## Genesis of the Lognormal Distribution

### Introductory text

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

Download: Format
dvi (=device indepentent) or
ps (=postscript))
### 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**

### Related Links

### 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., ABBT, M., STAHEL W.A., Lognormal distributions
across the sciences - keys and clues, 1998, submitted.
- 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.

### 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 Zuerich
, supervised by
Dr. Hans Hinterberger and
Dr. Eckhard Limpert.

The page follows some ideas of the
Leitprogramm by Prof. Dr. Karl Frey.