Benford's law

For the tongue-in-cheek "law" about controversy, see Benford's law of controversy.

Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time. This distribution of first digits is the same as the widths of gridlines on the logarithmic scale.

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.

The graph to the right shows Benford's law for base 10. There is a generalization of the law to numbers expressed in other bases (for example, base 16), and also a generalization to second digits and later digits.

It is named after physicist Frank Benford, who stated it in 1938,[1] although it had been previously stated by Simon Newcomb in 1881.[2]

Contents

Mathematical statement

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, …, 9}) occurs with probability

P(d)=\log_{10}(d%2B1)-\log_{10}(d)=\log_{10} \left(1%2B\frac{1}{d}\right).

Numerically, the leading digits have the following distribution in Benford's law, where d is the leading digit and P(d) the probability:

d P(d) Relative size of P(d)
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%
 

The quantity P(d) is proportional to the space between d and d + 1 on a logarithmic scale. Therefore, this is the distribution expected if the logarithms of the numbers (but not the numbers themselves) are uniformly and randomly distributed. For example, a one-digit number x starts with the digit 1 if 1 ≤ x < 2, and starts with the digit 9 if 9 ≤ x < 10. Therefore, x starts with the digit 1 if log 1 ≤ log x < log 2, or starts with 9 if log 9 ≤ log x < log 10. The interval [log 1, log 2] is much wider than the interval [log 9, log 10] (0.30 and 0.05 respectively); therefore if log x is uniformly and randomly distributed, it is much more likely to fall into the wider interval than the narrower interval, i.e. more likely to start with 1 than with 9. The probabilities are proportional to the interval widths, and this gives the equation above. (The above discussion assumed x is a one-digit number, but the result is the same no matter how many digits x has.)

An extension of Benford's law predicts the distribution of first digits in other bases besides decimal; in fact, any base b ≥ 2. The general form is:

P(d)=\log_{b}(d%2B1)-\log_{b}(d)=\log_{b} \left(1%2B\frac{1}{d}\right).

For b = 2 (the binary number system), Benford's law is true but trivial: All binary numbers (except for 0) start with the digit 1. (On the other hand, the generalization of Benford's law to second and later digits is not trivial, even for binary numbers.) Also, Benford's law does not apply to unary systems such as tally marks.

Benford's "law" is different from a typical mathematical theorem: It is an empirical statement about real-world datasets. It applies to some datasets but not all, and even when it applies it is at best only approximate, never exact.

Example

Examining a list of the heights of the 60 tallest structures in the world by category shows that 1 is by far the most common leading digit, irrespective of the unit of measurement:

Leading digit meters feet In Benford's law
Count  % Count  %
1 26 43.3% 18 30.0% 30.1%
2 7 11.7% 8 13.3% 17.6%
3 9 15.0% 8 13.3% 12.5%
4 6 10.0% 6 10.0% 9.7%
5 4 6.7% 10 16.7% 7.9%
6 1 1.7% 5 8.3% 6.7%
7 2 3.3% 2 3.3% 5.8%
8 5 8.3% 1 1.7% 5.1%
9 0 0.0% 2 3.3% 4.6%

History

The discovery of this fact goes back to 1881, when the American astronomer Simon Newcomb noticed that in logarithm tables (used at that time to perform calculations), the earlier pages (which contained numbers that started with 1) were much more worn than the other pages.[2] Newcomb's published result is the first known instance of this observation and includes a distribution on the second digit, as well. Newcomb proposed a law that the probability of a single number N being the first digit of a number was equal to log(N + 1) − log(N).

The phenomenon was rediscovered in 1938 by the physicist Frank Benford,[1] who checked it on a wide variety of data sets and was credited for it. In 1995, Ted Hill proved the result about mixed distributions mentioned below.[3] The discovery was named after Benford making it an example of Stigler's law.

Explanations

Benford's law has been explained in various ways.

Outcomes of exponential growth processes

The precise form of Benford's law can be explained if one assumes that the logarithms of the numbers are uniformly distributed; for instance that a number is 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 sets that grow exponentially such as incomes and stock prices, this is a reasonable assumption.

For example, if a quantity increases continuously and doubles every year, then it will be twice its original value after one year, four times its original value after two years, eight times its original value after three years, and so on. When this quantity reaches a value of 100, the value will have a leading digit of 1 for a year, reaching 200 at the end of the year. Over the course of the next year, the value increases from 200 to 400; it will have a leading digit of 2 for a little over seven months, and 3 for the remaining five months. In the third year, the leading digit will pass through 4, 5, 6, and 7, spending less and less time with each succeeding digit, reaching 800 at the end of the year. Early in the fourth year, the leading digit will pass through 8 and 9. The leading digit returns to 1 when the value reaches 1000, and the process starts again, taking a year to double from 1000 to 2000. From this example, it can be seen that if the value is sampled at uniformly distributed random times throughout those years, it is more likely to be measured when the leading digit is 1, and successively less likely to be measured with higher leading digits.

This example makes it plausible that data tables that involve measurements of exponentially growing quantities will agree with Benford's Law. But the law also appears to hold for many cases where an exponential growth pattern is not obvious.

Scale invariance

The law can alternatively be explained by the fact that, if it is indeed true that the first digits have a particular distribution, it must be independent of the measuring units used (otherwise the law would be an effect of the units, not the data). This means that if one converts from feet to yards (multiplication by a constant), for example, the distribution must be unchanged — it is scale invariant, and the only continuous distribution that fits this is one whose logarithm is uniformly distributed.

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 feet, yards, or anything else. But there are three feet in a yard, so the probability that the first digit of a length in yards is 1 must be the same as the probability that the first digit of a length in feet is 3, 4, or 5. Applying this to all possible measurement scales gives a logarithmic distribution, and combined with the fact that log10(1) = 0 and log10(10) = 1 gives Benford's law. That is, if there is a distribution of first digits, it must apply to a set of data regardless of what measuring units are used, and the only distribution of first digits that fits that is the Benford Law.

Multiple probability distributions

For numbers drawn from certain distributions, for example IQ scores, human heights or other variables following normal distributions, the law is not valid. However, if one "mixes" numbers from those distributions, for example by taking numbers from newspaper articles, Benford's law reappears. This can also be proven mathematically: if one repeatedly "randomly" chooses a probability distribution (from an uncorrelated set) and then randomly chooses a number according to that distribution, the resulting list of numbers will obey Benford's law.[4][3] Élise Janvresse and Thierry de la Rue from CNRS advanced as similar probabilistic explanation for the appearance of Benford's law in everyday-life numbers, by showing that it arises naturally when one considers mixtures of uniform distributions.[5]

Applications

In 1972, 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.[6] Following this idea, Mark Nigrini showed that Benford's law could be used in forensic accounting and auditing as an indicator of accounting and expenses fraud.[7] In the United States, evidence based on Benford's law is legally admissible in criminal cases at the federal, state, and local levels.[8]

Benford's law has been invoked as evidence of fraud in the 2009 Iranian elections.[9] However, other experts consider Benford's law essentially useless as a statistical indicator of election fraud in general.[10][11]

Limitations

Benford's law can only be applied to data that are distributed across multiple orders of magnitude. For instance, one might expect that Benford's law would apply to a list of numbers representing the populations of UK villages beginning with 'A', or representing the values of small insurance claims. But if a "village" is a settlement with population between 300 and 999, or a "small insurance claim" is a claim between $50 and $100, then Benford's law will not apply.[12][13]

Consider the probability distributions shown below, plotted on a log scale.[14] In each case, the total area in red is the relative probability that the first digit is 1, and the total area in blue is the relative probability that the first digit is 8.

For the left distribution, the size of the areas of red and blue are approximately proportional to the widths of each red and blue bar. Therefore the numbers drawn from this distribution will approximately follow Benford's law. On the other hand, for the right distribution, the ratio of the areas of red and blue is very different from the ratio of the widths of each red and blue bar. Rather, the relative areas of red and blue are determined more by the height of the bars than the widths. The heights, unlike the widths, do not satisfy the universal relationship of Benford's law; instead, they are determined entirely by the shape of the distribution in question. Accordingly, the first digits in this distribution do not satisfy Benford's law at all.[13]

Thus, real-world distributions that span several orders of magnitude rather smoothly like the left distribution (e.g. income distributions, or populations of towns and cities) are likely to satisfy Benford's law to a very good approximation. On the other hand, a distribution that covers only one or two orders of magnitude, like the right distribution (e.g. heights of human adults, or IQ scores) is unlikely to satisfy Benford's law well.[12][13]

Distributions that exactly satisfy Benford's law

Some well-known infinite integer sequences provably satisfy Benford's law exactly (in the asymptotic limit as more and more terms of the sequence are included). Among these are the Fibonacci numbers,[15][16] the factorials,[17] the powers of 2,[18][19] and the powers of almost any other number.[18]

Likewise, some continuous processes satisfy Benford's law exactly (in the asymptotic limit as the process continues longer and longer). One is an exponential growth or decay process: If a quantity is exponentially increasing or decreasing in time, then the percentage of time that it has each first digit satisfies Benford's law asymptotically (i.e., more and more accurately as the process continues for more and more time).

Generalization to digits beyond the first

It is possible to extend the law to digits beyond the first.[20] In particular, the probability of encountering a number starting with the string of digits n is given by:

\log_{10}\left(n%2B1\right)-\log_{10}\left(n\right)=\log_{10}\left(1 %2B\frac{1}{n}\right).

(For example, the probability that a number starts with the digits 3, 1, 4 is log10(1 + 1/314) ≈ 0.0014.) This result can be used to find the probability that a particular digit occurs at a given position within a number. For instance, the probability that a "2" is encountered as the second digit is[20]

\log_{10}\left(1 %2B\frac{1}{12}\right)%2B\log_{10}\left(1 %2B\frac{1}{22}\right)%2B\cdots%2B\log_{10}\left(1 %2B\frac{1}{92}\right) \approx 0.109.

And the probability that d (d = 0, 1, ..., 9) is encountered as the n-th (n > 1) digit is

\sum_{k=10^{n-2}}^{10^{n-1}-1} \log_{10}\left(1 %2B \frac{1}{10k%2Bd}\right).

The distribution of the n-th digit, as n increases, rapidly approaches a uniform distribution with 10% for each of the ten digits.[20]

In practice, applications of Benford's law for fraud detection routinely use more than the first digit.[7]

See also

Notes

  1. ^ a b Frank Benford (March 1938). "The law of anomalous numbers". Proceedings of the American Philosophical Society 78 (4): 551–572. JSTOR 984802.  (subscription required)
  2. ^ a b Simon Newcomb (1881). "Note on the frequency of use of the different digits in natural numbers". American Journal of Mathematics (American Journal of Mathematics, Vol. 4, No. 1) 4 (1/4): 39–40. doi:10.2307/2369148. JSTOR 2369148.  (subscription required)
  3. ^ a b Theodore P. Hill (1995). "A Statistical Derivation of the Significant-Digit Law" (PDF). Statistical Science 10: 354–363. http://www.tphill.net/publications/BENFORD%20PAPERS/statisticalDerivationSigDigitLaw1995.pdf. 
  4. ^ Theodore P. Hill (July–August 1998). "The first digit phenomenon" (PDF). American Scientist 86: 358. http://www.tphill.net/publications/BENFORD%20PAPERS/TheFirstDigitPhenomenonAmericanScientist1996.pdf. 
  5. ^ Élise Janvresse and Thierry de la Rue (2004), From Uniform Distributions to Benford's Law, Journal of Applied Probability, 41 1203-1210 (2004) preprint
  6. ^ Varian, Hal. "Benford's law". The American Statistician 26: 65. 
  7. ^ a b Mark J. Nigrini (May 1999). "I've Got Your Number". Journal of Accountancy. http://www.journalofaccountancy.com/Issues/1999/May/nigrini. 
  8. ^ "From Benford to Erdös". Radio Lab. 2009-09-30. No. 2009-10-09.
  9. ^ Stephen Battersby Statistics hint at fraud in Iranian election New Scientist 24 June 2009
  10. ^ Joseph Deckert, Mikhail Myagkov and Peter C. Ordeshook, (2010) The Irrelevance of Benford’s Law for Detecting Fraud in Elections, Caltech/MIT Voting Technology Project Working Paper No. 9
  11. ^ Charles R. Tolle, Joanne L. Budzien, and Randall A. LaViolette (2000) Do dynamical systems follow Benford?s law?, Chaos 10, 2, pp.331-336 (2000); DOI:10.1063/1.166498
  12. ^ a b See [1], in particular [2].
  13. ^ a b c Fewster, R. M. (2009). "A simple explanation of Benford's Law". The American Statistician 63 (1): 26–32. doi:10.1198/tast.2009.0005 
  14. ^ Note that if you have a regular probability distribution (on a linear scale), you have to multiply it by a certain function to get a proper probability distribution on a log scale: The log scale distorts the horizontal distances, so the height has to be changed also, in order for the area under each section of the curve to remain true to the original distribution. See, for example, [3]
  15. ^ L. C. Washington, "Benford's Law for Fibonacci and Lucas Numbers", The Fibonacci Quarterly, 19.2, (1981), 175–177
  16. ^ R. L. Duncan, "An Application of Uniform Distribution to the Fibonacci Numbers", The Fibonacci Quarterly, 5, (1967), 137–140
  17. ^ P. B. Sarkar, "An Observation on the Significant Digits of Binomial Coefficients and Factorials", Sankhya B, 35, (1973), 363–364
  18. ^ a b In general, the sequence k1, k2, k3, etc., satisfies Benford's law exactly, under the condition that log10 k is an irrational number. This is a straightforward consequence of the equidistribution theorem.
  19. ^ That the first 100 powers of 2 approximately satisfy Benford's law is mentioned by Ralph Raimi. Ralph A. Raimi, "The First Digit Problem", American Mathematical Monthly, 83, number 7 (August–September 1976), 521–538
  20. ^ a b c Theodore P. Hill, "The Significant-Digit Phenomenon", The American Mathematical Monthly, Vol. 102, No. 4, (Apr., 1995), pp. 322–327. Official web link (subscription required). Alternate, free web link.

References

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