Departure question
Read signals from chancePort 9 of 11Travel from dice and insurance to data, noise, and learning algorithms—measuring uncertainty and turning it into decisions.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
For data spanning several orders of magnitude and generated under suitable conditions, the leading digit can be 1 about 30% of the time. Some population and accounting datasets fit, while phone numbers or tightly bounded measurements may not. Auditors can use the law as a screening signal, but a mismatch alone does not prove fabrication.
Leading digit d | Probability P(d) | Visual |
|---|---|---|
| 1 | 30.1% | ██████████████ |
| 2 | 17.6% | ████████ |
| 3 | 12.5% | ██████ |
| 4 | 9.7% | █████ |
| 5 | 7.9% | ████ |
| 6 | 6.7% | ███ |
| 7 | 5.8% | ███ |
| 8 | 5.1% | ██ |
| 9 | 4.6% | ██ |
Unlike the uniform distribution (1/9 ≈ 11.1%), natural datasets show a pronounced asymmetry.
In real-world data, the leading digit is 1 about 30% of the time, 9 only 5%.
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
Astronomer Simon Newcomb noticed that pages for numbers beginning with smaller digits were more worn, suggesting a nonuniform distribution of leading digits.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearGE physicist Frank Benford confirmed the pattern across 20,229 observations from many datasets, and the law came to bear his name.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearHigh-profile corporate accounting scandals accelerated interest in statistical screening tools, and leading-digit analysis became a standard warning signal in forensic auditing.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearAccounting-fraud detection, election-data analysis, scientific-data validation, tax audits, and AI-generated image detection.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
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Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Travel from dice and insurance to data, noise, and learning algorithms—measuring uncertainty and turning it into decisions.
Open the number voyage
Travel from dice and insurance to data, noise, and learning algorithms—measuring uncertainty and turning it into decisions.
Open the number voyage
Travel from dice and insurance to data, noise, and learning algorithms—measuring uncertainty and turning it into decisions.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Benford's Law
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
