Content deleted Content added
m lk Ratio scale |
Taking the geometric mean of negative numbers might not make sense in general, so this formula could probably be removed |
||
Line 60:
additionally, if negative values of the <math>a_i</math> are allowed,
: <math>\left( \prod_{i=1}^n a_i \right)^\frac{1}{n} = \left(\left(-1\right)^m\right)^\frac{1}{n} \exp\left[\frac{1}{n}\sum_{i=1}^n \ln \left|a_i\right| \right],</math>
where {{math|''m''}} is the number of negative numbers.{{Dubious|reason=This formula returns a complex number when m is odd, and it is unclear if the geometric mean of negative numbers even makes sense}}
This is sometimes called the '''log-average''' (not to be confused with the [[logarithmic average]]). It is simply computing the [[arithmetic mean]] of the logarithm-transformed values of <math>a_i</math> (i.e., the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale, i.e., it is the [[generalised f-mean]] with <math>f(x) = \log x</math>. For example, the geometric mean of 2 and 8 can be calculated as the following, where <math>b</math> is any base of a [[logarithm]] (commonly 2, [[e (mathematical constant)|<math>e</math>]] or 10):
| |||