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Difference between revisions of "Geometric mean"

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''of positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044260/g0442601.png" />''
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''of positive numbers $a_1,\ldots,a_n''
  
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The number equal to the real positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044260/g0442602.png" />-th root of their product, i.e. to
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The number equal to the real positive $n$-th root of their product, i.e. to
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044260/g0442603.png" /></td> </tr></table>
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$$(a_1\ldots a_n)^{1/n}.$$
  
 
The geometric mean is always smaller than the [[Arithmetic mean|arithmetic mean]], except when all the numbers are equal (then these two means are equal). The geometric mean of two numbers is also known as the proportional mean.
 
The geometric mean is always smaller than the [[Arithmetic mean|arithmetic mean]], except when all the numbers are equal (then these two means are equal). The geometric mean of two numbers is also known as the proportional mean.

Revision as of 13:06, 9 April 2014

of positive numbers $a_1,\ldots,a_n'' The number equal to the real positive $n$-th root of their product, i.e. to

$$(a_1\ldots a_n)^{1/n}.$$

The geometric mean is always smaller than the arithmetic mean, except when all the numbers are equal (then these two means are equal). The geometric mean of two numbers is also known as the proportional mean.

How to Cite This Entry:
Geometric mean. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometric_mean&oldid=31434
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