Why do so many real-world numbers start with 1?
In electricity bills, river lengths and stock prices, about 30% of numbers begin with 1, and fraud investigators use that.
▶ Start the storyBecause many real-world quantities grow by multiplying rather than adding, which makes their logarithms, not the numbers themselves, spread evenly. The result, called Benford's law, is that about 30% of numbers in data like electricity bills, river lengths, populations and stock prices start with 1, while fewer than 5% start with 9. If digits were spread evenly, each would lead about 11% of the time.
How often a number starts with…
| Share of numbers | |
|---|---|
| 1 (Benford) | 30% |
| 9 (Benford) | 4.6% |
| Any digit, if even | 11.1% |
Why? Think of a price that changes by a small percentage each day. To go from 100 to 200 it has to double, which takes a long time, so it lingers among numbers starting with 1. To go from 900 to 1,000 it only needs to grow by about a ninth, so it zips through the 9s. Stretched over many orders of magnitude, numbers spend far more time starting with small digits.
The pattern was first spotted in 1881 by the astronomer Simon Newcomb, who noticed that the first pages of his logarithm tables, the ones for numbers starting with 1, were much more worn than the rest. Today it's a tool for fraud detection: people who invent figures tend to spread their digits too evenly.
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Recap
Things that grow by multiplying linger among numbers starting with 1, and fakers usually forget it.
Surprising fact · Greece's economic data submitted before joining the eurozone was later shown to be probably fraudulent using Benford's law.
Sources (3)
No source, no claim. Every fact in this lesson (15 claims) cites at least one of these.