# Benfords Law, by

## EN

In 1993 Wayne James Nelson was found guilty of trying to defraud the state of Arizona of nearly $2 million.Nelson, who worked for the Arizona State Treasurer, had carefully arranged apparently “random” payments to a fictitious supplier with the aim of pocketing the proceeds himself. Unfortunately for Nelson his crime was uncovered when his figures were tested against Benford’s law.

Benford’s law, named after physicist Frank Benford who stated it in 1938, regards the distribution of digits in certain real life situations.The law states that 1 occurs as the leading digit about 30% of the time, while larger digits occur in that position increasingly less frequently as the number increases.

For example the number 9 occurs as the first digit less than 5% of the time. So if you were to take a newspaper and list all of the numbers in it, roughly 30% of the leading digits in the numbers would be 1s and there would be roughly 5% 9s.

The idea was that there might be some sort of pattern to the occurrence of numbers came from an American astronomer called Simon Newcomb in 1881. Newcomb noticed that in logarithm tables (used at that time to perform calculations) the earlier pages, containing numbers that started with 1, were considerably more worn than the other pages. Newcomb published a distribution on the first and second digit. The phenomenon was again noted in 1938 by Frank Benford who investigated the idea more extensively than Newcomb. He tested it comprehensively using 20 data sets including the surface areas of 335 rivers; the sizes of 3259 US populations and the street addresses of the first 342 persons listed in American Men of Science.

But why do certain data sets satisfy this law? Although this seems like an unexplainable phenomenon, the following provides a simple and logical example where Benford's law would occur: Say you have a population of 100 rabbits. The number of rabbits doubles each month. Every day, you record the number of rabbits in the population. Then this list of recorded numbers will accurately follow Benford’s law.

Why? Remember, the number of rabbits is doubling each month. During the first month, the number of rabbits is increasing from 100 towards 200: The first digit is 1 for the whole month. In the second month, there are 200 rabbits increasing towards 400: The first digit passes through 2 and 3 spending more time as 2. The leading digits passes through the rest of the numbers spending less time with each digit. The number of rabbits will get to 1000 and follow the same pattern with the amount of time spent to get to the next digit decreasing as the digit increases. It is easy to see from this example that Benford’s law occurs very naturally in certain situations.

In 1972, it was suggested that the law could be used to detect unlikely entries and consequently possible fraud in lists of economic data. Based on the plausible assumption that people who make up figures tend to distribute their digits fairly uniformly, in an attempt to imitate how they think data is naturally distributed, by comparing the first digit frequency distribution of a data set with the expected distribution according to Benford’s law it is easy to identify any irregular data. Following this idea, Mark Nigrini, a faculty member at West Virginia University, showed that Benford's Law could be used in forensic data analysis to identify possible cases of fraud in accounting and expenses.

So when Wayne James Nelson entered in what he thought was a random and natural distribution of

payments, through a simple comparison with Benford’s Law his fraudulent payments were easily

detected. What Benford’s law shows us is that without the proper knowledge it is almost impossible

to imitate natural randomly distributed data. It is truly an incredible phenomenon.

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