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Difference between revisions of "Geometric mean"

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Given a set of ''n'' numbers, the '''Geometric Mean''' is the ''nth'' root of the product of the numbers. It is analogous to the [[Arithmetic Mean]], except with products.  
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The '''geometric mean''' of a collection of <math>n</math> [[positive]] [[real number]]s is the <math>n</math>th [[root]] of the product of the numbers. Note that if <math>n</math> is even, we take the positive <math>n</math>th root.  It is analogous to the [[arithmetic mean]] (with addition replaced by multiplication) in the following sense: the arithmetic mean of two numbers <math>b</math> and <math>c</math> is the number <math>a</math> such that <math>a + a = b + c</math>, while the geometric mean of the numbers <math>b</math> and <math>c</math> is the number <math>g</math> such that <math>g\cdot g = b\cdot c</math>.
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== Examples ==
 
== Examples ==
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Find the geometric mean of the numbers <math>x_1, x_2, x_3, x_4 ... x_n</math>
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The geometric mean of the numbers 6, 4, 1 and 2 is <math>\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}</math>.
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We want the nth root of the product of the n numbers. There are n numbers so our geometric mean would be <math>\sqrt[n]{x_1 x_2 x_3 x_4 ... x_n}</math>
 
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The geometric mean features prominently in the [[Arithmetic Mean-Geometric Mean Inequality]].
  
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Find the geometric mean of the numbers 6, 4, 1 and 2.
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The geometric mean arises in [[geometry]] in the following situation: if <math>AB</math> is a [[chord]] of [[circle]] <math>O</math> with [[midpoint]] <math>M</math> and <math>M</math> divides the [[diameter]] passing through it into pieces of length <math>a</math> and <math>b</math> then the length of [[line segment]] <math>AM</math> is the geometric mean of <math>a</math> and <math>b</math>.
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There are 4 numbers, so we want the 4th root. The numbers' product is 48, so our answer is <math>\sqrt[4]{48}=2\sqrt[4]{3}</math>
 
  
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The Geometric Mean is a component of the well-known [[Arithmetic Mean-Geometric Mean]] [[Inequality]].
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The geometric mean also arises in the following common [[word problem]]: if a driver travels half the distance of a trip at a speed of <math>a</math> miles per hour and the other half at a speed of <math>b</math> miles per hour, the average speed over the whole trip is the geometric mean of <math>a</math> and <math>b</math>.  (If the driver spent half the ''time'' of the trip at each speed, we would instead get the arithmetic mean.)
  
 
== Practice Problems ==
 
== Practice Problems ==

Revision as of 11:43, 2 April 2008

The geometric mean of a collection of $n$ positive real numbers is the $n$th root of the product of the numbers. Note that if $n$ is even, we take the positive $n$th root. It is analogous to the arithmetic mean (with addition replaced by multiplication) in the following sense: the arithmetic mean of two numbers $b$ and $c$ is the number $a$ such that $a + a = b + c$, while the geometric mean of the numbers $b$ and $c$ is the number $g$ such that $g\cdot g = b\cdot c$.

Examples

The geometric mean of the numbers 6, 4, 1 and 2 is $\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}$.

The geometric mean features prominently in the Arithmetic Mean-Geometric Mean Inequality.

The geometric mean arises in geometry in the following situation: if $AB$ is a chord of circle $O$ with midpoint $M$ and $M$ divides the diameter passing through it into pieces of length $a$ and $b$ then the length of line segment $AM$ is the geometric mean of $a$ and $b$.

The geometric mean also arises in the following common word problem: if a driver travels half the distance of a trip at a speed of $a$ miles per hour and the other half at a speed of $b$ miles per hour, the average speed over the whole trip is the geometric mean of $a$ and $b$. (If the driver spent half the time of the trip at each speed, we would instead get the arithmetic mean.)

Practice Problems

Introductory Problems

See Also

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