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Concept GuideFirst-Digit Distribution · log

Benford's Law

Benford's law is an empirical rule that, across much real-world numeric data, smaller leading digits appear more often. The probability that the first digit is d is log₁₀(1 + 1/d) — about 30.1% for 1 and only 4.6% for 9. It's observed broadly across Fibonacci numbers, powers, populations, and accounting data.

01Benford's Law

You'd expect the first digit to be uniform across 1–9 (11.1% each), but real data is nothing like that.

Benford's law: the probability the first digit is d is log₁₀(1 + 1/d). 1 is most common at about 30.1%, falling to 4.6% by 9.

The first digits of 1000 Fibonacci numbers (1·1·2·3·5·8·13·…) — remarkably, they almost match the Benford curve.

Powers of 2 (2·4·8·16·32·64·…) follow the same distribution.

Powers of 3 too. It's a property shared by numbers that grow multiplicatively across many digit widths.

So when accounting ledgers, election results, or scientific data deviate sharply from this distribution, tampering can be suspected — it's used for fraud detection.

Observed distributionBenford expected (dashed)
11.1
1
11.1
2
11.1
3
11.1
4
11.1
5
11.1
6
11.1
7
11.1
8
11.1
9
1 / 6

02 Understand It Simply

For Everyone
🔑How It Works

For a quantity growing at a constant rate, the leading digit needs a doubling to go from 1 to 2 but only about 1.1x to go from 9 to 10. It therefore spends far longer on small leading digits, and 1 appears most often.

💡In Plain Words

Data that grows multiplicatively across many digit widths spreads evenly on a log scale, so the leading digit follows the log₁₀(1+1/d) distribution — 1 is most common, 9 rarest.

The exact opposite of the intuition that they'd be uniform (11.1% each).

📍Where It's Used
  • –Detecting accounting fraud
  • –election tampering
  • –and scientific data fabrication; checking data quality; verifying whether data is naturally occurring

03 Frequently Asked Questions

FAQ
What is Benford's Law?+

Benford's law is an empirical rule that, across much real-world numeric data, smaller leading digits appear more often. The probability that the first digit is d is log₁₀(1 + 1/d) — about 30.1% for 1 and only 4.6% for 9. It's observed broadly across Fibonacci numbers, powers, populations, and accounting data.

Where is Benford's Law used?+

Detecting accounting fraud, election tampering, and scientific data fabrication; checking data quality; verifying whether data is naturally occurring.

What's a simple analogy for Benford's Law?+

For a quantity growing at a constant rate, the leading digit needs a doubling to go from 1 to 2 but only about 1.1x to go from 9 to 10. It therefore spends far longer on small leading digits, and 1 appears most often.