Benford's Law — Definition, Formula & Examples

Benford's Law is the observation that in many naturally occurring datasets, the leading (first) digit is far more likely to be 1 than 9. Specifically, the digit 1 appears as the leading digit about 30% of the time, while 9 appears less than 5% of the time.

Benford's Law states that for many collections of numbers arising from real-world sources, the probability that the first significant digit equals $d$ (where $d \in \{1, 2, \ldots, 9\}$ ) is $P(d) = \log_{10}\!\left(1 + \frac{1}{d}\right)$ . Data that span several orders of magnitude and are not artificially constrained tend to conform to this distribution.

Key Formula

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

Where:

$d$ = The leading digit, where d ∈ {1, 2, 3, …, 9}
$P(d)$ = The probability that the first significant digit is d

How It Works

To check whether a dataset follows Benford's Law, extract the leading digit of every value and tally the frequencies. Then compare your observed frequencies to the expected probabilities from the formula. A chi-squared goodness-of-fit test is commonly used to determine whether deviations from the expected distribution are statistically significant. Datasets that naturally span multiple orders of magnitude—such as city populations, financial transactions, or river lengths—tend to follow the law well. Datasets drawn from a narrow range (like human heights) typically do not.

Worked Example

Problem: What is the expected probability that the leading digit of a value in a Benford-distributed dataset is 1? What about 5?

For d = 1: Substitute d = 1 into the formula.

P(1) = \log_{10}\!\left(1 + \frac{1}{1}\right) = \log_{10}(2) \approx 0.301

For d = 5: Substitute d = 5 into the formula.

P(5) = \log_{10}\!\left(1 + \frac{1}{5}\right) = \log_{10}(1.2) \approx 0.079

Answer: The leading digit 1 is expected about 30.1% of the time, while the leading digit 5 is expected only about 7.9% of the time.

Visualization

Why It Matters

Forensic accountants and auditors use Benford's Law to flag potentially fabricated financial data—people who invent numbers tend to distribute leading digits too uniformly. It also appears in fraud detection coursework within introductory statistics and data science programs.

Common Mistakes

Mistake: Assuming every dataset should follow Benford's Law.

Correction: The law applies to datasets that span several orders of magnitude and are not artificially bounded. Datasets like exam scores (typically 0–100) or human heights do not follow it because their range is too narrow.