## Genesis of the Lognormal Distribution

### Introductory text

Life is log-normal! Science and art, life and statistics (by E. Limpert and W. Stahel)

### Model

START the applet illustrating the lognormal version of the "Galton Board". User manual:
The program is mostly self-explaining. When you start it, you see the triangle configuration for the lognormal distribution and some default parameters. The simulation is not yet running. You can now customize the model by switching to the normal distribution (Galton board) or by changing the default parameters to your own values. The important model parameters are:
• The lognormal-factor (or the increment in case of normal distribution), defining the relation of the horizontal position of the tips of triangles in two neighbour columns.
• The number of triangle rows.
You can also change the size of the balls, and you can move their entry point using the scrollbar. The dynamic behaviour of the model can be influenced by changing speed and frequency: The ball speed is not an exact quantity and therefore has no units. It corresponds approximately to the number of pixels (=unities) per 0.01 seconds. The ball frequency is the number of newly entered balls per second. Depending on the choice of some parameters, some other parameters have to be adjusted. To start the simulation, press the start-button. The stop-button allows you to interrupt the simulation: The current balls will move on to their end position, but no new balls will be entered. The start-stop cycle can be repeated. The simulation ends when the first bucket is full: Then the screen is frozen and current balls will not move on to their end position. Before, during and after the simulation, the curve of the corresponding distribution function can be displayed. Before the simulation, the curve is equal to the x-axis. The parameters cannot be changed during and after the simulation. To start a new simulation with new parameters, press the reset-button.

START

### Reference

1. AITCHISON, J., BROWN, J.A.C., The lognormal distribution, Cambridge University Press, Cambridge, 1957.
2. CROW, E.L., SHIMIZU, K., (Eds.), Lognormal Distributions: Theory and Application, Dekker, New York, 1988.
3. GALTON, F., Natural Inheritance, Macmillan, London, 1889.
4. LIMPERT, E., ABBT, M., STAHEL W.A., Lognormal distributions across the sciences - keys and clues, 1998, submitted.
5. MCALISTER D., Proc. Roy. Soc. 29 (1879), p.367.
6. WILRICH, P.-T., et al., Formeln und Tabellen der angewandten mathematischen Statistik, 3. Aufl., Springer, Berlin, 1987, p.37f.