Eckhard Limpert
and
Werner A. Stahel

Swiss Federal Institute of Technology Zurich

The figure at the cover page (Fig 1) provides a link between science
and
art and is not only interesting to look at. Moreover, it can lead to a
deeper comprehension of frequency distributions that are important for
life^{1,2}. The figure shows a physical model for the
distribution of particles that provides major clues to a better
understanding of the distribution, for instance of:

- Mineral resources in the earth crust,
- Pollutants in the air,
- The sensitivity of the individuals in a population to a chemical compound.
- Survival times after diagnosis of cancer.
- The abundance of plants, fish, birds and insects in ecology.

Analogous to the well known Galton board^{3} for normal
distributions, Fig. 1 demonstrates the genesis of a family of skewed,
so-called *lognormal* distributions. The flow of particles
falling from the funnel is deviated horizontally at each triangle in a
particular, * multiplicative * way. If a triangle tip is at the
horizontal position x, triangle tips to the right and to the left below
it are placed at x times c and x divided by c (c = constant). At the
same time, the model is a physical representation of the *
multiplicative central limit theorem * in mathematical statistics.
This theorem demonstrates how the lognormal distribution arises from
many small, * multiplicative random effects *.

The comparison made in Fig. 2 offers a new way of characterizing
lognormal data which is more informative than the established ways.
Often, distributions are summarized by mean and standard deviation which
is a poor description for skew distributions. If lognormal data are
subjected to the log transformation (Fig. 2b), a normal distribution
results, with mean mu and standard deviation sigma (e.g. 2,0.3). Back-
transforming these values to the original scale gives the geometric
mean^{4}, mu^{*}, and a standard deviation,
sigma^{*}, that is now *multiplicative*. These parameters
(100,2) indicate that 68% of the distribution are within the range of
100 x/ 2 (100 times/divide 2), and 95% within 100 x/ 2^{2}.

Fig. 2 (below): Two ways of characterizing lognormal distributions, in terms of the original data (a) and after log-transformation (b). ... and more, see paper in plosOne

The quantities mu^{*} and sigma^{*} allow to
characterize and compare lognormal data in terms of the original scale,
which is preferred by most people. The multiplicative standard
deviation sigma^{*}, which determines the skewness of the
distribution, is found to exhibit a typical value in many fields of
applications.

Table 1: Ubiquity of lognormal distribution in life^{5,6}

Disciplines | mu^{*} | sigma^{*} | |
---|---|---|---|

Medicine | Onset of Alzheimer disease | ~ 60 years | 1.2 |

Latent periods of infectious diseases | Hours to months | 1.5 | |

Survival time after diagnosis of cancer | Months to years | 3 | |

Environment | Air pollution in the U.S.A. | 40-110 PSI | 1.5-1.9 |

Rainfall | 80-200 m3 (x103) | 4-5 | |

Species abundance in ecology | - | 6-30 | |

Social sciences and linguistics | Income of employed persons | 6.700sFr | 1.5 |

Lengths of spoken words | 3-5 letters | 1.5 |

The model (Fig. 1), a computer application of which is in
preparation,
fills a 100 years old gap of demonstrating the genesis of these skewed
distributions. Their characterization in terms of the original data
makes
complicated things easy, from science to various applications and
everyday life. It's normal, that life is log-normal or - *
multiplicative normal *.

Acknowledgements: The support from COST Switzerland and the Swiss Institute of Technology (ETH) is gratefully acknowledged.

- Aitchison, J and Brown JAC, 1957.
*The lognormal distribution*, Cambridge University Press, Cambridge. - Crow EL and Shimizu K Eds, 1988.
*Lognormal Distributions: Theory and Application*, Dekker, New York. - Galton F, 1889.
*Natural Inheritance*, Macmillan, London. - McAlister D, 1879.
*Proc. Roy. Soc.***29**, 367 - Limpert E Abbt M and Stahel WA, 1998. Lognormal distributions across the sciences - keys and clues, subm.
- Limpert E, 1998. Fungicide sensitivity - towards improved
understanding of genetic variability.
*Modern Fungicides and Antifungal Compounds II*. Eds Lyr H, Russell PE and Sisler H. Intercept, Andover.179-185 in press.

Eckhard Limpert

Werner A. Stahel
is at the
Statistics Seminar
.
Swiss Federal Institute of Technology (ETH)
, CH-8092 Zurich, Switzerland.

E-mail addresses:
eckhard.limpert@bluewin.ch

werner.stahel@stat.math.ethz.ch

**Copyright © E. Limpert ETH Zurich, 1988**