The law of the first digit
Tonight's idea
You would expect the first digit of a random number to be 1 about one time in nine. For most real data it is not. Populations of towns, lengths of rivers, electricity bills, numbers in a newspaper, stock prices: the first digit is 1 about 30% of the time, and 9 only 4.6% of the time. This is Benford's law, noticed in 1881 by Simon Newcomb, who saw that the early pages of books of logarithm tables (the ones for numbers starting with 1) were much dirtier than the later pages.
The exact prediction for the first digit \(d\) is \(P(d) = \log_{10}\left(1 + \tfrac{1}{d}\right)\):
| first digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| % | 30.1 | 17.6 | 12.5 | 9.7 | 7.9 | 6.7 | 5.8 | 5.1 | 4.6 |
Why? Think about something that grows by multiplying, like a town or a savings account. To go from 100 to 200 it has to double. To go from 900 to 1000 it only has to grow by 11%. So it spends far longer with a leading 1 than with a leading 9, then at 1000 the whole thing starts again. On a logarithmic scale, the stretch from 1 to 2 takes up 30.1% of the distance from 1 to 10, from 2 to 3 takes 17.6%, and so on. Anything spread evenly on that scale obeys the law. That is the real condition, not “the numbers grow” or “the range is wide”: ask where each number lands inside its decade.
Accountants use this as a screening tool. People inventing numbers often spread the first digits out evenly, because that feels random, so a tally far from the table is a reason to look closer. It is only a reason to look, not proof. Before you accuse anyone, ask two questions.
1. Should these numbers follow the law at all? You need a model of where the numbers come from; a wide range alone does not settle it. Suppose every whole-dollar lunch receipt from $10 to $99 is equally likely. Then each first digit owns ten of the ninety amounts, and honest people produce a flat tally. (Equally likely is an assumption; honesty alone does not make it true.)
2. Could a cheat copy the table? Anyone who knows the law can invent numbers that obey it. Try the law on nine lists in the first-digit demo, and keep asking where the numbers came from.
Watch one: powers of 2
Powers of 2 grow by multiplying, so they should obey the law. List them: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, ... Take the first digits: 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1.
Already 1 is winning. Over the first 30 powers (\(2^0\) to \(2^{29}\)), 9 of them start with 1, which is exactly 30%, and none start with 7 or 9 yet. Keep going to a thousand powers and every digit settles onto the Benford percentages.
Your turn
- Find 50 numbers from the real world: an atlas, a newspaper, a cookbook, the backs of food packets. Record only the first digit of each. Tally them and compare your percentages with the table. If your tally does not match, do not assume you made a mistake: ask whether the numbers you chose spread across several orders of magnitude or were all bunched in one range.
- Check the claim above by writing out \(2^0\) to \(2^{29}\) and counting how many begin with each digit. (The doubling makes this quick: you only need the first two or three digits each time.) Then take the powers of 10, from \(10^0\) to \(10^{29}\). They also grow by multiplying and span thirty orders of magnitude. Predict their tally before you count. Where does each one land inside its decade?
- Add up the nine percentages in the table. Why must the total be 100%? Then show it exactly: write each \(\log_{10}(1 + \tfrac1d)\) as \(\log_{10}\frac{d+1}{d}\) and multiply all nine fractions together (adding logs is the same as multiplying the numbers inside, from the logarithms sheet).
- Which of these should not follow Benford's law, and why? Heights of adults in centimetres. Phone numbers. Lottery numbers. Populations of countries. House numbers on a very long street.
- An employee submits 100 expense claims. The first digits are: 1 appears 12 times, and each of the digits 2 to 9 appears 11 times. The claims are all team lunches, which the company caps at $99, and nobody bothers claiming under $10. What first-digit tally would honest whole-dollar lunch claims produce? Would you investigate? Now suppose instead the 100 claims ranged from $5 bus fares to $4,000 flights. Can the range alone tell you what tally to expect? Try two models: (a) every whole-dollar amount from $5 to $4,000 equally likely; (b) amounts spread evenly along the logarithmic scale. How many claims should start with 1 under each?
- Four lists of numbers: (i) the powers of 2 from question 2; (ii) one honest receipt for every whole-dollar amount from $10 to $99; (iii) 100 amounts a cheat invented by deliberately choosing first digits to match the table; (iv) the powers of 10 from question 2. Which lists pass the first-digit test? Which are honest? Write one sentence about what a pattern test can and cannot tell you.
Stretch: changing units, and the second digit
River lengths in kilometres obey the law. Convert every length to miles by multiplying by 0.621. Explain why the first digits still obey the law, even though every individual number changed. (On the logarithmic scale, what does multiplying everything by the same constant do?)
Benford's law has a version for the second digit too, but it is much flatter: second digit 0 occurs about 12.0% of the time and second digit 9 about 8.5%. Explain in a sentence why the second digit should be more evenly spread than the first.
Answers
- Results vary. A mixed bag from an atlas and a newspaper typically gives roughly 15 ones, 9 twos, 6 threes, 5 fours, and two or three each of the rest. Fifty prices from one supermarket aisle, all between $2 and $9, will not match, and that is honest data too: it simply does not spread across orders of magnitude.
- Powers of 2, digits 1 to 9: 9, 6, 3, 3, 3, 3, 0, 3, 0. Nine leading 1s out of 30 is 30%. Powers of 10: all thirty begin with 1, so the tally is 30, 0, 0, 0, 0, 0, 0, 0, 0. They grow by multiplying and span a huge range, yet fail the law completely, because every one lands at the very start of its decade. Growth and range were never the condition; even spread inside the decade is.
- Every number starts with some digit from 1 to 9, so the percentages must total 100%. The product \(\frac21 \times \frac32 \times \frac43 \times \cdots \times \frac{10}{9}\) telescopes: everything cancels except \(\frac{10}{1}\), and \(\log_{10} 10 = 1\).
- Heights: no, they are all around 150–190 cm, so nearly all start with 1 for a boring reason, never with 5 or 9. Phone numbers: no, they are assigned codes, not measurements. Lottery numbers: no, they are drawn uniformly. Country populations: yes, in practice they are spread fairly evenly along the logarithmic scale (spanning many orders of magnitude is a hint, not a guarantee: see the powers of 10). House numbers: no, they go 1, 2, 3, ... evenly.
- If every whole-dollar lunch from $10 to $99 is equally likely, each first digit gets exactly ten amounts (10–19, 20–29, and so on), so a flat tally of about 11 each is what honesty looks like under that model. No reason to investigate. For $5 to $4,000 the range alone cannot tell you. Model (a): of the 3,996 amounts, 1,110 start with 1 (10–19, 100–199, 1,000–1,999), and likewise 1,110 each for 2 and 3, but only 111 each for 4 to 9. Expect about 28 claims starting with 1, 28 with 2, 28 with 3 and about 3 each for the rest: honest, yet nowhere near Benford. Model (b): close to Benford’s 30, 18, 12, down to 5 (about 31 start with 1). Only once you have a reason to believe one model can a mismatch mean anything, and even then it is a reason to ask for receipts, not a verdict.
- Lists (i) and (iii) pass the test; lists (ii) and (iv) fail it. Lists (i), (ii) and (iv) are honest; list (iii) is the fake. So the test can fail honest data, even honest data that grows and spans a wide range, and it can pass invented data. A pattern test only means something when you have a good reason to expect the pattern in the first place, and even then a mismatch is a question, not an answer.
Stretch. On the logarithmic scale, multiplying every number by 0.621 slides every point the same distance left (by \(\log_{10} 0.621 \approx -0.207\)). A pattern that is spread evenly along that scale is still spread evenly after sliding, so the proportion landing in each first-digit band is unchanged. The second digit is more even because, within any first-digit band, the ten second-digit sub-bands are much closer in width on the log scale (for example 10–11 and 19–20 differ far less in log-width than 1–2 and 9–10 do).