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Benford's Law is an empirical ?law? governing the frequency of leading digits in numerical data sets. While for real-world data Benford's Law typically represents a rela- tively crude approximation to the actual frequencies, for mathematical sequences the predic- tions derived from it can be uncannily accurate.

What is Benford's Law? Benford's Law states that, in a naturally occurring set of numbers, the smaller digits appear disproportionately more often as the leading digits.

The first significant (leftmost nonzero) digit of seemingly random numbers often appears to conform to a logarithmic distribution, with more 1s than 2s, more 2s than 3s, and so forth, a phenomenon known as Benford's law. When humans try to produce random numbers, they often fail to conform to this distribution.

The law usually doesn't apply to data sets that have a stated minimum and maximum, like interest rates or hourly wages. If numbers are assigned, rather than naturally occurring, they will also not follow the law. Examples of assigned numbers include: zip codes, telephone numbers and Social Security numbers.

The number distribution is the number of particles counted of each size, shown as a differential across total number of counts. I.e. if there were nine particles spread evenly over three sizes the number differential would be 33.3% for each size.

In 1995, the phenomenon was finally proven by Hill. His proof was based on the fact that numbers in data series following the Benford's Law are, in effect, ?second generation? distributions, i.e. combinations of other distributions.

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.

One such test is to see how well the data follow Benford's Law. Benford's Law involves the distribution of lead digits in ?naturally occurring numbers,? e.g., vendor payments, customer invoices, and similar financial values that occur in the normal course of business.

For instance, Benford's Law will not hold true for data sets of human heights, human weights and intellectual quotient (IQ) scores. Another example would be small insurance claims (e.g., between US $50 and US $100). The theory also does not hold true when a data set covers only one or two orders of magnitude.

Benford's Law can recognize the probabilities of highly likely or highly unlikely frequencies of numbers in a data set. The probabilities are based on mathematical logarithms of the occurrence of digits in randomly generated numbers in large data sets.