**Benford’s law**, also called the **first-digit law**, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time.

**Mathematical statement**

… this is the distribution expected if the *logarithms* of the numbers (but not the numbers themselves) are uniformly and randomly distributed. For example, a one-digit number x starts with the digit 1 if 1 <= x < 2, and starts with the digit 9 if 9 <= x < 10. Therefore, x starts with the digit 1 if log 1 <= log x < log 2, or starts with 9 if log 9 <= log x < log 10. The interval [log 1, log 2] is much wider than the interval [log 9, log 10] (0.30 and 0.05 respectively); therefore if **log x** is uniformly and randomly distributed, it is much more likely to fall into the wider interval than the narrower interval, i.e. more likely to start with 1 than with 9.

**Explanations**

**Outcomes of exponential growth processes**

The precise form of Benford’s law can be explained if one assumes that the *logarithms* of the numbers are uniformly distributed; for instance that a number is just as likely to be between 100 and 1000 (logarithm between 2 and 3) as it is between 10,000 and 100,000 (logarithm between 4 and 5). For many sets of numbers, especially sets that **grow exponentially** such as incomes and stock prices, this is a reasonable assumption.

**Applications**

In 1972, Hal Varian suggested that the law could be used to detect possible fraud in lists of socio-economic data submitted in support of public planning decisions. Based on the plausible assumption that people who make up figures tend to distribute their digits fairly uniformly, a simple comparison of first-digit frequency distribution from the data with the expected distribution according to Benford’s law ought to show up any anomalous results.

Following this idea, Mark Nigrini showed that Benford’s law could be used in forensic accounting and auditing as an indicator of accounting and expenses fraud. In the United States, evidence based on Benford’s law is legally admissible in criminal cases at the federal, state, and local levels.

**Limitations**

Benford’s law can only be applied to data that are distributed across multiple orders of magnitude.

— Wikipedia on *Benford’s law*

2012.05.15 Tuesday ACHK