Distribution Of First Digits Research Articles

Benford's law in the natural sciences

More than 100 years ago it was predicted that the distribution of first digits of real world observations would not be uniform, but instead follow a trend where measurements with lower first digit (1,2,…) occur more frequently than those with higher first digits (…,8,9). This result has long been known but regarded largely as a mathematical curiosity and received little attention in the natural sciences. Here we show that the first digit rule is likely to be a widespread phenomenon and may provide new ways to detect anomalous signals in data. We test 15 sets of modern observations drawn from the fields of physics, astronomy, geophysics, chemistry, engineering and mathematics, and show that Benford's law holds for them all. These include geophysical observables such as the length of time between geomagnetic reversals, depths of earthquakes, models of Earth's gravity, geomagnetic and seismic structure. In addition we find it also holds for other natural science observables such as the rotation frequencies of pulsars; green‐house gas emissions, the masses of exoplanets as well as numbers of infectious diseases reported to the World Health Organization. The wide range of areas where it is manifested opens up new possibilities for exploitation. An illustration is given of how seismic energy from an earthquake can be detected from just the first digit distribution of displacement counts on a seismometer, i.e., without actually looking at the details of a seismogram at all. This led to the first ever detection of an earthquake using first digit information alone.

First-digit law in nonextensive statistics

Nonextensive statistics, characterized by a nonextensive parameter q, is a promising and practically useful generalization of the Boltzmann statistics to describe power-law behaviors from physical and social observations. We here explore the unevenness of the first-digit distribution of nonextensive statistics analytically and numerically. We find that the first-digit distribution follows Benford's law and fluctuates slightly in a periodical manner with respect to the logarithm of the temperature. The fluctuation decreases when q increases, and the result converges to Benford's law exactly as q approaches 2. The relevant regularities between nonextensive statistics and Benford's law are also presented and discussed.

Data mining journal entries for fraud detection: An exploratory study

Fraud detection has become a critical component of financial audits and audit standards have heightened emphasis on journal entries as part of fraud detection. This paper canvasses perspectives on applying data mining techniques to journal entries. In the past, the impediment to researching journal entry data mining is getting access to journal entry data sets, which may explain why the published research in this area is a null set. For this project, we had access to journal entry data sets for 29 different organizations. Our initial exploratory test of the data sets had interesting preliminary findings. (1) For all 29 entities, the distribution of first digits of journal dollar amounts differed from that expected by Benford's Law. (2) Regarding last digits, unlike first digits, which are expected to have a logarithmic distribution, the last digits would be expected to have a uniform distribution. Our test found that the distribution was not uniform for many of the entities. In fact, eight entities had one number whose frequency was three times more than expected. (3) We compared the number of accounts related to the top five most frequently occurring three last digit combinations. Four entities had a very high occurrences of the most frequent three digit combinations that involved only a small set of accounts, one entity had a low occurrences of the most frequent three digit combination that involved a large set of accounts and 24 had a low occurrences of the most frequent three digit combinations that involved a small set of accounts. In general, the first four entities would probably pose the highest risk of fraud because it could indicate that the fraudster is covering up or falsifying a particular class of transactions. In the future, we will apply more data mining techniques to discover other patterns and relationships in the data sets. We also want to seed the dataset with fraud indicators (e.g., pairs of accounts that would not be expected in a journal entry) and compare the sensitivity of the different data mining techniques to find these seeded indicators.

Stigler’s approach to recovering the distribution of first significant digits in natural data sets

Benford’s Law can be seen as one of the many first significant digit (FSD) distributions in a family of monotonically decreasing distributions. We examine the interrelationship between Benford and other monotonically decreasing distributions such as those arising from Stigler, Zipf, and the power laws. We examine the theoretical basis of the Stigler distribution and extend his reasoning by incorporating FSD first-moment information into information-theoretic methods. We present information-theoretic methods as a way to describe, connect, and unify these related distributions and thereby extend the reach of Benford’s Law and FSD research more generally.

Benford, Zipf and the blogosphere

The blogosphere is a relatively recent development on the Internet and, for this reason, has received limited empirical investigation. This short note investigates whether the popularity of the 500 most popular blogs during August and September of 2005, as measured by in-coming links, followed the first-digit distribution attributed to Benford (1938) and the rank-size distribution attributed to Zipf (1949). The evidence suggests that the blogs investigated were not characterized by either empirical regularity, consistent with blog popularity being caused by network externalities.

Assessing the Accuracy of Self-Reported Data: an Evaluation of the Toxics Release Inventory

Self-reported regulatory data are hard to verify. This article compares air emissions reported by plants in the Toxics Release Inventory with chemical concentration levels measured by EPA pollution monitors. We find that the large drops in air emissions reported by firms in the TRI are not always matched by similar reductions in measured concentrations from EPA monitors. When the first digits of the monitored chemical concentrations follow a monotonically decreasing distribution, we expect (via Benford's Law) a similar distribution of first digits for the TRI data. For lead and nitric acid the self-reported data do not follow the expected first digit pattern. This suggests that for these two heavily regulated chemicals plants are not reporting accurate estimates of their air emissions.

Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf

The distribution of first digits in numbers series obtained from very different origins shows a marked asymmetry in favor of small digits that goes under the name of Benford's law. We analyze in detail this property for different data sets and give a general explanation for the origin of the Benford's law in terms of multiplicative processes. We show that this law can be also generalized to series of numbers generated from more complex systems like the catalogs of seismic activity. Finally, we derive a relation between the generalized Benford's law and the popular Zipf's law which characterize the rank order statistics and has been extensively applied to many problems ranging from city population to linguistics.

The relationship between Zipf's law and the distribution of first digits

Summary Zipf 's experimental law states that, for a given large piece of text, the product of the relative frequency of a word and its order in descending frequency order is a constant, shown to be equal to 1 divided by the natural logarithm of the number of different words. It is shown to be approximately equal to Benford's logarithmic distribution of first significant digits in tables of numbers. Eleven samples allow comparison of observed and theoretical frequencies.

On the numbers of things and the distribution of first digits

I answer the question ‘‘Why are there more small things in the world than large things?’’ in terms of a probabilisitc model of partitioning a conserved quantity. Benford’s empirical rule for lists of numbers, that the proportion of numbers with first digit m is log10(m+1)−log10 m, is an exact consequence of the model.

70.5 The distribution of first significant digits

70.5 The distribution of first significant digits

On Scale-Invariant Distributions

The distribution of first digits in large sets of physical data given by Benford’s law had already been deduced from the assumptions of continuity and invariance under all changes of scale used to measure the data. In this paper we show that continuity is superfluous and only invariance under certain countable sets of scale factors is required. A criterion for the scale-invariance of any distribution is then derived in terms of the zeros of its characteristic function.

Distribution of Numbers and Distribution of Significant Figures

IN an earlier note1 it is shown that the distribution of first digits of the numbers in a table will obey a logarithmic law provided that a certain sum can be replaced by an integral. We shall here discuss this approximation in more detail. The notation used will be the same as before, f(x) giving the distribution of the numbers and F(p) the distribution of the significant figures. We shall also introduce the distribution of the logarithms of the numbers; setting y = In x, we have for the fraction of entries with logarithms between y and y + dy: