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>. |
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| + | 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> | ||
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| + | In the line of power of means, the harmonic mean is the mean of the -1 power. | ||
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| + | Harmonic mean is commonly used to find the average of rates. | ||
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| + | == Practice Problems == | ||
| + | * [[2002 AMC 12A Problems/Problem 11]] | ||
Latest revision as of 23:49, 5 January 2021
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 
: ![$\frac{x_1+x_2+\ldots+x_n}{n}\ge \sqrt[n]{x_1\cdot x_2 \cdots x_n}\ge \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$](http://latex.artofproblemsolving.com/3/4/8/3480680951e4be70f6e0c5299b90b863f375dade.png)
In the line of power of means, the harmonic mean is the mean of the -1 power.
Harmonic mean is commonly used to find the average of rates.
