### Berger, A and Hill, TP (2010)

#### Fundamental Flaws in Feller’s Classical Derivation of Benford’s Law

University of Alberta preprint; posted on math arXiv 14May 2010.

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

**Abstract:** Feller’s classic text An Introduction to Probability Theory and its Applications contains a
derivation of the well known significant-digit law called Benford’s law. More specifically,
Feller gives a sufficient condition (“large spread”) for a random variable X to be approxi-
mately Benford distributed, that is, for log10 X to be approximately uniformly distributed
modulo one. This note shows that the large-spread derivation, which continues to be widely
cited and used, contains serious basic errors. Concrete examples and a new inequality clearly
demonstrate that large spread (or large spread on a logarithmic scale) does not imply that
a random variable is approximately Benford distributed, for any reasonable definition of
“spread” or measure of dispersion.

**Bibtex:**

```
@ARTICLE{,
author = {Berger, Arno and Hill, Theodore P.},
title = "{Fundamental Flaws in Feller's Classical Derivation of Benford's Law}",
journal = {ArXiv e-prints},
archivePrefix = "arXiv",
eprint = {1005.2598},
year = 2010,
month = may,
}
```

**Reference Type:** Preprint

**Subject Area(s):** Statistics