Benford's law, also called the First-Digit Law, is a phenomenological
law about the frequency distribution of leading digits in many real-life sets of
numerical data. That law states that in many naturally occurring collections of
numbers the small digits occur disproportionately often as leading
significant digits. For example, in sets which obey the law the number 1 would
appear as the most significant digit about 30% of the time, while larger
digits would occur in that position less frequently: 9 would appear less than 5%
of the time. If all digits were distributed uniformly, they would each
occur about 11.1% of the time. Benford's law also concerns the expected
distribution for digits beyond the first, which approach a uniform
distribution. It has been shown that this result
applies 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. It
tends to be most accurate when values are distributed across multiple orders
of magnitude. The graph here shows Benford's law for
base 10. There is a generalization of the law to numbers expressed in other
bases, and also a generalization from leading 1 digit to leading n digits.
It is named after physicist Frank Benford, who stated it in 1938, although
it had been previously stated by Simon Newcomb in 1881.
Mathematical statement A set of numbers is said to satisfy
Benford's law if the leading digit d occurs with probability
Numerically, the leading digits have the following distribution in Benford's law,
where d is the leading digit and P(d) the probability:
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
mantissae of the logarithms of the numbers are uniformly and randomly
distributed. For example, a number x, constrained to lie between 1 and 10,
starts with the digit 1 if 1 ≤ x 0. The value of A depends on the value of ε and
the distribution of the random variable. A method of accounting fraud detection
based on bootstrapping and regression has been proposed.
Generalization to digits beyond the first
It is possible to extend the law to digits beyond the first. In particular,
the probability of encountering a number starting with the string of digits n is
given by: (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 And the probability that d is
encountered as the n-th digit is The distribution of the n-th digit, as n
increases, rapidly approaches a uniform distribution with 10% for each of the
ten digits. Four digits is often enough to assume a uniform distribution of 10%
as '0' appears 10.0176% of the time in the fourth digit while '9' appears
9.9824% of the time. Tests with common distributions
Benford's law was empirically tested against the numbers generated by a
number of important distributions, including the uniform distribution, the
exponential distribution, the half-normal distribution, the
right-truncated normal, the normal distribution, the chi square
distribution and the log normal distribution. In addition to these the
ratio distribution of two uniform distributions, the ratio distribution of
two exponential distributions, the ratio distribution of two half-normal
distributions, the ratio distribution of two right-truncated normal
distributions, the ratio distribution of two chi-square distributions and the log
normal distribution were tested. The uniform distribution as might be
expected does not obey Benford's law. In contrast, the ratio distribution of two
uniform distributions is well described by Benford's law. Benford's law also
describes the exponential distribution and the ratio distribution of two
exponential distributions well. Although the half-normal distribution does not
obey Benford's law, the ratio distribution of two half-normal
distributions does. Neither the right-truncated normal distribution nor
the ratio distribution of two right-truncated normal distributions are
well described by Benford's law. This is not surprising as this distribution is
weighted towards larger numbers. Neither the normal distribution nor the ratio
distribution of two normal distributions obey Benford's law. The fit of chi
square distribution depends on the degrees of freedom with good agreement
with df = 1 and decreasing agreement as the df increases. The F distribution is
fitted well for low degrees of freedom. With increasing dfs the fit decreases
but much more slowly than the chi square distribution. The fit of the log-normal
distribution depends on the mean and the variance of the distribution. The
variance has a much greater effect on the fit than does the mean. Larger
values of both parameters result in better agreement with the law. The ratio
of two log normal distributions is a log normal so this distribution was not
examined. Other distributions that have been
examined include the Muth distribution, Gompertz distribution, Weibull
distribution, gamma distribution, log-logistic distribution and the
exponential power distribution all of which show reasonable agreement with the
law. The Gumbel distribution – a density increases with increasing value of the
random variable – does not show agreement with this law.
Distributions known to obey Benford's law
Some well-known infinite integer sequences provably satisfy Benford's Law
exactly. Among these are the Fibonacci numbers, the factorials, the powers of
2, and the powers of almost any other number.
Likewise, some continuous processes satisfy Benford's Law exactly. 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.
Distributions known to not obey Benford's law
Square roots and reciprocals do not obey this law. The 1974 Vancouver, Canada
telephone book violates Benford's law because regulations require that
telephone numbers have a fixed number of digits and do not begin with 1.
Benford's law is violated by the populations of all places with
population at least 2500 from five US states according to the 1960 and 1970
censuses, where only 19% began with digit 1 but 20% began with digit 2, for
the simple reason that the truncation at 2500 introduces statistical bias. The
terminal digits in pathology reports violate Benford's law due to rounding,
and the fact that terminal digits are never expected to follow Benford's law
in the first place. Criteria for distributions expected and
not expected to obey Benford's Law A number of criteria—applicable
particularly to accounting data—have been suggested where Benford's Law can
be expected to apply and not to apply. = Distributions that can be expected to
obey Benford's Law = When the mean is greater than the median
and the skew is positive Numbers that result from mathematical
combination of numbers: e.g. quantity × price
Transaction level data: e.g. disbursements, sales
Numbers produced when doing any multiplicative calculations with an
Oughtred slide rule, since the answers naturally fall into the right
logarithmic distribution. = Distributions that would not be
expected to obey Benford's Law = Where numbers are assigned sequentially:
e.g. check numbers, invoice numbers Where numbers are influenced by human
thought: e.g. prices set by psychological thresholds
Accounts with a large number of firm-specific numbers: e.g. accounts set
up to record $100 refunds Accounts with a built-in minimum or
maximum Where no transaction is recorded
Moments Moments of random variables for the
digits 1 to 9 following this law have been calculated:
mean 3.440 variance 6.057
skewness 0.796 kurtosis -0.548
For the first and second digit distribution these values are also
known: mean 38.590
variance 621.832 skewness 0.772
kurtosis -0.547 A table of the exact probabilities for
the joint occurrence of the first two digits according to Benford's law is
available, as is the population correlation between the first and second
digits: ρ = 0.0561. See also
Fraud detection in predictive analytics Zipf's law
References Further reading
External links = General audience =
Benford Online Bibliography, an online bibliographic database on Benford's Law.
Companion website for Benford's Law by Mark Nigrini Website includes 15 data
sets, 10 Excel templates, photos, documents, and other miscellaneous items
related to Benford's Law Following Benford's Law, or Looking Out
for No. 1, 1998 article from The New York Times.
A further five numbers: number 1 and Benford's law, BBC radio segment by
Simon Singh From Benford to Erdös, Radio segment
from the Radiolab program Looking out for number one by Jon
Walthoe, Robert Hunt and Mike Pearson, Plus Magazine, September 1999
Video showing Benford's Law applied to Web Data
An illustration of Benford's Law, showing first-digit distributions of
various sequences evolve over time, interactive.
Generate your own Benford numbers A script for generating random numbers
compliant with Benford's Law. Testing Benford's Law An open source
project showing Benford's Law in action against publicly available datasets.
Testing Benford’s Law in OLAP Cubes Implementation with Microsoft Analysis
Services. Mould, Steve. "Number 1 and Benford's
Law". Numberphile. Brady Haran. A third of property values begin with a
1 An example of Benford's Law appearing in house price data.
Benford's Very Strange Law - Professor John D. Barrow, lecture on Benford's
Law. = More mathematical =
Weisstein, Eric W., "Benford's Law", MathWorld.
Benford’s law, Zipf’s law, and the Pareto distribution by Terence Tao
Country Data and Benford's Law, Benford's Law from Ratios of Random
Numbers at Wolfram Demonstrations Project.
Benford's Law Solved with Digital Signal Processing
Interactive graphic: Univariate Distribution Relationships