# Life is log-normal !

## Science and art, life and statistics

Eckhard Limpert and Werner A. Stahel
Swiss Federal Institute of Technology Zurich

### Filling a gap

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 life1,2. 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.
The figure, combined with the theory outlined below, can also help to understand and characterize the distribution of e.g. the age of marriage in human populations, and economical data such as the size of enterprises and personal income.

### Basic features

Analogous to the well known Galton board3 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 mean4, 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/ 22.

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 life5,6

Disciplines Medicine Environment mu* sigma* 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 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 Income of employed persons 6.700sFr 1.5 Lengths of spoken words 3-5 letters 1.5

### Conclusions and outlook

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.

### References

1. Aitchison, J and Brown JAC, 1957. The lognormal distribution , Cambridge University Press, Cambridge.
2. Crow EL and Shimizu K Eds, 1988. Lognormal Distributions: Theory and Application, Dekker, New York.
3. Galton F, 1889. Natural Inheritance, Macmillan, London.
4. McAlister D, 1879. Proc. Roy. Soc. 29, 367
5. Limpert E Abbt M and Stahel WA, 1998. Lognormal distributions across the sciences - keys and clues, subm.
6. 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.