{"id":65082,"date":"2025-08-01T08:07:47","date_gmt":"2025-08-01T05:07:47","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65082"},"modified":"2026-09-30T20:34:43","modified_gmt":"2026-09-30T17:34:43","slug":"harmonic-mean","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/harmonic-mean\/","title":{"rendered":"Harmonic Mean 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 the Harmonic Average of Numbers<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65082\" data-favorite-title=\"Harmonic Mean\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/harmonic-mean\/\" data-favorite-excerpt=\"Enter two or more positive numbers and this calculator returns their harmonic mean \u2014 the reciprocal-of-the-average-of-reciprocals, ideal for averaging rates and speeds.\" 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\/harmonic-mean\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65082\" data-post=\"65082\"><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 you want to calculate the harmonic 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=\"65082\">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=\"65082\" 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>Harmonic Mean 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\\[ H = \\frac{n}{\\dfrac{1}{x_1} + \\dfrac{1}{x_2} + \\cdots + \\dfrac{1}{x_n}} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(H\\) \u2014 the harmonic mean<\/li>\n<li>\\(n\\) \u2014 the number of values<\/li>\n<li>\\(x_1, x_2, \\ldots, x_n\\) \u2014 the individual values; each must be positive and non-zero<\/li>\n<\/ul>\n<p>The formula divides the count of values by the sum of their reciprocals. Equivalently, it is the reciprocal of the arithmetic mean of the reciprocals, which is why it gives greater weight to smaller 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>Harmonic Mean \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> you travel the same distance twice \u2014 first at 10 km\/h, then at 20 km\/h \u2014 and want your average speed for the whole trip.<\/p>\n<p>\\[ H = \\frac{2}{\\dfrac{1}{10} + \\dfrac{1}{20}} = \\frac{2}{0.1 + 0.05} = \\frac{2}{0.15} \\approx 13.33 \\]<\/p>\n<p><strong>Result:<\/strong> the harmonic mean of 10 and 20 is approximately <strong>13.33 km\/h<\/strong> \u2014 not 15, which the arithmetic mean would incorrectly suggest.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter how many values you want to average in the <strong>Number of values<\/strong> field (minimum 2). Additional input fields appear automatically.<\/li>\n<li>Type your positive numbers into <strong>Number 1<\/strong>, <strong>Number 2<\/strong>, and so on.<\/li>\n<li>The result appears instantly in the <strong>Average<\/strong> field \u2014 no button needed.<\/li>\n<li>If any value is zero or negative, the calculator shows an error, because the harmonic mean is defined only for positive numbers.<\/li>\n<\/ol>\n<h2>What Is the Harmonic Mean?<\/h2>\n<p>The harmonic mean is one of the three classical Pythagorean means, alongside the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/average-calculator\/\">arithmetic mean<\/a> and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/geometric-mean\/\">geometric mean<\/a>. For any set of positive numbers it always satisfies the ordering:<\/p>\n<p>\\[ H \\leq G \\leq A \\]<\/p>\n<p>with equality only when all values are identical. In practice, the harmonic mean is the right average whenever the quantities being combined are expressed as rates \u2014 that is, as something <em>per unit<\/em> of something else, such as kilometres per hour, litres per 100 km, or units produced per hour.<\/p>\n<p>The classic illustration is average speed. If you drive 100 km at 60 km\/h and the next 100 km at 40 km\/h, your true average speed is not 50 km\/h but the harmonic mean \\(2\/(1\/60 + 1\/40) = 48\\) km\/h, because you spend more time at the slower speed. Whenever each value applies to the <em>same distance, volume, or workload<\/em>, the harmonic mean \u2014 not the arithmetic mean \u2014 gives the correct overall rate.<\/p>\n<p>Because small values contribute disproportionately large reciprocals, the harmonic mean is pulled toward the smallest entries and is robust against very large outliers. This property is used in finance for averaging price\u2013earnings ratios of a portfolio, in machine learning in the F1 score (the harmonic mean of precision and recall), and in electrical engineering for resistors in parallel.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>All values must be positive.<\/strong> A single zero or negative number makes the harmonic mean undefined \u2014 the calculator will reject such input.<\/li>\n<li><strong>Use it for rates, not quantities.<\/strong> If you are averaging prices, weights, or scores directly, the arithmetic or <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/weighted-mean\/\">weighted mean<\/a> is usually the correct choice.<\/li>\n<li><strong>Equal weighting is assumed.<\/strong> Each value counts equally. If your rates apply to different distances or volumes, a weighted harmonic mean is needed.<\/li>\n<li><strong>One extreme small value dominates.<\/strong> The harmonic mean of \\(1, 100, 100\\) is about 2.94, not anywhere near 67. Make sure a single tiny value is genuinely part of your data.<\/li>\n<li>To compare all three classical means on the same data \u2014 or to work with the root-mean-square used in statistics \u2014 try the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-mean-calculator-root-mean-square\/\">quadratic mean (RMS) calculator<\/a>.<\/li>\n<\/ul>\n<h2>Harmonic Mean vs. Other Averages<\/h2>\n<table>\n<thead>\n<tr>\n<th>Average<\/th>\n<th>Best for<\/th>\n<th>Example use<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Harmonic mean<\/td>\n<td>Rates over equal units (speed, efficiency)<\/td>\n<td>Average speed over equal distances<\/td>\n<\/tr>\n<tr>\n<td>Geometric mean<\/td>\n<td>Multiplicative data, growth rates<\/td>\n<td>Average investment return over years<\/td>\n<\/tr>\n<tr>\n<td>Arithmetic mean<\/td>\n<td>Additive quantities<\/td>\n<td>Average of test scores<\/td>\n<\/tr>\n<tr>\n<td>Quadratic mean (RMS)<\/td>\n<td>Data varying around zero<\/td>\n<td>Effective voltage of AC signal<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>A related real-world application is <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/team-work-calculator\/\">computing how long two people take to complete a job together<\/a> \u2014 the underlying math is essentially a harmonic-mean calculation.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter two or more positive numbers and this calculator returns their harmonic mean \u2014 the reciprocal-of-the-average-of-reciprocals, ideal for averaging rates and speeds.<\/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-65082","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\/65082","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=65082"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65082\/revisions"}],"predecessor-version":[{"id":66249,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65082\/revisions\/66249"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65082"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65082"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65082"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}