## Benford's law, first digit law, leading digit phenomenon

Today's post is belongs to the "new thought of the day" category. It bears on issues of data quality, "favorite numbers," plausibility of results, etc., but it's also just an interesting phenomenon in itself. Some of you may have known about Benford's law already, but it was news to me. I stumbled into it while perusing online resources about FFT and DSP (see chapter 34 in Smith, available at http://www.dspguide.com/ch34/1.htm, see References below).

Hill (1995) provides a concise description of Benford's law

. . . in many naturally occurring tables of numerical data, the leading significant digits are not uniformly distributed as might be expected, but instead follow a particular logarithmic distribution.
and several mathematical examples, the simplest of which is
Prob(first significant digit=d) = log10(1 + 1/d) where d=1..9
but there are other variants you will want to see for yourself.

Wikipedia's entry describes the phenomenon and has some useful links, but fumbles the explanation somewhat. Benford's law doesn't have any mystical significance or require that there be a "natural logarithmic scaling" of things. Also, contrary to a discussion on Wolfram's WWW site, it doesn't have anything to do whether the numbers are dimensionless or not. Instead, it is just a consequence of how empirical data are pooled and analyzed, and it can be derived through a central-limit-like theorem (Hill 1995).

Durtschi et al. wrote a very readable article that includes some illustrations of when Benford's law might or might not apply--see
http://www.rtedwards.com/journals/JFA/V-1/17.pdf
Their examples were drawn from the field of forensic accounting, not neuroscience, but it's still a useful exercise to review them. And who says your own lab notebooks might not some day be examined by a forensic accountant . . .

### References

Durtschi, C., Hillison, W. and Pacini, C.. The effective use of Benford's law to assist in detecting fraud in accounting data. Journal of Forensic Accounting 5:17-34, 2004.

Hill, T.P.. A statistical derivation of the significant-digit law. Statistical Science 10:354–363, 1995.

Smith, S.W.. The Scientist and Engineer's Guide to Digital Signal Processing, available at http://www.dspguide.com/.