{"id":67007,"date":"2026-10-13T11:30:11","date_gmt":"2026-10-13T08:30:11","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=67007"},"modified":"2026-10-13T11:30:11","modified_gmt":"2026-10-13T08:30:11","slug":"multiple-linear-regression-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/multiple-linear-regression-calculator\/","title":{"rendered":"Multiple Linear Regression 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\">Fit an Outcome Using Multiple Predictors<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"67007\" data-favorite-title=\"Multiple Linear Regression\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/multiple-linear-regression-calculator\/\" data-favorite-excerpt=\"Enter your outcome values and two or more predictor columns to fit a multiple linear regression by least squares. The calculator returns the intercept, each predictor coefficient, R\u00b2, the fitted equation, and predicted values with residuals.\" 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\/multiple-linear-regression-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-67007\" data-post=\"67007\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Outcome values (y)<\/label><textarea id=\"field-1\" name=\"outcome_values_y\">7; 12; 11; 16; 15; 20; 19; 24<\/textarea><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Predictor columns (X)<\/label><textarea id=\"field-2\" name=\"predictor_columns_x\">1 0\n2 1\n3 0\n4 1\n5 0\n6 1\n7 0\n8 1<\/textarea><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"67007\">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> Regression Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Intercept<\/span><span class=\"result-term\">\u03b2\u2080<\/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 is-column\" id=\"output-2\"><span class=\"field-title\"><span>Predictor coefficients<\/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>R\u00b2<\/span><span class=\"result-term\">R\u00b2<\/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 is-column\" id=\"output-4\"><span class=\"field-title\"><span>Predicted values for each observation<\/span><span class=\"result-term\">\u0177<\/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>Model and fit details<\/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=\"67007\" 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>Multiple Linear Regression 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\\[ \\hat{y} = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2 + \\cdots + \\beta_p X_p \\\\\\quad R^2 = 1 - \\frac{\\sum_{i=1}^{n} \\left( y_i - \\hat{y}_i \\right)^2}{\\sum_{i=1}^{n} \\left( y_i - \\bar{y} \\right)^2} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(\\hat{y}\\) \u2014 the predicted value of the outcome<\/li>\n<li>\\(\\beta_0\\) \u2014 the intercept: the expected value of \\(y\\) when every predictor equals zero<\/li>\n<li>\\(\\beta_j\\) \u2014 the coefficient of predictor \\(X_j\\): the expected change in \\(y\\) when \\(X_j\\) increases by one unit while all other predictors are held fixed<\/li>\n<li>\\(p\\) \u2014 the number of predictors<\/li>\n<li>\\(n\\) \u2014 the number of observations<\/li>\n<li>\\(R^2\\) \u2014 the proportion of the variance in \\(y\\) explained by the model<\/li>\n<\/ul>\n<p>The coefficients are estimated by ordinary least squares (OLS), which chooses the \\(\\beta\\) values that minimize the sum of squared residuals \\(\\sum (y - \\hat{y})^2\\). In matrix form the solution is \\(\\hat{\\boldsymbol{\\beta}} = (\\mathbf{X}^T \\mathbf{X})^{-1} \\mathbf{X}^T \\mathbf{y}\\); this calculator solves the same least-squares problem with a numerically stable QR decomposition instead of forming \\(\\mathbf{X}^T\\mathbf{X}\\) explicitly.<\/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>Multiple Linear Regression \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> eight observations with outcome values 7; 12; 11; 16; 15; 20; 19; 24, and two predictors: \\(X_1\\) running from 1 to 8 and \\(X_2\\) alternating 0, 1, 0, 1, \u2026 \u2014 exactly the default data in the calculator.<\/p>\n<p>\\[ \\hat{y} = 5 + 2X_1 + 3X_2 \\]<\/p>\n<p><strong>Result:<\/strong> intercept \\(\\beta_0 = 5\\), coefficients \\(\\beta_1 = 2\\) and \\(\\beta_2 = 3\\), and \\(R^2 = 1\\). Every predicted value equals the actual outcome, so all residuals are 0 and the residual sum of squares is 0 out of a total sum of squares of 210. This dataset was constructed to fit perfectly, which is why \\(R^2\\) reaches its maximum of 1; real data almost always leave some residual variation and give \\(R^2 &lt; 1\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Enter your <strong>Outcome values (y)<\/strong> \u2014 one value per observation, separated by spaces, semicolons or new lines. Decimal commas are accepted.<\/li>\n<li>Enter the <strong>Predictor columns (X)<\/strong> \u2014 one observation per line, with that observation's predictor values separated by spaces or semicolons. Keep the columns in the same order on every line, and enter at least two columns.<\/li>\n<li>Check that the number of X lines matches the number of y values, and that every line has the same number of values. You need at least two more observations than predictors.<\/li>\n<li>Read the results: the <strong>Intercept<\/strong>, the <strong>Predictor coefficients<\/strong> table, <strong>R\u00b2<\/strong>, the table of <strong>Predicted values<\/strong> with residuals, and the fitted equation with fit details.<\/li>\n<\/ol>\n<h2>What is multiple linear regression?<\/h2>\n<p>Multiple linear regression models a continuous outcome as a weighted sum of two or more predictors plus a constant. It extends simple linear regression, which uses a single predictor, to situations where several variables may influence the outcome simultaneously \u2014 for example predicting a house price from floor area, number of rooms and age, or crop yield from rainfall, temperature and fertilizer amount.<\/p>\n<p>The key advantage over separate simple regressions is that each coefficient isolates the contribution of one predictor <em>holding the other included predictors fixed<\/em>. A coefficient of 2 for \\(X_1\\) means the outcome is expected to rise by 2 units for each one-unit increase in \\(X_1\\), regardless of where the other predictors stand. This is why the coefficients can differ substantially from the slopes you would get by regressing on each predictor alone \u2014 the model separates overlapping effects.<\/p>\n<p>R\u00b2 measures how well the fitted plane (or hyperplane) tracks the data: it is the fraction of the variance in \\(y\\) that the predictors account for. An R\u00b2 of 0.80 means 80% of the observed variation is reproduced by the model. Note that R\u00b2 can only stay the same or increase when you add predictors, so a high R\u00b2 with many predictors is not automatically evidence of a good model \u2014 it simply describes the fit to the data used for estimation.<\/p>\n<p>To explore the pairwise relationships between your variables before fitting a model, try the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/spearman-rank-correlation-calculator\/\">Spearman rank correlation calculator<\/a> or the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/kendalls-tau-calculator\/\">Kendall's tau calculator<\/a>.<\/p>\n<h2>Assumptions, limitations and practical tips<\/h2>\n<ul>\n<li><strong>Sample size:<\/strong> you need at least \\(p + 2\\) observations for \\(p\\) predictors and an intercept (the calculator enforces this). In practice, aim for 10\u201320 observations per predictor for stable estimates.<\/li>\n<li><strong>No redundant columns:<\/strong> the predictors must not be linearly dependent. If you encode a categorical variable with dummy 0\/1 columns, include one fewer dummy than there are categories \u2014 the intercept absorbs the baseline, just like the alternating 0\/1 column in the example.<\/li>\n<li><strong>Multicollinearity:<\/strong> predictors that are highly correlated with each other produce unstable, hard-to-interpret coefficients even though the overall fit may look fine. Check pairwise associations first and drop or combine overlapping variables.<\/li>\n<li><strong>Linearity:<\/strong> OLS assumes each predictor relates to the outcome linearly, with the other predictors held fixed. Curved relationships can often be accommodated by adding transformed columns (e.g. a squared term) as extra predictors.<\/li>\n<li><strong>Residual checks:<\/strong> the model assumes independent, roughly constant-variance errors. Inspecting the residuals in the prediction table helps; for time-ordered data, the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/autocorrelation-calculator\/\">autocorrelation calculator<\/a> can reveal leftover pattern in the residuals, and the <a href=\"https:\/\/wpcalc.com\/en\/statistics\/skewness-and-kurtosis-calculator\/\">skewness and kurtosis calculator<\/a> helps spot strongly non-normal data.<\/li>\n<li><strong>Association is not causation:<\/strong> coefficients describe associations within your data. Claiming a causal effect requires a study design that supports it, not just a good fit.<\/li>\n<li><strong>Stay inside your data's range:<\/strong> predictions far outside the range of the observed predictors are extrapolations and can be badly wrong.<\/li>\n<li><strong>No inference output:<\/strong> this calculator reports the fitted equation and in-sample fit only; for standard errors, p-values and confidence intervals, use full statistical software.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Enter your outcome values and two or more predictor columns to fit a multiple linear regression by least squares. The calculator returns the intercept, each predictor coefficient, R\u00b2, the fitted equation, and predicted values with residuals.<\/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":[78],"class_list":["post-67007","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\/67007","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=67007"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67007\/revisions"}],"predecessor-version":[{"id":67008,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67007\/revisions\/67008"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=67007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=67007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=67007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}