Benford’s law

The concept

There is one remarkable thing about randomness: Its existence is neither proved nor disproved it even appears everyday in science and in our everyday lives. Random walk is interesting for people who want to know more about the mystic character of this invisible companion.

This diploma thesis was created at the University of Applied Sciences in Mainz, Germany by Daniel A. Becker in 2009 and was supervised by Prof. Johannes Bergerhausen.

The visualizations

All visualizations of physical mathematical phenomena are not just illustrated representations; they are actually simulated with the help of „proce55ing“. True chaos and order within randomness has been demonstrated by the use of pure numerical data and software simulations. Therefore, the software not only presents the visualizations but it also uses the graphics to prove the phenomenon of randomness.

Benford’s law

Certain data, the values of which are not determined by intentional manipulation (e.g. the sizes of the world´s countries) have an interesting property: Observing only the first digits of the size, it can be observed, that the possible numbers 1, 2, 3 and so on until 9 lie in a certain proportion to each other. This proportion can also be observed with completely different data, such as DAX index values. This phenomenon concerning many real-life sources of data can often be observed by Benford´s law, also called the first-digit law. Perhaps, because the data is not intentionally influenced by human hands, it satisfies this rule. If the data is manipulated, this can be shown with some clarity with the help of this law.

All the world’s countries with their area sizes are depicted on the right hand scale of the visualization. Taking only the first digit of its’s size, a line connects each country to it’s respective group on the left hand scale. Each of the groups on the left hand scale grows with every new line joining it. Finally, the distribution to each digit is shown for each group. Remarkably, even with such a comparably small dataset, Benford´s law can be verified quite clearly.

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