# The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

# Lognormal Distribution

• P. E. Hart
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1089

## Abstract

If there is a number, ϴ, such that Y = loge(Xϴ) is normally distributed, the distribution of X is lognormal. The important special case of ϴ = 0 gives the two parameter lognormal distribution, X ~ Λ(μ, σ2) with Y ~ N(μ, σ2), where μ and σ2 denote the mean and variance of logeX. The classic work on the subject is by Aitchison and Brown (1957). A useful survey is provided by Johnson et al. (1994, ch. 14). They also summarize the history of this distribution: the pioneer contributions by Galton (1879) on its genesis, and by McAlister (1879) on its measures of location and dispersion, were followed by Kapteyn (1903), who studied its genesis in more detail and also devised an analogue machine to generate it. Gibrat’s (1931) study of economic size distributions was a most important development because of his law of proportionate effect. Since then there has been an immense number of applications of the lognormal distribution in the natural, behavioural and social sciences.

## Keywords

Central limit theorems Gibrat’s Law Lognormal distribution

## JEL Classifications

C1

If there is a number, ϴ, such that Y = loge(Xϴ) is normally distributed, the distribution of X is lognormal. The important special case of ϴ = 0 gives the two parameter lognormal distribution, X ~ Λ(μ, σ2) with Y ~ N(μ, σ2), where μ and σ2 denote the mean and variance of logeX. The classic work on the subject is by Aitchison and Brown (1957). A useful survey is provided by Johnson et al. (1994, ch. 14). They also summarize the history of this distribution: the pioneer contributions by Galton (1879) on its genesis, and by McAlister (1879) on its measures of location and dispersion, were followed by Kapteyn (1903), who studied its genesis in more detail and also devised an analogue machine to generate it. Gibrat’s (1931) study of economic size distributions was a most important development because of his law of proportionate effect. Since then there has been an immense number of applications of the lognormal distribution in the natural, behavioural and social sciences.

Why does the lognormal distribution appear to occur so frequently? One plausible answer is based on the central limit theorems used to explain the genesis of a normal curve. If a large number of random shocks, some positive, some negative, change the size of a particular variable, X, in an additive fashion, the distribution of that variable will tend to become normal as the number of shocks increases. But if these shocks act multiplicatively, changing the value of X by randomly distributed proportions instead of absolute amounts, the central limit theorems apply to Y = logeX which tends to be normally distributed. Hence X has a lognormal distribution.

The substitution of multiplicative for additive random shocks generates a positively skew, leptokurtic, lognormal distribution instead of the symmetric, mesokurtic normal curve. But the degree of skewness and kurtosis of the two-parameter lognormal curve depends solely on σ2, so if this is low enough, the lognormal approximates the normal curve. The important difference is that X cannot take zero or negative values which may make the lognormal distribution a more appropriate representation of variables, such as height and weight, which must take positive values. Clearly, the widespread occurrence of positive variables in practice, coupled with the great flexibility of the shape of the lognormal, provide further reasons for its frequent application.

## Bibliography

1. Aitchison, J., and J.A.C. Brown. 1957. The lognormal distribution. Cambridge: Cambridge University Press.Google Scholar
2. Galton, F. 1879. The geometric mean in vital and social statistics. Proceedings of the Royal Society of London 29: 365–367.
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5. Gibrat, R. 1931. Les inégalites économiques. Paris: Libraire du Recueil Sirey.Google Scholar
6. Johnson, N., S. Kotz, and L. Balakrishnan. 1994. Continuous Univariate Distributions. Vol. 1. New York: John Wiley.Google Scholar
7. Kapteyn, J.C. 1903. Skew frequency curves in biology and statistics. Astronomical Laboratory, Groningen: Noordhoff.Google Scholar
8. McAlister, D. 1879. The law of the geometric mean. Proceedings of the Royal Society of London 29: 367–375.