Benford's Law, also known as the First-Digit Law, states that in many naturally occurring sets of numbers, the first digit is more likely to be a smaller number (1, 2, 3) than a larger one (8, 9). This means that in a random dataset, numbers starting with 1 will occur about 30% of the time, while numbers starting with 9 will only occur about 5% of the time.
Benford's Law has been observed in many different types of datasets, including population numbers, stock prices, and even the lengths of rivers. For example, in a dataset of the populations of all countries in the world, the first digit of each country's population is more likely to be a 1 than any other number.
Benford's Law can be used as a tool for detecting fraud or errors in datasets. If a dataset does not follow the expected distribution of first digits according to Benford's Law, it may indicate that the data has been manipulated or is inaccurate. This has been used in forensic accounting, election fraud detection, and other fields.
While Benford's Law holds true for many datasets, there are also cases where it does not apply. For example, man-made datasets, such as financial data, tend to follow Benford's Law more closely than naturally occurring datasets. Additionally, datasets with a limited range of numbers, such as phone numbers or social security numbers, may not follow the expected distribution of first digits.
Benford's Law was first observed by astronomer and mathematician Simon Newcomb in the late 19th century, but it was not widely recognized until mathematician Frank Benford published a paper on the subject in 1938. Benford's Law gained further attention in the 1990s with the publication of a book by physicist and mathematician Theodore Hill, which explored the phenomenon in more detail and coined the term "Benford's Law".