## Monday, May 17, 2010

### The Peculiar Distribution of Last Color Names

Michael H. Brill and Karen E. Linder

This column might seem an excuse for a tangent on mathematics, but be patient---the color-related topic will re-appear…MHB]

Here’s a color-related experiment you can do with the phone book: Record the number of people whose last names are colors. The distribution is far from uniform. The 2010 Residential White Pages for Princeton/Suburban-Trenton NJ shows the following incidence of last names that are color names: Brown (555), White (228), Green (154), Gray (73) [Grey (3)], Black (47), Blue (14). As a check on the geographic specificity of this result, we tried Marquis Who’s Who in the World 2001 and got a similar ranking: Brown (140), White (51), Green (30), Black (25), Gray (23) [Grey (2)], Blue (4). Neither source has any last names Red, Orange, Yellow, Indigo, Violet, We didn’t try any other color names. Why does this regularity exist? Mathematicians have taken such observations as points of departure for century-long theorizing. For example, in 1881, Simon Newcomb [1] noted, “that the ten digits do not occur with equal frequency must be evident to any one making use of logarithmic tables, and noticing how much faster the first pages wear out than the last ones. The first digit is oftener 1 than any other digit, and the frequency diminishes up to 9.” Benford rediscovered the tendency (henceforth called Benford’s law) in 1938. Others [2] have given theoretical explanations, and it is discussed in the context of the mathematics of fractals [3].

The probability of first digit d works out to log10(1 + 1/d), which summed over d gives 1. Proof of plausibility: Require the probability density function of a continuous number to be scale-invariant [constant in log10(x)]; note that implies a/x is the density function in x; note the integral of a/x diverges, so consider first only over one decade, from 10m to 10m+1\, and then realize the numbers don’t change when 1 decade is expanded to n decades. Over one decade of x (10m to 10m+1), the probability density is f(x) = 1/x, and the probability of first digit d is log10(d+1) – log10(d) = log10(1 + 1/d); Finally, realize that, although starting or stopping in the middle of a decade produces an artifact, the artifact gets vanishingly small when the number of decades n gets larger and larger. Another view of this argument and its limitations appears in [4].

You can confirm Newcomb’s observation by an experiment with a phone book: Tally the first digits of street-address numbers, and observe that Newcomb, Raimi, et al. were right. You can stop after one or two pages…

By the way, Benford’s law is not just a curiosity, but is now used in detecting fraudulent random-guess data in income tax returns and other financial reports (http://www.rexswain.com/benford.html, and also [5]).

A related effect is called Zipf’s law (http://en.wikipedia.org/wiki/Zipf's_law), which has the same form in a variety of venues: The most frequent word in a natural language occurs about twice as often as the second-most-frequent word, about three times as often as the third-most-frequent word, etc. The same kind of relationship applies to the populations of cities in a country versus their population ranking.

Can such roots be found in the peculiar distribution of last color names? The last-color-name distribution is similar to Zipf’s law, but the decrease is too steep. Have we forgotten some common last-color-names that would fill in the gaps? Perhaps a cross-cultural study (e.g., performed by previous authors of this column) might fill in some of the gaps or give a key insight. We hope any such insights might find happier uses than detecting tax evasion.

1. S. Newcomb, Note on the frequency of use of the different digits in natural numbers, Amer. J. Math., 4, 39-40 (1881).
2. R. A. Raimi, The peculiar distribution of first significant digits, Sci. Amer. 221, 109-120 (December, 1969).
3. B. Mandelbrot, Fractals: Form, Chance, and Dimension, W. H. Freeman, 1977.
4. http://mathworld.wolfram.com/BenfordsLaw.html
5. T. P. Hill. "The First-Digit Phenomenon", American Scientist< 86, p. 358 (July-August 1998).