Benford's Law Mathematical Statement Explanation Applications

More precisely, Benford's Law states that the leading digit n (n = 1, ..., 9) occurs with probability log₁₀(n + 1) − log₁₀(n), or

Leading digit	Probability
1	30.1 %
2	17.6 %
3	12.5 %
4	9.7 %
5	7.9 %
6	6.7 %
7	5.8 %
8	5.1 %
9	4.6 %

One can also formulate a law for the first two digits: the probability that the first two-digit block is equal to n (n = 10, ..., 99) is log₁₀(n+1) − log₁₀(n), and similarly for three-blocks without leading zeros and longer blocks.

2 Explanation

That in general the leading digit 1 should be more common than the other digits can be understood as follows: start counting from 1: 1, 2, 3, ... As you reach 9, every digit will have been equally likely. But then, from 10 to 19, you only have the leading digit 1, so 1 gets a huge head start. Only when you reach 99 will all digits be equally likely again. But then 1 gets another huge head start from 100 to 199. And so it continues: 1 has always a lead, except for very rare exceptions (9, 99, 999, 9999, ...). This is not particularly satisfactory as an explanation, unless some probability of stopping counting at some point is also included.

Perhaps somewhat more precisely, suppose (capital) X is a random variable whose probability of being equal to any positive integer (lower-case) x is a constant times x^−s, where s > 1. The aforementioned "constant" must then be 1/ζ(s), where ζ is the Riemann zeta function (see zeta distribution). The probability that the first digit of X is n approaches log₁₀(n + 1) − log₁₀(n) as s approaches 1.

The precise form of Benford's law can be explained if one assumes that the logarithms of the numbers are uniformly distributed; this means that a number is for instance just as likely to be between 100 and 1000 (logarithm between 2 and 3) as it is between 10,000 and 100,000 (logarithm between 4 and 5). For many sets of numbers, especially ones that grow exponentially such as incomes and stock prices, this is a reasonable assumption.

Another explanation is that if a distribution of first digits exists, it should be scale invariant. For example the first (non-zero) digit of the lengths or distances of objects should have the same distribution whether the unit of measurement is planck lengths, inches, feet, yards, metres, miles, light yearA light year abbreviated ly is the distance light travels in one year: roughly 9. 46 × 1012 kilometres (9. 46 petametres, or about 5. 88 × 1012 miles). More specifically, a light year is defined as the distance that a photon would travel, in free space ans, or anything else. But, for example, there are three feet in a yard, so the probability that the first digit of a length (e.g. in yards) is 1 must be the same as the probability that the first digit of a length (e.g. in feet) starts 3, 4, 5, 6, 7, or 8. Applying this to all possible measurement scales gives a logarithmic distribution, and combined with the fact that log₁(1)=0 and log₁₀(1)=1 gives Benford's law.

Note that for numbers drawn from many distributions, for example IQ scores, human heights or other variables following normal distributionProbability density function of Gaussian distribution (bell curve). The normal distribution is an extremely important probability distribution in many fields. It is also called the Gaussian distribution especially in physics and engineering. It is actualls, the law is not valid. However, if one "mixes" number from those distributions, for example by taking numbers from newspaper articles, Benford's law reappears. This can be proven mathematically: if one repeatedly "randomly" chooses a probability distributionIn mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra and then randomly chooses a number according to that distribution, the resulting list of numbers will obey Benford's law.

3 Applications and limitations

In 19721972 is a leap year starting on Saturday (click link for calendar). Events January events January 2 the Pierre Hotel Heist Six men rob the safety deposit boxes of the Pierre Hotel in New York City. Loot is at least $4 million January 5 President of the Un, Hal Varian suggested that the law could be used to detect possible fraud in lists of socio-economic data submitted in support of public planning decisions. Based on the plausible assumption that people who make up figures tend to distribute their digits fairly uniformly, a simple comparison of first-digit frequency distribution from the data with the expected distribution according to Benford's law ought to show up any anomalous results. In the same vein, Benford's law can be used to analyse insurance, accounting or expenses data and identify possible fraud.

Care must be taken with these applications, however. Strictly speaking, only a set of numbers chosen at random (from a given probability distributionIn mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra) is sure to obey the law. A set of real-life data may or may not obey the law, depending on the extent to which the distribution of numbers it contains are skewed by the category of data under consideration.

For instance, one might well expect a list of numbers representing 'populations of UK villages beginning with A' or 'small insurance claims' to obey Benford's law. But if it turns out that the definition of a 'village' in this case is 'settlement with population between 300 and 999', or that the definition of a 'small insurance claim' in this case is 'claim between $50 and $100', then Benford's law would be manifestly false because certain numbers have been excluded by the definition of the data category.

4 History

The discovery of this fact goes back to 1881, when the American astronomer Simon NewcombSimon Newcomb ( March 12, 1835 July 11, 1909) was an astronomer and mathematician. Born the city of Wallace, Nova Scotia, Newcomb appears to have enjoyed no formal education beyond his short apprenticeship to a charlatan herbalist in 1851. Life Son of Emi noticed that the first pages of logarithm books (used at that time to perform calculations), the ones containing numbers that started with 1, were much more worn than the other pages. However, it has been argued that any book that is sequentially accessed would show more wear and tear on the earlier pages. This story might thus be apocryphal, just like Newton's supposed discovery of gravity from observation of a falling apple.

The phenomenon was rediscovered in 1938 by the physicist Frank Benford , who checked it on a wide variety on data sets and was credited for it. In 1996, Ted HillTed Hill is the mathematician who proved Benford's law. External links . proved the result about mixed distributions mentioned above.

5 References

Frank Benford: The law of anomalous numbers, Proceedings of the American Philosophical Society, 78 (1938), p. 551
Ted Hill: The first digit phenomenon, American Scientist 86 (July-August 1998), p. 358.
Hal Varian: Benford's law, American Statistician 26, p.65.

6 See also

Zipf's lawOriginally the term Zipf's law meant the observation of Harvard linguist George Kingsley Zipf ( SAMPA: [zIf]) that the frequency of use of the n''th-most-frequently-used word in any natural language is approximately inversely proportional to n''. Zipf's l

7 External links

Benford's Law and Zipf's Law

Statistics

<Hide>

Home > Benford's law

1 Mathematical statement