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In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Likewise, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.[1] The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.[1]

Probability density function
Plot of the Lognormal PDF
Some log-normal density functions with identical parameter but differing parameters
Cumulative distribution function
Plot of the Lognormal CDF
Cumulative distribution function of the log-normal distribution (with )
Ex. kurtosis
MGFdefined only for numbers with a non-positive real part, see text
CFrepresentation is asymptotically divergent but sufficient for numerical purposes
Fisher information

A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain. The log-normal distribution is the maximum entropy probability distribution for a random variate X for which the mean and variance of ln(X) are specified.[2]



Given a log-normally distributed random variable   and two parameters   and   that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of   is normally distributed, and we can write   as


with   a standard normal variable.

Relation between normal and lognormal distribution. If   is normally distributed, then   is lognormally distributed.

This relationship is true regardless of the base of the logarithmic or exponential function. If   is normally distributed, then so is  , for any two positive numbers  . Likewise, if   is log-normally distributed, then so is  , where  .

The two parameters   and   are not location and scale parameters for a lognormally distributed random variable X, but they are respectively location and scale parameters for the normally distributed logarithm ln(X). The quantity eμ is a scale parameter for the family of lognormal distributions.

In contrast, the mean and variance of the non-logarithmized sample values are respectively denoted  , and   in this article. The two sets of parameters can be related as (see also Arithmetic moments below)[3]



Probability density functionEdit

A positive random variable X is log-normally distributed if the logarithm of X is normally distributed,


Let   and   be respectively the cumulative probability distribution function and the probability density function of the N(0,1) distribution.

Then we have[1]


Cumulative distribution functionEdit

The cumulative distribution function is


where   is the cumulative distribution function of the standard normal distribution (i.e. N(0,1)).

This may also be expressed as follows:


where erfc is the complementary error function.

Characteristic function and moment generating functionEdit

All moments of the log-normal distribution exist and


This can be derived by letting   within the integral. However, the expected value   is not defined for any positive value of the argument   as the defining integral diverges. In consequence the moment generating function is not defined.[4] The last is related to the fact that the lognormal distribution is not uniquely determined by its moments.

The characteristic function   is defined for real values of t but is not defined for any complex value of t that has a negative imaginary part, and therefore the characteristic function is not analytic at the origin. In consequence, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.[5] In particular, its Taylor formal series diverges:


However, a number of alternative divergent series representations have been obtained[5][6][7][8]

A closed-form formula for the characteristic function   with   in the domain of convergence is not known. A relatively simple approximating formula is available in closed form and given by[9]


where   is the Lambert W function. This approximation is derived via an asymptotic method but it stays sharp all over the domain of convergence of  .


Let   denote the geometric mean, and   the geometric standard deviation of the random variable  , and let   and   be the arithmetic mean, or expected value, and the arithmetic standard deviation as usual.

Geometric momentsEdit

The geometric mean of the log-normal distribution is  , and the geometric standard deviation is  .[10][11] By analogy with the arithmetic statistics, one can define a geometric variance,  , and a geometric coefficient of variation,[10]  .

Because the log-transformed variable   is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median,  .[12]

Note that the geometric mean is less than the arithmetic mean. This is due to the AM–GM inequality, and corresponds to the logarithm being convex down. In fact,


In finance the term   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 for geometric Brownian motion.

Arithmetic momentsEdit

For any real or complex number n, the n-th moment of a log-normally distributed variable X is given by[1]


Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are given by



The parameters μ and σ can be obtained if the arithmetic mean and the arithmetic variance are known:


A probability distribution is not uniquely determined by the moments E[Xn] = e + 1/2n2σ2 for n ≥ 1. That is, there exist other distributions with the same set of moments.[1] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.[citation needed]

Mode and medianEdit

Comparison of mean, median and mode of two log-normal distributions with different skewness.

The mode is the point of global maximum of the probability density function. In particular, it solves the equation  :


The median (the value   for which  ) is


Arithmetic coefficient of variationEdit

The arithmetic coefficient of variation   is the ratio   (on the natural scale). For a log-normal distribution it is equal to


Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

Partial expectationEdit

The partial expectation of a random variable   with respect to a threshold   is defined as


Alternatively, and using the definition of conditional expectation, it can be written as  . For a log-normal random variable the partial expectation is given by:


where   is the normal cumulative distribution function. The derivation of the formula is provided in the discussion of this Wikipedia entry. 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 expectationEdit

The conditional expectation of a lognormal random variable   with respect to a threshold   is its partial expectation divided by the cumulative probability of being in that range:


Alternative parameterizationsEdit

There are multiple ways how the lognormal distribution can be parameterized. ProbOnto, the knowledge base and ontology of probability distributions[14][15] lists seven such forms:

Overview of parameterizations of the log-normal distributions.
  • LogNormal2(μ,υ) with mean, μ, and variance, υ, both on the log-scale
  • LogNormal3(m,σ) with median, m, on the natural scale and standard deviation, σ, on the log-scale[16]
  • LogNormal5(μ,τ) with mean, μ, and precision, τ, both on the log-scale[17]
  • LogNormal7(μNN) with mean, μN, and standard deviation, σN, both on the natural scale[19]
Examples for re-parameterizationEdit

Consider the situation when one would like to run a model using two different optimal design tools, e.g. PFIM[20] and PopED.[21] 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   following formulas hold  .

For the transition   following formulas hold  .

All remaining re-parameterisation formulas can be found in the specification document on the project website.[22]


A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).[23]

The harmonic  , geometric   and arithmetic   means of this distribution are related;[24] such relation is given by


Log-normal distributions are infinitely divisible,[25] but they are not stable distributions, which can be easily drawn from.[26]

Occurrence and applicationsEdit

The log-normal distribution is important in the description of natural phenomena. This follows, because many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale. If the effect of any one change is negligible, the central limit theorem says that the distribution of their sum is more nearly normal than that of the summands. When back-transformed onto the original scale, it makes the distribution of sizes approximately log-normal (though if the standard deviation is sufficiently small, the normal distribution can be an adequate approximation).

This multiplicative version of the central limit theorem is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies.[27] If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if that's not true, the size distributions at any age of things that grow over time tends to be log-normal.

Examples include the following:

  • Human behaviors
    • The length of comments posted in Internet discussion forums follows a log-normal distribution.[28]
    • Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.[29]
    • The length of chess games tends to follow a log normal distribution.[30]
    • Onset durations of acoustic comparison stimuli that are matched to a standard stimulus follow a log-normal distribution.[13]
  • In biology and medicine,
    • Measures of size of living tissue (length, skin area, weight);[31]
    • For highly communicable epidemics, such as SARS in 2003, if publication intervention is involved, the number of hospitalized cases is shown to satisfy the lognormal 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.[citation needed]
    • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth;[citation needed]
    • The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.
    • Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations)[32]
    • In neuroscience, the distribution of firing rates across a population of neurons is often approximately lognormal. This has been first observed in the cortex and striatum [33] and later in hippocampus and entorhinal cortex,[34] and elsewhere in the brain.[35][36] Also, intrinsic gain distributions and synaptic weight distributions appear to be lognormal[37] as well.
  • In colloidal chemistry and polymer chemistry

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.

Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see distribution fitting
  • 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.[38]
The image on the right, made with CumFreq, 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.[39]
The rainfall data are represented by plotting positions as part of a cumulative frequency analysis.
  • In social sciences and demographics
  • Technology
    • In reliability analysis, the lognormal distribution is often used to model times to repair a maintainable system.[46]
    • In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution."[47] Also, the random obstruction of radio signals due to large buildings and hills, called shadowing, is often modeled as a lognormal distribution.
    • Particle size distributions produced by comminution with random impacts, such as in ball milling
    • The file size distribution of publicly available audio and video data files (MIME types) follows a log-normal distribution over five orders of magnitude.[48]
    • In computer networks and Internet traffic analysis, lognormal 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 Interent 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.[49]

Extremal principle of entropy to fix the free parameter Edit

  • In applications,   is a parameter to be determined. For growing processes balanced by production and dissipation, the use of a extremal principle of Shannon entropy shows that
  • This value can then be used to give some scaling relation between the inflexion point and maximum point of the lognormal distribution.[50] It is shown that this relationship is determined by the base of natural logarithm,  , and exhibits some geometrical similarity to the minimal surface energy principle.
  • These scaling relations are shown to be 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 instance, the lognormal function with such   fits well with the size of secondary produced droplet during droplet impact [51] and the spreading of one epidemic disease.[52]
  • The value   is used to provide a probabilistic solution for the Drake equation.[53]

Maximum likelihood estimation of parametersEdit

For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. To avoid repetition, we observe that


where   is the density function of the normal distribution  . Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:


Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions,   and  , reach their maximum with the same   and  . Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the log-normal distribution it holds that


Multivariate log-normalEdit

If   is a multivariate normal distribution then   has a multivariate log-normal distribution[54][55] with mean


and covariance matrix


Related distributionsEdit

  • If   is a normal distribution, then  
  • If   is distributed log-normally, then   is a normal random variable.
  • If   are   independent log-normally distributed variables, and  , then   is also distributed log-normally:
  • Let   be independent log-normally distributed variables with possibly varying   and   parameters, and  . The distribution of   has no closed-form expression, but can be reasonably approximated by another log-normal distribution   at the right tail.[56] Its probability density function at the neighborhood of 0 has been characterized[26] 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 mathematical justified by Marlow[57]) is obtained by matching the mean and variance of another lognormal distribution:
In the case that all   have the same variance parameter  , these formulas simplify to

For a more accurate approximation one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and right tail. [58][59]

  • If   then   is said to have a Three-parameter log-normal distribution with support  .[60]  ,  .
  • If   then  
  • If   then  
  • If   then   for  
  • Lognormal distribution is a special case of semi-bounded Johnson distribution
  • If   with  , then   (Suzuki distribution)
  • A substitute for the log-normal whose integral can be expressed in terms of more elementary functions[61] can be obtained based on the logistic distribution to get an approximation for the CDF
This is a log-logistic distribution.

See alsoEdit


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Further readingEdit

  • Crow, Edwin L.; Shimizu, Kunio, eds. (1988), Lognormal Distributions, Theory and Applications, Statistics: Textbooks and Monographs, 88, New York: Marcel Dekker, Inc., pp. xvi+387, ISBN 978-0-8247-7803-3, MR 0939191, Zbl 0644.62014
  • Aitchison, J. and Brown, J.A.C. (1957) The Lognormal Distribution, Cambridge University Press.
  • Limpert, E; Stahel, W; Abbt, M (2001). "Lognormal distributions across the sciences: keys and clues". BioScience. 51 (5): 341–352. doi:10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2.
  • Eric W. Weisstein et al. Log Normal Distribution at MathWorld. Electronic document, retrieved October 26, 2006.
  • Holgate, P. (1989). "The lognormal characteristic function". Communications in Statistics - Theory and Methods. 18 (12): 4539–4548. doi:10.1080/03610928908830173.
  • Brooks, Robert; Corson, Jon; Donal, Wales (1994). "The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion". Advances in Futures and Options Research. 7. SSRN 5735.

External linksEdit