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No. 120: Nov-Dec 1998

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The First Digit Phenomenon

Back in 1881, Simon Newcomb, the renowned Canadian-American scientist, published a provocative conjecture that was promptly forgotten by everyone. Newcomb had noticed that books of logarithms in the libraries were always much dirtier at the beginning. Hmmm! Were his fellow scientists looking up the logarithms of numbers beginning with 1 more frequently than 2, 3, etc.? It certainly seemed like it. He formalized his suspicions in a conjecture:

p = log10 (1 + 1/d)

Where p = the probability that the first significant digit is d.

This (unproven) equation states that about 30% of the numbers in a table or group will begin with 1. Only about 4.6% will begin with 9. This result certainly clashes with our expectation that the nine digits should occur with equal probability.

Fifty-seven years later, F. Benford, a GE physicist, unaware of Newcomb's paper, observed the same dirty early pages in the logarithm tables. He came up with exactly the same conjecture. Benford didn't stop there. He spent several years collecting diverse data sets -- 20,229 sets, to be exact. He included baseball statistics, atomic weights, river areas, the numbers appearing in Reader's Digest articles, etc. He concluded that his (and Newcomb's) conjecture fit his data very well. There were notable exceptions, though. Telephone directories and square-root tables didn't support the conjecture.

Interestingly, the second digits in numbers are more equitably distributed; the third, even more so.

Mathematicians have never been able to prove the Newcomb-Benford conjecture. How could they if it doesn't apply to all tables? Nevertheless, it works for most data sets, and that's still hard enough to explain.

(Hill, T.P.; "The First Digit Phenomenon," American Scientist, 86:358, 1998.)

From Science Frontiers #120, NOV-DEC 1998. 1998-2000 William R. Corliss

Science Frontiers Sourcebook Project Reviewed in:

Quotes

  • "A sourcebook of unexplained phenomena is therefore a valuable addition to a collection of scientific literature. William R. Corliss has provided this in the past with his source books of scientific anomalies in several subjects, and now he has provided it for astronomy. He has done an excellent job of collecting and editing a large amount of material, taken in part from scientific journals and in part from scientific reporting in the popular or semi-scientific press." -- "The Mysterious Universe: A Handbook of Astronomical Anomalies", reviwed by Thomas Gold, Cornell University, in Icarus, vol.41, 1980

  • "An interesting, systematic presentation of unusual weather [..] This book is recommended for a general audience" --"Corliss, William R., Tornados, Dark Days, Anomalous Precipitation, and Related Weather Phenomena, Sourcebook Project, 1983.", revieweed in Choice, September 1983
  • "..the science is necessarily somewhat speculative, but Corliss's symthesis is based on reputable sources." -- "Corliss, William R. (Compiler). Lightning, Auroras, Nocturnal Lights, and Related Luminous Phenomena" reviwed by Joseph M. Moran, Univ. of Wisconsin in Science Books and Films, Sep/Oct 1983

  • "Before opening the book, I set certain standards that a volume which treads into dangerous grounds grounds like this must meet. The author scrupulously met, or even exceeded those standards. Each phenomenon is exhaustively documented, with references to scientific journals [..] and extensive quotations" -- "Book Review: The moon and planets: a catalog of astronomical anomalies", The Sourcebook Project, 1985., Corliss, W. R., Journal of the Royal Astronomical Society of Canada, Vol. 81, no. 1 (1987), p. 24., 02/1987