Benford's Law: Why the First Digit Is a 1 About 30% of the Time

What Benford's law actually says

Write any positive number in scientific notation, x = m × 10^k with 1 ≤ m < 10. The number m is the significand (or mantissa), and the leading digit D₁ is just ⌊m⌋. Benford's law is a statement about how D₁ is distributed:

P(D₁ = d) = log₁₀(1 + 1/d), for d = 1, 2, …, 9.

Because the nine probabilities are the widths of the intervals [1,2), [2,3), …, [9,10) measured on a base-10 logarithmic scale, they automatically sum to log₁₀10 = 1. The full table:

Leading digit d	log10(1 + 1/d)	Uniform guess
1	30.103%	11.1%
2	17.609%	11.1%
3	12.494%	11.1%
4	9.691%	11.1%
5	7.918%	11.1%
6	6.695%	11.1%
7	5.799%	11.1%
8	5.115%	11.1%
9	4.576%	11.1%

A 1 is 6.58 times more likely to lead than a 9, and digits 1 and 2 together take 47.7% of all numbers. The stronger, modern form of the law is a statement about the whole significand, not just its first digit: P(m ≤ t) = log₁₀ t for 1 ≤ t < 10, i.e. log₁₀ m is uniform on [0, 1). Every digit statement — first digit, second digit, first two digits — is a corollary of that single distributional claim.

Two hypotheses matter and are usually left unsaid. First, Benford's law is a property of a distribution, not of any individual number; asking whether 6,650 “obeys Benford’s law” is meaningless. Second, there is no uniform probability measure on all positive integers, so “a random number starts with 1 thirty percent of the time” is not a well-formed statement until you specify the measure or, for a sequence, switch to natural density.

Newcomb's worn logarithm tables and Benford's 20,229 numbers

In 1881 the astronomer Simon Newcomb published a two-page note in the American Journal of Mathematics (vol. 4, no. 1, pp. 39–40) titled “Note on the Frequency of Use of the Different Digits in Natural Numbers.” His evidence was physical wear: in shared books of logarithm tables, the early pages — the ones you turn to for numbers beginning with 1 — were visibly dirtier and more thumbed than the later pages. From that observation he wrote down P(D₁ = d) = log(d+1) − log(d), and even gave the second-digit law. The note attracted essentially no attention.

Fifty-seven years later the General Electric physicist Frank Benford rediscovered it and did the empirical work. “The Law of Anomalous Numbers,” Proceedings of the American Philosophical Society 78(4), 1938, pp. 551–572, assembled 20,229 observations across 20 heterogeneous tables: the surface areas of 335 rivers, 3,259 populations, 104 physical constants, 1,800 molecular weights, 5,000 entries from a mathematical handbook, 308 numbers scraped out of an issue of Reader's Digest, street addresses of the first 342 people listed in American Men of Science, death rates, and baseball statistics. Individually the tables wobbled; pooled, they landed almost exactly on the logarithmic curve. The law carries Benford's name rather than Newcomb's, which makes it a textbook instance of Stigler's law of eponymy — itself, fittingly, not due to Stigler.

Rigour arrived much later. Roger Pinkham (1961) showed that if a leading-digit law is scale-invariant, it must be Benford's. Theodore P. Hill (1995, Statistical Science 10(4), 354–363) built the significand σ-algebra properly, proved the base-invariance characterisation, and proved a random-mixture theorem: if you draw distributions at random and then sample from them, the pooled digits converge to Benford. That theorem is the honest explanation of why Benford's own grab-bag of twenty unrelated tables fit the curve better than any single table did.

The mechanism: a logarithmic ruler, and why units cannot matter

Lay the significands 1 through 10 on a ruler where distance is log₁₀. The digits are no longer evenly spaced. The stretch of ruler owned by a leading digit of 1 — from 1 to 2 — has length log₁₀ 2 = 0.30103. The stretch owned by 9 — from 9 to 10 — has length log₁₀(10/9) = 0.04576. That is the entire picture in the video: the strips are the probabilities, and the strip for 1 is the widest simply because doubling is a bigger multiplicative step than going from 9 to 10.

That reframes the question. “Why do leading digits follow log₁₀(1 + 1/d)?” becomes “why would data land uniformly on that logarithmic ruler?” The answer is scale invariance. Suppose your data has some leading-digit law, and you now measure everything in feet instead of metres. Every value is multiplied by 3.28084, which on the logarithmic ruler is a rigid shift by log₁₀ 3.28084 = 0.51599, wrapping around at the ends. If the law is to be the same in feet as in metres, the distribution of log₁₀ m modulo 1 must be unchanged by every such shift — and the unique shift-invariant (Haar) probability measure on the circle ℝ/ℤ is the uniform one. Uniform in log means Benford in digits. That is Pinkham's argument, made rigorous by Hill.

Two honest caveats. First, the ruler picture is an illustration of the conclusion, not a proof that any particular dataset is log-uniform; nothing forces river lengths to be scale-invariant, it is an empirical fact about them that they nearly are. Second, exact scale invariance cannot hold for a real finite dataset — it is a limiting idealisation, in the same way that “the errors are normal” is. What the theorem delivers is a conditional: if a significant-digit law is scale-invariant, it is Benford, and no other candidate law survives the test.

A practical rule of thumb comes from the lognormal family. If log₁₀ x is normal with standard deviation σ, the resulting digit distribution is visually indistinguishable from Benford once σ ≳ 1 — that is, once the data spans more than about a decade of spread — and drifts away from it as σ shrinks toward 0. Spread over orders of magnitude is the operative condition.

Sequences that provably obey it: 2ⁿ, Fibonacci, and n!

For deterministic sequences the law can be proved outright, and the proof is Weyl's equidistribution theorem (1916): the sequence {nα} of fractional parts is equidistributed modulo 1 if and only if α is irrational.

Take the powers of two. log₁₀ 2ⁿ = n·log₁₀ 2, and log₁₀ 2 = 0.301029995… is irrational (if it were p/q then 2^q = 10^p, impossible by unique factorisation, since the right side has a factor of 5). So the fractional parts equidistribute on [0,1), and the proportion of n ≤ N with leading digit d converges to log₁₀(1 + 1/d). Counting the first hundred powers of two gives the digits

[30, 17, 13, 10, 7, 7, 6, 5, 5] against a Benford prediction of [30.1, 17.6, 12.5, 9.7, 7.9, 6.7, 5.8, 5.1, 4.6].

Exactly 30 of them start with a 1, ending at 2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376. The Fibonacci numbers work the same way: F_n = (φⁿ − ψⁿ)/√5 with ψⁿ → 0, so log₁₀ F_n ≈ n·log₁₀φ − log₁₀√5 and log₁₀φ = 0.20899… is irrational; the first 100 Fibonacci numbers give [30, 18, 13, 9, 8, 6, 5, 7, 4]. Factorials are harder because log₁₀(n!) is not an arithmetic progression, but Persi Diaconis settled it in 1977 (Annals of Probability 5(1), 72–81): n! is Benford. The first 100 factorials give [30, 18, 13, 7, 7, 7, 3, 10, 5].

Say precisely what has been proved. These are statements about natural density — the limit of a counting fraction — not about a probability measure, because there is no uniform probability measure on ℕ. And the counterexample matters: 10ⁿ spans unlimited orders of magnitude yet always leads with a 1, because log₁₀10 = 1 is an integer and the fractional part never moves. Spanning many decades is necessary but not sufficient; the multiplier's logarithm must be irrational.

Where Benford's law fails

Benford's law is a hypothesis about data, and plenty of data refutes it. It tends to fail when any of these hold:

• The range is too narrow. Adult heights in centimetres run roughly 150–200, so D₁ = 1 almost always. Body temperature in °F is always 9 or 1. Less than one decade of spread leaves no room for the logarithmic ruler to wrap.
• The numbers were assigned, not generated. Telephone numbers inside one area code, sequential invoice IDs, ZIP codes, ISBNs. These carry structure imposed by a naming scheme.
• The distribution is uniform or symmetric by construction. Lottery draws, dice sums, random digits, standardised IQ scores centred at 100 with SD 15.
• There is a strong human anchor. Prices ending in .99, round-number thresholds, tax brackets, and any quantity clustered at a reporting cut-off.
• The generating multiplier is degenerate. The powers of ten, as above.

The positive conditions are equally concrete. Benford conformity is expected when the data is positive, spans several orders of magnitude, has no built-in minimum or maximum near the bulk, and arises from multiplicative or mixed processes — growth, compounding, pooling of many different sources. Benford's own 1938 result is best read as a demonstration of Hill's mixture theorem: any one of his tables was mediocre, but twenty pooled tables were excellent.

Beyond the first digit: second digits and first-two-digit tests

The general significant-digit law extends to any block of leading digits. For an n-digit leading block, P(first n digits = k) = log₁₀(1 + 1/k), so the first two digits satisfy P(D₁D₂ = 10) = log₁₀(1.1) = 4.139% down to P(D₁D₂ = 99) = log₁₀(100/99) = 0.4365%.

Marginalising gives the second-digit law, P(D₂ = d) = Σ_k=1⁹ log₁₀(1 + 1/(10k + d)):

Second digit	0	1	2	3	4	5	6	7	8	9
Probability	11.97%	11.39%	10.88%	10.43%	10.03%	9.67%	9.34%	9.04%	8.76%	8.50%

The second digit is much flatter than the first, and by the third digit the law is within 0.18 percentage points of uniform (10.178% for a 0 down to 9.827% for a 9) — which is why digit tests beyond the first two carry almost no signal.

In practice the workhorse is the first-two-digits test: bin a dataset into the 90 buckets 10–99, compare observed to expected, and score the fit. Nigrini's mean absolute deviation (MAD) thresholds for that test are the ones auditors quote: below 0.0012 is close conformity, 0.0012–0.0018 acceptable, 0.0018–0.0022 marginal, above 0.0022 nonconformity. A χ² goodness-of-fit statistic with 89 degrees of freedom is the classical alternative, but it is famously sensitive to sample size: with 500,000 records, a deviation far too small to matter will still reject at p < 0.001. The base change is straightforward too — in base b, P(d) = log_b(1 + 1/d) — and note the degenerate case: in base 2 the only possible leading digit is 1, with probability 1.

Forensic accounting, elections, and what the evidence can and cannot show

The practical payoff is that fabricated numbers are usually too flat. People inventing figures reach for digits roughly evenly, avoid repeats, and cluster just under thresholds; genuine multiplicative data does not. Mark Nigrini turned this into an audit technique in his 1992 University of Cincinnati dissertation and a sequence of papers with Linda Mittermaier (Auditing: A Journal of Practice & Theory, 1997), and digit analysis is now a standard module in commercial audit software. It is used on tax returns, expense claims, journal entries and insurance claims.

Macroeconomic data has been tested the same way. Rauch, Göttsche, Brähler and Engel, “Fact and Fiction in EU-Governmental Economic Data” (German Economic Review 12(3), 2011, 243–255), applied the first-digit test to figures reported to Eurostat and found Greece's deficit and debt data showed the largest deviation from Benford of any EU member state — a finding that landed during the sovereign debt crisis and is still the most-cited applied result in the field. Election forensics is the contested case: Benford tests have been applied to precinct-level vote counts, notably in the 2009 Iranian presidential election, but Joseph Deckert, Mikhail Myagkov and Peter Ordeshook argued in Political Analysis (2011) that first-digit Benford tests on vote counts are close to useless, because precinct sizes are bounded and do not span the required decades. That criticism is methodologically sound and applies to any bounded count data.

State the limits plainly. A Benford deviation is a screening signal, not proof. Non-conformity can arise from legitimate causes — price points, rounding policy, bounded ranges, subsidy thresholds, small samples — and conformity does not clear anyone, since a fraudster who knows the law can generate Benford-compliant numbers. Every serious deployment treats the test as a way to rank accounts for human examination, and the underlying claim is always conditional: if this account's legitimate activity would span several orders of magnitude and be generated multiplicatively, then the observed flatness deserves an explanation.

Which datasets obey Benford's law, and which ignore it

Dataset	Orders of magnitude spanned	Follows Benford?	Why
Powers of two, 2¹…2¹⁰⁰	≈30 (2 up to 1.27 × 10³⁰)	Yes — provably, in natural density	log₁₀ 2 is irrational, so the mantissas equidistribute (Weyl, 1916). Exactly 30 of the 100 start with a 1.
National populations	≈6 (≈10³ to 1.4 × 10⁹)	Yes, closely	Grows multiplicatively, spreads over many decades, and has no preferred unit of size
Adult heights in centimetres	<1 (≈150–200 cm)	No — badly	The range is narrower than one decade, so nearly every value begins with a 1
Lottery draws and assigned IDs	1–2	No	Uniform or assigned by design; nothing multiplicative generated them
Powers of ten, 10ⁿ	unbounded	No	The significand is frozen at 1.000…, so the leading digit is always 1 — spanning many decades is necessary, not sufficient