Benford's Law is the observation that in a set of numbers taken from real life, numbers with smaller leading digits occur at a higher percentage, and the percentage decreases as the leading digit increases.

What is Benford's Law?

Benford's Law, also known as the first-digit law or the Newcomb-Benford law, states that in a dataset consisting of numbers, the leading digits follow a specific distribution pattern.

In simpler terms, Benford's Law reveals that in a dataset such as population figures of countries, the number of countries with populations starting with the digit 1 is the highest, and this number decreases as you move towards the digit 9. This dataset considers only the first digit of the population number. Whether a country’s population is 10,000, 100,000, or 1,000,000, they all count as starting with 1. The same applies to other digits, resulting in a graph showing decreasing percentages from 1 to 9.

According to Benford's research, if the starting number is a whole number and the probability is shown as a percentage out of 100, the distribution is expected to be:

What makes Benford's Law so intriguing is that if the values in any dataset are naturally occurring, they approximately follow this distribution. 

Where is Benford's Law Used?

Although Benford's Law can be used to check the accuracy of data in various fields, it is most prominently used to detect fraud in financial data. Since naturally occurring data is expected to follow this law, any discrepancy found in financial data may indicate potential fraud. However, not following Benford's Law alone does not prove that the data is incorrect. It merely serves as a warning sign for experts to investigate further.

Discovery of Benford's Law

The discovery of the mathematical expression known as Benford's Law dates back to the 17th century. Canadian-American astronomer and mathematician Simon Newcomb (1835 - 1909) discovered what is now known as Benford's Law in 1881. While conducting his astronomical research, Newcomb used logarithm books for calculations as calculators were not yet invented. Newcomb noticed that the logarithm books' early pages were much more worn out compared to later pages. This observation led him to discover what continues to fascinate mathematicians to this day. He deduced that pages containing smaller digits were used more frequently, evidenced by the wear and tear left by users. Newcomb proposed that the probability of a number N being the first digit of a number could be found using the formula 
log

Despite Newcomb’s work, the law is named after physicist Frank Benford. In 1938, Benford revisited this mathematical phenomenon in his studies. He tested the law with over 20,000 datasets, including river lengths, U.S. population figures, molecular weights, physical constants, street names, and death rates, confirming its validity. Thus, it is known as Benford’s Law or Benford’s claim.

The significance of Benford's Law in detecting fraud and errors in financial data was further emphasized by Mark Nigrini at West Virginia University. Nigrini demonstrated that Benford’s Law could be actively used to detect fraud, solidifying its importance as a tool for examining financial data for inconsistencies even today."