Benford’s law (also called the first-digit law or Newcomb–Benford law) says that in many real-world collections of numbers the leading digit is far more often small than large. About 30.1% of values start with 1, 17.6% with 2, and only 4.6% with 9 – not the 11.1% each you might expect. The pattern shows up in populations of towns, river lengths, stock prices, invoice amounts, physical constants and the Fibonacci numbers.
The reason is scale: data that spread over several orders of magnitude and grow multiplicatively spend more of their “logarithmic time” between 1 and 2 than between 9 and 10. The distribution is the only one that stays the same when you change units (dollars to euros, metres to feet).
Auditors and forensic accountants use the law as a quick screen. Made-up or manipulated numbers rarely follow it, so a large gap at a particular digit is a reason to look closer – not proof of fraud. Data with a narrow range (heights of adults), assigned numbers (phone numbers, ZIP codes) or prices set to psychological thresholds should not be expected to follow Benford.
How to use the Benford’s Law Calculator
- What do you have?
- Data set (numbers) — Paste numbers separated by commas, spaces or new lines. Zeros are skipped.
- Values starting with 1
- Values starting with 2
- Values starting with 3
- Values starting with 4
- Values starting with 5
- Values starting with 6
- Values starting with 7
- Values starting with 8
- Values starting with 9
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Each non-zero value is written in scientific notation and its first significant digit is counted (signs and decimal points are ignored). The observed share of each digit is compared with log₁₀(1 + 1/d), the chi-square statistic is turned into a p-value with 8 degrees of freedom, and the mean absolute deviation is graded against Nigrini’s thresholds for first-digit tests.
Worked example
An auditor counts the first digits of 500 invoice amounts: 152, 86, 64, 49, 39, 33, 29, 26 and 22 for digits 1 to 9. Benford’s law expects 500 × 0.30103 = 150.5 ones, 88.0 twos, 62.5 threes and so on. Adding (O − E)²/E over the nine digits gives χ² = 0.16 with 8 degrees of freedom, a p-value above 99.9%. The mean absolute deviation is 0.18 percentage points – close conformity – so these invoices show no warning signs.
Switch to raw numbers and the first 30 Fibonacci numbers give 9 ones (30.0%, expected 30.1%) and 5 twos (16.7%, expected 17.6%), but with only 30 values the test has little power.
Expected first-digit frequencies
| Digit | Benford share | Expected in 1,000 values |
|---|---|---|
| 1 | 30.10% | 301 |
| 2 | 17.61% | 176 |
| 3 | 12.49% | 125 |
| 4 | 9.69% | 97 |
| 5 | 7.92% | 79 |
| 6 | 6.69% | 67 |
| 7 | 5.80% | 58 |
| 8 | 5.12% | 51 |
| 9 | 4.58% | 46 |
Formula
P(first digit = d) = log₁₀(1 + 1/d), d = 1 … 9 · χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ with 8 degrees of freedom · MAD = Σ |observed share − expected share| / 9
Worked example
Inputs — What do you have?: Counts of each leading digit; Values starting with 1: 152; Values starting with 2: 86; Values starting with 3: 64; Values starting with 4: 49; Values starting with 5: 39; Values starting with 6: 33; Values starting with 7: 29; Values starting with 8: 26; Values starting with 9: 22.
Results — p-value of the chi-square test = 100 %; Conclusion = No significant departure from Benford's law (p ≥ 0.05).; First-digit table = Digit Count Observed Benford Expected count Diff (pts)
────────────────────────────────────────────────────────────────
1 152 30.4% 30.1% 150.51 +0.3
2 86 17.2% 17.61% 88.05 −0.41
3 64 12.8% 12.49% 62.47 +0.31
4 49 9.8% 9.69% 48.46 +0.11
5 39 7.8% 7.92% 39.59 −0.12
6 33 6.6% 6.69% 33.47 −0.09
7 29 5.8% 5.8% 29 +0
8 26 5.2% 5.12% 25.58 +0.08
9 22 4.4% 4.58% 22.88 −0.18; Chi-square statistic (df = 8) = 0.162; Mean absolute deviation (percentage points) = 0.177; Nigrini conformity (by MAD) = 0.
Key terms
- Leading (first significant) digit
- The first non-zero digit of a number, e.g. 3 for 0.0372 and 1 for 1,948.
- Goodness-of-fit test
- A test of whether observed counts match the counts a model predicts.
- Mean absolute deviation (MAD)
- The average absolute difference between observed and expected digit shares.
- Scale invariance
- The property that a distribution looks the same after multiplying every value by a constant.
Frequently asked questions
What does Benford's law predict for each digit?
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%. Each value is log₁₀(1 + 1/d).
How many numbers do I need?
At least 50, and preferably several hundred. With 30 values a single number moves a digit's share by 3.3 percentage points, so random noise dominates.
Does a failed test prove fraud?
No. It shows the digits are unusual. Many honest data sets – narrow ranges, rounded values, prices like 9.99 – fail too. Use it to choose what to audit.
Which data follow Benford's law?
Data spanning several orders of magnitude that arise from growth or multiplication: populations, sales, account balances, stock volumes, physical constants, Fibonacci and powers of 2.
Which data do not?
Human heights and test scores (narrow range), identification numbers, lottery numbers, values with a fixed minimum or maximum, and anything assigned rather than measured.
What is the MAD test?
Mark Nigrini's mean absolute deviation: the average gap between observed and expected shares. For first digits, below 0.6 points is close conformity, 0.6–1.2 acceptable, 1.2–1.5 marginal and above 1.5 nonconformity.
Why is the chi-square test so strict with big data?
With thousands of values even tiny, harmless departures become statistically significant, which is why auditors also look at the MAD.
Do negative numbers and decimals count?
Yes – the sign is ignored and the first significant digit is used, so −0.0453 counts as a 4.