Log-normal distribution
Probability density function
Plot of the Lognormal PDF
Identical parameter but differing parameters
Cumulative distribution function
Plot of the Lognormal CDF
Parameters (logarithm of location),
(logarithm of scale)
Excess kurtosis
MGF  defined only for numbers with a
 non-positive real part, see text
CF  representation
 is asymptotically divergent, but adequate
 for most numerical purposes
Fisher information
Method of moments

Expected shortfall [1]

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.[2][3] Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y) , has a log-normal distribution. A random variable that is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in the natural sciences, engineering, as well as medicine, economics and other fields. It can be applied to diverse quantities such as energies, concentrations, lengths, prices of financial instruments, and other metrics, while acknowledging the inherent uncertainty in all measurements.

The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.[4] The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.[4]

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 (sometimes called Gibrat's law). 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.[5]


Generation and parameters

Let be a standard normal variable, and let and be two real numbers, with . Then, the distribution of the random variable

is called the log-normal distribution with parameters and . These are the expected value (or mean) and standard deviation of the variable's natural logarithm, not the expectation and standard deviation of itself.

Relation between normal and log-normal distribution. If is normally distributed, then is log-normally 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 .

In order to produce a distribution with desired mean and variance one uses and

Alternatively, the "multiplicative" or "geometric" parameters and can be used. They have a more direct interpretation: is the median of the distribution, and is useful for determining "scatter" intervals, see below.

Probability density function

A positive random variable is log-normally distributed (i.e., ), if the natural logarithm of is normally distributed with mean and variance

Let and be respectively the cumulative probability distribution function and the probability density function of the standard normal distribution, then we have that[2][4] the probability density function of the log-normal distribution is given by:

Cumulative distribution function

The cumulative distribution function is

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

This may also be expressed as follows:[2]

where erfc is the complementary error function.

Multivariate log-normal

If is a multivariate normal distribution, then has a multivariate log-normal distribution.[6][7] The exponential is applied elementwise to the random vector . The mean of is

and its covariance matrix is

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

This can be derived by letting within the integral. However, the log-normal distribution is not determined by its moments.[8] This implies that it cannot have a defined moment generating function in a neighborhood of zero.[9] Indeed, the expected value is not defined for any positive value of the argument , since the defining integral diverges.

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 hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.[10] In particular, its Taylor formal series diverges:

However, a number of alternative divergent series representations have been obtained.[10][11][12][13]

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 is given by[14]

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 .


a. is a log-normal variable with . is computed by transforming to the normal variable , then integrating its density over the domain defined by (blue regions), using the numerical method of ray-tracing.[15] b & c. The pdf and cdf of the function 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.[15] (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.[15] (Matlab code)

Geometric or multiplicative moments

The geometric or multiplicative mean of the log-normal distribution is . It equals the median. The geometric or multiplicative standard deviation is .[16][17]

By analogy with the arithmetic statistics, one can define a geometric variance, , and a geometric coefficient of variation,[16] , 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 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,


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 moments

For any real or complex number n, the n-th moment of a log-normally distributed variable X is given by[4]

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are respectively given by:[2]

The arithmetic coefficient of variation is the ratio . For a log-normal distribution it is equal to[3]

This estimate is sometimes referred to as the "geometric CV" (GCV),[19][20] 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 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.[4] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.[citation needed]

Mode, median, quantiles

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, by solving the equation , we get that:

Since the log-transformed variable has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of are

where is the quantile of the standard normal distribution.

Specifically, the median of a log-normal distribution is equal to its multiplicative mean,[21]

Partial expectation

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

Alternatively, by 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 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 —with respect to a threshold —is its partial expectation divided by the cumulative probability of being in that range:

Alternative parameterizations

In addition to the characterization by or , here are multiple ways how the log-normal distribution can be parameterized. ProbOnto, the knowledge base and ontology of probability distributions[22][23] lists seven such forms:

Overview of parameterizations of the log-normal distributions.

Examples for re-parameterization

Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM[28] and PopED.[29] 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 and .

For the transition following formulas hold and .

All remaining re-parameterisation formulas can be found in the specification document on the project website.[30]

Multiple, reciprocal, power

Multiplication and division of independent, log-normal random variables

If two independent, log-normal variables and are multiplied [divided], the product [ratio] is again log-normal, with parameters [] and , where . This is easily generalized to the product of such variables.

More generally, if are independent, log-normally distributed variables, then

Multiplicative central limit theorem

See also: Gibrat's law

The geometric or multiplicative mean of independent, identically distributed, positive random variables shows, for , approximately a log-normal distribution with parameters and , assuming is finite.

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of 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.


A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).[31]

The harmonic , geometric and arithmetic means of this distribution are related;[32] such relation is given by

Log-normal distributions are infinitely divisible,[33] but they are not stable distributions, which can be easily drawn from.[34]

Related distributions

For a more accurate approximation, one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.[37][38]

The sum of correlated log-normally distributed random variables can also be approximated by a log-normal distribution[citation needed]

Statistical inference

Estimation of parameters

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. Note that where is the density function of the normal distribution . Therefore, the log-likelihood function is

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, the maximum likelihood estimators are identical to those for a normal distribution for the observations ,

For finite n, the estimator for is unbiased, but the one for is biased. As for the normal distribution, an unbiased estimator for can be obtained by replacing the denominator n by n−1 in the equation for .

When the individual values are not available, but the sample's mean and standard deviation s is, then the Method of moments can be used. The corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation and variance for and :

Interval estimates

Further information: Reference range § Log-normal distribution

The most efficient way to obtain interval estimates 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 intervals: For the normal distribution, the interval contains approximately two thirds (68%) of the probability (or of a large sample), and contain 95%. Therefore, for a log-normal distribution, contains 2/3, and 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 μ*

Using the principle, note that a confidence interval for is , where is the standard error and q is the 97.5% quantile of a t distribution with n-1 degrees of freedom. Back-transformation leads to a confidence interval for (the median), is: with

Confidence interval for μ

This section needs expansion with: adding the relevant formulas (based on the existing references). You can help by adding to it. (May 2024)

The literature discusses several options for calculating the confidence interval for (the mean of the log-normal distribution). These include bootstrap as well as various other methods.[42][43]

Extremal principle of entropy to fix the free parameter σ

In applications, 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[44]

This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution.[44] 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 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 fits well with the size of secondarily produced droplets during droplet impact [45] and the spreading of an epidemic disease.[46]

The value is used to provide a probabilistic solution for the Drake equation.[47]

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". This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies.[48] 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] 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]

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.[49] contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas.[50] is a review article on log-normal distributions in neuroscience, with annotated bibliography.

Human behavior

Biology and medicine


Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see distribution fitting


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.[68]
The rainfall data are represented by plotting positions as part of a cumulative frequency analysis.

Social sciences and demographics


See also


  1. ^ Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF). Annals of Operations Research. 299 (1–2). Springer: 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1. S2CID 254231768. Archived (PDF) from the original on 2021-04-18. Retrieved 2023-02-27 – via stonybrook.edu.
  2. ^ a b c d Weisstein, Eric W. "Log Normal Distribution". mathworld.wolfram.com. Retrieved 2020-09-13.
  3. ^ a b " Lognormal Distribution". www.itl.nist.gov. U.S. National Institute of Standards and Technology (NIST). Retrieved 2020-09-13.
  4. ^ a b c d e Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), "14: Lognormal Distributions", Continuous univariate distributions. Vol. 1, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-58495-7, MR 1299979
  5. ^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics. 150 (2): 219–230, esp. Table 1, p. 221. CiteSeerX doi:10.1016/j.jeconom.2008.12.014. Archived from the original (PDF) on 2016-03-07. Retrieved 2011-06-02.
  6. ^ Tarmast, Ghasem (2001). Multivariate Log–Normal Distribution (PDF). ISI Proceedings: 53rd Session. Seoul. Archived (PDF) from the original on 2013-07-19.
  7. ^ Halliwell, Leigh (2015). The Lognormal Random Multivariate (PDF). Casualty Actuarial Society E-Forum, Spring 2015. Arlington, VA. Archived (PDF) from the original on 2015-09-30.
  8. ^ Heyde, CC. (2010), "On a Property of the Lognormal Distribution", Journal of the Royal Statistical Society, Series B, vol. 25, no. 2, pp. 392–393, doi:10.1007/978-1-4419-5823-5_6, ISBN 978-1-4419-5822-8
  9. ^ Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Hoboken, N.J.: Wiley. p. 415. ISBN 978-1-118-12237-2. OCLC 780289503.
  10. ^ a b Holgate, P. (1989). "The lognormal characteristic function, vol. 18, pp. 4539–4548, 1989". Communications in Statistics - Theory and Methods. 18 (12): 4539–4548. doi:10.1080/03610928908830173.
  11. ^ Barakat, R. (1976). "Sums of independent lognormally distributed random variables". Journal of the Optical Society of America. 66 (3): 211–216. Bibcode:1976JOSA...66..211B. doi:10.1364/JOSA.66.000211.
  12. ^ Barouch, E.; Kaufman, GM.; Glasser, ML. (1986). "On sums of lognormal random variables" (PDF). Studies in Applied Mathematics. 75 (1): 37–55. doi:10.1002/sapm198675137. hdl:1721.1/48703.
  13. ^ Leipnik, Roy B. (January 1991). "On Lognormal Random Variables: I – The Characteristic Function" (PDF). Journal of the Australian Mathematical Society, Series B. 32 (3): 327–347. doi:10.1017/S0334270000006901.
  14. ^ S. Asmussen, J.L. Jensen, L. Rojas-Nandayapa (2016). "On the Laplace transform of the Lognormal distribution", Methodology and Computing in Applied Probability 18 (2), 441-458. Thiele report 6 (13).
  15. ^ a b c Das, Abhranil (2021). "A method to integrate and classify normal distributions". Journal of Vision. 21 (10): 1. arXiv:2012.14331. doi:10.1167/jov.21.10.1. PMC 8419883. PMID 34468706.
  16. ^ a b Kirkwood, Thomas BL (Dec 1979). "Geometric means and measures of dispersion". Biometrics. 35 (4): 908–9. JSTOR 2530139.
  17. ^ 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.
  18. ^ a b Heil P, Friedrich B (2017). "Onset-Duration Matching of Acoustic Stimuli Revisited: Conventional Arithmetic vs. Proposed Geometric Measures of Accuracy and Precision". Frontiers in Psychology. 7: 2013. doi:10.3389/fpsyg.2016.02013. PMC 5216879. PMID 28111557.
  19. ^ Sawant, S.; Mohan, N. (2011) "FAQ: Issues with Efficacy Analysis of Clinical Trial Data Using SAS" Archived 24 August 2011 at the Wayback Machine, PharmaSUG2011, Paper PO08
  20. ^ Schiff, MH; et al. (2014). "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". Ann Rheum Dis. 73 (8): 1–3. doi:10.1136/annrheumdis-2014-205228. PMC 4112421. PMID 24728329.
  21. ^ Daly, Leslie E.; Bourke, Geoffrey Joseph (2000). Interpretation and Uses of Medical Statistics. Vol. 46 (5th ed.). Oxford, UK: Wiley-Blackwell. p. 89. doi:10.1002/9780470696750. ISBN 978-0-632-04763-5. PMC 1059583; ((cite book)): |journal= ignored (help) print edition. Online eBook ISBN 9780470696750
  22. ^ "ProbOnto". Retrieved 1 July 2017.
  23. ^ Swat, MJ; Grenon, P; Wimalaratne, S (2016). "ProbOnto: ontology and knowledge base of probability distributions". Bioinformatics. 32 (17): 2719–21. doi:10.1093/bioinformatics/btw170. PMC 5013898. PMID 27153608.
  24. ^ a b Forbes et al. Probability Distributions (2011), John Wiley & Sons, Inc.
  25. ^ Lunn, D. (2012). The BUGS book: a practical introduction to Bayesian analysis. Texts in statistical science. CRC Press.
  26. ^ Limpert, E.; Stahel, W. A.; Abbt, M. (2001). "Log-normal 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.
  27. ^ Nyberg, J.; et al. (2012). "PopED - An extended, parallelized, population optimal design tool". Comput Methods Programs Biomed. 108 (2): 789–805. doi:10.1016/j.cmpb.2012.05.005. PMID 22640817.
  28. ^ Retout, S; Duffull, S; Mentré, F (2001). "Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs". Comp Meth Pro Biomed. 65 (2): 141–151. doi:10.1016/S0169-2607(00)00117-6. PMID 11275334.
  29. ^ The PopED Development Team (2016). PopED Manual, Release version 2.13. Technical report, Uppsala University.
  30. ^ ProbOnto website, URL: http://probonto.org
  31. ^ Damgaard, Christian; Weiner, Jacob (2000). "Describing inequality in plant size or fecundity". Ecology. 81 (4): 1139–1142. doi:10.1890/0012-9658(2000)081[1139:DIIPSO]2.0.CO;2.
  32. ^ Rossman, Lewis A (July 1990). "Design stream flows based on harmonic means". Journal of Hydraulic Engineering. 116 (7): 946–950. doi:10.1061/(ASCE)0733-9429(1990)116:7(946).
  33. ^ Thorin, Olof (1977). "On the infinite divisibility of the lognormal distribution". Scandinavian Actuarial Journal. 1977 (3): 121–148. doi:10.1080/03461238.1977.10405635. ISSN 0346-1238.
  34. ^ a b Gao, Xin (2009). "Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables". International Journal of Mathematics and Mathematical Sciences. 2009: 1–28. doi:10.1155/2009/630857.
  35. ^ Asmussen, S.; Rojas-Nandayapa, L. (2008). "Asymptotics of Sums of Lognormal Random Variables with Gaussian Copula" (PDF). Statistics and Probability Letters. 78 (16): 2709–2714. doi:10.1016/j.spl.2008.03.035.
  36. ^ Marlow, NA. (Nov 1967). "A normal limit theorem for power sums of independent normal random variables". Bell System Technical Journal. 46 (9): 2081–2089. doi:10.1002/j.1538-7305.1967.tb04244.x.
  37. ^ Botev, Z. I.; L'Ecuyer, P. (2017). "Accurate computation of the right tail of the sum of dependent log-normal variates". 2017 Winter Simulation Conference (WSC), 3rd–6th Dec 2017. Las Vegas, NV, USA: IEEE. pp. 1880–1890. arXiv:1705.03196. doi:10.1109/WSC.2017.8247924. ISBN 978-1-5386-3428-8.
  38. ^ Asmussen, A.; Goffard, P.-O.; Laub, P. J. (2016). "Orthonormal polynomial expansions and lognormal sum densities". arXiv:1601.01763v1 [math.PR].
  39. ^ Sangal, B.; Biswas, A. (1970). "The 3-Parameter Lognormal Distribution Applications in Hydrology". Water Resources Research. 6 (2): 505–515. doi:10.1029/WR006i002p00505.
  40. ^ Johnson, N. L. (1949). "Systems of Frequency Curves Generated by Methods of Translation". Biometrika. 36 (1/2): 149–176. doi:10.2307/2332539. JSTOR 2332539. PMID 18132090.
  41. ^ Swamee, P. K. (2002). "Near Lognormal Distribution". Journal of Hydrologic Engineering. 7 (6): 441–444. doi:10.1061/(ASCE)1084-0699(2002)7:6(441).
  42. ^ Olsson, Ulf. "Confidence intervals for the mean of a log-normal distribution." Journal of Statistics Education 13.1 (2005).pdf html
  43. ^ 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
  44. ^ a b Wu, Ziniu; Li, Juan; Bai, Chenyuan (2017). "Scaling Relations of Lognormal Type Growth Process with an Extremal Principle of Entropy". Entropy. 19 (56): 1–14. Bibcode:2017Entrp..19...56W. doi:10.3390/e19020056.
  45. ^ Wu, Zi-Niu (2003). "Prediction of the size distribution of secondary ejected droplets by crown splashing of droplets impinging on a solid wall". Probabilistic Engineering Mechanics. 18 (3): 241–249. Bibcode:2003PEngM..18..241W. doi:10.1016/S0266-8920(03)00028-6.
  46. ^ Wang, WenBin; Wu, ZiNiu; Wang, ChunFeng; Hu, RuiFeng (2013). "Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model". Science China Physics, Mechanics and Astronomy. 56 (11): 2143–2150. arXiv:1304.5603. Bibcode:2013SCPMA..56.2143W. doi:10.1007/s11433-013-5321-0. ISSN 1674-7348. PMC 7111546. PMID 32288765.
  47. ^ Bloetscher, Frederick (2019). "Using predictive Bayesian Monte Carlo- Markov Chain methods to provide a probabilistic solution for the Drake equation". Acta Astronautica. 155: 118–130. Bibcode:2019AcAau.155..118B. doi:10.1016/j.actaastro.2018.11.033. S2CID 117598888.
  48. ^ Sutton, John (Mar 1997). "Gibrat's Legacy". Journal of Economic Literature. 32 (1): 40–59. JSTOR 2729692.
  49. ^ a b c Limpert, Eckhard; Stahel, Werner A.; Abbt, Markus (2001). "Log-normal Distributions across the Sciences: Keys and Clues". BioScience. 51 (5): 341. doi:10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2. ISSN 0006-3568.
  50. ^ a b Buzsáki, György; Mizuseki, Kenji (2017-01-06). "The log-dynamic brain: how skewed distributions affect network operations". Nature Reviews. Neuroscience. 15 (4): 264–278. doi:10.1038/nrn3687. ISSN 1471-003X. PMC 4051294. PMID 24569488.
  51. ^ a b c Pawel, Sobkowicz; et al. (2013). "Lognormal distributions of user post lengths in Internet discussions - a consequence of the Weber-Fechner law?". EPJ Data Science.
  52. ^ Yin, Peifeng; Luo, Ping; Lee, Wang-Chien; Wang, Min (2013). Silence is also evidence: interpreting dwell time for recommendation from psychological perspective. ACM International Conference on KDD.
  53. ^ "What is the average length of a game of chess?". chess.stackexchange.com. Retrieved 14 April 2018.
  54. ^ Huxley, Julian S. (1932). Problems of relative growth. London. ISBN 978-0-486-61114-3. OCLC 476909537.
  55. ^ Sartwell, Philip E. "The distribution of incubation periods of infectious disease." American journal of hygiene 51 (1950): 310-318.
  56. ^ S. K. Chan, Jennifer; Yu, Philip L. H. (2006). "Modelling SARS data using threshold geometric process". Statistics in Medicine. 25 (11): 1826–1839. doi:10.1002/sim.2376. PMID 16345017. S2CID 46599163.
  57. ^ Ono, Yukiteru; Asai, Kiyoshi; Hamada, Michiaki (2013-01-01). "PBSIM: PacBio reads simulator—toward accurate genome assembly". Bioinformatics. 29 (1): 119–121. doi:10.1093/bioinformatics/bts649. ISSN 1367-4803. PMID 23129296.
  58. ^ Makuch, Robert W.; D.H. Freeman; M.F. Johnson (1979). "Justification for the lognormal distribution as a model for blood pressure". Journal of Chronic Diseases. 32 (3): 245–250. doi:10.1016/0021-9681(79)90070-5. PMID 429469.
  59. ^ Lacey, L. F.; Keene, O. N.; Pritchard, J. F.; Bye, A. (1997-01-01). "Common noncompartmental pharmacokinetic variables: are they normally or log-normally distributed?". Journal of Biopharmaceutical Statistics. 7 (1): 171–178. doi:10.1080/10543409708835177. ISSN 1054-3406. PMID 9056596.
  60. ^ Scheler, Gabriele; Schumann, Johann (2006-10-08). Diversity and stability in neuronal output rates. 36th Society for Neuroscience Meeting, Atlanta.
  61. ^ Mizuseki, Kenji; Buzsáki, György (2013-09-12). "Preconfigured, skewed distribution of firing rates in the hippocampus and entorhinal cortex". Cell Reports. 4 (5): 1010–1021. doi:10.1016/j.celrep.2013.07.039. ISSN 2211-1247. PMC 3804159. PMID 23994479.
  62. ^ Wohrer, Adrien; Humphries, Mark D.; Machens, Christian K. (2013-04-01). "Population-wide distributions of neural activity during perceptual decision-making". Progress in Neurobiology. 103: 156–193. doi:10.1016/j.pneurobio.2012.09.004. ISSN 1873-5118. PMC 5985929. PMID 23123501.
  63. ^ Scheler, Gabriele (2017-07-28). "Logarithmic distributions prove that intrinsic learning is Hebbian". F1000Research. 6: 1222. doi:10.12688/f1000research.12130.2. PMC 5639933. PMID 29071065.
  64. ^ Morales-Gregorio, Aitor; van Meegen, Alexander; van Albada, Sacha (2023). "Ubiquitous lognormal distribution of neuron densities in mammalian cerebral cortex". Cerebral Cortex. 33 (16): 9439–9449. doi:10.1093/cercor/bhad160. PMC 10438924. PMID 37409647.
  65. ^ Polizzi, Stefano; Laperrousaz, Bastien; Perez-Reche, Francisco J; Nicolini, Franck E; Satta, Véronique Maguer; Arneodo, Alain; Argoul, Françoise (2018-05-29). "A minimal rupture cascade model for living cell plasticity". New Journal of Physics. 20 (5): 053057. Bibcode:2018NJPh...20e3057P. doi:10.1088/1367-2630/aac3c7. hdl:2164/10561. ISSN 1367-2630.
  66. ^ Ahrens, L. H. (1954-02-01). "The lognormal distribution of the elements (A fundamental law of geochemistry and its subsidiary)". Geochimica et Cosmochimica Acta. 5 (2): 49–73. Bibcode:1954GeCoA...5...49A. doi:10.1016/0016-7037(54)90040-X. ISSN 0016-7037.
  67. ^ Oosterbaan, R.J. (1994). "6: Frequency and Regression Analysis" (PDF). In Ritzema, H.P. (ed.). Drainage Principles and Applications, Publication 16. Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp. 175–224. ISBN 978-90-70754-33-4.
  68. ^ CumFreq, free software for distribution fitting
  69. ^ Clementi, Fabio; Gallegati, Mauro (2005) "Pareto's law of income distribution: Evidence for Germany, the United Kingdom, and the United States", EconWPA
  70. ^ Wataru, Souma (2002-02-22). "Physics of Personal Income". In Takayasu, Hideki (ed.). Empirical Science of Financial Fluctuations: The Advent of Econophysics. Springer. arXiv:cond-mat/0202388. doi:10.1007/978-4-431-66993-7.
  71. ^ Black, F.; Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy. 81 (3): 637. doi:10.1086/260062. S2CID 154552078.
  72. ^ Mandelbrot, Benoit (2004). The (mis-)Behaviour of Markets. Basic Books. ISBN 9780465043552.
  73. ^ Bunchen, P., Advanced Option Pricing, University of Sydney coursebook, 2007
  74. ^ Thelwall, Mike; Wilson, Paul (2014). "Regression for citation data: An evaluation of different methods". Journal of Informetrics. 8 (4): 963–971. arXiv:1510.08877. doi:10.1016/j.joi.2014.09.011. S2CID 8338485.
  75. ^ Sheridan, Paul; Onodera, Taku (2020). "A Preferential Attachment Paradox: How Preferential Attachment Combines with Growth to Produce Networks with Log-normal In-degree Distributions". Scientific Reports. 8 (1): 2811. arXiv:1703.06645. doi:10.1038/s41598-018-21133-2. PMC 5809396. PMID 29434232.
  76. ^ Eeckhout, Jan (2004). "Gibrat's Law for (All) Cities". American Economic Review. 94 (5): 1429–1451. doi:10.1257/0002828043052303. JSTOR 3592829 – via JSTOR.
  77. ^ Kault, David (1996). "The Shape of the Distribution of the Number of Sexual Partners". Statistics in Medicine. 15 (2): 221–230. doi:10.1002/(SICI)1097-0258(19960130)15:2<221::AID-SIM148>3.0.CO;2-Q. PMID 8614756.
  78. ^ O'Connor, Patrick; Kleyner, Andre (2011). Practical Reliability Engineering. John Wiley & Sons. p. 35. ISBN 978-0-470-97982-2.
  79. ^ "Shadowing". www.WirelessCommunication.NL. Archived from the original on January 13, 2012.
  80. ^ Dexter, A. R.; Tanner, D. W. (July 1972). "Packing Densities of Mixtures of Spheres with Log-normal Size Distributions". Nature Physical Science. 238 (80): 31–32. Bibcode:1972NPhS..238...31D. doi:10.1038/physci238031a0. ISSN 2058-1106.
  81. ^ Gros, C; Kaczor, G.; Markovic, D (2012). "Neuropsychological constraints to human data production on a global scale". The European Physical Journal B. 85 (28): 28. arXiv:1111.6849. Bibcode:2012EPJB...85...28G. doi:10.1140/epjb/e2011-20581-3. S2CID 17404692.
  82. ^ Douceur, John R.; Bolosky, William J. (1999-05-01). "A large-scale study of file-system contents". ACM SIGMETRICS Performance Evaluation Review. 27 (1): 59–70. doi:10.1145/301464.301480. ISSN 0163-5999.
  83. ^ Alamsar, Mohammed; Parisis, George; Clegg, Richard; Zakhleniuk, Nickolay (2019). "On the Distribution of Traffic Volumes in the Internet and its Implications". arXiv:1902.03853 [cs.NI].
  84. ^ ASTM D3654, Standard Test Method for Shear Adhesion on Pressure-Sensitive Tapesw
  85. ^ ASTM D4577, Standard Test Method for Compression Resistance of a container Under Constant Load>\

Further reading