{"id":66985,"date":"2026-10-13T10:54:41","date_gmt":"2026-10-13T07:54:41","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66985"},"modified":"2026-10-13T10:54:41","modified_gmt":"2026-10-13T07:54:41","slug":"two-proportion-z-test-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/two-proportion-z-test-calculator\/","title":{"rendered":"Two-Proportion Z-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 Two Proportions or Conversion Rates<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66985\" data-favorite-title=\"Two-Proportion Z-Test\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/two-proportion-z-test-calculator\/\" data-favorite-excerpt=\"Compare two proportions or conversion rates with the two-proportion z-test. Enter successes and sample sizes for each group to get the z statistic, p-value, decision at your significance level, and a confidence interval for the difference.\" 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\/two-proportion-z-test-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66985\" data-post=\"66985\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Group 1<\/span><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Successes<\/label><div class=\"term\">x\u2081<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"successes\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 45\" min=\"0\" value=\"45\"\/><small id=\"calc-info-2\">Number of subjects with the outcome of interest (e.g. conversions, responders) in group 1.<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Sample size<\/label><div class=\"term\">n\u2081<\/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\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 500\" min=\"1\" value=\"500\"\/><small id=\"calc-info-3\">Total number of subjects observed in group 1.<\/small><\/div><div class=\"is-single field\" id=\"input-4\"><span class=\"font-semibold\">Group 2<\/span><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">Successes<\/label><div class=\"term\">x\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"successes\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 25\" min=\"0\" value=\"25\"\/><small id=\"calc-info-5\">Number of subjects with the outcome in group 2.<\/small><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">Sample size<\/label><div class=\"term\">n\u2082<\/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\" id=\"field-6\" aria-describedby=\"calc-info-6\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 500\" min=\"1\" value=\"500\"\/><small id=\"calc-info-6\">Total number of subjects observed in group 2.<\/small><\/div><div class=\"field\" id=\"input-7\"><label for=\"field-7\">Alternative hypothesis<\/label><div class=\"grid grid-cols-1\"><select id=\"field-7\" name=\"alternative_hypothesis\"><option value=\"two\" selected>p\u2081 \u2260 p\u2082 (two-sided)<\/option><option value=\"greater\">p\u2081 &gt; p\u2082 (one-sided)<\/option><option value=\"less\">p\u2081 &lt; p\u2082 (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-8\"><label for=\"field-8\">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-8\" aria-describedby=\"calc-info-8\" 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-8\">Type I error rate \u03b1, the threshold below which the p-value counts as statistically significant. 0.05 is the conventional choice.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66985\">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> Two-Proportion Z-Test Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Proportion in group 1<\/span><span class=\"result-term\">p\u0302\u2081<\/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>Proportion in group 2<\/span><span class=\"result-term\">p\u0302\u2082<\/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>Difference in proportions (p\u0302\u2081 \u2212 p\u0302\u2082)<\/span><span class=\"result-term\">\u0394p\u0302<\/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>z statistic<\/span><span class=\"result-term\">z<\/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>p-value<\/span><span class=\"result-term\">p<\/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>Decision at significance level \u03b1<\/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>Confidence interval for p\u2081 \u2212 p\u2082<\/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 is-column\" id=\"output-8\"><span class=\"field-title\"><span>How the test works and interpretation<\/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><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=\"66985\" 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>Two-Proportion Z-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\\[ \\hat p_1 = \\frac{x_1}{n_1}, \\qquad \\hat p_2 = \\frac{x_2}{n_2}, \\qquad \\hat p = \\frac{x_1 + x_2}{n_1 + n_2} \\]\n\\[ z = \\frac{\\hat p_1 - \\hat p_2}{\\sqrt{\\hat p (1 - \\hat p) \\left( \\frac{1}{n_1} + \\frac{1}{n_2} \\right)}} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(x_1, x_2\\) \u2014 number of successes (conversions, responders, defects, etc.) in each group<\/li>\n<li>\\(n_1, n_2\\) \u2014 sample sizes of the two groups<\/li>\n<li>\\(\\hat p_1, \\hat p_2\\) \u2014 sample proportions in each group<\/li>\n<li>\\(\\hat p\\) \u2014 pooled proportion, the best estimate of the common proportion under the null hypothesis \\(H_0: p_1 = p_2\\)<\/li>\n<li>\\(z\\) \u2014 test statistic, compared against the standard normal distribution<\/li>\n<\/ul>\n<p>The z statistic measures how far the observed difference \\(\\hat p_1 - \\hat p_2\\) is from zero, in units of its standard error computed under the assumption that the two population proportions are equal. Large absolute values of \\(z\\) indicate that the observed difference is unlikely to be due to sampling variation alone.<\/p>\n<p>The confidence interval for the difference uses the unpooled standard error:<\/p>\n<p>\\[ (\\hat p_1 - \\hat p_2) \\pm z_{\\alpha\/2} \\sqrt{ \\frac{\\hat p_1 (1 - \\hat p_1)}{n_1} + \\frac{\\hat p_2 (1 - \\hat p_2)}{n_2} } \\]\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>Two-Proportion Z-Test \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> an A\/B test compares two landing pages. Version A (group 1) records 45 conversions out of 500 visitors; version B (group 2) records 25 conversions out of 500 visitors. We test \\(H_0: p_1 = p_2\\) against \\(p_1 \\neq p_2\\) at \\(\\alpha = 0.05\\).<\/p>\n<p>\\[ \\hat p_1 = \\frac{45}{500} = 0.09, \\qquad \\hat p_2 = \\frac{25}{500} = 0.05, \\qquad \\hat p = \\frac{45 + 25}{500 + 500} = 0.07 \\]<\/p>\n<p>\\[ z = \\frac{0.09 - 0.05}{\\sqrt{0.07 \\times 0.93 \\times \\left( \\frac{1}{500} + \\frac{1}{500} \\right)}} = \\frac{0.04}{0.0161} = 2.4788 \\]<\/p>\n<p>The two-sided p-value is \\(2\\left(1 - \\Phi(2.4788)\\right) = 0.0132\\), which is below \\(\\alpha = 0.05\\).<\/p>\n<p><strong>Result:<\/strong> \\(z = 2.4788\\), \\(p = 0.0132\\); reject \\(H_0\\). The conversion rates differ significantly (9.00% vs 5.00%, a +4.00 percentage-point difference). The 95% confidence interval for \\(p_1 - p_2\\) runs from +0.85 to +7.15 percentage points.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Under <strong>Group 1<\/strong>, enter the number of <strong>Successes<\/strong> (e.g. 45) and the <strong>Sample size<\/strong> (e.g. 500). Successes cannot exceed the sample size.<\/li>\n<li>Under <strong>Group 2<\/strong>, enter the same information for the second group (e.g. 25 successes out of 500).<\/li>\n<li>Choose the <strong>Alternative hypothesis<\/strong>: two-sided (\\(p_1 \\neq p_2\\)) for any difference, or one-sided if you predicted the direction in advance.<\/li>\n<li>Set the <strong>Significance level<\/strong> \\(\\alpha\\) \u2014 0.05 is the conventional choice; 0.01 or 0.10 are common alternatives.<\/li>\n<li>Read the results: sample proportions, their difference, the z statistic, the p-value, the decision at your \\(\\alpha\\), and a confidence interval for \\(p_1 - p_2\\).<\/li>\n<\/ol>\n<h2>What is the two-proportion z-test?<\/h2>\n<p>The two-proportion z-test answers the question: <em>is the difference between two proportions large enough to conclude the underlying population proportions differ?<\/em> It is the standard tool when you observe a binary outcome in two independent groups \u2014 conversion rates in an A\/B test, response rates in two treatment arms, defect rates from two production lines, or support for a policy among two demographics.<\/p>\n<p>The logic is straightforward. Even if both populations had exactly the same proportion, the two sample proportions would rarely be identical because of random sampling variation. The test quantifies how much of a difference sampling variation alone would typically produce (the standard error) and compares the observed difference against that yardstick. The resulting z statistic follows approximately a standard normal distribution, which converts it into a p-value \u2014 the probability of seeing a difference at least this large if the null hypothesis of equal proportions were true.<\/p>\n<p>Interpreting the output: if the p-value is at or below your chosen \\(\\alpha\\), the difference is statistically significant and the null hypothesis is rejected. The confidence interval is just as informative: it gives the plausible range for the true difference \\(p_1 - p_2\\). If it excludes zero, the difference is significant; its width tells you how precisely the data pin down the effect. A significant but tiny difference (e.g. 0.2 percentage points) may matter commercially less than a non-significant but large one \u2014 significance and practical importance are different things.<\/p>\n<p>For a quick check of how to read p-values in general, see our <a href=\"https:\/\/wpcalc.com\/en\/statistics\/p-value-calculator\/\">P-Value Calculator<\/a>.<\/p>\n<h2>Assumptions, limitations and practical tips<\/h2>\n<ul>\n<li><strong>Independent samples.<\/strong> The two groups must consist of different, unrelated subjects. If the same subjects are measured twice (e.g. before\/after), use <a href=\"https:\/\/wpcalc.com\/en\/statistics\/mcnemars-test-calculator\/\">McNemar's Test<\/a> instead.<\/li>\n<li><strong>Binary outcome.<\/strong> Each observation must be a success or failure \u2014 a count of conversions, cures, defects, and so on. Successes must be whole numbers not exceeding the sample size.<\/li>\n<li><strong>Normal approximation.<\/strong> The z-test is an approximation that works when both groups have enough expected successes and failures under \\(H_0\\) (at least 5 each is a common rule of thumb, at least 10 for a stricter standard). With small samples or rare events, <a href=\"https:\/\/wpcalc.com\/en\/statistics\/fishers-exact-test-calculator\/\">Fisher's Exact Test<\/a> is more appropriate and does not rely on this approximation.<\/li>\n<li><strong>Choose the alternative before looking at the data.<\/strong> A one-sided test is more powerful but only valid if the direction was genuinely predicted in advance. Switching to one-sided after seeing a result inflates the Type I error rate.<\/li>\n<li><strong>Watch for zero standard error.<\/strong> If every observation has the same outcome (all successes or all failures), the pooled proportion is 0 or 1, the standard error collapses to zero, and the z-test is undefined \u2014 the calculator will flag this.<\/li>\n<li><strong>Significance \u2260 importance.<\/strong> With very large samples, trivial differences become significant. Judge the effect size through the confidence interval, and complement it with the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/odds-ratio-and-relative-risk-calculator\/\">Odds Ratio and Relative Risk Calculator<\/a> for other ways to express the strength of the association.<\/li>\n<li><strong>Related methods.<\/strong> The chi-square test of independence on the underlying 2\u00d72 table gives an equivalent two-sided result; see the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/chi-square-test-calculator\/\">Chi-Square Test Calculator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Compare two proportions or conversion rates with the two-proportion z-test. Enter successes and sample sizes for each group to get the z statistic, p-value, decision at your significance level, and a confidence interval for the difference.<\/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-66985","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\/66985","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=66985"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66985\/revisions"}],"predecessor-version":[{"id":66986,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66985\/revisions\/66986"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66985"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66985"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}