Genesis Lognormal Distribution

Modelling the Genesis of Normal and Log-Normal Distributions

A computer simulation on the web to visualize the genesis of normal and log-normal distributions

Christian Gut Eckhard Limpert Hans Hinterberger

Background and objectives

Model

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 0.1 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. For better visualization every 10. ball will be black and every 100. ball green. 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 suppressed. 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, use the reload/activate functionality of your browser.

START
(screen size 800x600)
(screen size 1024x768)

Basic mathematical properties [Postscript]

Reference

AITCHISON, J., BROWN, J.A.C., The lognormal distribution, Cambridge University Press, Cambridge, 1957.
CROW, E.L., SHIMIZU, K., (Eds.), Lognormal Distributions: Theory and Application, Dekker, New York, 1988.
GALTON, F., Natural Inheritance, Macmillan, London, 1889.
LIMPERT, E., STAHEL, W.A., ABBT, M., Lognormal distributions across the sciences: keys and clues, Bioscience 51 (5) (2001), p. 341-352 [PDF] [Postscript].
MCALISTER D., Proc. Roy. Soc. 29 (1879), p.367.
WILRICH, P.-T., et al., Formeln und Tabellen der angewandten mathematischen Statistik, 3. Aufl., Springer, Berlin, 1987, p.37f.

Contact

Dr. Eckhard Limpert

eckhard.limpert@ipw.agrl.ethz.ch

Dr. Werner A. Stahel

werner.stahel@stat.math.ethz.ch

About this page

The page and specially the model were developped by Christian Gut as a Semesterarbeit at the Institute of Scientific Computing of the ETH Zürich , supervised by Prof. Dr. Hans Hinterberger and Dr. Eckhard Limpert. The page follows some ideas of the Leitprogramm by Prof. Dr. Karl Frey.