The Mysteries of Benford’s Law

Benford’s Law, in the most elementary form of understanding, states that the number “1” transpires as the leading digit 30% of the time compared to higher digits such as 9 which occurs 5% of the time. This occurs for all kinds of data sets ranging from electricity bills, street addresses, stock prices, to even physical and mathematical constants. Yes, that’s right, the physical and mathematical constants of the universe follow this mysterious law.

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This graph showcases the percentage to be expected based on the results of Benford’s Law. The number 1 representing 30% frequency rate as the leading digit to the number 9 representing a mere 4.6%. Even more strange, the percentage decrease in order from 1 to 9.

 

The law has been used in court cases to detect fraud based on the ‘plausible’ assumption that people who make up numbers evenly distribute them. It has been used to detect election/voting frauds, fraudulent macroeconomic data, and even scientific fraud.

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This graph showcases Benford’s law in relation to physical constants. The results are absolutely remarkable to me. Under what logic does this occur? How does a seemingly random measurement such as physical constants end up following the rules of the Benford’s law? Well, the explanation (albeit, the best possible) is that many data sets seem to follow logarithmic scaling, thus allowing for a distribution of leading numbers as found in the law. To understand how this works, the following image showcases how numbers are distributed in logarithmic scaling.

 

 

Probability Distribution

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Clearly, there may lie some logic behind this seemingly random occurrence in life.

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