Editing Log-normal distribution (section)

===Geometric or multiplicative moments===

The [[geometric mean|geometric or multiplicative mean]] of the log-normal distribution is <math>\operatorname{GM}[X] = e^\mu = \mu^*</math>. It equals the median. The [[geometric standard deviation|geometric or multiplicative standard deviation]] is <math>\operatorname{GSD}[X] = e^{\sigma} = \sigma^*</math>.<ref name="ReferenceA">{{cite journal | last1 = Kirkwood | first1 = Thomas BL | title = Geometric means and measures of dispersion | journal = Biometrics | date = Dec 1979 | volume = 35 | issue = 4 | pages = 908–909 | jstor = 2530139 }}</ref><ref>{{cite journal | last1 = Limpert | first1 = E | last2 = Stahel | first2 = W | last3 = Abbt | first3 = M | title = Lognormal distributions across the sciences: keys and clues | journal = BioScience | year = 2001 | volume = 51 | issue = 5 | pages = 341–352 | doi = 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2 | doi-access = free }}</ref>

By analogy with the arithmetic statistics, one can define a geometric variance, <math>\operatorname{GVar}[X] = e^{\sigma^2}</math>, and a [[Coefficient of variation#Log-normal data|geometric coefficient of variation]],<ref name="ReferenceA" /> <math>\operatorname{GCV}[X] = e^{\sigma} - 1</math>, has been proposed. This term was intended to be ''analogous'' to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of <math>\operatorname{CV}</math> itself (see also [[Coefficient of variation]]).

Note that the geometric mean is smaller than the arithmetic mean. This is due to the [[AM–GM inequality]] and is a consequence of the logarithm being a [[concave function]]. In fact,<ref name="Acoustic Stimuli Revisited 2016">{{cite journal | last1 = Heil P | first1 = Friedrich B | title = Onset-Duration Matching of Acoustic Stimuli Revisited: Conventional Arithmetic vs. Proposed Geometric Measures of Accuracy and Precision | journal = Frontiers in Psychology | volume = 7 | page = 2013 | doi = 10.3389/fpsyg.2016.02013 | pmid = 28111557 | pmc = 5216879 | year = 2017 | doi-access = free}}</ref>

<math display="block">\operatorname{E}[X] = e^{\mu + \frac12 \sigma^2} = e^{\mu} \cdot \sqrt{e^{\sigma^2}} = \operatorname{GM}[X] \cdot \sqrt{\operatorname{GVar}[X]}.</math>

In finance, the term <math>e^{-\sigma^2/2}</math> is sometimes interpreted as a [[convexity correction]]. From the point of view of [[stochastic calculus]], this is the same correction term as in [[Itō's lemma#Geometric Brownian motion|Itō's lemma for geometric Brownian motion]].