Benford's Law: Universal Truth Or Mathematical Myth?

Benford's Law, also known as the Newcomb-Benford Law, is a mathematical theory that states that in many real-life sets of numerical data, the leading digit is likely to be small. The law is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881. The law holds true for data sets that grow exponentially, but also appears to hold true for many cases in which an exponential growth pattern is not obvious. It is best applied to data sets that go across multiple orders of magnitude, such as populations of towns or cities, or income distributions. Benford's Law has been used in a variety of fields, including accounting, computer science, and politics, to detect fraud.

Benford's Law and fraud detection

Benford's Law, also known as the Newcomb-Benford Law, is a mathematical theory of leading digits. It states that the first digit in naturally occurring collections of numbers is more likely to be small than large. In other words, the number 1 is the first digit about 30% of the time, while 9 is the leading digit less than 5% of the time.

The law was first observed by American astronomer Simon Newcomb in 1881. He noticed that pages of logarithm tables beginning with the number 1 were more worn and dirty than later pages. He concluded that they were used more. Several decades later, physicist Frank Benford made a similar observation. He found that the pages of logarithm tables covering numbers with initial digits 1 and 2 were more worn than pages covering 7, 8, and 9.

Benford's Law can be a useful tool for fraud detection. Offenders rarely consider the law when creating false transaction documents. It can be used to uncover fictitious numbers in random data sets because it detects manual intervention in otherwise automated transaction activity.

A simple example of applying Benford's Law to detect fraud is as follows. Suppose a bank's policy is to refer loans at or above $50,000 to a loan committee. A loan officer might then have the opportunity to discover loan frauds just below this threshold. In this case, a Benford's Law test of the leading digit (specifically, the number 4) or two leading digits (specifically, 49) could uncover the fraud.

Benford's Law has been used to detect fraud in various contexts, including accounting and expenses fraud, voter fraud, and tax fraud. It has also been legally admitted as evidence in criminal cases at the federal, state, and local levels in the United States.

Benford's Law and election fraud

Benford's Law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. The law is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.

Benford's Law has been used to detect fraud in various contexts, including accounting and financial fraud, and has been deemed admissible for use in a court of law for such purposes. It has also been used to detect fraud in elections, such as the 2009 Iranian presidential election, where it was found that the second digits in vote counts for the winner, President Mahmoud Ahmadinejad, tended to differ significantly from the expectations of Benford's Law, suggesting ballot stuffing.

However, the use of Benford's Law to detect election fraud is not without controversy. Some have argued that it is problematic and misleading as a statistical indicator of election fraud, as it may not be applicable to all types of election data. For example, precinct-level votes generally do not span multiple orders of magnitude, which is a necessary condition for a data set to follow Benford's Law. As such, the application of Benford's Law to precinct-level vote counts in the 2020 United States presidential election was criticised as a misapplication of the law.

Additionally, even if Benford's Law is applicable, the presence of statistical irregularities does not necessarily imply voter fraud. There may be other reasons for vote totals to fail a particular statistical test, such as relatively strong or weak third-party performance, high or low voter turnout, or disruptions caused by a pandemic.

In conclusion, while Benford's Law can be a useful tool for detecting fraud in some contexts, its application to election fraud is more complex and requires careful consideration of the specific data being analysed.

Benford's Law and data sets

Benford's Law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time.

The law is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881. Benford's law has been shown to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, and physical and mathematical constants.

Benford's law tends to be most accurate when values are distributed across multiple orders of magnitude, especially if the process generating the numbers is described by a power law. It holds true for a data set that grows exponentially, but also appears to hold true for many cases in which an exponential growth pattern is not obvious. It is best applied to data sets that go across multiple orders of magnitude, such as populations of towns or cities, or income distributions.

However, it is important to note that Benford's law does not hold true for all data sets. For example, it does not apply to data sets in which digits are predisposed to begin with a limited set of digits, such as human heights, weights, or IQ scores. It also does not hold true when a data set covers only one or two orders of magnitude.

Benford's law has been used in a variety of applications, including fraud detection, election forensics, and scientific fraud detection. It can be a useful tool for auditors and investigators to identify potential anomalies or manipulated data.

Benford's Law and scale invariance

Benford's Law, also known as the Newcomb-Benford Law, the law of anomalous numbers, or the first-digit law, states that in many real-life sets of numerical data, the leading digit is likely to be small. The law is named after physicist Frank Benford, who stated it in 1938, although it was first discovered by Simon Newcomb in 1881.

The law holds true for data sets that grow exponentially, but also appears to hold true for many cases in which an exponential growth pattern is not obvious. It is best applied to data sets that span multiple orders of magnitude, such as populations of towns or cities, or income distributions.

The law tends to apply most accurately to data that span several orders of magnitude. As a rule of thumb, the more orders of magnitude that the data evenly covers, the more accurately Benford's Law applies.

When the distribution of the first digits of a data set is scale-invariant (independent of the units that the data are expressed in), it is always given by Benford's Law. This means that the probability of the first digit of a number in a given set will be the same, regardless of the units used to measure it.

For example, the height of an adult human will almost always start with a 1 or 2 when measured in metres, and almost always start with a 4, 5, 6, or 7 when measured in feet. However, in a list of lengths spread evenly over many orders of magnitude, it is reasonable to expect the distribution of first digits to be the same, whether the lengths are in metres or feet.

Benford's Law can be used to detect fraud in various contexts, including accounting, elections, and scientific research.

Benford's Law and exponential growth

Benford's Law, also known as the Newcomb-Benford Law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. The law is named after physicist Frank Benford, who stated it in 1938, although it was previously observed by Simon Newcomb in 1881.

Benford's Law holds true for a dataset that grows exponentially, such as one that doubles and then doubles again in the same timespan. It is best applied to datasets that span multiple orders of magnitude, like populations of towns or cities, or income distributions. The law tends to be most accurate when values are distributed across multiple orders of magnitude, especially if the process generating the numbers is described by a power law, which is common in nature.

Exponential growth occurs when the growth rate of a given quantity is proportional to the quantity's current value. When exponential growth data is plotted as a simple histogram, disregarding the time dimension, it fits the positively skewed k/x distribution, where the small is numerous and the big is rare. This quantitative preference for the small corresponds to Benford's Law, which predicts that the first significant digit on the left-most side of numbers in typical real-life data is proportioned between all possible 1 to 9 digits, with low digits occurring much more frequently than high digits.

Exponential growth series with high growth rates are nearly perfectly Benford, given that plenty of elements are considered. An additional constraint is that the logarithm of the growth factor must be an irrational number. This is because irrational numbers vastly outnumber rational numbers. However, this explanation is too simplistic, and a more complex explanation is provided in a study by Alex Ely Kossovsky.

Benford's Law can be used to detect fraud in lists of socio-economic data submitted in support of public planning decisions. It has also been used as evidence in criminal trials in the United States, including the 2009 Iranian elections, where it was used to uncover ballot stuffing.

Frequently asked questions