Editing Log-normal distribution (section)

==Occurrence and applications==
The log-normal distribution is important in the description of natural phenomena. Many natural growth processes are driven by the accumulation of many small percentage changes which become additive on a log scale. Under appropriate regularity conditions, the distribution of the resulting accumulated changes will be increasingly well approximated by a log-normal, as noted in the section above on "[[#Multiplicative Central Limit Theorem|Multiplicative Central Limit Theorem]]". This is also known as [[Gibrat's law]], after Robert Gibrat (1904–1980) who formulated it for companies.<ref>{{cite journal | jstor = 2729692 | last = Sutton | first = John | date = Mar 1997 | title = Gibrat's Legacy | journal = Journal of Economic Literature | volume = 32 | issue = 1 | pages = 40–59}}</ref> If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if this assumption is not true, the size distributions at any age of things that grow over time tends to be log-normal.{{citation needed | date = January 2023}} Consequently, [[reference ranges]] for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.{{citation needed | date = January 2023}}

A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases.

Specific examples are given in the following subsections.<ref name=":0">{{Cite journal | last1 = Limpert | first1 = Eckhard | last2 = Stahel | first2 = Werner A. | last3 = Abbt | first3 = Markus | date = 2001 | title = Log-normal Distributions across the Sciences: Keys and Clues | journal = BioScience | language = en | volume = 51 | issue = 5 | pages = 341 | doi = 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2 | issn = 0006-3568 | doi-access = free }}</ref> contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas.<ref name=":4" /> is a review article on log-normal distributions in neuroscience, with annotated bibliography.

=== Human behavior ===

* The length of comments posted in Internet discussion forums follows a log-normal distribution.<ref name=":3">{{cite journal | last1 = Pawel | first1 = Sobkowicz | title = Lognormal distributions of user post lengths in Internet discussions – a consequence of the Weber-Fechner law? | journal = EPJ Data Science | year = 2013 | display-authors = etal}}</ref>
* Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.<ref>{{cite conference | last1 = Yin | first1 = Peifeng | last2 = Luo | first2 = Ping | last3 = Lee | first3 = Wang-Chien | last4 = Wang | first4 = Min | title = Silence is also evidence: interpreting dwell time for recommendation from psychological perspective | conference = ACM International Conference on KDD | year = 2013 | url = http://mldm.ict.ac.cn/platform/pweb/academicDetail.htm?id=16}}</ref>
* The length of [[chess]] games tends to follow a log-normal distribution.<ref>{{cite web | url = http://chess.stackexchange.com/questions/2506/what-is-the-average-length-of-a-game-of-chess/4899#4899 | title = What is the average length of a game of chess? | website = chess.stackexchange.com | access-date = 14 April 2018}}</ref>
* Onset durations of acoustic comparison stimuli that are matched to a standard stimulus follow a log-normal distribution.<ref name="Acoustic Stimuli Revisited 2016"/>

=== Biology and medicine ===

* Measures of size of living tissue (length, skin area, weight).<ref>{{cite book | last = Huxley | first = Julian S. | year = 1932 | title = Problems of relative growth | publisher = London | oclc = 476909537 | isbn = 978-0-486-61114-3 }}</ref>
* Incubation period of diseases.<ref>Sartwell, Philip E. "The distribution of incubation periods of infectious disease." ''American journal of hygiene'' 51 (1950): 310–318.</ref>
* Diameters of banana leaf spots, powdery mildew on barley.<ref name=":0" />
* For highly communicable epidemics, such as SARS in 2003, if public intervention control policies are involved, the number of hospitalized cases is shown to satisfy the log-normal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of [[entropy production]].<ref>{{cite journal | last1 = S. K. Chan | first1 = Jennifer | last2 = Yu | first2 = Philip L. H. | title = Modelling SARS data using threshold geometric process | journal = Statistics in Medicine | date = 2006 | volume = 25 | issue = 11 | pages = 1826–1839 | doi = 10.1002/sim.2376 | pmid = 16345017 | s2cid = 46599163 }}</ref>
* The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth.{{Citation needed | date = February 2011}}
* The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.
* The [[Pacific Biosciences|PacBio]] sequencing read length follows a log-normal distribution.<ref>{{Cite journal | last1 = Ono | first1 = Yukiteru | last2 = Asai | first2 = Kiyoshi | last3 = Hamada | first3 = Michiaki | date = 2013-01-01 | title = PBSIM: PacBio reads simulator—toward accurate genome assembly | url = https://academic.oup.com/bioinformatics/article/29/1/119/273243 | journal = Bioinformatics | language = en | volume = 29 | issue = 1 | pages = 119–121 | doi = 10.1093/bioinformatics/bts649 | pmid = 23129296 | issn = 1367-4803 | doi-access = free}}</ref>
* Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).<ref>{{cite journal | last = Makuch | first = Robert W. | author2 = D.H. Freeman | author3 = M.F. Johnson | title = Justification for the lognormal distribution as a model for blood pressure | journal = Journal of Chronic Diseases | year = 1979 | volume = 32 | issue = 3 | pages = 245–250 | doi = 10.1016/0021-9681(79)90070-5 | pmid = 429469 }}</ref>
*Several [[Pharmacokinetics|pharmacokinetic]] variables, such as [[Cmax (pharmacology)|C<sub>max</sub>]], [[Biological half-life|elimination half-life]] and the [[elimination rate constant]].<ref>{{Cite journal | last1 = Lacey | first1 = L. F. | last2 = Keene | first2 = O. N. | last3 = Pritchard | first3 = J. F. | last4 = Bye | first4 = A. | date = 1997-01-01 | title = Common noncompartmental pharmacokinetic variables: are they normally or log-normally distributed? | url = https://www.tandfonline.com/doi/full/10.1080/10543409708835177 | journal = Journal of Biopharmaceutical Statistics | language = en | volume = 7 | issue = 1 | pages = 171–178 | doi = 10.1080/10543409708835177 | pmid = 9056596 | issn = 1054-3406| url-access = subscription }}</ref>
* In neuroscience, the distribution of firing rates across a population of neurons is often approximately log-normal. This has been first observed in the cortex and striatum <ref>{{Cite conference | last1 = Scheler | first1 = Gabriele | last2 = Schumann | first2 = Johann | title = Diversity and stability in neuronal output rates | conference = 36th Society for Neuroscience Meeting, Atlanta | date = 2006-10-08}}</ref> and later in hippocampus and entorhinal cortex,<ref>{{Cite journal | last1 = Mizuseki | first1 = Kenji | last2 = Buzsáki | first2 = György | date = 2013-09-12 | title = Preconfigured, skewed distribution of firing rates in the hippocampus and entorhinal cortex | journal = Cell Reports | volume = 4 | issue = 5 | pages = 1010–1021 | doi = 10.1016/j.celrep.2013.07.039 | issn = 2211-1247 | pmc = 3804159 | pmid = 23994479}}</ref> and elsewhere in the brain.<ref name=":4">{{Cite journal | last1 = Buzsáki | first1 = György | last2 = Mizuseki | first2 = Kenji | date = 2017-01-06 | title = The log-dynamic brain: how skewed distributions affect network operations | journal = Nature Reviews. Neuroscience | volume = 15 | issue = 4 | pages = 264–278 | doi = 10.1038/nrn3687 | issn = 1471-003X | pmc = 4051294 | pmid = 24569488}}</ref><ref>{{Cite journal | last1 = Wohrer | first1 = Adrien | last2 = Humphries | first2 = Mark D. | last3 = Machens | first3 = Christian K. | date = 2013-04-01 | title = Population-wide distributions of neural activity during perceptual decision-making | journal = Progress in Neurobiology | volume = 103 | pages = 156–193 | doi = 10.1016/j.pneurobio.2012.09.004 | issn = 1873-5118 | pmid = 23123501 | pmc = 5985929}}</ref> Also, intrinsic gain distributions and synaptic weight distributions appear to be log-normal<ref>{{Cite journal | last = Scheler | first = Gabriele | title = Logarithmic distributions prove that intrinsic learning is Hebbian | journal = F1000Research | doi = 10.12688/f1000research.12130.2 | date = 2017-07-28 | pmid = 29071065 | volume = 6 | pmc = 5639933 | page = 1222 | doi-access = free }}</ref> as well.
*Neuron densities in the cerebral cortex, due to the noisy cell division process during neurodevelopment.<ref>{{cite journal | last1 = Morales-Gregorio | first1 = Aitor | last2 = van Meegen | first2 = Alexander | last3 = van Albada | first3 = Sacha | year = 2023 | title = Ubiquitous lognormal distribution of neuron densities in mammalian cerebral cortex | journal = Cerebral Cortex | volume = 33 | issue = 16 | pages = 9439–9449 | doi = 10.1093/cercor/bhad160 | pmid = 37409647 | pmc = 10438924 }}</ref>
*In operating-rooms management, the distribution of [[Predictive methods for surgery duration|surgery duration]].
*In the size of avalanches of fractures in the cytoskeleton of living cells, showing log-normal distributions, with significantly higher size in cancer cells than healthy ones.<ref>{{Cite journal | last1 = Polizzi | first1 = Stefano | last2 = Laperrousaz | first2 = Bastien | last3 = Perez-Reche | first3 = Francisco J | last4 = Nicolini | first4 = Franck E | last5 = Satta | first5 = Véronique Maguer | last6 = Arneodo | first6 = Alain | last7 = Argoul | first7 = Françoise | date = 2018-05-29 | title = A minimal rupture cascade model for living cell plasticity | url = https://iopscience.iop.org/article/10.1088/1367-2630/aac3c7 | journal = New Journal of Physics | volume = 20 | issue = 5 | pages = 053057 | doi = 10.1088/1367-2630/aac3c7 | bibcode = 2018NJPh...20e3057P | issn = 1367-2630 | hdl = 2164/10561 |hdl-access = free }}</ref>

=== Chemistry ===

* [[Particle size distribution]]s and [[molar mass distribution]]s.
* The concentration of rare elements in minerals.<ref>{{Cite journal | last = Ahrens | first = L. H. | date = 1954-02-01 | title = The lognormal distribution of the elements (A fundamental law of geochemistry and its subsidiary) | url = https://www.sciencedirect.com/science/article/abs/pii/001670375490040X | journal = Geochimica et Cosmochimica Acta | language = en | volume = 5 | issue = 2 | pages = 49–73 | doi = 10.1016/0016-7037(54)90040-X | bibcode = 1954GeCoA...5...49A | issn = 0016-7037| url-access = subscription }}</ref>
* Diameters of crystals in ice cream, oil drops in mayonnaise, pores in cocoa press cake.<ref name=":0" />

[[File:FitLogNormDistr.tif|thumb|Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see [[distribution fitting]] ]]

=== Physical sciences ===

*In [[hydrology]], the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.<ref>{{cite book | last = Oosterbaan | first = R.J. | editor-last = Ritzema | editor-first = H.P. | chapter = 6: Frequency and Regression Analysis | year = 1994 | title = Drainage Principles and Applications, Publication 16 | publisher = International Institute for Land Reclamation and Improvement (ILRI) | location = Wageningen, The Netherlands | pages = [https://archive.org/details/drainageprincipl0000unse/page/175 175–224] | chapter-url = http://www.waterlog.info/pdf/freqtxt.pdf | isbn = 978-90-70754-33-4 | url = https://archive.org/details/drainageprincipl0000unse/page/175 }}</ref>
**The image on the right illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]].<ref>[https://www.waterlog.info/cumfreq.htm CumFreq, free software for distribution fitting]</ref>
**The rainfall data are represented by [[plotting position]]s as part of a [[cumulative frequency analysis]].
*In [[physical oceanography]], the sizes of icebergs in the midwinter Southern Atlantic Ocean were found to follow a log-normal size distribution. The iceberg sizes, measured visually and by radar from the F.S. ''Polarstern'' in 1986, were thought to be controlled by wave action in heavy seas causing them to flex and break.<ref>{{cite journal | last1 = Wadhams | first1 = Peter | date = 1988 | title = Winter observations of iceberg frequencies and sizes in the South Atlantic Ocean | journal = Journal of Geophysical Research: Oceans | volume = 93 | issue = C4 | pages = 3583–3590 | doi = 10.1029/JC093iC04p03583 }}</ref>
*In [[atmospheric science]], log-normal distributions (or distributions made by combining multiple log-normal functions) have been used to characterize both measurements and models of the sizes and concentrations of many different types of particles, from volcanic ash, to clouds and rain, to airborne microbes.<ref>{{cite journal | last1 = Heintzenberg | first1 = Jost | date =1994 | title = Properties of the Log-Normal Particle Size Distribution | journal = Aerosol Science and Technology | volume = 21 | issue = 1 | pages = 46–48 | doi = 10.1080/02786829408959695 | doi-access= free }}</ref><ref>{{cite journal | last1 = Limpert | first1 = Eckhard | last2 = Stahel | first2 =Werner A. | last3 = Abbt | first3 = Markus | date =2001 | title = Log-normal Distributions across the Sciences: Keys and Clues | journal = BioScience | volume = 51 | issue = 5 | pages = 341–352 | doi = 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2 }}</ref><ref>{{cite journal | last1 = Di Giorgio | first1 = C | last2 =Krempff | first2 =A | last3 =Guiraud | first3 =H | last4 =Binder | first4 =P | last5 =Tiret | first5 = C | last6 =Dumenil | first6 = G | date =1996 | title = Atmospheric pollution by airborne microorganisms in the city of Marseilles | journal = Atmospheric Environment | volume = 30 | pages = 155–160 | doi = 10.1016/1352-2310(95)00143-M }}</ref><ref>{{cite journal | last1 =Sheridan | first1 =M.F. | last2 =Wohletz | first2 =K.H. | last3 =Dehn | first3 =J. | date =1987 | title = Discrimination of grain-size subpopulations in pyroclastic deposits | journal = Geology | volume = 15 | issue =4 | pages = 367–370 | doi = 10.1130/0091-7613(1987)15<367:DOGSIP>2.0.CO;2 }}</ref> The log-normal distribution is strictly empirical, so more physically-based distributions have been adopted to better understand processes controlling size distributions of particles such as volcanic ash.<ref>{{cite journal | last1 = Wohletz | first1 = K.H. | last2 = Sheridan | first2 = M.F. | last3 = Brown | first3 = W.K. | date = 1989 | title = Particle size distributions and the sequential fragmentation/transport theory applied to volcanic ash | journal = Journal of Geophysical Research: Solid Earth | volume = 94 | issue = B11 | pages = 15703–15721 | doi = 10.1029/JB094iB11p15703 }}</ref>

=== 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>

=== Technology ===
* In [[Reliability (statistics)|reliability]] analysis, the log-normal distribution is often used to model times to repair a maintainable system.<ref>{{cite book | last1 = O'Connor | first1 = Patrick | last2 = Kleyner | first2 = Andre | year = 2011 | title = Practical Reliability Engineering | publisher = John Wiley & Sons | isbn = 978-0-470-97982-2 | page = 35 }}</ref>
* In [[wireless communication]], "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution."<ref>{{cite web | title = Shadowing | website = www.WirelessCommunication.NL | url = http://wireless.per.nl/reference/chaptr03/shadow/shadow.htm |url-status = dead | archive-url = https://web.archive.org/web/20120113201345/http://wireless.per.nl/reference/chaptr03/shadow/shadow.htm | archive-date = January 13, 2012 }}</ref> Also, the random obstruction of radio signals due to large buildings and hills, called [[Fading|shadowing]], is often modeled as a log-normal distribution.
* Particle size distributions produced by comminution with random impacts, such as in [[ball mill]]ing.<ref>{{Cite journal | last1 = Dexter | first1 = A. R. | last2 = Tanner | first2 = D. W. | date = July 1972 | title = Packing Densities of Mixtures of Spheres with Log-normal Size Distributions | url = https://www.nature.com/articles/physci238031a0 | journal = Nature Physical Science | language = en | volume = 238 | issue = 80 | pages = 31–32 | doi = 10.1038/physci238031a0 | bibcode = 1972NPhS..238...31D | issn = 2058-1106| url-access = subscription }}</ref>
* The [[file size]] distribution of publicly available audio and video data files ([[MIME types]]) follows a log-normal distribution over five [[orders of magnitude]].<ref>
{{cite journal | last1 = Gros | first1 = C | last2 = Kaczor | first2 = G. | last3 = Markovic | first3 = D | title = Neuropsychological constraints to human data production on a global scale | journal = The European Physical Journal B | year = 2012 | volume = 85 | issue = 28 | pages = 28 | doi = 10.1140/epjb/e2011-20581-3 | arxiv = 1111.6849 | bibcode = 2012EPJB...85...28G | s2cid = 17404692 }}</ref>
* File sizes of 140 million files on personal computers running the Windows OS, collected in 1999.<ref>{{Cite journal | last1 = Douceur | first1 = John R. | last2 = Bolosky | first2 = William J. | date = 1999-05-01 | title = A large-scale study of file-system contents | journal = ACM SIGMETRICS Performance Evaluation Review | volume = 27 | issue = 1 | pages = 59–70 | doi = 10.1145/301464.301480 | issn = 0163-5999 | doi-access = free }}</ref><ref name=":3" />
* Sizes of text-based emails (1990s) and multimedia-based emails (2000s).<ref name=":3" />
* In computer networks and [[Internet traffic]] analysis, log-normal is shown as a good statistical model to represent the amount of traffic per unit time. This has been shown by applying a robust statistical approach on a large groups of real Internet traces. In this context, the log-normal distribution has shown a good performance in two main use cases: (1) predicting the proportion of time traffic will exceed a given level (for service level agreement or link capacity estimation) i.e. link dimensioning based on bandwidth provisioning and (2) predicting 95th percentile pricing.<ref>{{cite arXiv | last1 = Alamsar | first1 = Mohammed | last2 = Parisis | first2 = George | last3 = Clegg | first3 = Richard | last4 = Zakhleniuk | first4 = Nickolay | year = 2019 | title = On the Distribution of Traffic Volumes in the Internet and its Implications | eprint = 1902.03853 | class = cs.NI }}</ref>
* in [[physical test]]ing when the test produces a time-to-failure of an item under specified conditions, the data is often best analyzed using a lognormal distribution.<ref>ASTM D3654, Standard Test Method for Shear Adhesion on Pressure-Sensitive Tapesw</ref><ref>ASTM D4577, Standard Test Method for Compression Resistance of a container Under Constant Load>\</ref>