{"id":148,"date":"2018-03-22T16:59:04","date_gmt":"2018-03-22T15:59:04","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=148"},"modified":"2018-04-29T17:58:54","modified_gmt":"2018-04-29T15:58:54","slug":"lart-de-moyenner","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/03\/22\/lart-de-moyenner\/","title":{"rendered":"L’art de moyenner"},"content":{"rendered":"

Quand on veut calculer la taille moyenne des Fran\u00e7ais, le principe est simple. On mesure la taille de chaque fran\u00e7ais de plus de 18 ans (les mesurer depuis la naissance fausserait la moyenne), on fait le total de ces tailles et on divise par le nombre total de Fran\u00e7ais adultes. On trouve un nombre comme 176 cm qui est donc la taille moyenne des Fran\u00e7ais adultes. On peut recommencer avec les Fran\u00e7aises, on trouve 163 cm.<\/p>\n

\"\"
Pour calculer la moyenne de la taille des girafes, on ne retient que la taille des adultes. @ Herv\u00e9 Lehning<\/figcaption><\/figure>\n

La moyenne arithm\u00e9tique<\/h2>\n

En math\u00e9matiques, on parle de moyenne arithm\u00e9tique. Par exemple, la moyenne arithm\u00e9tique des dix nombres du tableau ci-dessous est \u00e9gale \u00e0 leur somme 618 divis\u00e9e par 10 soit 61,8.<\/p>\n\n\n\n
\n

82<\/p>\n<\/td>\n

71<\/td>\n98<\/td>\n64<\/td>\n77<\/td>\n39<\/td>\n86<\/td>\n69<\/td>\n22<\/td>\n\n

10<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

La vitesse moyenne<\/h2>\n

Prenons l\u2019exemple du calcul d\u2019une vitesse moyenne sous la forme d\u2019une petite \u00e9nigme\u00a0:<\/p>\n

Deux villes A et B sont distantes de 100 km, un automobiliste effectue le trajet de A \u00e0 B en une heure et le retour en deux heures. Quelle est sa vitesse moyenne\u00a0?<\/p>\n

Comme le premier trajet s\u2019effectue \u00e0 100 km\/h de moyenne et le retour \u00e0 50 km\/h, on peut \u00eatre tent\u00e9 de faire la moyenne arithm\u00e9tique des deux nombres et r\u00e9pondre 75 km\/h. En fait, ce r\u00e9sultat est faux. Un raisonnement plus correct consiste \u00e0 dire que l\u2019automobiliste a parcouru 200 km en trois heures et donc que sa vitesse moyenne a \u00e9t\u00e9 de 200 \/ 3 = 67 km\/h (en arrondissant). La diff\u00e9rence est notable.<\/p>\n

La moyenne harmonique<\/h2>\n

Cette nouvelle moyenne, ad\u00e9quate pour calculer les vitesses, est appel\u00e9e la moyenne harmonique. Si on consid\u00e8re une suite finie de n<\/em> nombres a<\/em>, b<\/em>, etc. les moyennes arithm\u00e9tique et harmonique A et B sont donn\u00e9es par les formules\u00a0:<\/p>\n

A\u00a0=\u00a0(a<\/em>\u00a0+\u00a0b<\/em>\u00a0+\u00a0\u2026)\u00a0\/\u00a0n<\/em> \u00a0 et \u00a0 1\u00a0\/\u00a0H\u00a0=\u00a0(1\/a<\/em>\u00a0+\u00a01\/b<\/em>\u00a0+\u00a0\u2026)\u00a0\/\u00a0n<\/em><\/p>\n

Il existe toute sorte d\u2019autres moyennes correspondantes chacune \u00e0 la nature des quantit\u00e9s \u00e0 moyenner. On ne moyenne pas de m\u00eame des longueurs, des poids, des vitesses, des temp\u00e9ratures, etc.<\/p>\n","protected":false},"excerpt":{"rendered":"

Quand on veut calculer la taille moyenne des Fran\u00e7ais, le principe est simple. On mesure la taille de chaque fran\u00e7ais de plus de 18 ans (les mesurer depuis la naissance fausserait la moyenne), on fait le total de ces tailles et on divise par le nombre total de Fran\u00e7ais adultes. On trouve un nombre comme … Continuer la lecture de L’art de moyenner<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":153,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[29],"tags":[61,40,60,62],"class_list":["post-148","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths-2","tag-calcul","tag-mathematiques","tag-moyenne","tag-temperature"],"yoast_head":"\nL'art de moyenner, par Herv\u00e9 Lehning<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.futura-sciences.com\/lehning\/2018\/03\/22\/lart-de-moyenner\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"L'art de moyenner, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"Quand on veut calculer la taille moyenne des Fran\u00e7ais, le principe est simple. On mesure la taille de chaque fran\u00e7ais de plus de 18 ans (les mesurer depuis la naissance fausserait la moyenne), on fait le total de ces tailles et on divise par le nombre total de Fran\u00e7ais adultes. 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