{"id":65465,"date":"2026-09-29T12:26:45","date_gmt":"2026-09-29T09:26:45","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65465"},"modified":"2026-09-29T15:19:18","modified_gmt":"2026-09-29T12:19:18","slug":"pearson-correlation-coefficient-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/pearson-correlation-coefficient-calculator\/","title":{"rendered":"Pearson Correlation Coefficient 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\">Calculate Pearson Correlation<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65465\" data-favorite-title=\"Correlation Coefficient\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/pearson-correlation-coefficient-calculator\/\" data-favorite-excerpt=\"Enter two paired data sets and instantly get Pearson&#039;s correlation coefficient r, the coefficient of determination r\u00b2, covariance and a plain-English interpretation of the relationship&#039;s strength and direction.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-wpcalc-regression-plot\">\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\/pearson-correlation-coefficient-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65465\" data-post=\"65465\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Data set X<\/label><textarea id=\"field-1\" name=\"data_set_x\">1, 2, 3, 4, 5, 6, 7, 8<\/textarea><\/div><div class=\"field is-single\" id=\"input-2\"><label for=\"field-2\">Data set Y<\/label><textarea id=\"field-2\" name=\"data_set_y\">53, 61, 68, 74, 80, 85, 90, 95<\/textarea><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Covariance type<\/label><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"covariance_type\"><option value=\"sample\" selected>Sample (divide by n \u2212 1)<\/option><option value=\"population\">Population (divide by n)<\/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=\"65465\">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> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Pearson correlation coefficient<\/span><span class=\"result-term\">r<\/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>Coefficient of determination<\/span><span class=\"result-term\">r\u00b2<\/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>Covariance<\/span><span class=\"result-term\">cov(X,Y)<\/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>Number of pairs<\/span><span class=\"result-term\">n<\/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>Interpretation<\/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=\"65465\" 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>Correlation Coefficient 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\\[ r = \\frac{\\sum_{i=1}^{n} (x_i - \\bar{x})(y_i - \\bar{y})}{\\sqrt{\\sum_{i=1}^{n} (x_i - \\bar{x})^2 \\cdot \\sum_{i=1}^{n} (y_i - \\bar{y})^2}} \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(r\\) \u2014 Pearson correlation coefficient, always between \\(-1\\) and \\(+1\\)<\/li>\n<li>\\(x_i, y_i\\) \u2014 the paired data values<\/li>\n<li>\\(\\bar{x}, \\bar{y}\\) \u2014 the means of the two data sets<\/li>\n<li>\\(n\\) \u2014 the number of pairs<\/li>\n<\/ul>\n<p>The numerator is the sum of cross-products of deviations (the covariance numerator); the denominator rescales it by the product of the spreads of X and Y, so the result becomes a dimensionless number. This is why r is unaffected by the units or scale of your data.<\/p>\n<p>The calculator also reports the coefficient of determination \\( r^2 \\) and the covariance:<\/p>\n<p>\\[ r^2 = r \\times r, \\qquad \\operatorname{cov}(X,Y) = \\frac{\\sum_{i=1}^{n} (x_i - \\bar{x})(y_i - \\bar{y})}{n - 1 \\ \\text{(sample)} \\ \\text{or} \\ n \\ \\text{(population)}} \\]\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>Correlation Coefficient \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a researcher measures study hours (X) and test scores (Y) for eight students: X = 1, 2, 3, 4, 5, 6, 7, 8 and Y = 53, 61, 68, 74, 80, 85, 90, 95.<\/p>\n<p>\\[ \\bar{x} = \\frac{1+2+\\cdots+8}{8} = 4.5, \\qquad \\bar{y} = \\frac{53+61+\\cdots+95}{8} = 75.75 \\]<\/p>\n<p>\\[ \\sum (x_i - \\bar{x})(y_i - \\bar{y}) = 248, \\qquad \\sum (x_i - \\bar{x})^2 = 42, \\qquad \\sum (y_i - \\bar{y})^2 = 1475.5 \\]<\/p>\n<p>\\[ r = \\frac{248}{\\sqrt{42 \\times 1475.5}} = \\frac{248}{248.94} \\approx 0.9962 \\]<\/p>\n<p><strong>Result:<\/strong> r \u2248 0.9962, a very strong positive linear relationship. The coefficient of determination is r\u00b2 \u2248 0.9925, meaning about 99.25% of the variation in test scores is explained by study hours. With the \"Sample\" covariance type, cov(X, Y) = 248 \/ 7 \u2248 35.43; with \"Population\", cov(X, Y) = 248 \/ 8 = 31. There are n = 8 pairs.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter your first variable's values in the <strong>Data set X<\/strong> field. Separate numbers with commas, spaces, semicolons or new lines \u2014 whichever you prefer.<\/li>\n<li>Enter the matching values in <strong>Data set Y<\/strong>, in the same order as X. Each X value must be paired with exactly one Y value, and both sets must contain the same number of values.<\/li>\n<li>Choose the <strong>Covariance type<\/strong>: \"Sample (divide by n \u2212 1)\" if your pairs are a sample of a larger population, or \"Population (divide by n)\" if you have data for the entire population. This choice affects only the reported covariance \u2014 the correlation coefficient r is identical either way.<\/li>\n<li>Read the results: Pearson's <strong>r<\/strong>, the coefficient of determination <strong>r\u00b2<\/strong>, the <strong>covariance<\/strong>, the number of pairs <strong>n<\/strong>, and a plain-language interpretation of the relationship.<\/li>\n<\/ol>\n<h2>What Is the Pearson Correlation Coefficient?<\/h2>\n<p>The Pearson correlation coefficient (usually written as <em>r<\/em>) measures the strength and direction of a <strong>linear<\/strong> relationship between two numeric variables. It always lies between \\(-1\\) and \\(+1\\):<\/p>\n<ul>\n<li><strong>+1<\/strong> \u2014 a perfect positive linear relationship: every increase in X comes with a proportional increase in Y.<\/li>\n<li><strong>0<\/strong> \u2014 no linear association between the variables.<\/li>\n<li><strong>\u22121<\/strong> \u2014 a perfect negative linear relationship: as X increases, Y decreases proportionally.<\/li>\n<\/ul>\n<p>The sign tells you the direction of the relationship, while the absolute value tells you how tightly the data points cluster around a straight line. Because r is standardized, it does not depend on the units of measurement: the correlation between height and weight is the same whether you measure in kilograms and centimeters or pounds and inches.<\/p>\n<p>The coefficient of determination <strong>r\u00b2<\/strong> answers a related question: how much of the variation in Y is explained by its linear relationship with X. For example, r = 0.7 gives r\u00b2 = 0.49, so roughly 49% of the variation in Y is accounted for by the linear trend. This is often a more intuitive and honest way to report the practical size of a relationship.<\/p>\n<p>The <strong>covariance<\/strong> reported alongside r is the unscaled version of the same idea. Covariance describes the direction of the joint variation but its magnitude depends on the units, which is exactly why the standardized r is usually the more useful statistic.<\/p>\n<h2>Interpreting the Result: A Rough Guide<\/h2>\n<p>The interpretation scale used by this calculator is a common descriptive convention. Treat the boundaries as approximate, not strict rules \u2014 context matters:<\/p>\n<table>\n<thead>\n<tr>\n<th>|r|<\/th>\n<th>Descriptive strength<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0.90 \u2013 1.00<\/td>\n<td>Very strong<\/td>\n<\/tr>\n<tr>\n<td>0.70 \u2013 0.89<\/td>\n<td>Strong<\/td>\n<\/tr>\n<tr>\n<td>0.50 \u2013 0.69<\/td>\n<td>Moderate<\/td>\n<\/tr>\n<tr>\n<td>0.30 \u2013 0.49<\/td>\n<td>Weak<\/td>\n<\/tr>\n<tr>\n<td>0.00 \u2013 0.29<\/td>\n<td>Very weak to none<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Assumptions, Limitations and Practical Tips<\/h2>\n<ul>\n<li><strong>Linearity:<\/strong> Pearson's r only detects straight-line patterns. A relationship can be very strong but curved (for example, Y = X\u00b2) and still yield r close to 0. Always plot your data before drawing conclusions from r alone.<\/li>\n<li><strong>Correlation is not causation:<\/strong> a high r shows that two variables move together, not that one causes the other. A hidden third variable may drive both.<\/li>\n<li><strong>Outliers:<\/strong> r is sensitive to extreme values. A single unusual point can inflate or deflate the coefficient dramatically, especially with small samples.<\/li>\n<li><strong>Paired data:<\/strong> the calculation requires that each X value is paired with one Y value (same source or unit). Two unrelated lists of numbers produce a meaningless r.<\/li>\n<li><strong>Sample size:<\/strong> with very few pairs, even a large r can arise by chance. With n = 2 the coefficient is always exactly +1 or \u22121, which is why at least 2 pairs are required and more are strongly recommended.<\/li>\n<li><strong>Continuous data:<\/strong> Pearson's r is designed for continuous, roughly numeric data. For ordinal (ranked) data, Spearman's rank correlation is usually the more appropriate choice.<\/li>\n<\/ul>\n<p>Finally, remember the difference between the covariance options: the divisor (n \u2212 1 versus n) changes the covariance value but never changes r or r\u00b2, since the same factor appears in the numerator and denominator of the correlation formula.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter two paired data sets and instantly get Pearson's correlation coefficient r, the coefficient of determination r\u00b2, covariance and a plain-English interpretation of the relationship's strength and direction.<\/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":[78],"class_list":["post-65465","post","type-post","status-publish","format-standard","hentry","category-statistics","tag-correlation-regression"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65465","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=65465"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65465\/revisions"}],"predecessor-version":[{"id":65709,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65465\/revisions\/65709"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65465"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65465"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65465"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}