Benfords Law Calculator

Understanding Benford's Law (First-Digit Law)

Benford's Law, also known as the First-Digit Law, is an empirical statistical law describing the frequency distribution of leading digits in naturally occurring numerical datasets. Contrary to intuition, the leading digits are not uniformly distributed (where each digit 1-9 would appear 11.1% of the time).

Benford's Law Logarithmic Formula

The probability $P(d)$ that a number begins with digit $d \in \{1, 2, \dots, 9\}$ is given by:

$$P(d) = \log_{10}\left(1 + \frac{1}{d}\right)$$

Expected leading digit percentages:

• Digit 1: 30.1%
• Digit 2: 17.6%
• Digit 3: 12.5%
• Digit 4: 9.7%
• Digit 5: 7.9%
• Digit 6: 6.7%
• Digit 7: 5.8%
• Digit 8: 5.1%
• Digit 9: 4.6%

Applications in Forensic Accounting and Fraud Detection

Because human-fabricated data tends to distribute digits uniformly or bias toward specific round numbers, auditors, tax authorities, and forensic accountants use Benford's Law analysis to detect anomalies in financial ledgers, tax returns, and scientific publications.

For further statistical analysis, explore our ANOVA Calculator and Z-Score Calculator.