{"id":65088,"date":"2025-08-01T08:24:50","date_gmt":"2025-08-01T05:24:50","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65088"},"modified":"2026-09-30T20:46:54","modified_gmt":"2026-09-30T17:46:54","slug":"quadratic-mean-calculator-root-mean-square","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-mean-calculator-root-mean-square\/","title":{"rendered":"Quadratic Mean Calculator (Root Mean Square)"},"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 the Root Mean Square of Any Data Set<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65088\" data-favorite-title=\"Quadratic Mean\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-mean-calculator-root-mean-square\/\" data-favorite-excerpt=\"Calculate the quadratic mean (root mean square, RMS) of any set of numbers. Enter how many values you have and the values themselves \u2014 the calculator squares each one, averages the squares, and takes the square root.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-arithmetic\">\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\/mathematics\/quadratic-mean-calculator-root-mean-square\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65088\" data-post=\"65088\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Number of values<\/label><input type=\"text\" name=\"number_of_values\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"2\" value=\"2\"\/><small id=\"calc-info-1\">Enter how many values to calculate the geometric mean for<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Number 1<\/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=\"number\" name=\"number_1\" id=\"field-2\" step=\"any\" value=\"10\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Number 2<\/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=\"number\" name=\"number_2\" id=\"field-3\" step=\"any\" value=\"20\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65088\">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>Average<\/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><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=\"65088\" 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>Quadratic Mean (RMS) 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\\[ x_{\\mathrm{RMS}} = \\sqrt{\\frac{x_1^2 + x_2^2 + \\cdots + x_n^2}{n}} = \\sqrt{\\frac{1}{n}\\sum_{i=1}^{n} x_i^2} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(x_1, x_2, \\ldots, x_n\\) \u2014 the individual values in the data set<\/li>\n<li>\\(n\\) \u2014 the number of values<\/li>\n<\/ul>\n<p>The formula squares every value, takes the arithmetic mean of those squares, and then takes the square root of the result. Because squaring removes negative signs and then the square root brings the result back to the original scale, the quadratic mean is always non-negative and is expressed in the same units as the input values.<\/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>Quadratic Mean \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> find the quadratic mean of the two default values used by the calculator, 10 and 20.<\/p>\n<p>\\[ x_{\\mathrm{RMS}} = \\sqrt{\\frac{10^2 + 20^2}{2}} = \\sqrt{\\frac{100 + 400}{2}} = \\sqrt{\\frac{500}{2}} = \\sqrt{250} \\approx 15.81 \\]<\/p>\n<p><strong>Result:<\/strong> the quadratic mean (RMS) of 10 and 20 is approximately <strong>15.81<\/strong>.<\/p>\n<p>For comparison, the ordinary arithmetic mean of the same numbers is \\(\\frac{10 + 20}{2} = 15\\), which is always less than or equal to the quadratic mean.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Quadratic Mean Calculator<\/h2>\n<ol>\n<li>Enter the total count of your values in the <strong>Number of values<\/strong> field (minimum 2). The calculator will automatically add or remove input fields to match.<\/li>\n<li>Type your first value into <strong>Number 1<\/strong> and your second value into <strong>Number 2<\/strong>.<\/li>\n<li>If you entered more than two values, fill in the additional fields (<strong>Number 3<\/strong>, <strong>Number 4<\/strong>, and so on) that appear.<\/li>\n<li>The result, shown in the <strong>Average<\/strong> field, updates instantly \u2014 no button to press.<\/li>\n<li>To start over, reset the form; the extra fields will be removed automatically.<\/li>\n<\/ol>\n<h2>What Is the Quadratic Mean?<\/h2>\n<p>The quadratic mean, better known as the <strong>root mean square (RMS)<\/strong>, is a type of average that measures the effective magnitude of a set of values. Instead of averaging the values directly, it averages their squares and then takes the square root. This makes it the natural choice whenever both positive and negative values occur, since squaring eliminates the sign \u2014 an alternating current of +10 A and \u221210 A is far from \"zero\" in effect, and the RMS value captures that.<\/p>\n<p>A key mathematical property is that the quadratic mean is never smaller than the arithmetic mean; the two are equal only when all values are identical. The more the values spread out, the more the RMS exceeds the simple average, because larger deviations are squared and therefore weigh more heavily. In symbols:<\/p>\n<p>\\[ x_{\\mathrm{RMS}} \\ge \\bar{x} \\quad \\text{for any set of real numbers} \\]<\/p>\n<p>This sensitivity to spread is exactly why the RMS appears in error analysis: the root mean square error (RMSE) of predictions is just the quadratic mean of the individual errors, and it penalizes large mistakes more than small ones.<\/p>\n<p>The quadratic mean is one of the classical Pythagorean means, alongside the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/average-calculator\/\">arithmetic mean<\/a>, the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/geometric-mean\/\">geometric mean<\/a>, and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/harmonic-mean\/\">harmonic mean<\/a>. Each is suited to a different kind of data: the arithmetic mean for additive quantities, the geometric mean for growth rates and ratios, the harmonic mean for rates and speeds, and the quadratic mean for quantities whose sign varies or whose magnitude matters more than their direction.<\/p>\n<h2>Where RMS Is Used<\/h2>\n<ul>\n<li><strong>Electrical engineering:<\/strong> RMS voltage and current describe the effective power delivered by an AC signal. The 230 V or 120 V of household mains electricity is an RMS value, not a peak value.<\/li>\n<li><strong>Physics:<\/strong> the RMS speed of gas molecules determines temperature and pressure in the kinetic theory of gases.<\/li>\n<li><strong>Statistics and machine learning:<\/strong> RMSE and its square, the mean squared error, are standard measures of prediction accuracy.<\/li>\n<li><strong>Signal processing:<\/strong> RMS levels quantify the average power of audio and vibration signals over time.<\/li>\n<\/ul>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Signs are ignored:<\/strong> because values are squared, the RMS of \u22125 and 5 equals the RMS of 5 and 5. If the sign of your data carries meaning, the quadratic mean is not the right summary statistic on its own.<\/li>\n<li><strong>Outliers are amplified:<\/strong> a single very large value raises the RMS noticeably, since it is squared before averaging. Use this to your advantage when large deviations matter, but keep it in mind with noisy data.<\/li>\n<li><strong>Units are preserved:<\/strong> the result is in the same units as the inputs. If your data are in meters, the RMS is in meters too \u2014 unlike the variance, which is in squared units.<\/li>\n<li><strong>All values count equally:<\/strong> if some observations should carry more weight, a <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/weighted-mean\/\">weighted mean<\/a> is more appropriate.<\/li>\n<li><strong>Minimum of two values:<\/strong> the calculator requires at least two numbers; for a single value the quadratic mean simply equals its absolute value.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the quadratic mean (root mean square, RMS) of any set of numbers. Enter how many values you have and the values themselves \u2014 the calculator squares each one, averages the squares, and takes the square root.<\/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":[7],"tags":[72],"class_list":["post-65088","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-arithmetic-calculators"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65088","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=65088"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65088\/revisions"}],"predecessor-version":[{"id":66281,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65088\/revisions\/66281"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65088"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65088"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65088"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}