This is a preview of subscription content, access via your institution.

## References

- [AP1]
Aldous, D., and Phan, T. (2009), “When Can One Test an Explanation? Compare and Contrast Benford’s Law and the Fuzzy CLT”, Class project report dated May 11, 2009, Statistics Department, UC Berkeley; accessed on May 14, 2010, at [BH].

- [AP2]
Aldous, D., and Phan, T. (2010), “When Can One Test an Explanation? Compare and Contrast Benford’s Law and the Fuzzy CLT”, Preprint dated Jan. 3, 2010, Statistics Department, UC Berkeley; accessed on May 14, 2010, at [BH].

- [Ben]
Benford, F. (1938), “The Law of Anomalous Numbers”,

*Proc. Amer. Philosophical Soc*. 78, 551–572. - [Ber]
Berger, A. (2010), “Large Spread Does Not Imply Benford’s Law”, Preprint; accessed on May 14, 2010, at http://www.math.ualberta.ca/~aberger/Publications.html.

- [BBH]
Berger, A., Bunimovich, L., and Hill, T.P. (2005), “One-dimensional Dynamical Systems and Benford’s Law”,

*Trans. Amer. Math. Soc*. 357, 197–219. - [BH]
Berger, A., and Hill, T.P. (2009),

*Benford Online Bibliography*; accessed May 14, 2010, at http://www.benfordonline.net. - [Fel]
Feller, W. (1966),

*An Introduction to Probability Theory and Its Applications*vol. 2, 2nd ed., J. Wiley, New York. - [Few]
Fewster, R. (2009), “A Simple Explanation of Benford’s Law”,

*American Statistician*63(1), 20–25. - [Ga]
Gardner, M. (1959), “Mathematical Games: Problems involving questions of probability and ambiguity”,

*Scientific American*201, 174–182. - [GD]
Gauvrit, N., and Delahaye, J.P. (2009), “Loi de Benford générale”,

*Mathématiques et sciences humaines*186, 5–15; accessed May 14, 2010, at http://msh.revues.org/document11034.html. - [H1]
Hill, T.P. (1995), “Base-Invariance Implies Benford’s Law”,

*Proc. Amer. Math. Soc*. 123(3), 887–895. - [H2]
Hill, T.P. (1995), “A Statistical Derivation of the Significant-Digit Law”,

*Statistical Science*10(4), 354–363. - [K]
Knuth, D. (1997),

*The Art of Computer Programming*, pp. 253-264, vol. 2, 3rd ed, Addison-Wesley, Reading, MA. - [N]
Newcomb, S. (1881), “Note on the Frequency of Use of the Different Digits in Natural Numbers”,

*Amer. J. Math*. 4(1), 39–40. - [P]
Pinkham, R. (1961), “On the Distribution of First Significant Digits”,

*Annals of Mathematical Statistics*32(4), 1223–1230. - [R1]
Raimi, R. (1976), “The First Digit Problem”,

*Amer. Mathematical Monthly*83(7), 521–538. - [R2]
Raimi, R. (1985), “The First Digit Phenomenon Again”,

*Proc. Amer. Philosophical Soc*. 129, 211–219. - [S]
Speed, T. (2009), “You Want Proof?”,

*Bull. Inst. Math. Statistics*38, p 11. - [W]
Wagon, S. (2010), “Benford’s Law and Data Spread”; accessed May 14, 2010, at http://demonstrations.wolfram.com/BenfordsLawAndDataSpread.

## Acknowledgment

The authors are grateful to Rachel Fewster, Kent Morrison, and Stan Wagon for excellent suggestions that helped to improve the exposition.

## Author information

### Affiliations

### Corresponding author

## Additional information

Arno Berger was supported by an NSERC Discovery Grant.

## Rights and permissions

## About this article

### Cite this article

Berger, A., Hill, T.P. Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem.
*Math Intelligencer* **33, **85–91 (2011). https://doi.org/10.1007/s00283-010-9182-3

Published:

Issue Date:

### Keywords

- Large Spread
- Decimal Digit
- Positive Random Variable
- Mathematical Intelligencer Figure
- Easy Derivation