Difference between revisions of "Harmonic mean"
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− | The harmonic mean (frequently | + | The '''harmonic mean''' (frequently abbreviated HM) is a special kind of mean (like [[arithmetic mean]] and [[geometric mean]]). The harmonic mean of a [[set]] of <math>n</math> [[positive]] [[real number]]s <math> x_1, x_2... x_n </math> is defined to be: <math> \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}</math>. |
− | The harmonic mean is a part of a frequently used inequality, the [[RMS-AM-GM-HM | Arithmetic mean-Geometric mean-Harmonic mean inequality]]. The Inequality states that for a set of positive numbers <math>x_1, x_2,\ldots,x_n</math>: <math>\frac{x_1+x_2+\ldots+x_n}{n}\ge \sqrt[n]{x_1\cdot x_2 \ | + | The restriction to positive numbers is necessary to avoid division by zero. For instance, if we tried to take the harmonic mean of the set <math>\{-2, 3, 6\}</math> we would be trying to calculate <math>\frac 3{\frac 13 + \frac 16 - \frac 12} = \frac 30</math>, which is obviously problematic. |
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+ | The harmonic mean is a part of a frequently used inequality, the [[RMS-AM-GM-HM | Arithmetic mean-Geometric mean-Harmonic mean inequality]]. The Inequality states that for a set of positive numbers <math>x_1, x_2,\ldots,x_n</math>: <math>\frac{x_1+x_2+\ldots+x_n}{n}\ge \sqrt[n]{x_1\cdot x_2 \cdots x_n}\ge \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}} </math> |
Revision as of 10:48, 26 July 2006
The harmonic mean (frequently abbreviated HM) is a special kind of mean (like arithmetic mean and geometric mean). The harmonic mean of a set of positive real numbers is defined to be: .
The restriction to positive numbers is necessary to avoid division by zero. For instance, if we tried to take the harmonic mean of the set we would be trying to calculate , which is obviously problematic.
The harmonic mean is a part of a frequently used inequality, the Arithmetic mean-Geometric mean-Harmonic mean inequality. The Inequality states that for a set of positive numbers :