Journal · Short film
Why 30% of numbers start with 1
Benford's Law in 98 seconds: what it says, why it happens, and what a bad fit can and can't tell an auditor.
Captions are on. The described version stops at each scene while a voice says what's on screen; that voice is synthetic, and the narration is ours. There's no sign-language version yet. The full transcript is below.
The pattern
Take a big pile of real-world numbers, such as town populations or the areas of rivers, and keep only the first digit of each. You might expect every digit from 1 to 9 to turn up about equally, roughly 11% each. It doesn't. About 30% start with 1, and fewer than 5% start with 9.
Frank Benford published this in 1938, working from about twenty sets of data that included the areas of 335 rivers and the populations of 3,259 places. The share for each digit d is log10(1 + 1/d):
| First digit | Share |
|---|---|
| 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% |
Why it happens
Think of anything that grows by percentages. Getting from 100 to 200 means doubling. Getting from 900 to 1,000 takes just 11% more. So a growing number spends a long time with a 1 in front and races through the 9s.
The pattern holds when numbers spread across several orders of magnitude, with tens, hundreds, thousands and millions all in one pile. Squeeze them into a narrow range and it breaks.
Why accountants care
Forensic auditors chart the first digits in a set of accounts and look hard at whatever doesn't fit the curve. Real data is lopsided in a predictable way, so a set that isn't lopsided is a place to start asking questions.
What this doesn't prove
A bad fit isn't fraud. Plenty of honest data breaks the curve: prices set at amounts like $9.99, amounts capped by a policy limit, a narrow range of values, or a small sample.
A good fit doesn't prove the numbers are genuine either. It's often said that people who invent numbers fail this test. The research is more mixed than that: some studies found that made-up numbers with a real-world meaning fit the first-digit curve reasonably well (Diekmann, 2007). We cut that claim from the film for this reason.
The chart tells you where to look, not what you'll find when you look.
Transcript
Narration as spoken, with what's on screen described in the indented lines. Company names and amounts in the film are made up.
On screen: Bar chart of first digits 1 to 9 following Benford's Law. 1 is 30.1 percent, falling to 4.6 percent for 9.
This is Benford's Law, and it's weird. Take a big pile of real-world numbers.
On screen: Eight made-up numbers. The first digit of each turns gold and is underlined: 1, 3, 1, 2, 8, 1, 4, 1.
Town populations. The areas of rivers. Ignore everything except the first digit.
On screen: Nine equal bars at 11.1 percent each, labelled What you'd expect. They reshape into the Benford curve, labelled What you actually get, with a dashed line at 11.1 percent.
You'd bet each digit through 9 gets a fair share. About 11% each. Nope. About 30% start with a 1. Fewer than 5% start with a 9.
On screen: Election-style result: 1 wins with 30.1 percent. 9 has 4.6 percent. log10 of 2 is 0.301; log10 of 10 over 9 is 0.046.
The 1s win by a landslide. Here's a way to feel it.
On screen: Two bars. 100 to 200 needs +100 percent growth, a full bar. 900 to 1,000 needs +11 percent, a short bar.
For anything that grows by percentages, getting from 100 to 200 means doubling. Getting from 900 to 1,000 takes just 11% more.
On screen: A ruler from 1 to 10 on a growth scale. The stretch from 1 to 2 is widest, 30 percent; 9 to 10 is narrowest, 5 percent. A marker moves at a steady speed while a counter grows from 100 to 1,000: it lingers in the 100s and races through the 900s.
So numbers spend a long time starting with 1 and racing through the 9s.
On screen: Numbers appear one by one, each bigger: 10, 100, 1,000, 1,000,000.
It works when numbers spread across several orders of magnitude. Tens, hundreds, thousands, millions, all in one pile. Here's why accountants care.
On screen: Illustrative, made-up data. Bars for first digits in a set of accounts with the Benford curve drawn as dots. Digit 5 sits at 13.1 percent, above the curve, turns red and is marked Look here.
So forensic auditors chart the first digits in a set of accounts and look hard at whatever doesn't fit this pattern.
On screen: Computed chart: first digits of a value growing 7 percent a year, bars against the Benford curve drawn as dots. They match: 30.7 percent start with 1, 4.6 percent with 9.
Real data is lopsided in a predictable way. A bad fit isn't necessarily fraud.
On screen: Illustrative. A shop full of $9.99 prices: 90 percent of prices start with 9. Breaks the curve, no fraud.
Prices set at $9.99 amounts, capped by policy,
On screen: Illustrative. Expense claims of $10 to $49.99 under a $50 cap: only digits 1 to 4 appear, and a red CAP line drops between 4 and 5. Nothing can start with 5 to 9.
a narrow range or a small sample will all break the curve innocently with no malintent.
On screen: Card: What this doesn't prove. Fraud. A bad fit tells you where to look, not what you'll find.
It tells you where to look, not what you will find when you look.
On screen: End card: the Highland Willow willow-tree logo grows from its trunk and roots into its canopy, then the words Highland Willow, highlandwillow.com/journal, and We publish our working.
Sources
- Benford, F. (1938). "The Law of Anomalous Numbers." Proceedings of the American Philosophical Society 78(4), 551–572.
- Nigrini, M. (2012). Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection. Wiley.
- Diekmann, A. (2007). "Not the First Digit! Using Benford's Law to Detect Fraudulent Scientific Data." Journal of Applied Statistics.
Narrated and produced by Highland Willow. The audio-description voice is synthetic (macOS text-to-speech). The animation is drawn in code. None of the film's data comes from any client.