{"id":406,"date":"2026-10-03T19:07:44","date_gmt":"2026-10-03T18:07:44","guid":{"rendered":"https:\/\/wpcalc.com\/pt\/media-harmonica\/"},"modified":"2026-10-03T19:07:44","modified_gmt":"2026-10-03T18:07:44","slug":"media-harmonica","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pt\/matematica\/media-harmonica\/","title":{"rendered":"Calculadora de M\u00e9dia Harm\u00f3nica"},"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\">Calcular a M\u00e9dia Harm\u00f3nica de N\u00fameros<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"406\" data-favorite-title=\"M\u00e9dia Harm\u00f3nica\" data-favorite-url=\"https:\/\/wpcalc.com\/pt\/matematica\/media-harmonica\/\" data-favorite-excerpt=\"Introduza dois ou mais n\u00fameros positivos e esta calculadora devolve a respetiva m\u00e9dia harm\u00f3nica \u2014 o rec\u00edproco da m\u00e9dia dos rec\u00edprocos, ideal para mediar taxas e velocidades.\" aria-label=\"Adicionar aos favoritos\" 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\/pt\/matematica\/media-harmonica\/\" method=\"POST\" class=\"calculator\" id=\"calculator-406\" data-post=\"406\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Campos de entrada<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Quantidade de valores<\/label><input type=\"text\" name=\"quantidade_de_valores\" 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\">Informe quantos valores voc\u00ea deseja usar para calcular a m\u00e9dia harm\u00f4nica<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">N\u00famero 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=\"nmero_1\" id=\"field-2\" step=\"any\" value=\"10\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">N\u00famero 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=\"nmero_2\" id=\"field-3\" step=\"any\" value=\"20\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalcular\" id=\"calculate-button\" data-post=\"406\">Calcular<\/button><button type=\"reset\">Redefinir<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calcular automaticamente<\/label><small>Se ativado, o resultado \u00e9 atualizado automaticamente quando voc\u00ea altera qualquer valor.<\/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> Seus resultados<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>M\u00e9dia<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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\">Copiar link para este c\u00e1lculo<\/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>Deixar um coment\u00e1rio<\/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=\"406\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">O que voc\u00ea achou deste resultado?<\/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\">Curti <\/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\">\u00datil <\/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\">Confuso <\/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\">Decepcionado <\/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\">Impreciso <\/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>F\u00f3rmula da M\u00e9dia Harm\u00f3nica<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">F\u00f3rmula<\/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>Onde:<\/strong><\/p>\n<ul>\n<li>\\(H\\) \u2014 a m\u00e9dia harm\u00f3nica<\/li>\n<li>\\(n\\) \u2014 o n\u00famero de valores<\/li>\n<li>\\(x_1, x_2, \\ldots, x_n\\) \u2014 os valores individuais; cada um deve ser positivo e diferente de zero<\/li>\n<\/ul>\n<p>A f\u00f3rmula divide a quantidade de valores pela soma dos seus rec\u00edprocos. De forma equivalente, \u00e9 o rec\u00edproco da m\u00e9dia aritm\u00e9tica dos rec\u00edprocos, e \u00e9 por isso que d\u00e1 maior peso aos valores mais pequenos.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.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>M\u00e9dia Harm\u00f3nica \u2013 Exemplo de C\u00e1lculo<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemplo:<\/strong> percorre a mesma dist\u00e2ncia duas vezes \u2014 primeiro a 10 km\/h, depois a 20 km\/h \u2014 e quer saber a velocidade m\u00e9dia de toda a viagem.<\/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>Resultado:<\/strong> a m\u00e9dia harm\u00f3nica de 10 e 20 \u00e9 aproximadamente <strong>13,33 km\/h<\/strong> \u2014 e n\u00e3o 15, como a m\u00e9dia aritm\u00e9tica sugeriria incorretamente.<\/p>\n<p>\n\r\n<\/section>\n<h2>Como Utilizar a Calculadora<\/h2>\n<ol>\n<li>Introduza quantos valores quer mediar no campo <strong>N\u00famero de valores<\/strong> (m\u00ednimo 2). Os campos de introdu\u00e7\u00e3o adicionais aparecem automaticamente.<\/li>\n<li>Introduza os seus n\u00fameros positivos em <strong>N\u00famero 1<\/strong>, <strong>N\u00famero 2<\/strong>, e assim por diante.<\/li>\n<li>O resultado aparece de imediato no campo <strong>M\u00e9dia<\/strong> \u2014 sem precisar de clicar em nenhum bot\u00e3o.<\/li>\n<li>Se algum valor for zero ou negativo, a calculadora mostra um erro, porque a m\u00e9dia harm\u00f3nica s\u00f3 est\u00e1 definida para n\u00fameros positivos.<\/li>\n<\/ol>\n<h2>O Que \u00c9 a M\u00e9dia Harm\u00f3nica?<\/h2>\n<p>A m\u00e9dia harm\u00f3nica \u00e9 uma das tr\u00eas m\u00e9dias pitag\u00f3ricas cl\u00e1ssicas, a par da <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-media\/\">m\u00e9dia aritm\u00e9tica<\/a> e da <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/media-geometrica\/\">m\u00e9dia geom\u00e9trica<\/a>. Para qualquer conjunto de n\u00fameros positivos, verifica-se sempre a ordena\u00e7\u00e3o:<\/p>\n<p>\\[ H \\leq G \\leq A \\]<\/p>\n<p>com igualdade apenas quando todos os valores s\u00e3o id\u00eanticos. Na pr\u00e1tica, a m\u00e9dia harm\u00f3nica \u00e9 a m\u00e9dia certa sempre que as grandezas combinadas s\u00e3o expressas como taxas \u2014 ou seja, como algo <em>por unidade<\/em> de outra coisa, como quil\u00f3metros por hora, litros por 100 km ou unidades produzidas por hora.<\/p>\n<p>A ilustra\u00e7\u00e3o cl\u00e1ssica \u00e9 a velocidade m\u00e9dia. Se conduzir 100 km a 60 km\/h e os 100 km seguintes a 40 km\/h, a sua verdadeira velocidade m\u00e9dia n\u00e3o \u00e9 50 km\/h, mas sim a m\u00e9dia harm\u00f3nica \\(2\/(1\/60 + 1\/40) = 48\\) km\/h, porque passa mais tempo \u00e0 velocidade mais lenta. Sempre que cada valor se aplica \u00e0 <em>mesma dist\u00e2ncia, volume ou carga de trabalho<\/em>, \u00e9 a m\u00e9dia harm\u00f3nica \u2014 e n\u00e3o a aritm\u00e9tica \u2014 que d\u00e1 a taxa global correta.<\/p>\n<p>Como os valores pequenos contribuem com rec\u00edprocos desproporcionalmente grandes, a m\u00e9dia harm\u00f3nica \u00e9 puxada para os valores mais pequenos e \u00e9 robusta face a valores at\u00edpicos muito grandes. Esta propriedade \u00e9 usada em finan\u00e7as para medir r\u00e1cios pre\u00e7o\/lucro de uma carteira, em aprendizagem autom\u00e1tica na pontua\u00e7\u00e3o F1 (a m\u00e9dia harm\u00f3nica da precis\u00e3o e da revoca\u00e7\u00e3o) e em engenharia eletrot\u00e9cnica para resist\u00eancias em paralelo.<\/p>\n<h2>Pressupostos, Limita\u00e7\u00f5es e Dicas<\/h2>\n<ul>\n<li><strong>Todos os valores devem ser positivos.<\/strong> Um \u00fanico zero ou n\u00famero negativo torna a m\u00e9dia harm\u00f3nica indefinida \u2014 a calculadora rejeitar\u00e1 essa introdu\u00e7\u00e3o.<\/li>\n<li><strong>Use-a para taxas, n\u00e3o para quantidades.<\/strong> Se est\u00e1 a fazer a m\u00e9dia de pre\u00e7os, pesos ou pontua\u00e7\u00f5es diretamente, a m\u00e9dia aritm\u00e9tica ou <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/media-ponderada\/\">m\u00e9dia ponderada<\/a> \u00e9 normalmente a escolha correta.<\/li>\n<li><strong>Assume-se pondera\u00e7\u00e3o igual.<\/strong> Cada valor vale o mesmo. Se as suas taxas se aplicam a dist\u00e2ncias ou volumes diferentes, \u00e9 necess\u00e1ria uma m\u00e9dia harm\u00f3nica ponderada.<\/li>\n<li><strong>Um \u00fanico valor extremamente pequeno domina.<\/strong> A m\u00e9dia harm\u00f3nica de \\(1, 100, 100\\) \u00e9 cerca de 2,94, e n\u00e3o algo pr\u00f3ximo de 67. Certifique-se de que um valor min\u00fasculo isolado faz realmente parte dos seus dados.<\/li>\n<li>Para comparar as tr\u00eas m\u00e9dias cl\u00e1ssicas nos mesmos dados \u2014 ou para trabalhar com a m\u00e9dia quadr\u00e1tica usada em estat\u00edstica \u2014 experimente a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-media-quadratica\/\">calculadora de m\u00e9dia quadr\u00e1tica (RMS)<\/a>.<\/li>\n<\/ul>\n<h2>M\u00e9dia Harm\u00f3nica vs. Outras M\u00e9dias<\/h2>\n<table>\n<thead>\n<tr>\n<th>M\u00e9dia<\/th>\n<th>Ideal para<\/th>\n<th>Exemplo de utiliza\u00e7\u00e3o<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>M\u00e9dia harm\u00f3nica<\/td>\n<td>Taxas sobre unidades iguais (velocidade, efici\u00eancia)<\/td>\n<td>Velocidade m\u00e9dia em dist\u00e2ncias iguais<\/td>\n<\/tr>\n<tr>\n<td>M\u00e9dia geom\u00e9trica<\/td>\n<td>Dados multiplicativos, taxas de crescimento<\/td>\n<td>Rendimento m\u00e9dio de um investimento ao longo de anos<\/td>\n<\/tr>\n<tr>\n<td>M\u00e9dia aritm\u00e9tica<\/td>\n<td>Grandezas aditivas<\/td>\n<td>M\u00e9dia de notas de testes<\/td>\n<\/tr>\n<tr>\n<td>M\u00e9dia quadr\u00e1tica (RMS)<\/td>\n<td>Dados que variam em torno de zero<\/td>\n<td>Tens\u00e3o eficaz de um sinal alternado<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Uma aplica\u00e7\u00e3o real relacionada \u00e9 <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-tempo-trabalho-equipe\/\">calcular quanto tempo duas pessoas demoram a completar um trabalho em conjunto<\/a> \u2014 a matem\u00e1tica subjacente \u00e9 essencialmente um c\u00e1lculo de m\u00e9dia harm\u00f3nica.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introduza dois ou mais n\u00fameros positivos e esta calculadora devolve a respetiva m\u00e9dia harm\u00f3nica \u2014 o rec\u00edproco da m\u00e9dia dos rec\u00edprocos, ideal para mediar taxas e velocidades.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[19],"class_list":["post-406","post","type-post","status-publish","format-standard","hentry","category-matematica","tag-calculadoras-aritmetica"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/406","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/comments?post=406"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/406\/revisions"}],"predecessor-version":[{"id":2860,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/406\/revisions\/2860"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/media?parent=406"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/categories?post=406"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/tags?post=406"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}