Benford's Law: How Aunt Linda Caught Gary's Made-Up Receipts
The short version: in many real lists of numbers that span several orders of magnitude, such as bills, populations or expenses, the first digit is 1 about 30% of the time and 9 only about 4.6%. People who make numbers up spread first digits much more evenly and avoid 1s. That's Benford's law, and auditors use it as a first screen for invented figures. It's a reason to look closer, not proof of anything, and it doesn't apply to numbers with fixed ranges, such as heights or prices set at $9.99.
Uncle Gary was elected treasurer of the allotment society, mainly because he wasn't at the meeting. At the end of the year he submitted 120 receipts for seed, compost, string and "sundries", several of them written on beer mats.
Aunt Linda, who audits the accounts because nobody else will, laid the receipts out on the kitchen table. She didn't check a single item. She just counted the first digit of every amount. Then she asked Gary which receipts he'd made up. Gary asked how she knew. Linda said that the numbers had told her. Gary said that numbers can't talk. Linda said these ones had been very chatty.
Real numbers love the digit 1
You'd think the first digit of an amount is equally likely to be anything from 1 to 9. In many real-world collections of numbers, it isn't. The pattern, called Benford's law, goes like this:
- First digit 1: about 30.1% of the time.
- 2: 17.6%. 3: 12.5%. 4: 9.7%. 5: 7.9%.
- 6: 6.7%. 7: 5.8%. 8: 5.1%. 9: just 4.6%.
It shows up in electricity bills, river lengths, town populations, company accounts and expense claims: anything that grows by percentages and spans several orders of magnitude, from tens to thousands.
Share of numbers starting with each digit, under Benford's law
Why 1 wins
Think of an amount growing steadily by a percentage, like a savings pot. To get from 100 to 200 it has to double. To get from 900 to 1,000 it only has to grow 11%. So it spends a long time with a first digit of 1 and races through the 8s and 9s. Then it reaches 1,000, and it's back to 1 again for ages. Collect lots of numbers like that and the 1s pile up.
Why made-up numbers don't
When people invent amounts, they try to look random, and to most people random means "spread out evenly". They avoid round numbers, avoid repeating themselves, and pick a lot of middling first digits like 4, 5, 6 and 7. They rarely start with 1, because 1 feels too small and too obvious.
Gary's 120 receipts: first digit 1 on only 10 of them, about 8%. First digit 6 or 7 on 41. The real ones, for compost and string, looked Benford-ish. The beer mats did not. The beer mats, said Linda, were what statisticians call an outlier, and what she calls "Thursday".
When Benford's law works, and when it doesn't
1. It needs range. The numbers should span several orders of magnitude: from tens to thousands, not all between 150 and 200.
2. It doesn't apply to assigned or capped numbers. Phone numbers, adult heights, shoe sizes, prices set at $9.99 and payments capped at a limit won't follow it.
3. It's a reason to look, not a verdict. Auditors use it as a first screen to decide where to dig. A small set can deviate by chance, so a deviation needs a sample of hundreds and a closer look before anyone is accused of anything.
Where it costs you money
Accounting figures in real companies usually follow Benford's law closely. Researchers and auditors have used first-digit tests to flag accounts that deserve a closer look, and a suspicious pattern is one more reason to read a company's numbers carefully rather than trust the headline. It won't pick your investments, but it is a cheap way to notice when numbers look invented, including in a too-good-to-be-true track record.
The moral
Gary confessed to seven beer mats and was allowed to remain treasurer, on the condition that all future receipts be printed by a till. The following year his receipts followed Benford's law beautifully. Linda found this suspicious too. It turned out Gary had read about Benford's law and was now making up receipts that start with 1. Linda has since moved on to the second digit, which also has a pattern. Gary does not yet know this.
Benford percentages are the standard first-digit frequencies, log10(1 + 1/d). Gary's receipts are fictional. A first-digit pattern is a screening tool, not evidence of wrongdoing. Educational content, not financial advice.