### Ciofalo, M (2009)

#### Entropy, Benford’s first digit law, and the distribution of everything

Unpublished manuscript.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**Abstract:** The history of the so called “Benford’s Law”, which concerns the distribution of the first significant digits in “natural” sets of measurements, is summarized, and its relation with exponential rank-size distributions (associated with geometric progressions of naturally-occurring quantities) is outlined. The physical significance of alternative distributions is then discussed by considering also the associated probability density functions, and it is shown that – under appropriate assumptions – exponential rank-size distributions can be derived from a maximum-entropy principle (in the information-theory sense as introduced by Shannon). Finally, naturally-occurring samples (e.g., surface areas of islands) are considered in detail and it is shown that they closely follow exponential rank-size distributions and satisfy both Benford’s law and appropriately formulated principles of uniform probability density and maximum information entropy.

**Bibtex:**

```
@misc{,
AUTHOR = {Michele Ciofalo},
YEAR = {2009},
TITLE = {Entropy, Benford’s first digit law, and the distribution of everything},
}
```

**Reference Type:** Not specified

**Subject Area(s):** Applied Mathematics, Statistics