output-6\"><span class=\"field-title\"><span>Approximate p-value<\/span><span class=\"result-term\">p<\/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\" id=\"output-7\"><span class=\"field-title\"><span>Decision at significance level \u03b1<\/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 class=\"field-result\" id=\"output-8\"><span class=\"field-title\"><span>Median of the paired differences<\/span><\/span><span class=\"field-value\" id=\"result-8\"><\/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-9\"><span class=\"field-title\"><span>Differences and signed ranks<\/span><\/span><span class=\"field-value\" id=\"result-9\"><\/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-10\"><span class=\"field-title\"><span>How the test works and interpretation<\/span><\/span><span class=\"field-value\" id=\"result-10\"><\/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=\"66987\" 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>Wilcoxon Signed-Rank Test Formula<\/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\\[\n\\begin{aligned}\nd_i &= y_i - x_i, \\quad n = \\#\\{d_i \\ne 0\\} \\\\\nR_i &= \\text{rank of } |d_i| \\text{ (mid-ranks for ties)} \\\\\nW^+ &= \\sum_{d_i > 0} R_i, \\qquad W^- = \\sum_{d_i < 0} R_i \\\\\nz &= \\frac{\\operatorname{sgn}(W^+ - \\mu_W)\\left(\\left|W^+ - \\mu_W\\right| - 0.5\\right)}{\\sigma_W}, \\quad \\mu_W = \\frac{n(n+1)}{4}, \\quad \\sigma_W^2 = \\frac{n(n+1)(2n+1)}{24} - \\frac{\\sum (t^3 - t)}{48}\n\\end{aligned}\n\\]\n<\/div>\r\n\t<\/figure>\r\n\t<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(d_i\\) \u2014 paired difference \\(y_i - x_i\\) (second measurement minus first)<\/li>\n<li>\\(n\\) \u2014 number of non-zero differences (pairs with \\(d_i = 0\\) are dropped)<\/li>\n<li>\\(t\\) \u2014 size of each group of tied absolute differences<\/li>\n<li>\\(z\\) \u2014 standardized statistic referred to the standard normal distribution; the \\(-0.5\\) term is the optional continuity correction<\/li>\n<\/ul>\n<p>Under the null hypothesis of no systematic shift, positive and negative ranks should balance, so \\(W^+\\) is expected to equal \\(\\mu_W\\). A \\(W^+\\) far from \\(\\mu_W\\) produces a large \\(|z|\\) and a small p-value. The tie correction \\(\\sum(t^3-t)\/48\\) reduces the variance when several differences share the same absolute value.<\/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>Wilcoxon Signed-Rank Test \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> ten students scored 85, 92, 78, 88, 95, 82, 90, 76, 89, 84 on a first test (Sample 1) and 91, 90, 85, 95, 99, 86, 88, 79, 92, 82 on a second test (Sample 2). Testing the two-sided hypothesis at \\(\\alpha = 0.05\\) with continuity correction:<\/p>\n<p>\\[ d_i = y_i - x_i = 6;\\ -2;\\ 7;\\ 7;\\ 4;\\ 4;\\ -2;\\ 3;\\ 3;\\ -2 \\quad \\Rightarrow \\quad n = 10 \\]<\/p>\n<p>\\[ |d| = 2,2,2,3,3,4,4,6,7,7 \\;\\Rightarrow\\; \\text{ranks } 2,2,2,4.5,4.5,6.5,6.5,8,9.5,9.5 \\]<\/p>\n<p>\\[ W^+ = 8+9.5+9.5+6.5+6.5+4.5+4.5 = 49, \\qquad W^- = 2+2+2 = 6 \\]<\/p>\n<p>\\[ \\mu_W = \\frac{10 \\cdot 11}{4} = 27.5, \\qquad \\sigma_W^2 = \\frac{10 \\cdot 11 \\cdot 21}{24} - \\frac{42}{48} = 95.375 \\]<\/p>\n<p>\\[ z = \\frac{49 - 27.5 - 0.5}{\\sqrt{95.375}} = \\frac{21}{9.7658} = 2.1493 \\]<\/p>\n<p>\\[ p = 2\\left(1 - \\Phi(2.1493)\\right) = 0.0316 \\]<\/p>\n<p><strong>Result:<\/strong> \\(W^+ = 49\\), \\(W^- = 6\\), \\(W = 6\\), \\(z = 2.1493\\), \\(p \\approx 0.0316 &lt; 0.05\\). Reject \\(H_0\\): the second test scores are systematically higher. The median paired difference is 3.5 points.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Choose the <strong>Data format<\/strong>: \u201cTwo paired samples\u201d if you have both measurements for each pair, or \u201cDifferences only\u201d if you have already computed \\(d_i\\).<\/li>\n<li>Enter the first measurement of every pair into <strong>Sample 1 (first \/ before)<\/strong>, one value per pair, separated by spaces, new lines or semicolons.<\/li>\n<li>Enter the matching values into <strong>Sample 2 (second \/ after)<\/strong>, in the same order and the same count. Differences are computed as second \u2212 first. If you chose \u201cDifferences only\u201d, type them into <strong>Paired differences (second \u2212 first)<\/strong> instead.<\/li>\n<li>Pick the <strong>Alternative hypothesis<\/strong>: two-sided for any systematic change, one-sided if you expect the second measurement to tend higher (&gt; 0) or lower (&lt; 0) than the first.<\/li>\n<li>Set the <strong>Significance level<\/strong> \\(\\alpha\\) (0.05 by default) and leave <strong>Apply continuity correction<\/strong> checked for a more accurate approximate p-value.<\/li>\n<li>Read the results: \\(n\\), \\(W^+\\), \\(W^-\\), the test statistic \\(W\\), \\(z\\), the p-value, the decision at \\(\\alpha\\), and the median of the paired differences, together with a full table of signed ranks.<\/li>\n<\/ol>\n<h2>What the Wilcoxon signed-rank test is<\/h2>\n<p>The Wilcoxon signed-rank test is a non-parametric test for comparing two <em>paired<\/em> measurements \u2014 before and after a treatment, two methods applied to the same subjects, matched pairs \u2014 when you cannot assume the differences are normally distributed. It is the distribution-free counterpart of the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/t-test-calculator\/\">paired t-test<\/a>.<\/p>\n<p>The test works entirely on the differences \\(d_i\\). It asks not just whether differences tend to be positive or negative (that would be the weaker sign test) but also how <em>large<\/em> they are: each non-zero difference is ranked by absolute size, and the sign is attached to the rank. If the null hypothesis of no systematic shift is true, the positive and negative ranks should roughly balance, so \\(W^+ \\approx W^- \\approx n(n+1)\/4\\). A strong imbalance \u2014 a large \\(W^+\\) with a small \\(W^-\\), or vice versa \u2014 is evidence of a consistent direction of change. Because the test uses ranks, it is robust to outliers and works for ordinal as well as numeric data with meaningful differences.<\/p>\n<p>The p-value answers: if there were truly no systematic difference, how likely is a rank imbalance at least as extreme as the one observed? A small p-value (at or below \\(\\alpha\\)) means the observed pattern is hard to explain by chance. Note that significance tells you a shift <em>exists<\/em>, not that it is large \u2014 always judge practical importance alongside the reported median of the paired differences and the raw data. If you want to quantify effect size rather than test a hypothesis, the paired structure of your data still applies, and the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/cohens-d-calculator\/\">Cohen's d calculator<\/a> is a common companion for the parametric version.<\/p>\n<h2>Assumptions, limitations and practical tips<\/h2>\n<ul>\n<li><strong>Pairing is essential.<\/strong> The observations must come in pairs measured on the same or matched units; two independent samples cannot be tested this way. For independent samples, a rank-sum test or the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/t-test-calculator\/\">t-test<\/a> for independent groups is appropriate.<\/li>\n<li><strong>Independence between pairs.<\/strong> Each pair must be independent of the others. Repeated measurements on the same subject over several time points violate this and need a different approach.<\/li>\n<li><strong>Symmetry of the differences.<\/strong> The test assumes the distribution of the \\(d_i\\) is roughly symmetric, so the hypothesis is genuinely about the median. With strongly skewed differences, the test still runs but the median interpretation weakens; check with the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/skewness-and-kurtosis-calculator\/\">skewness and kurtosis calculator<\/a>.<\/li>\n<li><strong>Zeros and ties.<\/strong> Pairs whose difference is exactly 0 are dropped, reducing \\(n\\). Tied absolute differences receive mid-ranks, and the variance is corrected by \\(\\sum(t^3 - t)\/48\\). With very heavy ties the corrected variance can shrink to zero, in which case the normal approximation is undefined.<\/li>\n<li><strong>Small samples.<\/strong> The p-value here uses the normal approximation with continuity correction, which is accurate from about \\(n \\ge 10\\) but rough below that; with fewer non-zero differences, prefer an exact signed-rank test from a statistics package. The calculator warns you when \\(n &lt; 10\\).<\/li>\n<li><strong>One-sided hypotheses.<\/strong> \u201cMedian difference &gt; 0\u201d means the second measurement tends to be larger than the first. Choose a one-sided test only if that direction was specified before looking at the data.<\/li>\n<li><strong>Binary paired outcomes<\/strong> (e.g. success\/failure before and after) do not fit this test \u2014 use the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/mcnemars-test-calculator\/\">McNemar's test calculator<\/a> instead.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Enter two paired samples (or ready-made differences) and this Wilcoxon signed-rank test calculator computes W\u207a, W\u207b, the z statistic with tie correction and continuity correction, the p-value, and the decision at your chosen significance level.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[74],"tags":[79],"class_list":["post-66987","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\/66987","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=66987"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66987\/revisions"}],"predecessor-version":[{"id":67104,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66987\/revisions\/67104"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66987"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66987"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66987"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}