Editing Log-normal distribution (section)

=== Social sciences and demographics ===

* In [[economics]], there is evidence that the [[income]] of 97–99% of the population is distributed log-normally.<ref>Clementi, Fabio; [[Mauro Gallegati|Gallegati, Mauro]] (2005) [http://ideas.repec.org/p/wpa/wuwpmi/0505006.html "Pareto's law of income distribution: Evidence for Germany, the United Kingdom, and the United States"], EconWPA</ref> (The distribution of higher-income individuals follows a [[Pareto distribution]]).<ref>{{cite conference | arxiv = cond-mat/0202388 | title= Physics of Personal Income | last = Wataru | first = Souma | date= 2002-02-22 | publisher= Springer | book-title= Empirical Science of Financial Fluctuations: The Advent of Econophysics | doi = 10.1007/978-4-431-66993-7 | editor-last= Takayasu | editor-first= Hideki }}</ref>
* If an income distribution follows a log-normal distribution with standard deviation <math>\sigma</math>, then the [[Gini coefficient]], commonly use to evaluate income inequality, can be computed as <math>G = \operatorname{erf}\left(\frac{\sigma }{2 }\right)</math> where <math>\operatorname{erf}</math> is the [[error function]], since <math> G = 2 \Phi{\left(\frac{\sigma }{\sqrt{2}}\right)} - 1</math>, where <math>\Phi(x)</math> is the cumulative distribution function of a standard normal distribution.
* In [[finance]], in particular the [[Black–Scholes model]], changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal<ref>{{Cite journal | doi = 10.1086/260062 | title = The Pricing of Options and Corporate Liabilities | journal = Journal of Political Economy | volume = 81 | issue = 3 | pages = 637 | year = 1973 | last1 = Black | first1 = F. | last2 = Scholes | first2 = M. | s2cid = 154552078}}</ref> (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as [[Benoit Mandelbrot]] have argued <ref>{{cite book | last = Mandelbrot | first = Benoit | title = The (mis-)Behaviour of Markets | year = 2004 | url = https://books.google.com/books?id=9w15j-Ka0vgC | publisher = Basic Books | isbn = 9780465043552}}</ref> that [[Lévy skew alpha-stable distribution|log-Lévy distributions]], which possesses [[heavy tails]] would be a more appropriate model, in particular for the analysis for [[stock market crash]]es. Indeed, stock price distributions typically exhibit a [[fat tail]].<ref>Bunchen, P., ''Advanced Option Pricing'', University of Sydney coursebook, 2007</ref> The fat tailed distribution of changes during stock market crashes invalidate the assumptions of the [[central limit theorem]].
* In [[scientometrics]], the number of citations to journal articles and patents follows a discrete log-normal distribution.<ref>{{cite journal | last1 = Thelwall | first1 = Mike | last2 = Wilson | first2 = Paul | title = Regression for citation data: An evaluation of different methods | journal = Journal of Informetrics | year = 2014 | volume = 8 | issue = 4 | pages = 963–971 | doi = 10.1016/j.joi.2014.09.011 | arxiv = 1510.08877 | s2cid = 8338485}}</ref><ref>{{cite journal | last1 = Sheridan | first1 = Paul | last2 = Onodera | first2 = Taku | title = A Preferential Attachment Paradox: How Preferential Attachment Combines with Growth to Produce Networks with Log-normal In-degree Distributions | journal = Scientific Reports | year = 2020 | volume = 8 | issue = 1 | page = 2811 | doi = 10.1038/s41598-018-21133-2 | pmid = 29434232 | pmc = 5809396 | arxiv = 1703.06645}}</ref>
* [[Historical urban community sizes|City sizes]] (population) satisfy Gibrat's Law.<ref>{{Cite journal | last = Eeckhout | first = Jan | date = 2004 | title = Gibrat's Law for (All) Cities | url = https://www.jstor.org/stable/3592829 | journal = American Economic Review | volume = 94 | issue = 5 | pages = 1429–1451 | doi = 10.1257/0002828043052303 | jstor = 3592829| url-access = subscription }}</ref> The growth process of city sizes is proportionate and invariant with respect to size. From the [[central limit theorem]] therefore, the log of city size is normally distributed.
* The number of sexual partners appears to be best described by a log-normal distribution.<ref>{{Cite journal | last = Kault | first = David | title = The Shape of the Distribution of the Number of Sexual Partners | url = https://onlinelibrary.wiley.com/doi/10.1002/(SICI)1097-0258(19960130)15:2%3C221::AID-SIM148%3E3.0.CO;2-Q | journal = Statistics in Medicine | year = 1996 | volume = 15 | issue = 2 | pages = 221–230 | doi = 10.1002/(SICI)1097-0258(19960130)15:2<221::AID-SIM148>3.0.CO;2-Q | pmid = 8614756 | url-access = subscription }}</ref>