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{{Short description|Probability distribution}}
{{Infobox probability distribution
 | name = Log-normal distribution
 | type = continuous
 | pdf_image = [[Image:Log-normal-pdfs.png|300px|Plot of the Lognormal PDF]]<br/><small>Identical parameter <math> \mu </math> but differing parameters <math>\sigma</math></small>
 | cdf_image = [[Image:Log-normal-cdfs.png|300px|Plot of the Lognormal CDF]]<br/><small><math> \mu = 0 </math></small>
 | notation = <math> \operatorname{Lognormal}\left( \mu,\,\sigma^2 \right) </math>
 | parameters = {{plainlist |
* <math> \mu \in ( -\infty, +\infty ) </math> (logarithm of [[location parameter|location]]),
* <math> \sigma > 0 </math> (logarithm of [[scale parameter|scale]])
}}
 | support = <math> x \in ( 0, +\infty ) </math>
 | pdf = <math> \frac{ 1 }{ x \sigma \sqrt{2\pi } } \exp\left( - \frac{ \left( \ln x - \mu \right)^2}{ 2 \sigma^2 } \right)</math>
 | cdf = <math>\begin{align}
&\frac{ 1 }{2}\left[1 + \operatorname{erf}\left( \frac{ \ln x - \mu }{\sigma\sqrt{2}} \right)\right] \\[1ex]
&= \Phi{\left(\frac{\ln x -\mu}{\sigma} \right)}
\end{align}</math>
 | quantile = <math>\begin{align}
&\exp\left( \mu + \sqrt{2\sigma^2}\operatorname{erf}^{-1}(2 p - 1) \right) \\[1ex]
& = \exp(\mu + \sigma \Phi^{-1}(p))
\end{align}</math>
 | mean = <math> \exp\left( \mu + \frac{\sigma^2}{2} \right) </math>
 | median = <math> \exp( \mu ) </math>
 | mode = <math> \exp\left( \mu - \sigma^2 \right) </math>
 | variance = <math> \left[ \exp(\sigma^2) - 1 \right] \exp\left( 2 \mu + \sigma^2\right ) </math>
 | skewness = <math> \left[ \exp\left( \sigma^2 \right) + 2 \right] \sqrt{\exp(\sigma^2) - 1 }</math>
 | kurtosis = <math> \exp\left( 4 \sigma^2 \right) + 2 \exp\left( 3 \sigma^2 \right) + 3 \exp\left( 2\sigma^2 \right) - 6 </math>
 | entropy = <math> \log_2 \left( \sqrt{2\pi e} \, \sigma e^{ \mu } \right) </math>
 | mgf = defined only for numbers with a {{nowrap|non-positive}} real part, see text
 | char = representation <math> \sum_{n=0}^{\infty} \frac{ {\left(i t\right)}^n }{ n! }e^{ n \mu + n^2 \sigma^2/2} </math> is asymptotically divergent, but adequate for most numerical purposes
 | fisher = <math> \frac{1}{\sigma^2} \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} </math>
 | moments = <math> \mu = \ln \operatorname{E}[X] - \frac{1}{2} \ln\left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right),</math>
<br/>
<math> \sigma = \sqrt{ \ln \left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right) } </math>
 | ES = <math>\begin{align}
 &\frac{ e^{ \mu + \frac{ \sigma^2 }{2}} }{ 2p } \left[ 1 + \operatorname{erf} \left( \frac{ \sigma }{ \sqrt{2} } + \operatorname{erf}^{-1}(2p-1) \right) \right] \\[0.5ex]
 &= \frac{e^{ \mu + \frac{ \sigma^2 }{2}}}{1-p} \left[1 - \Phi(\Phi^{-1}(p) - \sigma)\right]
\end{align}</math><ref name="norton">{{cite journal | last1 = Norton | first1 = Matthew | last2 = Khokhlov | first2 = Valentyn | last3 = Uryasev | first3 = Stan | year = 2019 | title = Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation | journal = [[Annals of Operations Research]] | volume = 299 | issue = 1–2 | pages = 1281–1315 | publisher = Springer | doi = 10.1007/s10479-019-03373-1 | arxiv = 1811.11301 | s2cid = 254231768 | url = http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf | archive-url = https://web.archive.org/web/20210418151110/http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf | archive-date = 2021-04-18 | url-status = live | access-date = 2023-02-27 | via = stonybrook.edu }} </ref>
 }} <!-- end "infobox" -->

In [[probability theory]], a '''log-normal''' (or '''lognormal''') '''distribution''' is a continuous [[probability distribution]] of a [[random variable]] whose [[logarithm]] is [[normal distribution|normally distributed]]. Thus, if the random variable {{mvar|X}} is log-normally distributed, then {{math|1=''Y'' = ln ''X''}} has a normal distribution.<ref name=":1">{{Cite web | last = Weisstein | first = Eric W. | title = Log Normal Distribution | website = mathworld.wolfram.com | language = en | url = https://mathworld.wolfram.com/LogNormalDistribution.html | access-date = 2020-09-13 }}</ref><ref name=":2">{{cite web | title = 1.3.6.6.9. Lognormal Distribution | website = www.itl.nist.gov | publisher = U.S. [[National Institute of Standards and Technology]] (NIST) | url = https://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm |access-date = 2020-09-13 }}</ref> Equivalently, if {{mvar|Y}} has a normal distribution, then the [[exponential function]] of {{mvar|Y}}, {{math|1=''X'' = exp(''Y'')}}, has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and [[engineering]] sciences, as well as [[medicine]], [[economics]] and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).

The distribution is occasionally referred to as the '''Galton distribution''' or '''Galton's distribution''', after [[Francis Galton]].<ref name="JKB"/> The log-normal distribution has also been associated with other names, such as [[Donald MacAlister#log-normal|McAlister]], [[Gibrat's law|Gibrat]] and [[Cobb–Douglas]].<ref name="JKB"/>

A log-normal process is the statistical realization of the multiplicative [[mathematical product|product]] of many [[statistical independence|independent]] [[random variable]]s, each of which is positive. This is justified by considering the [[central limit theorem]] in the log domain (sometimes called [[Gibrat's law]]). The log-normal distribution is the [[maximum entropy probability distribution]] for a random variate {{mvar|X}}—for which the mean and variance of {{math|ln ''X''}} are specified.<ref>{{cite journal | last1 = Park | first1 = Sung Y. | last2 = Bera | first2 = Anil K. | year = 2009 | title = Maximum entropy autoregressive conditional heteroskedasticity model | journal = Journal of Econometrics | volume = 150 | issue = 2 | pages = 219–230, esp. Table&nbsp;1, p.&nbsp;221 | citeseerx = 10.1.1.511.9750 | doi = 10.1016/j.jeconom.2008.12.014 | url = http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf | url-status = dead | archive-url = https://web.archive.org/web/20160307144515/http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf | archive-date = 2016-03-07 | access-date = 2011-06-02}}</ref>

==Definitions==
===Generation and parameters===

Let <math> Z </math> be a [[Normal distribution#Standard normal distribution|standard normal variable]], and let <math>\mu</math> and <math>\sigma</math> be two real numbers, with {{nowrap|<math>\sigma > 0</math>.}} Then, the distribution of the random variable

<math display="block"> X = e^{\mu + \sigma Z} </math>

is called the log-normal distribution with parameters <math>\mu</math> and {{nowrap|<math>\sigma</math>.}} These are the [[expected value]] (or [[mean]]) and [[standard deviation]] of the variable's natural [[logarithm]], {{nowrap|<math>\ln X </math>,}} ''not'' the expectation and standard deviation of <math> X </math> itself.
[[File:Lognormal Distribution.svg|thumb|upright=1.5|Relation between normal and log-normal distribution. If <math> Y = \mu + \sigma Z </math> is normally distributed, then <math> X \sim e^Y </math> is log-normally distributed.]]

This relationship is true regardless of the base of the logarithmic or exponential function: If <math>\log_a X </math> is normally distributed, then so is <math>\log_b X </math> for any two positive numbers {{nowrap|<math> a, b \neq 1</math>.}} Likewise, if <math> e^Y </math> is log-normally distributed, then so is {{nowrap|<math> a^Y </math>,}} where {{nowrap|<math>0 < a \neq 1</math>.}}

In order to produce a distribution with desired mean <math>\mu_X</math> and variance {{nowrap|<math> \sigma_X^2</math>,}} one uses <math> \mu = \ln \frac{ \mu_X^2 }{ \sqrt{ \mu_X^2 + \sigma_X^2 } } </math> and {{nowrap|<math> \sigma^2 = \ln\left( 1 + \frac{ \sigma_X^2 }{ \mu_X^2 } \right) </math>.}}

Alternatively, the "multiplicative" or "geometric" parameters <math> \mu^* = e^\mu </math> and <math> \sigma^* = e^\sigma </math> can be used. They have a more direct interpretation: <math> \mu^* </math> is the '''''[[median]]''''' of the distribution, and <math> \sigma^* </math> is useful for determining "scatter" intervals, see below.

=== Probability density function ===

A positive random variable <math> X </math> is log-normally distributed (i.e., {{nowrap|<math display="inline"> X \sim \operatorname{Lognormal} \left( \mu, \sigma^2 \right) </math>),}} if the natural logarithm of <math> X </math> is normally distributed with mean <math> \mu</math> and variance {{nowrap|<math> \sigma^2</math>:}}

<math display="block"> \ln X \sim \mathcal{N}(\mu,\sigma^2)</math>

Let <math> \Phi </math> and <math> \varphi </math> be respectively the cumulative probability distribution function and the probability density function of the <math> \mathcal{N}( 0, 1 ) </math> standard normal distribution, then we have that<ref name=":1"/><ref name="JKB"/> the [[probability density function]] of the log-normal distribution is given by:

<math display="block">\begin{align}
f_X(x) & = \frac{d}{dx} \Pr\nolimits_X\left[ X \le x \right] \\[6pt]
& = \frac{d}{dx} \Pr\nolimits_X\left[ \ln X \le \ln x \right] \\[6pt]
& = \frac{d}{dx} \Phi{\left( \frac{ \ln x -\mu }{ \sigma } \right)} \\[6pt]
& = \varphi{\left( \frac{\ln x - \mu} \sigma \right)} \frac{d}{dx} \left( \frac{ \ln x - \mu }{ \sigma }\right) \\[6pt]
& = \varphi{\left( \frac{ \ln x - \mu }{ \sigma } \right)} \frac{ 1 }{ \sigma x } \\[6pt]
& = \frac{ 1 }{ x \sigma\sqrt{2 \pi } } \exp\left( -\frac{ (\ln x-\mu)^2 }{2 \sigma^2} \right) ~.
\end{align}</math>

=== Cumulative distribution function ===
<!-- erf changed into erfc because then the formula is slightly shorter, and besides the expression with erf is already present in the floating box -->
The [[cumulative distribution function]] is

<math display="block"> F_X(x) = \Phi{\left( \frac{\ln x - \mu} \sigma \right)} </math>

where <math> \Phi </math> is the cumulative distribution function of the standard normal distribution (i.e., {{nowrap|<math> \operatorname\mathcal{N}( 0, 1 ) </math>).}}

This may also be expressed as follows:<ref name=":1" />

<math display="block"> \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) \right] = \frac12 \operatorname{erfc} \left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) </math>

where {{big|{{math|erfc}}}} is the [[complementary error function]].

=== Multivariate log-normal ===
If <math>\boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma)</math> is a [[multivariate normal distribution]], then <math>Y_i = \exp(X_i)</math> has a multivariate log-normal distribution.<ref>{{Cite conference | last = Tarmast | first = Ghasem | year = 2001 | title = Multivariate Log–Normal Distribution | url = http://isi.cbs.nl/iamamember/CD2/pdf/329.PDF | archive-url = https://web.archive.org/web/20130719214220/http://isi.cbs.nl/iamamember/CD2/pdf/329.PDF | archive-date = 2013-07-19 | url-status = live | conference = ISI Proceedings: 53rd Session | location = Seoul}}</ref><ref>{{Cite conference | last = Halliwell | first = Leigh | year = 2015 | title = The Lognormal Random Multivariate | url = http://www.casact.org/pubs/forum/15spforum/Halliwell.pdf | archive-url = https://web.archive.org/web/20150930111908/http://www.casact.org/pubs/forum/15spforum/Halliwell.pdf | archive-date = 2015-09-30 | url-status = live | conference = Casualty Actuarial Society E-Forum, Spring 2015 | location = Arlington, VA}}</ref> The exponential is applied element-wise to the random vector <math>\boldsymbol X</math>. The mean of <math>\boldsymbol Y</math> is

<math display="block">\operatorname{E}[\boldsymbol Y]_i = e^{\mu_i + \frac{1}{2} \Sigma_{ii}} ,</math>

and its [[covariance matrix]] is

<math display="block">\operatorname{Var}[\boldsymbol Y]_{ij} = e^{\mu_i + \mu_j + \frac{1}{2}(\Sigma_{ii} + \Sigma_{jj}) } \left( e^{\Sigma_{ij}} - 1\right) . </math>

Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the [[univariate distribution]].

=== Characteristic function and moment generating function ===

All moments of the log-normal distribution exist and

<math display="block">\operatorname{E}[X^n] = e^{n\mu+n^2\sigma^2/2}</math>

This can be derived by letting <math display="inline">z = \tfrac{\ln x - \mu}{\sigma} - n \sigma</math> within the integral. However, the log-normal distribution is not determined by its moments.<ref name="Heyde">{{Citation | last = Heyde | first = CC. | title = On a Property of the Lognormal Distribution | work = Journal of the Royal Statistical Society, Series B | date = 2010 | volume = 25 | issue = 2 | pages = 392–393 | doi = 10.1007/978-1-4419-5823-5_6 | isbn = 978-1-4419-5822-8 | doi-access = free}}</ref> This implies that it cannot have a defined moment generating function in a neighborhood of zero.<ref>{{Cite book | last = Billingsley | first = Patrick | title = Probability and Measure | date = 2012 | publisher = Wiley | isbn = 978-1-118-12237-2 | edition = Anniversary | location = Hoboken, N.J. | pages = 415 | oclc = 780289503}}</ref> Indeed, the expected value <math>\operatorname{E}[e^{t X}]</math> is not defined for any positive value of the argument <math>t</math>, since the defining integral diverges.

The [[characteristic function (probability theory)|characteristic function]] <math>\operatorname{E}[e^{i t X}]</math> is defined for real values of {{mvar|t}}, but is not defined for any complex value of {{mvar|t}} that has a negative imaginary part, and hence the characteristic function is not [[Analytic function|analytic]] at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.<ref name="Holgate">{{Cite journal | last = Holgate | first = P. | year = 1989 | title = The lognormal characteristic function, vol. 18, pp. 4539–4548, 1989 | journal = Communications in Statistics – Theory and Methods | volume = 18 | issue = 12 | pages = 4539–4548 | doi = 10.1080/03610928908830173}}</ref> In particular, its Taylor [[formal series]] diverges:

<math display="block">\sum_{n=0}^\infty \frac{{\left(it\right)}^n}{n!} e^{n\mu + n^2\sigma^2/2}</math>

However, a number of alternative [[divergent series]] representations have been obtained.<ref name="Holgate" /><ref name="Barakat">{{Cite journal | last = Barakat | first = R. | year = 1976 | title = Sums of independent lognormally distributed random variables | journal = Journal of the Optical Society of America | volume = 66 | issue = 3 | pages = 211–216 | bibcode = 1976JOSA...66..211B | doi = 10.1364/JOSA.66.000211}}</ref><ref name="Barouch">{{Cite journal | last1 = Barouch | first1 = E. | last2 = Kaufman | first2 = GM. | last3 = Glasser | first3 = ML. | year = 1986 | title = On sums of lognormal random variables | url = http://dspace.mit.edu/bitstream/handle/1721.1/48703/onsumsoflognorma00baro.pdf | journal = Studies in Applied Mathematics | volume = 75 | issue = 1 | pages = 37–55 | doi = 10.1002/sapm198675137 | hdl = 1721.1/48703 | hdl-access = free }}</ref><ref name="Leipnik">{{Cite journal | last = Leipnik | first = Roy B. | date = January 1991 | title = On Lognormal Random Variables: I – The Characteristic Function | url = https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F1563B5AD8918EF2CD51092F82EB0B73/S0334270000006901a.pdf/div-class-title-on-lognormal-random-variables-i-the-characteristic-function-div.pdf | journal = Journal of the Australian Mathematical Society, Series B | volume = 32 | issue = 3 | pages = 327–347 | doi = 10.1017/S0334270000006901 | doi-access = free }}</ref>

A closed-form formula for the characteristic function <math>\varphi(t)</math> with <math>t</math> in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by<ref name="Asmussen">S. Asmussen, J.L. Jensen, L. Rojas-Nandayapa (2016). "On the Laplace transform of the Lognormal distribution",
[https://link.springer.com/article/10.1007/s11009-014-9430-7 Methodology and Computing in Applied Probability 18 (2), 441-458.]
[http://data.imf.au.dk/publications/thiele/2013/math-thiele-2013-06.pdf Thiele report 6 (13).]</ref>

<math display="block">\varphi(t) \approx \frac{\exp\left(-\frac{W^2(-it\sigma^2e^\mu) + 2W(-it\sigma^2e^\mu)}{2\sigma^2} \right)}{\sqrt{1 + W{\left(-it\sigma^2e^\mu\right)}}}</math>

where <math>W</math> is the [[Lambert W function]]. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of <math>\varphi</math>.

== Properties ==

[[File:Probabilities of log normal.png|thumb|right|a. <math>y</math> is a log-normal variable with {{nowrap|<math>\mu = 1</math>,}} {{nowrap|<math>\sigma = 0.5</math>.}} <math>p(\sin y>0)</math> is computed by transforming to the normal variable <math>x = \ln y</math>, then integrating its density over the domain defined by <math>\sin e^x>0</math> (blue regions), using the numerical method of ray-tracing.<ref name="Das" /> b & c. The pdf and cdf of the function <math> \sin y</math> of the log-normal variable can also be computed in this way.]]

===Probability in different domains===
The probability content of a log-normal distribution in any arbitrary domain can be computed to desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method.<ref name="Das">{{cite journal | last = Das | first = Abhranil | arxiv = 2012.14331 | title = A method to integrate and classify normal distributions | journal = Journal of Vision | date = 2021 | volume = 21 | issue = 10 | page = 1 | doi = 10.1167/jov.21.10.1 | pmid = 34468706 | pmc = 8419883 }}</ref> ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code])

===Probabilities of functions of a log-normal variable===
Since the probability of a log-normal can be computed in any domain, this means that the cdf (and consequently pdf and inverse cdf) of any function of a log-normal variable can also be computed.<ref name="Das"/> ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code])

===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]].

===Arithmetic moments===

For any real or complex number {{mvar|n}}, the {{mvar|n}}-th [[moment (mathematics)|moment]] of a log-normally distributed variable {{mvar|X}} is given by<ref name="JKB"/>
<math display="block">\operatorname{E}[X^n] = e^{n\mu + \frac{1}{2}n^2\sigma^2}.</math>

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable {{mvar|X}} are respectively given by:<ref name=":1" />

<math display="block">\begin{align}
 \operatorname{E}[X] & = e^{\mu + \tfrac{1}{2}\sigma^2}, \\[4pt]
 \operatorname{E}[X^2] & = e^{2\mu + 2\sigma^2}, \\[4pt]
 \operatorname{Var}[X] & = \operatorname{E}[X^2] - \operatorname{E}[X]^2
 = {\left(\operatorname{E}[X]\right)}^2 \left(e^{\sigma^2} - 1\right) \\[2pt]
 &= e^{2\mu + \sigma^2} \left(e^{\sigma^2} - 1\right), \\[4pt]
 \operatorname{SD}[X] & = \sqrt{\operatorname{Var}[X]}
 = \operatorname{E}[X] \sqrt{e^{\sigma^2} - 1} \\[2pt]
 &= e^{\mu + \tfrac{1}{2}\sigma^2} \sqrt{e^{\sigma^2} - 1},
\end{align}</math>

The arithmetic [[coefficient of variation]] <math>\operatorname{CV}[X]</math> is the ratio <math>\tfrac{\operatorname{SD}[X]}{\operatorname{E}[X]}</math>. For a log-normal distribution it is equal to<ref name=":2" />
<math display="block">\operatorname{CV}[X] = \sqrt{e^{\sigma^2} - 1}.</math>
This estimate is sometimes referred to as the "geometric CV" (GCV),<ref>Sawant, S.; Mohan, N. (2011) [http://pharmasug.org/proceedings/2011/PO/PharmaSUG-2011-PO08.pdf "FAQ: Issues with Efficacy Analysis of Clinical Trial Data Using SAS"] {{webarchive | url = https://web.archive.org/web/20110824094357/http://pharmasug.org/proceedings/2011/PO/PharmaSUG-2011-PO08.pdf | date = 24 August 2011 }}, ''PharmaSUG2011'', Paper PO08</ref><ref>{{cite journal | last1 = Schiff | first1 = MH | display-authors = etal | year = 2014 | title = Head-to-head, randomised, crossover study of oral versus subcutaneous methotrexate in patients with rheumatoid arthritis: drug-exposure limitations of oral methotrexate at doses >=15 mg may be overcome with subcutaneous administration | journal = Ann Rheum Dis | volume = 73 | issue = 8 | pages = 1–3 | doi = 10.1136/annrheumdis-2014-205228 | pmid = 24728329 | pmc = 4112421}}</ref> due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

The parameters {{math|''μ''}} and {{math|''σ''}} can be obtained, if the arithmetic mean and the arithmetic variance are known:

<math display="block">\begin{align}
\mu &= \ln \frac{\operatorname{E}[X]^2}{\sqrt{\operatorname{E}[X^2]}}
= \ln \frac{\operatorname{E}[X]^2}{\sqrt{\operatorname{Var}[X] + \operatorname{E}[X]^2}}, \\[1ex]
 \sigma^2 &= \ln \frac{\operatorname{E}[X^2]}{\operatorname{E}[X]^2}
= \ln \left(1 + \frac{\operatorname{Var}[X]}{\operatorname{E}[X]^2}\right).
 \end{align}</math>

A probability distribution is not uniquely determined by the moments {{math|1=E[''X''<sup>''n''</sup>] = e<sup>''nμ'' + {{sfrac|1|2}}''n''<sup>2</sup>''σ''<sup>2</sup></sup>}} for {{math|''n'' ≥ 1}}. That is, there exist other distributions with the same set of moments.<ref name="JKB"/> In fact, there is a whole family of distributions with the same moments as the log-normal distribution.{{Citation needed|date=March 2012}}

=== Mode, median, quantiles ===
[[File:Comparison mean median mode.svg|thumb|upright=1.25|Comparison of [[mean]], [[median]] and [[mode (statistics)|mode]] of two log-normal distributions with different [[skewness]].]]
The [[mode (statistics)|mode]] is the point of global maximum of the probability density function. In particular, by solving the equation <math>(\ln f)'=0</math>, we get that:

<math display="block">\operatorname{Mode}[X] = e^{\mu - \sigma^2}.</math>

Since the [[logarithm transformation|log-transformed]] variable <math>Y = \ln X</math> has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of <math>X</math> are

<math display="block">q_X(\alpha) = \exp\left[\mu + \sigma q_\Phi(\alpha)\right] = \mu^* (\sigma^*)^{q_\Phi(\alpha)},</math>

where <math>q_\Phi(\alpha)</math> is the quantile of the standard normal distribution.

Specifically, the median of a log-normal distribution is equal to its multiplicative mean,<ref>{{cite book | first1 = Leslie E. | last1 = Daly | first2 = Geoffrey Joseph | last2 = Bourke | year = 2000 | title = Interpretation and Uses of Medical Statistics | journal = Journal of Epidemiology and Community Health | volume = 46 | issue = 3 | edition = 5th | place = Oxford, UK | publisher = Wiley-Blackwell | isbn = 978-0-632-04763-5 | page = 89 | doi = 10.1002/9780470696750 | pmc = 1059583 <!-- | journal = Journal of Epidemiology and Community Health | volume = 46 | issue = 3 --- not found in WorldCat.org --> | postscript = ; }} print edition. Online eBook {{ISBN|9780470696750}}</ref>

<math display="block">\operatorname{Med}[X] = e^\mu = \mu^* ~.</math>

=== Partial expectation ===

The partial expectation of a random variable <math>X</math> with respect to a threshold <math>k</math> is defined as

<math display="block"> g(k) = \int_k^\infty x \, f_X(x)\, dx . </math>

Alternatively, by using the definition of [[conditional expectation]], it can be written as <math>g(k) = \operatorname{E}[X\mid X>k] \Pr(X>k)</math>. For a log-normal random variable, the partial expectation is given by:

<math display="block">\begin{align}
g(k) &= \int_k^\infty x f_X(x)\, dx \\[1ex]
&= e^{\mu+\tfrac{1}{2} \sigma^2}\, \Phi{\left(\frac{\mu-\ln k}{\sigma} + \sigma\right)}
\end{align} </math>

where <math>\Phi</math> is the [[normal cumulative distribution function]]. The derivation of the formula is provided in the [[Talk:Log-normal distribution|Talk page]]. The partial expectation formula has applications in [[insurance]] and [[economics]], it is used in solving the partial differential equation leading to the [[Black–Scholes formula]].

=== Conditional expectation ===

The conditional expectation of a log-normal random variable <math>X</math>—with respect to a threshold <math>k</math>—is its partial expectation divided by the cumulative probability of being in that range:

<math display="block">\begin{align}
\operatorname{E}[X\mid X<k] & = e^{\mu +\frac{\sigma^2}{2}} \cdot \frac{\Phi {\left[\frac{\ln k - \mu}{\sigma} - \sigma \right]}}{\Phi {\left[\frac{\ln k-\mu}{\sigma} \right]}} \\[8pt]
\operatorname{E}[X \mid X \geq k] &= e^{\mu +\frac{\sigma^2}{2}} \cdot \frac{\Phi {\left[\frac{\mu - \ln k}{\sigma} + \sigma \right]}}{1 - \Phi {\left[\frac{\ln k -\mu}{\sigma}\right]}} \\[8pt]
\operatorname{E}[X\mid X\in [k_1,k_2]] &= e^{\mu +\frac{\sigma^2}{2}} \cdot 
\frac{
 \Phi{\left[\frac{\ln k_2 - \mu}{\sigma} - \sigma \right]} - \Phi{\left[\frac{\ln k_1 - \mu}{\sigma} - \sigma\right]}
}{
 \Phi \left[\frac{\ln k_2 - \mu}{\sigma}\right]-\Phi \left[\frac{\ln k_1 - \mu}{\sigma}\right]
}
\end{align}</math>

=== Alternative parameterizations ===
In addition to the characterization by <math>\mu, \sigma</math> or <math>\mu^*, \sigma^*</math>, here are multiple ways how the log-normal distribution can be parameterized. [[ProbOnto]], the knowledge base and ontology of [[probability distribution]]s<ref>{{cite web | url = http://www.probonto.org | title = ProbOnto |access-date = 1 July 2017}}</ref><ref>{{cite journal | pmid = 27153608 | doi = 10.1093/bioinformatics/btw170 | pmc = 5013898 | volume = 32 | issue = 17 | pages = 2719–2721 | title = ProbOnto: ontology and knowledge base of probability distributions | year = 2016 | journal = Bioinformatics | last1 = Swat | first1 = MJ | last2 = Grenon | first2 = P | last3 = Wimalaratne | first3 = S}}</ref> lists seven such forms: [[File:LogNormal17.jpg|thumb|400px|Overview of parameterizations of the log-normal distributions.]]

* {{math|LogNormal1(''μ'',''σ'')}} with [[mean]], {{math|''μ''}}, and [[standard deviation]], {{math|''σ''}}, both on the log-scale <ref name="Forbes">Forbes et al. Probability Distributions (2011), John Wiley & Sons, Inc.</ref> <math display="block">P(x;\boldsymbol\mu,\boldsymbol\sigma) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp\left[-\frac{(\ln x - \mu)^2}{2 \sigma^2}\right]</math>
* {{math|LogNormal2(''μ'',''υ'')}} with mean, {{math|''μ''}}, and variance, {{math|''υ''}}, both on the log-scale <math display="block">P(x;\boldsymbol\mu,\boldsymbol {v}) = \frac{1}{x \sqrt{v} \sqrt{2 \pi}} \exp\left[-\frac{(\ln x - \mu)^2}{2 v}\right]</math>
* {{math|LogNormal3(''m'',''σ'')}} with [[median]], {{math|''m''}}, on the natural scale and standard deviation, {{math|''σ''}}, on the log-scale<ref name="Forbes" /> <math display="block">P(x;\boldsymbol m,\boldsymbol \sigma) =\frac{1}{x \sigma \sqrt{2 \pi}} \exp\left[-\frac{\ln^2(x/m)}{2 \sigma^2}\right]</math>
* {{math|LogNormal4(''m'',cv)}} with median, {{math|''m''}}, and [[coefficient of variation]], {{math|cv}}, both on the natural scale <math display="block">P(x;\boldsymbol m,\boldsymbol {cv}) = \frac{1}{x \sqrt{\ln(cv^2+1)} \sqrt{2 \pi}} \exp\left[-\frac{\ln^2(x/m)}{2\ln(cv^2+1)}\right]</math>
* {{math|LogNormal5(''μ'',''τ'')}} with mean, {{math|''μ''}}, and [[Precision (statistics)|precision]], {{math|''τ''}}, both on the log-scale<ref>Lunn, D. (2012). The BUGS book: a practical introduction to Bayesian analysis. Texts in statistical science. CRC Press.</ref> <math display="block">P(x;\boldsymbol\mu,\boldsymbol \tau) = \sqrt{\frac{\tau}{2 \pi}} \frac{1}{x} \exp\left[-\frac{\tau}{2}(\ln x-\mu)^2\right]</math>
* {{math|LogNormal6(''m'',''σ<sub>g</sub>'')}} with median, {{math|''m''}}, and [[geometric standard deviation]], {{math|''σ<sub>g</sub>''}}, both on the natural scale<ref>{{cite journal | last1 = Limpert | first1 = E. | last2 = Stahel | first2 = W. A. | last3 = Abbt | first3 = M. | year = 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 | doi-access = free }}</ref> <math display="block">
P(x;\boldsymbol m,\boldsymbol {\sigma_g}) = \frac{1}{x \sqrt{2 \pi} \, \ln\sigma_g} \exp\left[-\frac{\ln^2(x/m)}{2 \ln^2(\sigma_g)}\right]</math>
* {{math|LogNormal7(''μ<sub>N</sub>'',''σ<sub>N</sub>'')}} with mean, {{math|''μ<sub>N</sub>''}}, and standard deviation, {{math|''σ<sub>N</sub>''}}, both on the natural scale<ref>{{cite journal | last1 = Nyberg | first1 = J. | display-authors = etal | year = 2012 | title = PopED – An extended, parallelized, population optimal design tool | journal = Comput Methods Programs Biomed | volume = 108 | issue = 2 | pages = 789–805 | doi = 10.1016/j.cmpb.2012.05.005 | pmid = 22640817 }}</ref> <math display="block">P(x;\boldsymbol {\mu_N},\boldsymbol {\sigma_N}) = \frac{1}{x \sqrt{2 \pi \ln\left(1+\sigma_N^2/\mu_N^2\right)}} \exp\left[-\frac{\left( \ln x - \ln\frac{\mu_N}{\sqrt{1 + \sigma_N^2/\mu_N^2}}\right)^2}{2 \ln\left(1 + \frac{\sigma_N^2}{\mu_N^2}\right)}\right]</math>

==== Examples for re-parameterization ====
Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM<ref>{{cite journal | last1 = Retout | first1 = S | last2 = Duffull | first2 = S | last3 = Mentré | first3 = F | year = 2001 | title = Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs | journal = Comp Meth Pro Biomed | volume = 65 | issue = 2 | pages = 141–151 | doi = 10.1016/S0169-2607(00)00117-6 | pmid = 11275334 }}</ref> and PopED.<ref>The PopED Development Team (2016). PopED Manual, Release version 2.13. Technical report, Uppsala University.</ref> The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results.

For the transition <math>\operatorname{LN2}(\mu, v) \to \operatorname{LN7}(\mu_N, \sigma_N)</math> following formulas hold <math display="inline">\mu_N = \exp(\mu+v/2) </math> and <math display="inline">\sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}</math>.

For the transition <math>\operatorname{LN7}(\mu_N, \sigma_N) \to \operatorname{LN2}(\mu, v)</math> following formulas hold <math display="inline">\mu = \ln \mu_N - \frac{1}{2} v </math> and <math display="inline"> v = \ln(1+\sigma_N^2/\mu_N^2)</math>.

All remaining re-parameterisation formulas can be found in the specification document on the project website.<ref name="probontoWebsite">ProbOnto website, URL: http://probonto.org</ref>

=== Multiple, reciprocal, power ===
* Multiplication by a constant: If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> then <math>a X \sim \operatorname{Lognormal}( \mu + \ln a, \sigma^2)</math> for <math> a > 0. </math>
* Reciprocal: If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> then <math>\tfrac{1}{X} \sim \operatorname{Lognormal}(-\mu, \sigma^2).</math>
* Power: If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> then <math>X^a \sim \operatorname{Lognormal}(a\mu, a^2 \sigma^2)</math> for <math>a \neq 0.</math>

=== Multiplication and division of independent, log-normal random variables ===
If two [[statistical independence|independent]], log-normal variables <math>X_1</math> and <math>X_2</math> are multiplied [divided], the product [ratio] is again log-normal, with parameters <math>\mu = \mu_1 + \mu_2</math> {{nowrap|[<math>\mu = \mu_1-\mu_2</math>]}} and {{nowrap|<math>\sigma</math>,}} where {{nowrap|<math>\sigma^2 = \sigma_1^2 + \sigma_2^2</math>.}}

More generally, if <math>X_j \sim \operatorname{Lognormal} (\mu_j, \sigma_j^2)</math> are <math>n</math> independent, log-normally distributed variables, then <math display="inline">Y = \prod_{j=1}^n X_j \sim \operatorname{Lognormal} \Big( \sum_{j=1}^n\mu_j, \sum_{j=1}^n \sigma_j^2 \Big).</math>

=== <span class="anchor" id="Multiplicative Central Limit Theorem"></span>Multiplicative central limit theorem ===
{{See also|Gibrat's law}}

The geometric or multiplicative mean of <math>n</math> independent, identically distributed, positive random variables <math>X_i</math> shows, for <math>n \to \infty</math>, approximately a log-normal distribution with parameters <math>\mu = \operatorname{E}[\ln X_i]</math> and <math>\sigma^2 = \operatorname{var}[\ln X_i ]/n</math>, assuming <math>\sigma^2</math> is finite.

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of <math>\ln X_i</math> to all have finite variance and satisfy the other conditions of any of the many variants of the [[central limit theorem]].

This is commonly known as [[Gibrat's law]].

=== Heavy-tailness of the Log-Normal ===
Whether a Log-Normal can be considered or not a true heavy-tail distribution is still debated. The main reason is that its variance is always finite, differently from what happen with certain Pareto distributions, for instance. However a recent study has shown how it is possible to create a Log-Normal distribution with infinite variance using Robinson Non-Standard Analysis.<ref>{{Cite journal | last1 = Cococcioni | first1 = Marco | last2 = Fiorini | first2 = Francesco | last3 = Pagano | first3 = Michele | date = 2023-04-06 | title = Modelling Heavy Tailed Phenomena Using a LogNormal Distribution Having a Numerically Verifiable Infinite Variance | journal = Mathematics | language = en | volume = 11 | issue = 7 | page = 1758 | doi = 10.3390/math11071758 | doi-access = free | issn = 2227-7390 | hdl = 11568/1216554 | hdl-access = free }}</ref>

=== Other ===

A set of data that arises from the log-normal distribution has a symmetric [[Lorenz curve]] (see also [[Lorenz asymmetry coefficient]]).<ref name="EcolgyArticle">{{cite journal | doi = 10.1890/0012-9658(2000)081[1139:DIIPSO]2.0.CO;2 | last1 = Damgaard | first1 = Christian | first2 = Jacob | last2 = Weiner | title = Describing inequality in plant size or fecundity | journal = Ecology | year = 2000 | volume = 81 | issue = 4 | pages = 1139–1142 }}</ref>

The harmonic <math>H</math>, geometric <math>G</math> and arithmetic <math>A</math> means of this distribution are related;<ref name="Rossman1990">{{cite journal | last = Rossman | first = Lewis A | date = July 1990 | title = Design stream flows based on harmonic means | journal = Journal of Hydraulic Engineering | volume = 116 | issue = 7 | pages = 946–950 | doi = 10.1061/(ASCE)0733-9429(1990)116:7(946)}}</ref> such relation is given by

<math display="block">H = \frac{G^2} A.</math>

Log-normal distributions are [[infinite divisibility (probability)|infinitely divisible]],<ref name="OlofThorin1978LNInfDivi"/> but they are not [[stable distribution]]s, which can be easily drawn from.<ref name="Gao"/>

== Related distributions ==
* If <math>X \sim \mathcal{N}(\mu, \sigma^2)</math> is a [[normal distribution]], then <math>\exp(X) \sim \operatorname{Lognormal}(\mu, \sigma^2).</math>
* If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> is distributed log-normally, then <math>\ln X \sim \mathcal{N}(\mu, \sigma^2)</math> is a normal random variable.
* Let <math>X_j \sim \operatorname{Lognormal}(\mu_j, \sigma_j^2)</math> be independent log-normally distributed variables with possibly varying <math>\sigma</math> and <math>\mu</math> parameters, and <math display="inline">Y = \sum_{j = 1}^n X_j</math>. The distribution of <math>Y</math> has no closed-form expression, but can be reasonably approximated by another log-normal distribution <math>Z</math> at the right tail.<ref name="Asmussen2">{{cite journal | first1 = S. | last1 = Asmussen | first2 = L. | last2 = Rojas-Nandayapa | title = Asymptotics of Sums of Lognormal Random Variables with Gaussian Copula | journal = Statistics and Probability Letters | volume = 78 | issue = 16 | pages = 2709–2714 | year = 2008 | doi = 10.1016/j.spl.2008.03.035 | url = https://hal.archives-ouvertes.fr/hal-00595951/file/PEER_stage2_10.1016%252Fj.spl.2008.03.035.pdf }}</ref> Its probability density function at the neighborhood of 0 has been characterized<ref name = Gao/> and it does not resemble any log-normal distribution. A commonly used approximation due to L.F. Fenton (but previously stated by R.I. Wilkinson and mathematically justified by Marlow<ref name="Marlow">{{cite journal | first = NA. | last = Marlow | title = A normal limit theorem for power sums of independent normal random variables | journal = Bell System Technical Journal | volume = 46 | issue = 9 | pages = 2081–2089 | date = Nov 1967 | doi = 10.1002/j.1538-7305.1967.tb04244.x}}</ref>) is obtained by matching the mean and variance of another log-normal distribution: <math display="block">\begin{align}
 \sigma^2_Z &= \ln\!\left[ \frac{\sum_j e^{2\mu_j+\sigma_j^2} \left(e^{\sigma_j^2} - 1\right)}{{\left(\sum_j e^{\mu_j + \sigma_j^2/2}\right)}^2} + 1\right], \\[1ex]
 \mu_Z &= \ln\!\left[ \sum_j e^{\mu_j+\sigma_j^2/2} \right] - \frac{\sigma^2_Z}{2}.
\end{align}</math> In the case that all <math>X_j</math> have the same variance parameter {{nowrap|<math>\sigma_j = \sigma</math>,}} these formulas simplify to <math display="block">\begin{align}
 \sigma^2_Z &= \ln\!\left[ \left(e^{\sigma^2} - 1\right) \frac{\sum_j e^{2\mu_j}}{{\left(\sum_j e^{\mu_j}\right)}^2} + 1\right], \\[1ex]
 \mu_Z &= \ln\!\left[ \sum_j e^{\mu_j} \right] + \frac{\sigma^2}{2} - \frac{\sigma^2_Z}{2}.
\end{align}</math>
For a more accurate approximation, one can use the [[Monte Carlo method]] to estimate the cumulative distribution function, the pdf and the right tail.<ref name="BotLec2017">{{cite conference | title = Accurate computation of the right tail of the sum of dependent log-normal variates | last1 = Botev | first1 = Z. I. | last2 = L'Ecuyer | first2 = P. | date = 2017 | publisher = IEEE | isbn = 978-1-5386-3428-8 | book-title= 2017 Winter Simulation Conference (WSC), 3rd–6th Dec 2017 | pages = 1880–1890 | location = Las Vegas, NV | doi= 10.1109/WSC.2017.8247924 | arxiv = 1705.03196}}
</ref><ref name="AGL2016">{{cite arXiv | last1 = Asmussen | first1 = A. | last2 = Goffard | first2 = P.-O. | last3 = Laub | first3 = P. J. | date = 2016 | title = Orthonormal polynomial expansions and lognormal sum densities | eprint = 1601.01763v1 | class = math.PR }} </ref> The cdf and pdf of the sum of correlated log-normally distributed random variables can also be approximated by Monte Carlo simulation.<ref name="Botev2019">{{cite journal 
 | last1 = Botev 
 | first1 = Z.I. 
 | last2 = Salomone 
 | first2 = R. 
 | last3 = Mackinlay 
 | first3 = D. 
 | title = Fast and accurate computation of the distribution of sums of dependent log-normals 
 | journal = Annals of Operations Research 
 | volume = 280 
 | pages = 19–46 
 | year = 2019 
 | doi = 10.1007/s10479-019-03161-x 
 | url = https://doi.org/10.1007/s10479-019-03161-x 
| url-access = subscription 
 }}</ref>
* If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> then <math>X+c</math> is said to have a ''Three-parameter log-normal'' distribution with support {{nowrap|<math>x\in (c, +\infty)</math>.}}<ref name="Sangal1970">{{cite journal | first1 = B. | last1 = Sangal | first2 = A. | last2 = Biswas | title = The 3-Parameter Lognormal Distribution Applications in Hydrology | journal = Water Resources Research | volume = 6 | issue = 2 | pages = 505–515 | year = 1970 | doi = 10.1029/WR006i002p00505}}</ref> {{nowrap|<math>\operatorname{E}[X+c] = \operatorname{E}[X] + c</math>,}} {{nowrap|<math>\operatorname{Var}[X+c] = \operatorname{Var}[X]</math>.}}
* The log-normal distribution is a special case of the semi-bounded [[Johnson's SU-distribution]].<ref name="Johnson1949">{{cite journal |author-link = Norman Lloyd Johnson | last = Johnson | first = N. L. | date = 1949 | title = Systems of Frequency Curves Generated by Methods of Translation | journal=[[Biometrika]] | volume = 36 | issue = 1/2 | pages = 149–176 | jstor = 2332539 | doi = 10.2307/2332539 | pmid = 18132090 }}</ref>
* If <math>X\mid Y \sim \operatorname{Rayleigh}(Y)</math> with <math> Y \sim \operatorname{Lognormal}(\mu, \sigma^2)</math>, then <math> X \sim \operatorname{Suzuki}(\mu, \sigma)</math> ([[Suzuki distribution]]).
* A substitute for the log-normal whose integral can be expressed in terms of more elementary functions<ref>{{Cite journal | last1 = Swamee | first1 = P. K. | title = Near Lognormal Distribution | doi = 10.1061/(ASCE)1084-0699(2002)7:6(441) | journal = Journal of Hydrologic Engineering | volume = 7 | issue = 6 | pages = 441–444 | year = 2002 }}</ref> can be obtained based on the [[logistic distribution]] to get an approximation for the [[Cumulative distribution function|CDF]] <math display="block"> F(x;\mu,\sigma) = \left[\left(\frac{e^\mu}{x}\right)^{\pi/(\sigma \sqrt{3})} + 1\right]^{-1}.</math> This is a [[log-logistic distribution]].

== Statistical inference ==
===Estimation of parameters===
==== Maximum likelihood estimator ====

For determining the [[maximum likelihood]] estimators of the log-normal distribution parameters {{math|''μ''}} and {{math|''σ''}}, we can use the [[normal distribution#Estimation of parameters|same procedure]] as for the [[normal distribution]]. Note that
<math display="block">L(\mu, \sigma) = \prod_{i=1}^n \frac 1 {x_i} \varphi_{\mu,\sigma} (\ln x_i),</math>
where <math>\varphi</math> is the density function of the normal distribution <math>\mathcal N(\mu,\sigma^2)</math>. Therefore, the log-likelihood function is
<math display="block">
\ell (\mu,\sigma \mid x_1, x_2, \ldots, x_n) = - \sum _i \ln x_i + \ell_N (\mu, \sigma \mid \ln x_1, \ln x_2, \dots, \ln x_n).</math>

Since the first term is constant with regard to ''μ'' and ''σ'', both logarithmic likelihood functions, <math>\ell</math> and <math>\ell_N</math>, reach their maximum with the same <math>\mu</math> and <math>\sigma</math>. Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations <math>\ln x_1, \ln x_2, \dots, \ln x_n)</math>,
<math display="block">\widehat \mu = \frac {\sum_i \ln x_i}{n}, \qquad \widehat \sigma^2 = \frac {\sum_i {\left( \ln x_i - \widehat \mu \right)}^2} {n}.</math>

For finite ''n'', the estimator for <math>\mu</math> is unbiased, but the one for <math>\sigma</math> is biased. As for the normal distribution, an unbiased estimator for <math>\sigma</math> can be obtained by replacing the denominator ''n'' by ''n''−1 in the equation for <math>\widehat\sigma^2</math>.

From this, the MLE for the expectancy of x is:<ref>Shen, Wei-Hsiung. "Estimation of parameters of a lognormal distribution." ''Taiwanese Journal of Mathematics'' 2.2 (1998): 243–250. [https://projecteuclid.org/journalArticle/Download?urlid=10.11650%2Ftwjm%2F1500406934 pdf] </ref>
<math>
\widehat{\theta}_\text{MLE} = \widehat{\operatorname{E}[X]}_\text{MLE} = e^{\hat \mu + {\hat{\sigma}^2}/{2}}
</math>

==== Method of moments ====
When the individual values <math>x_1, x_2, \ldots, x_n</math> are not available, but the sample's mean <math>\bar x</math> and [[standard deviation]] ''s'' is, then the [[Method of moments (statistics)|method of moments]] can be used. The corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation <math>\operatorname{E}[X]</math> and variance <math>\operatorname{Var}[X]</math> for <math>\mu</math> and <math>\sigma</math>:<ref>Henry (https://math.stackexchange.com/users/6460/henry), Method of moments estimator for lognormal distribution, URL (version: 2022-01-12): https://math.stackexchange.com/q/4355343</ref>
<math display="block"> \begin{align}
\mu &= \ln \frac{ \bar x} {\sqrt{1+\widehat\sigma^2/\bar x^2} } , \\[1ex]
\sigma^2 &= \ln\left(1 + {\widehat\sigma^2} / \bar x^2 \right).
\end{align}</math>

==== Other estimators ====
Other estimators also exist, such as Finney's [[UMVUE]] estimator,<ref>Finney, D. J. "On the distribution of a variate whose logarithm is normally distributed." ''Supplement to the Journal of the Royal Statistical Society'' 7.2 (1941): 155–161.</ref> the "Approximately Minimum Mean Squared Error Estimator", the "Approximately Unbiased Estimator" and "Minimax Estimator",<ref>Longford, Nicholas T. "Inference with the lognormal distribution." ''Journal of Statistical Planning and Inference'' 139.7 (2009): 2329–2340.</ref> also "A Conditional Mean Squared Error Estimator",<ref>Zellner, Arnold. "Bayesian and non-Bayesian analysis of the log-normal distribution and log-normal regression." ''Journal of the American Statistical Association'' 66.334 (1971): 327–330.</ref> and other variations as well.<ref>Tang, Qi. "Comparison of different methods for estimating log-normal means". MS thesis. East Tennessee State University, 2014. [https://www.proquest.com/docview/1547379248?pq-origsite=gscholar&fromopenview=true link]https://dc.etsu.edu/cgi/viewcontent.cgi?article=3728&context=etd#page=12.13 pdf]</ref><ref>Kwon, Yeil. "An alternative method for estimating lognormal means." ''Communications for Statistical Applications and Methods'' 28.4 (2021): 351–368. [http://www.csam.or.kr/journal/view.html?doi=10.29220/CSAM.2021.28.4.351 link]</ref>

===Interval estimates===
{{further|Reference range#Log-normal distribution}}

The most efficient way to obtain [[interval estimate]]s when analyzing log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate.

====Prediction intervals====
A basic example is given by [[prediction interval]]s: For the normal distribution, the interval <math>[\mu-\sigma,\mu+\sigma]</math> contains approximately two thirds (68%) of the probability (or of a large sample), and <math>[\mu-2\sigma,\mu+2\sigma]</math> contain 95%. Therefore, for a log-normal distribution,
* <math>[\mu^*/\sigma^*,\mu^*\cdot\sigma^*]=[\mu^* {}^\times\!\!/ \sigma^*]</math> contains 2/3, and
* <math>[\mu^*/(\sigma^*)^2,\mu^*\cdot(\sigma^*)^2] = [\mu^* {}^\times\!\!/ (\sigma^*)^2]</math> contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals.

====Confidence interval for ''e<sup>μ</sup>''====
Using the principle, note that a [[confidence interval]] for <math>\mu</math> is <math>[\widehat\mu \pm q \cdot \widehat\mathop{se}]</math>, where <math>\mathop{se} = \widehat\sigma / \sqrt{n}</math> is the standard error and ''q'' is the 97.5% quantile of a [[Student's t-distribution|t distribution]] with ''n-1'' degrees of freedom. Back-transformation leads to a confidence interval for <math>\mu^* = e^\mu</math> (the median), is:
<math display="block">[\widehat\mu^* {}^\times\!\!/ (\operatorname{sem}^*)^q]</math> with <math>\operatorname{sem}^*=(\widehat\sigma^*)^{1/\sqrt{n}}</math>

====Confidence interval for {{math|E(''X'')}}====

The literature discusses several options for calculating the [[confidence interval]] for <math>\mu</math> (the mean of the log-normal distribution). These include [[Bootstrapping (statistics)|bootstrap]] as well as various other methods.<ref name = "Olsson2005">Olsson, Ulf. "Confidence intervals for the mean of a log-normal distribution." ''Journal of Statistics Education'' 13.1 (2005).[https://www.tandfonline.com/doi/pdf/10.1080/10691898.2005.11910638 pdf] [https://jse.amstat.org/v13n1/olsson.html html]</ref><ref>user10525, How do I calculate a confidence interval for the mean of a log-normal data set?, URL (version: 2022-12-18): https://stats.stackexchange.com/q/33395</ref>

The Cox Method{{efn|The Cox Method was quoted as "personal communication" in Land, 1971,<ref>Land, C. E. (1971), "Confidence intervals for linear functions of the normal mean and variance," Annals of Mathematical Statistics, 42, 1187–1205.</ref> and was also given in CitationZhou and Gao (1997)<ref>Zhou, X-H., and Gao, S. (1997), "Confidence intervals for the log-normal mean," ''Statistics in Medicine'', 16, 783–790.</ref> and Olsson 2005<ref name = "Olsson2005" />{{rp|Section 3.3}}}} proposes to plug-in the estimators <math display="block">\widehat \mu = \frac {\sum_i \ln x_i}{n}, \qquad S^2 = \frac {\sum_i \left( \ln x_i - \widehat \mu \right)^2} {n-1}</math>

and use them to construct [[Confidence_interval#Approximate_confidence_intervals|approximate confidence intervals]] in the following way:
<math>\mathrm{CI}(\operatorname{E}(X)) : \exp\left(\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)}} \right)</math>

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=[Proof]}}
We know that {{nowrap|<math>\operatorname{E}(X) = e^{\mu + \frac{\sigma^2}{2}}</math>.}} Also, <math>\widehat \mu</math> is a normal distribution with parameters: <math>\widehat \mu \sim N\left(\mu, \frac{\sigma^2}{n}\right)</math>

<math>S^2</math> has a [[chi-squared distribution]], which is [[Chi-squared_distribution#Related_distributions|approximately]] normally distributed (via [[Central limit theorem|CLT]]), with [[Variance#Distribution of the sample variance|parameters]]: {{nowrap|<math>S^2 \dot \sim N\left(\sigma^2, \frac{2\sigma^4}{n-1}\right)</math>.}} Hence, {{nowrap|<math>\frac{S^2}{2} \dot \sim N\left(\frac{\sigma^2}{2}, \frac{\sigma^4}{2(n-1)}\right)</math>.}}

Since the sample mean and variance are independent, and the sum of normally distributed variables is [[Normal distribution#Operations on two independent normal variables|also normal]], we get that:
<math>\widehat \mu + \frac{S^2}{2} \dot \sim N\left(\mu + \frac{\sigma^2}{2}, \frac{\sigma^2}{n} + \frac{\sigma^4}{2(n-1)}\right)</math>
Based on the above, standard [[Normal distribution#Confidence intervals|confidence intervals]] for <math>\mu + \frac{\sigma^2}{2}</math> can be constructed (using a [[Pivotal quantity]]) as: <math>\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)} } </math>
And since confidence intervals are preserved for monotonic transformations, we get that:
<math>\mathrm{CI}\left(\operatorname{E}[X] = e^{\mu + \frac{\sigma^2}{2}}\right): \exp\left(\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)}} \right)</math>

As desired.

{{hidden end}}

Olsson 2005, proposed a "modified Cox method" by replacing <math>z_{1-\frac{\alpha}{2}}</math> with <math>t_{n-1, 1-\frac{\alpha}{2}}</math>, which seemed to provide better coverage results for small sample sizes.<ref name = "Olsson2005" />{{rp|Section 3.4}}

====Confidence interval for comparing two log normals====

Comparing two log-normal distributions can often be of interest, for example, from a treatment and control group (e.g., in an [[A/B testing|A/B test]]). We have samples from two independent log-normal distributions with parameters <math>(\mu_1, \sigma_1^2)</math> and <math>(\mu_2, \sigma_2^2)</math>, with sample sizes <math>n_1</math> and <math>n_2</math> respectively.

Comparing the medians of the two can easily be done by taking the log from each and then constructing straightforward confidence intervals and transforming it back to the exponential scale.

<math display="block">\mathrm{CI}(e^{\mu_1-\mu_2}): \exp\left(\hat \mu_1 - \hat \mu_2 \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S_1^2}{n} + \frac{S_2^2}{n} } \right)</math>

These CI are what's often used in epidemiology for calculation the CI for [[relative-risk]] and [[odds-ratio]].<ref>[https://sphweb.bumc.bu.edu/otlt/MPH-Modules/PH717-QuantCore/PH717-Module8-CategoricalData/PH717-Module8-CategoricalData5.html?fbclid=IwY2xjawFeH3JleHRuA2FlbQIxMAABHbmxa15uyyzJuzEwh9PIUr_m2Jsc9NGiPuS6IwfA36Ca5r1wV1EoPEz3MQ_aem_03PRd_jlRfbsnr6xCPkZmw Confidence Intervals for Risk Ratios and Odds Ratios]</ref> The way it is done there is that we have two approximately Normal distributions (e.g., p<sub>1</sub> and p<sub>2</sub>, for RR), and we wish to calculate their ratio.{{efn|The issue is that we don't know how to do it directly, so we take their logs, and then use the [[delta method]] to say that their logs is itself (approximately) normal. This trick allows us to pretend that their exp was log normal, and use that approximation to build the CI. Notice that in the RR case, the median and the mean in the base distribution (i.e., before taking the log), is actually identical (since they are originally normal, and not log normal).
For example, <math>\hat p_1 \dot \sim N(p_1, p_1(1-p1)/n)</math> and <math>\ln \hat{p}_1 \dot \sim N(\ln p_1, (1-p1)/(p_1 n))</math> Hence, building a CI based on the log and than back-transform will give us <math>CI(p_1): e^{\ln \hat{p}_1 \pm (1 - \hat{p}_1)/(\hat{p}_1 n))}</math>. So while we expect the CI to be for the median, in this case, it's actually also for the mean in the original distribution.
i.e., if the original <math>\hat p_1</math> was log-normal, we'd expect that <math>\operatorname{E}[\hat p_1] = e^{\ln p_1 + \tfrac{1}{2} (1 - p1)/(p_1 n)}</math>. But in practice, we KNOW that <math>\operatorname{E}[\hat p_1] = e^{\ln p_1} = p_1</math>. Hence, the approximation we have is in the second step (of the delta method), but the CI are actually for the expectation (not just the median). This is because we are starting from a base distribution that is normal, and then using another approximation after the log again to normal. This means that a big approximation part of the CI is from the delta method.
}}

However, the ratio of the expectations (means) of the two samples might also be of interest, while requiring more work to develop. The ratio of their means is:

<math display="block">\frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} = \frac{e^{\mu_1 + \sigma_1^2 / 2}}{e^{\mu_2 + \sigma_2^2 /2}}
= e^{(\mu_1 - \mu_2) + \frac{1}{2} \left(\sigma_1^2 - \sigma_2^2\right)}</math>

Plugin in the estimators to each of these parameters yields also a log normal distribution, which means that the Cox Method, discussed above, could similarly be used for this use-case:

<math display="block">\mathrm{CI}\left( \frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} = \frac{e^{\mu_1 + \sigma_1^2 / 2}}{e^{\mu_2 + \sigma_2^2 / 2}} \right):
\exp\left(\left(\hat \mu_1 - \hat \mu_2 + \tfrac{1}{2}S_1^2 - \tfrac{1}{2}S_2^2\right) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } \right)</math>

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=[Proof]}}

To construct a confidence interval for this ratio, we first note that <math>\hat \mu_1 - \hat \mu_2</math> follows a normal distribution, and that both <math>S_1^2</math> and <math>S_2^2</math> has a [[chi-squared distribution]], which is [[Chi-squared distribution#Related distributions|approximately]] normally distributed (via [[Central limit theorem|CLT]], with the relevant [[Variance#Distribution of the sample variance|parameters]]).

This means that
<math display="block">(\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \sim N\left((\mu_1 - \mu_2) + \frac{1}{2}(\sigma_1^2 - \sigma_2^2), \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} + \frac{\sigma_1^4}{2(n_1-1)} + \frac{\sigma_2^4}{2(n_2-1)} \right)</math>

Based on the above, standard [[Normal distribution#Confidence intervals|confidence intervals]] can be constructed (using a [[Pivotal quantity]]) as: <math>(\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } </math>
And since confidence intervals are preserved for monotonic transformations, we get that:
<math>CI\left( \frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} = \frac{e^{\mu_1 + \frac{\sigma_1^2}{2}}}{e^{\mu_2 + \frac{\sigma_2^2}{2}}} \right):e^{\left((\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } \right)}</math>

As desired.

{{hidden end}}

It's worth noting that naively using the [[Maximum likelihood estimation|MLE]] in the ratio of the two expectations to create a [[ratio estimator]] will lead to a [[Consistency (statistics)|consistent]], yet biased, point-estimation (we use the fact that the estimator of the ratio is a log normal distribution):{{efn|The formula can found by just treating the estimated means and variances as approximetly normal, which indicates the terms is itself a log-normal, enabling us to quickly get the expectation. The bias can be partially minimized by using:

<math display="block">\begin{align}
\widehat \left[ \frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} \right] &= \left[ \frac{\widehat \operatorname{E}(X_1)}{\widehat \operatorname{E}(X_2)} \right] \frac{2}{\widehat \left( \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} + \frac{\sigma_1^4}{2(n_1-1)} + \frac{\sigma_2^4}{2(n_2-1)} \right)} \\
&\approx \left[e^{(\widehat \mu_1 - \widehat \mu_2) + \frac{1}{2}\left(S_1^2 - S_2^2\right)}\right] \frac{2}{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)}}
\end{align} </math>}}{{citation needed|date=December 2024}}

<math display="block">\begin{align}
\operatorname{E}\left[ \frac{\widehat \operatorname{E}(X_1)}{\widehat \operatorname{E}(X_2)} \right]
&= \operatorname{E}\left[\exp\left(\left(\widehat \mu_1 - \widehat \mu_2\right) + \tfrac{1}{2} \left(S_1^2 - S_2^2\right)\right)\right] \\
&\approx \exp\left[{(\mu_1 - \mu_2) + \frac{1}{2}(\sigma_1^2 - \sigma_2^2) + \frac{1}{2}\left( \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} + \frac{\sigma_1^4}{2(n_1-1)} + \frac{\sigma_2^4}{2(n_2-1)} \right) }\right]
\end{align} </math>

=== Extremal principle of entropy to fix the free parameter ''σ''===

In applications, <math>\sigma</math> is a parameter to be determined. For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that<ref name="bai">{{cite journal | last1 = Wu | first1 = Ziniu | last2 = Li | first2 = Juan | last3 = Bai | first3 = Chenyuan | title = Scaling Relations of Lognormal Type Growth Process with an Extremal Principle of Entropy | journal = Entropy | volume = 19 | issue = 56 | year = 2017 | pages = 1–14 | doi = 10.3390/e19020056 | bibcode = 2017Entrp..19...56W | doi-access = free}}</ref>
<math display="block">\sigma = \frac 1 \sqrt{6} </math>

This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution.<ref name = bai/> This relationship is determined by the base of natural logarithm, <math>e = 2.718\ldots</math>, and exhibits some geometrical similarity to the minimal surface energy principle.
These scaling relations are useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.).
For example, the log-normal function with such <math>\sigma</math> fits well with the size of secondarily produced droplets during droplet impact <ref name="wu"/> and the spreading of an epidemic disease.<ref name="Wang">{{cite journal | last1 = Wang | first1 = WenBin | last2 = Wu | first2 = ZiNiu | last3 = Wang | first3 = ChunFeng | last4 = Hu | first4 = RuiFeng | title = Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model | journal = Science China Physics, Mechanics and Astronomy | volume = 56 | issue = 11 | year = 2013 | pages = 2143–2150 | issn = 1674-7348 | doi = 10.1007/s11433-013-5321-0 | pmid = 32288765 | pmc = 7111546 | arxiv = 1304.5603 | bibcode = 2013SCPMA..56.2143W}}</ref>

The value <math display="inline">\sigma = 1 \big/ \sqrt{6}</math> is used to provide a probabilistic solution for the Drake equation.<ref name="Bloetscher">{{cite journal | last1 = Bloetscher | first1 = Frederick | title = Using predictive Bayesian Monte Carlo- Markov Chain methods to provide a probabilistic solution for the Drake equation | journal = Acta Astronautica | volume = 155 | year = 2019 | pages = 118–130 | doi = 10.1016/j.actaastro.2018.11.033 | bibcode = 2019AcAau.155..118B | s2cid = 117598888}}</ref>

==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 used 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 possess [[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>

== See also ==
* [[Heavy-tailed distribution]]
* [[Log-distance path loss model]]
* [[Modified lognormal power-law distribution]]
* [[Fading]]

== Notes ==
{{notelist}}

== References ==
{{Reflist|30em|refs=

<ref name="JKB">{{Citation | last1 = Johnson | first1 = Norman L. | last2 = Kotz | first2 = Samuel | last3 = Balakrishnan | first3 = N. | title = Continuous univariate distributions. Vol. 1 | publisher=[[John Wiley & Sons]] | location = New York | edition = 2nd | series = Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics | isbn = 978-0-471-58495-7 | mr = 1299979 | year = 1994 | chapter = 14: Lognormal Distributions}}</ref>

<ref name="Gao">{{cite journal | doi = 10.1155/2009/630857 | volume = 2009 | title = Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables | year = 2009 | journal = International Journal of Mathematics and Mathematical Sciences | pages = 1–28 | last1 = Gao | first1 = Xin | doi-access = free }}</ref>

<ref name="wu">{{cite journal | doi = 10.1016/S0266-8920(03)00028-6 | volume = 18 | issue = 3 | title = Prediction of the size distribution of secondary ejected droplets by crown splashing of droplets impinging on a solid wall | year = 2003 | journal = Probabilistic Engineering Mechanics | pages = 241–249 | last1 = Wu | first1 = Zi-Niu | bibcode = 2003PEngM..18..241W }}</ref>

<ref name="OlofThorin1978LNInfDivi">{{cite journal | last1 = Thorin | first1 = Olof | title = On the infinite divisibility of the lognormal distribution | journal = Scandinavian Actuarial Journal | volume = 1977 | issue = 3 | year = 1977 | pages = 121–148 | issn = 0346-1238 | doi = 10.1080/03461238.1977.10405635}}</ref>

}}

== Further reading ==
* {{Citation | editor-last = Crow | editor-first = Edwin L. | editor-last2 = Shimizu | editor-first2 = Kunio | title = Lognormal Distributions, Theory and Applications | place = New York | publisher = Marcel Dekker, Inc. | series = Statistics: Textbooks and Monographs | volume = 88 | year = 1988 | pages = xvi+387 | isbn = 978-0-8247-7803-3 | mr = 0939191 | zbl = 0644.62014}}
* {{cite book | last1 = Aitchison | first1 = J. | last2 = Brown | first2 = J.A.C. | year = 1957 | title = The Lognormal Distribution | publisher = Cambridge University Press }}
* {{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 }}
* {{Cite journal | last1 = Holgate | first1 = P. | title = The lognormal characteristic function | doi = 10.1080/03610928908830173 | journal = Communications in Statistics – Theory and Methods | volume = 18 | issue = 12 | pages = 4539–4548 | year = 1989 }}
* {{cite journal | last1 = Brooks | first1 = Robert | author-link3 = Jimbo Wales | last2 = Corson | first2 = Jon | last3 = Donal | first3 = Wales | title = The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion | ssrn = 5735 | journal = Advances in Futures and Options Research | volume = 7 | year = 1994 }}

==External links==
{{Commons category}}
*[https://stat.ethz.ch/~stahel/talks/lognormal.pdf The normal distribution is the log-normal distribution]

{{ProbDistributions|continuous-semi-infinite}}
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{{DEFAULTSORT:Log-Normal Distribution}}
[[Category:Continuous distributions]]
[[Category:Normal distribution]]
[[Category:Exponential family distributions]]
[[Category:Infinitely divisible probability distributions]]