Benford's law

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Benford’s Law
The Strange Mathematics of First Digits

Benford’s law, also known as the Newcomb–Benford law or the first-digit law, is a remarkable mathematical phenomenon concerning the frequency with which digits appear at the beginning of numbers in many real-world datasets.
One might naturally expect the digits 1 through 9 to appear as the first digit with approximately equal probability. If that were true, each digit would occur about $11.1%$ of the time.
Surprisingly, this is often not what happens.
In many naturally occurring collections of numbers, values beginning with 1 appear much more frequently than values beginning with 9.

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The Law
According to Benford’s law, the probability that the first significant digit of a number is $d$ is

$P(d)=\log_{10}\left(1+\frac{1}{d}\right)$

where

$d=1,2,\ldots,9$

For example,

$P(1)=\log_{10}(2)\approx0.301$

so approximately 30.1% of the numbers in a Benford-distributed dataset begin with the digit 1.
For the digit 9,

$P(9)=\log_{10}\left(\frac{10}{9}\right)\approx0.046$

so only about 4.6% begin with 9.
This means that numbers beginning with 1 can occur roughly six times as often as numbers beginning with 9.

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First-Digit Probabilities
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 probability steadily decreases as the leading digit becomes larger.

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Why Does This Happen?
The explanation is closely related to logarithmic scales.
On a logarithmic scale, the interval occupied by numbers beginning with 1 is considerably larger than the interval occupied by numbers beginning with 9.
For example, numbers with leading digit 1 occupy the logarithmic interval from

$\log_{10}(1)$ to $\log_{10}(2)$

whose length is

$\log_{10}(2)-\log_{10}(1)=\log_{10}(2)\approx0.301.$

Numbers beginning with 9 occupy only the interval from

$\log_{10}(9)$ to $\log_{10}(10)$

whose length is approximately

$0.046.$

Thus, if the logarithms of the values are distributed approximately uniformly, Benford’s law appears naturally.

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Where Does Benford’s Law Appear?
The law tends to work particularly well for datasets that span several orders of magnitude.
Examples can include:
Population figures
Financial and accounting data
Company revenues
Geographical measurements
Physical quantities
Economic statistics
Scientific measurements
Certain mathematical sequences
Interestingly, some purely mathematical sequences also approach Benford’s distribution.
Examples include the Fibonacci numbers, factorials and powers of 2.

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A Historical Curiosity
The phenomenon was first noticed by the astronomer and mathematician Simon Newcomb in 1881.
Newcomb noticed something peculiar about books containing logarithm tables.
The pages containing logarithms of numbers beginning with small digits—especially 1—were considerably more worn than pages corresponding to numbers beginning with larger digits.
He reasoned that people must therefore be looking up numbers beginning with 1 much more frequently than numbers beginning with 9.
The observation was largely forgotten.
More than fifty years later, physicist Frank Benford independently rediscovered the phenomenon.
In 1938, Benford published a large investigation involving more than 20,000 numerical observations taken from many different sources.
The phenomenon subsequently became known as Benford’s law.

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Fraud Detection
One of the most fascinating modern applications of Benford’s law is in forensic accounting and fraud detection.
Suppose that a large collection of financial transactions would normally be expected to follow Benford’s law.
An auditor can compare the actual frequencies of the first digits with the theoretical frequencies predicted by Benford’s distribution.
Large or systematic deviations may identify data that deserve further investigation.
For this reason, Benford analysis has been used in areas such as:
Accounting audits
Tax investigations
Financial fraud detection
Economic statistics
Scientific datasets
Election-data analysis
However, there is an extremely important limitation.
Failure to follow Benford’s law is not proof of fraud.
It merely indicates that the dataset may deserve closer examination.

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When Benford’s Law Does NOT Apply
Not every collection of numbers should follow Benford’s law.
It generally performs poorly when numbers are artificially restricted to a narrow interval or assigned according to human-designed systems.
Examples include:
Human heights
IQ scores
Telephone numbers
Postal codes
Identification numbers
Sequential invoice numbers
Numbers with predetermined minimum or maximum values
For example, the heights of adults might mostly lie between roughly 150 and 200 centimetres. Such values do not span several orders of magnitude, so there is no reason to expect them to follow Benford’s law.
This limitation is particularly important when Benford analysis is used to make claims about fraud or manipulation.

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Why Is Benford’s Law So Interesting?
Benford’s law is fascinating because it demonstrates that something as apparently arbitrary as the first digit of a number can exhibit a strong mathematical pattern.
It connects several areas of mathematics and statistics, including:
Probability theory
Mathematical statistics
Number theory
Logarithms
Data analysis
Forensic statistics
Perhaps most surprisingly, the law shows that in many datasets the digits 1 through 9 are far from equally likely.
The humble digit 1 appears at the beginning of numbers approximately one-third of the time.

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Key Takeaways
1. In many naturally generated datasets, first digits are not uniformly distributed.
2. About 30.1% of Benford-distributed numbers begin with 1.
3. Only about 4.6% begin with 9.
4. The distribution is described by

$P(d)=\log_{10}\left(1+\frac{1}{d}\right).$

5. The phenomenon is fundamentally connected with logarithmic scales.
6. Benford’s law is useful for detecting statistical anomalies, particularly in financial data.
7. A deviation from Benford’s law is not, by itself, evidence of fraud.

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Source:
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