{"id":67005,"date":"2026-10-13T11:29:15","date_gmt":"2026-10-13T08:29:15","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=67005"},"modified":"2026-10-13T11:29:15","modified_gmt":"2026-10-13T08:29:15","slug":"statistical-power-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/statistical-power-calculator\/","title":{"rendered":"Statistical Power 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\">Estimate the Power of a Planned Two-Group Study<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"67005\" data-favorite-title=\"Statistical Power\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/statistical-power-calculator\/\" data-favorite-excerpt=\"Estimate the statistical power of a planned two-group study from Cohen&#039;s d, sample size per group, significance level, and test direction. See the probability of detecting a real effect and the risk of a Type II error.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-wpcalc-sample-size\">\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\/statistical-power-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-67005\" data-post=\"67005\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Effect size (Cohen\u2019s d)<\/label><div class=\"term\">d<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"effect_size_cohens_d\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0.001\" value=\"0.5\"\/><small id=\"calc-info-1\">The expected difference between group means divided by their common standard deviation. Enter a positive value.<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Sample size per group<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"sample_size_per_group\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"2\" max=\"100000\" value=\"64\"\/><small id=\"calc-info-2\">Enter the planned number of independent observations in each group.<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Significance level (\u03b1)<\/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-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"1.0E-6\" max=\"0.5\" value=\"0.05\"\/><small id=\"calc-info-3\">The false-positive threshold; 0.05 is a common choice.<\/small><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">Test direction<\/label><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"test_direction\"><option value=\"two\" selected>Two-sided<\/option><option value=\"one\">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><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"67005\">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> Estimated Study Power<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Estimated statistical power<\/span><span class=\"result-term\">1 \u2212 \u03b2<\/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>Probability of a Type II error<\/span><span class=\"result-term\">\u03b2<\/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>Degrees of freedom<\/span><span class=\"result-term\">df<\/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>Noncentrality parameter<\/span><span class=\"result-term\">\u03b4<\/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 is-column\" id=\"output-5\"><span class=\"field-title\"><span>How to interpret this estimate<\/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><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=\"67005\" 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>Statistical Power<\/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\\[ \\delta = d\\sqrt{\\frac{n}{2}}, \\qquad \\nu = 2n - 2 \\\\\n\\lambda = 1 - \\beta = P\\!\\left(T_{\\nu,\\,\\delta} &gt; t_{\\alpha\/2}\\right) + P\\!\\left(T_{\\nu,\\,\\delta} &lt; -t_{\\alpha\/2}\\right) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(d\\) \u2014 effect size (Cohen's d), the expected difference between group means divided by the common standard deviation<\/li>\n<li>\\(n\\) \u2014 sample size per group<\/li>\n<li>\\(\\nu\\) \u2014 degrees of freedom of the independent-samples t test<\/li>\n<li>\\(\\delta\\) \u2014 noncentrality parameter, which scales the effect by the precision of the study<\/li>\n<li>\\(T_{\\nu,\\delta}\\) \u2014 a random variable following the noncentral t distribution with \\(\\nu\\) degrees of freedom and noncentrality \\(\\delta\\)<\/li>\n<li>\\(t_{\\alpha\/2}\\) \u2014 the critical value of the (central) t distribution for a two-sided test at level \\(\\alpha\\)<\/li>\n<\/ul>\n<p>The formula gives the probability that a two-sided independent-samples t test rejects the null hypothesis when the true standardized difference between the groups is \\(d\\). For a one-sided test, the second term is dropped and \\(t_{\\alpha\/2}\\) is replaced by the one-sided critical value \\(t_{\\alpha}\\).<\/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>Statistical Power \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a researcher plans a study comparing two groups with 64 participants each, expects a medium effect of \\(d = 0.5\\), and will use a two-sided t test at \\(\\alpha = 0.05\\).<\/p>\n<p>\\[ \\delta = 0.5\\sqrt{\\frac{64}{2}} = 0.5\\sqrt{32} \\approx 2.828 \\]<\/p>\n<p>\\[ \\nu = 2 \\times 64 - 2 = 126 \\]<\/p>\n<p>The two-sided critical value at \\(\\alpha = 0.05\\) with 126 degrees of freedom is \\(t_{0.025} \\approx 1.979\\). Evaluating the noncentral t distribution with \\(\\nu = 126\\) and \\(\\delta = 2.828\\):<\/p>\n<p>\\[ 1 - \\beta \\approx P(T &gt; 1.979) + P(T &lt; -1.979) \\approx 0.801 \\]<\/p>\n<p><strong>Result:<\/strong> the estimated power is about <strong>80.1%<\/strong>, and the probability of a Type II error is <strong>\\(\\beta \\approx 19.9%\\)<\/strong>.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Enter the expected standardized difference in <strong>Effect size (Cohen's d)<\/strong>. If you have pilot data or published means and standard deviations, the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/cohens-d-calculator\/\">Cohen's d calculator<\/a> can compute this value for you.<\/li>\n<li>Enter the planned number of observations in each arm in <strong>Sample size per group<\/strong>. Both groups are assumed to have the same size.<\/li>\n<li>Set the <strong>Significance level (\u03b1)<\/strong>, most commonly 0.05.<\/li>\n<li>Choose the <strong>Test direction<\/strong>: two-sided if either group could come out ahead, one-sided only if the direction of the effect was predicted before the data were collected.<\/li>\n<li>Read the estimated power, the Type II error probability, the degrees of freedom, the noncentrality parameter, and a plain-language interpretation of the results.<\/li>\n<\/ol>\n<h2>What statistical power means<\/h2>\n<p>Statistical power is the probability that a study will detect a real effect when one exists \u2014 that is, the chance of obtaining a statistically significant result if the true difference between the groups equals the effect size you specified. Its complement \\(\\beta\\) is the probability of a Type II error: missing a genuine effect because the sample was too small, the effect smaller than expected, or the measurements too noisy.<\/p>\n<p>Power is conventionally targeted at 80% or higher. A study with 80% power that tests a true effect of the assumed size will find it about four times out of five; the remaining 20% of replications would produce a non-significant result despite the effect being real. Low-powered studies are problematic in two ways: they often miss real effects, and the significant results they do produce tend to exaggerate the effect size.<\/p>\n<p>Power depends on three quantities, and the calculator lets you explore how they interact. Larger effects are easier to detect, larger samples shrink the standard error and increase power, and stricter significance thresholds (smaller \u03b1) reduce power. Increasing the sample size is usually the only lever a researcher controls directly: with \\(d = 0.5\\) at \\(\\alpha = 0.05\\), 64 per group yields about 80% power, but 26 per group yields only about 42%.<\/p>\n<p>The interpretation line tells you the estimated probability of rejecting the null hypothesis under your assumptions. Treat it as a planning estimate, not a guarantee: the true power depends on the actual effect size, which is never known exactly in advance.<\/p>\n<h2>Assumptions and limitations<\/h2>\n<ul>\n<li><strong>Equal, independent groups:<\/strong> the calculation assumes two independent samples of equal size. For paired designs, the effective sample size and standard deviation differ, and this tool will overestimate or underestimate power.<\/li>\n<li><strong>Normality and equal variances:<\/strong> the result assumes the observations are normally distributed in each group and that the two populations share a common variance, as the standard independent-samples t test does. The t test is fairly robust to moderate departures from normality when \\(n\\geq 30\\) per group.<\/li>\n<li><strong>Effect size must be specified honestly:<\/strong> power is only as good as the \\(d\\) you enter. Overly optimistic assumptions inflate the estimated power, so it is wise to compute power for a range of plausible effect sizes.<\/li>\n<li><strong>One-sided tests require pre-registration:<\/strong> choosing a one-sided test after looking at the data inflates the effective false-positive rate. Only select \"One-sided\" if the direction was fixed in advance.<\/li>\n<li><strong>Designed for means, not proportions or survival:<\/strong> this calculator addresses the two-group t test. For comparing two proportions, use the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/two-proportion-z-test-calculator\/\">two-proportion z-test calculator<\/a>; for a single mean against a known standard, see the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/z-test-for-a-mean-calculator\/\">z-test for a mean calculator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Estimate the statistical power of a planned two-group study from Cohen's d, sample size per group, significance level, and test direction. See the probability of detecting a real effect and the risk of a Type II error.<\/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":[80],"class_list":["post-67005","post","type-post","status-publish","format-standard","hentry","category-statistics","tag-sample-size"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67005","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=67005"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67005\/revisions"}],"predecessor-version":[{"id":67006,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67005\/revisions\/67006"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=67005"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=67005"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=67005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}