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Difference between revisions of "Harmonic mean"
 
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The harmonic mean (frequently referred to as HM) is a special kind of mean (like [[Arithmetic mean]], [[Geometric mean]]). The harmonic mean of n numbers <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 '''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 harmonic mean is a part of a frequently used inequality, the [[Arithmetic mean-Gemetric 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 \ldots x_n}\ge \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 ==
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* [[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 $n$ positive real numbers $x_1, x_2... x_n$ is defined to be: $\frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$.

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 $\{-2, 3, 6\}$ we would be trying to calculate $\frac 3{\frac 13 + \frac 16 - \frac 12} = \frac 30$, 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 $x_1, x_2,\ldots,x_n$: $\frac{x_1+x_2+\ldots+x_n}{n}\ge \sqrt[n]{x_1\cdot x_2 \cdots x_n}\ge \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$

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.

Practice Problems

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