https://wikimd.org/index.php?action=history&feed=atom&title=Geometric_meanGeometric mean - Revision history2026-05-12T21:17:25ZRevision history for this page on the wikiMediaWiki 1.44.2https://wikimd.org/index.php?title=Geometric_mean&diff=6314300&oldid=prevPrab: CSV import2025-02-18T02:11:24Z<p>CSV import</p>
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<p><b>New page</b></p><div>'''Geometric mean''' is a type of [[mean]] or average, which indicates the central tendency or typical value of a set of numbers. Unlike the [[arithmetic mean]], which uses addition and division, the geometric mean is calculated by multiplying the numbers together and then taking the square root (for two numbers), cube root (for three numbers), etc.<br />
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== Definition ==<br />
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The geometric mean of a data set {a1, a2, ..., an} is given by the nth root of the product of the numbers, i.e.,<br />
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: (a1 * a2 * ... * an)^(1/n)<br />
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This can also be expressed in logarithmic form:<br />
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: exp((1/n) * (ln(a1) + ln(a2) + ... + ln(an)))<br />
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== Applications ==<br />
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The geometric mean has many applications in various fields, including [[mathematics]], [[statistics]], [[finance]], [[biology]], [[engineering]], and [[computer science]]. It is particularly useful when dealing with data that varies exponentially or multiplicatively, such as growth rates, ratios, and geometric sequences.<br />
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In finance, the geometric mean is used to calculate the [[compound annual growth rate]] (CAGR), which represents the average annual growth rate over a specified period of time.<br />
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In biology, the geometric mean is often used in the analysis of [[proportional data]] and [[log-normal distribution]]s.<br />
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== Properties ==<br />
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The geometric mean has several important properties:<br />
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* It is always less than or equal to the arithmetic mean, with equality only when all the numbers are the same.<br />
* It is invariant under change of scale, i.e., multiplying all the numbers by a constant factor does not change the geometric mean.<br />
* It is the limit of the [[arithmetic-geometric mean]] as the two means converge.<br />
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== See also ==<br />
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* [[Arithmetic mean]]<br />
* [[Harmonic mean]]<br />
* [[Arithmetic-geometric mean]]<br />
* [[Log-normal distribution]]<br />
* [[Compound annual growth rate]]<br />
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[[Category:Mathematics]]<br />
[[Category:Statistics]]<br />
[[Category:Finance]]<br />
[[Category:Biology]]<br />
[[Category:Engineering]]<br />
[[Category:Computer Science]]<br />
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