Editing Geometric mean (section)

==={{anchor|Log-average}}Formulation using logarithms ===<!--"Log-average" redirects here-->
The geometric mean can also be expressed as the exponential of the arithmetic mean of logarithms.<ref>{{cite book |title=Statistics: An Introduction using R |first=Michael J. |last=Crawley |publisher=John Wiley & Sons Ltd. |year=2005 |isbn=9780470022986 }}</ref> By using [[logarithmic identities]] to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication:

When <math>a_1, a_2, \dots, a_n > 0</math>
: <math>\biggl( \prod_{i=1}^n a_i \biggr)^\frac{1}{n} = \exp\biggl(\frac{1}{n} \sum_{i=1}^n \ln a_i\biggr),</math> 
since <math>\textstyle \vphantom\Big| \ln \sqrt[n]{a_1a_2\cdots a_n \vphantom{t}}
= \frac1n\ln(a_1a_2\cdots a_n)
= \frac1n(\ln a_1 + \ln a_2 + \cdots + \ln a_n).
</math> 

This is sometimes called the '''log-average''' (not to be confused with the [[logarithmic average]]).  It is simply the [[arithmetic mean]] of the logarithm-transformed values of <math>a_i</math> (i.e., the arithmetic mean on the log scale), using the exponentiation to return to the original scale, i.e., it is the [[generalised f-mean|generalized f-mean]] with <math>f(x) = \log x</math>. A logarithm of any base can be used in place of the natural logarithm. For example, the geometric mean of {{tmath|1}}, {{tmath|2}}, {{tmath|8}}, and {{tmath|16}} can be calculated using logarithms base 2:
:<math>\sqrt[4]{1 \cdot 2 \cdot 8 \cdot 16}
= 2^{(\log_2\! 1 \,+\, \log_2\!2 \,+\, \log_2\!8 \,+\, \log_2\!16)/4}
= 2^{(0 \,+\, 1 \,+\, 3 \,+\, 4)/4} = 2^2 = 4.</math>

Related to the above, it can be seen that for a given sample of points <math>a_1, \ldots, a_n</math>, the geometric mean is the minimizer of 
:<math>f(a) = \sum_{i=1}^n (\log a_i - \log a )^2 = \sum_{i=1}^n \left(\log \frac{a_i}{a} \right)^2</math>, 
whereas the arithmetic mean is the minimizer of 
:<math>f(a) = \sum_{i=1}^n (a_i - a)^2</math>. 
Thus, the geometric mean provides a summary of the samples whose exponent best matches the exponents of the samples (in the least squares sense).

In computer implementations, naïvely multiplying many numbers together can cause [[arithmetic overflow]] or [[arithmetic underflow|underflow]]. Calculating the geometric mean using logarithms is one way to avoid this problem.