The Surprising “Benford Law” of Leading Digits

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Benford’s Law applies to nature as well. Image by mzacha

Benford’s Law, More than a Mere Statistical Quirk

Benford’s Law is a bit unusual in mathematics, in that it started with a statistical quirk rather than a logical deduction from axioms through proofs. It also comes as a surprise to most people.

It is also rare in science for a law to be named for the second of three “discoverers”. Usually the first publisher claims the prize. Just what is this law?

A Quick Statement of Benford’s Law

Benford’s Law states that, for a large collection of numbers with a wide range of values, the probability that the leading digit is n, is log((n+1)/n).

The Graph of Benford’s Law

“Graph of Benford Law” Image by Mike DeHaan

This spreadsheet shows the graph of Benford’s Law. As you can see, it shows that the number “1” would be the leading digit about 30% of the time; “2” over 17%; declining to “9” at under 5%.

Who Developed Benford’s Law?

Three people share the honours for developing Benford’s Law: Newcomb, Benford and Pinkham.

Simon Newcomb Suggested the Formula

Simon Newcomb (1835-1909) simply noticed that the well-used book of logarithm values was more “tatty” at the front than at the back. He assumed the reason was that the wear and tear related to how frequently people used those pages. The implication was that numbers starting with lower digits were more often used in the calculations. Perhaps without much further research, he proposed the “log((n+1)/n)” rule.

Simon Newcomb, 1871: Image credit U.S. Naval Observatory

Newcomb’s career was devoted to the mathematics of astronomy. Besides being a professor of mathematics in Washington, DC, he also was a leader at the astronomical observatory and for the US Navy’s Nautical Almanac.

These interests were actually tied together: astronomical observations plus diligent mathematical calculations lead to accurate tide tables. He was also busy enough that he did not exert himself to pursue his “log((n+1)/n)” finding.

Frank Benford Compiled the Statistics

Frank Benford also saw this probability pattern for leading digits, but pursued the study “scientifically”. He gathered data from baseball statistics, geographic areas and other diverse sources. The pattern held. Benford publicized his findings in 1938, about five decades after Newcomb, but could not explain them either.

Benford lived from 1887 to 1948. He worked as a physicist; developed a tool to measure the refractivity of glass; and was employed by General Electric in the year he died.  Please visit S9.com to see Frank Benford’s portrait.

Roger Pinkham Proved the Mathematics

In 1961, Roger Pinkham began working out two rules about Benford’s Law. First, the law should apply regardless of the “scale”. Secondly, any similar law that fulfills the first condition must be precisely Benford’s Law. The next main section will examine these ideas.

It is remarkably tricky to find reliable biographic information about Professor Pinkham. His faculty portrait at the Stevens Institute is circa 2007-2008.

Pinkham’s Proof of Benford’s Law

Pinkham divided the proof into two major segments: invariance of scale and uniqueness of this law.

His paper is available online as a PDF article which only runs for eight pages. However, he often appeals to other authors’ proofs, so the full explanation would be much longer.

Invariance of Scale for Benford’s Law

Let’s say that a list of distances between major cities, as measured in miles, conforms to Benford’s Law. What happens if we convert to metric by multiplying those mileage figures by 1.6? This is known as “varying the scale”.

“Benford Invariance” Image by Mike DeHaan

As this graph shows, the distribution remains pretty solid even though after converting from miles to kilometres. I started with “miles” following Benford’s Law; multiplied by 1.6 to convert to kilometres; and counted the leading digits.

Of course, “pretty solid” results from a completely artificial data set does not prove anything, let alone the specific argument about “invariance”.

The statistical argument is simple. Suppose that Benford’s Law were merely an artifact of the scales that were chosen. Surely some data sets should have yielded other frequently-chosen leading digits. Then Benford would not have found the statistical confirmation.

Let’s define the function L(v[i]) to give the leading digit of the i-th value, v[i], in a large set with wide-ranging values.

We could then define the distribution function, “D(n)” = ( sum for all “i” where L(v[i]) = “n” ) / ( sum for all “i” of L(v[i]) ).

We find, empirically, that D(n) did indeed follow Benford’s Law for that set of values.

Changing the scale means that we multiply each v[i] by a conversion factor, “c”. So now we have L(c*v[i]) = “n[c]”. In other words, we change the scale by conversion factor “c” and find a new set of leading digits, the set of “n[c])”.

Invariance of scale basically says that D(n[c]) still follows Benford’s Law. The astute reader notes that this is not a proof; it just outlines what Pinkham does prove.

Uniqueness of Benford’s Law

Pinkham notes that the proof that only Benford’s Law is invariant under a change of scale is “a non-trivial mathematical result”. Feel free to check Pinkham’s paper for his discussion.

His proof seems to depend on using the derivative of the logarithm function after a scaling factor.

Consider a simple logarithmic function, f(x) = log(x). Then f(c*x) = log(c*x) = log(c) + log(x), and “log(c)” is a constant value. Taking the derivative of f(x), the constant “log(c)” disappears.

Once again, this is far from being a complete proof; it only gives a glimpse of the approach Pinkham took.

Equal Initial Distributions are Not Invariant under Scaling

This spreadsheet shows an initial even distribution of leading digits 1-9 in the first column. In the next 8 columns, that first column is multiplied “Times 2” through “Times 9”. The “Count-#” column has the number of leading 1s, 2s, etc. from all 10 earlier columns. The “As %” column shows how these values are strongly skewed towards leading with “1” or “2”.

“Benford Multiplication” Image by Mike DeHaan

In the initial even distribution, each digit has 11.11% of the lead, or 1/9. Applying any scaling factor skews that distribution, and brings it roughly towards Benford’s distribution.

Applications for Benford’s Law

Dr. Mark Nigrini uses Benford’s Law to find situations where fraud has probably been committed. Someone “cooking the books” will likely use an even distribution of leading digits rather than leading with low digits. On the other hand, prices are more likely to be cut down from $10.00 to $9.99; so a sales catalogue may be a poor choice for testing this law.

This law provides a “smell test” for almost any large sets of numbers, whether from scientific experiments, demographic surveys or the numbers of voters in polling stations. If the sample is large and has large variances, then the leading digit should approximately follow the law.

As Sambridge et al reported, “We test(ed) 15 sets of modern observations drawn from the fields of physics, astronomy, geophysics, chemistry, engineering and mathematics…the rotation frequencies of pulsars…the masses of exoplanets as well as numbers of infectious diseases reported to the World Health Organization”. Benford’s Law held “for them all”.

Others have suggested that these results can extend to non-numeric situations. One example is to count the initial letters of names or words in a given language, then rank them by frequency. So “L[1]” would be the most common letter and “L[26]” the least. Does that set of letters conform to an alphabetic version of the law: P(i) = log((i+1)/i)?

References:

Joe Walthoe, +Plus, “Looking Out for Number One“, Sept. 1, 1999, referenced July 2, 2011.
Dictionary of Canadian Biography Online, University of Toronto, “Simon Newcomb“, 2000, referenced July 2, 2011.
Roger S. Pinkham, The Annals of Mathematical Statistics, “On the Distribution of First Significant Digits“, Dec. 1961, PDF referenced July 2, 2011.

Sambridge, M., H. Tkalčić, and A. Jackson,Geophys. Res. Letters(37, L22301, doi:10.1029/2010GL044830), “Benford’s law in the natural sciences“, 2010, referenced July 4, 2011.

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