Benford Law Calculator
Get the Benford share and expected count for any first digit 1 to 9.
What it does
First digits are not democratic. In river lengths, city populations, physical constants, and many payment ledgers, small leading digits appear far more often than large ones. Digit 1 opens about three in ten numbers while 9 opens fewer than one in twenty. That skew is Benford law, and auditors use it as a first pass screen.
The tool above gives the expectation for one digit at a time. Choose d and a sample size N to get the expected share, the expected count, and the ratio against a flat 11.11% split. Defaults ask about 1s in 1000 records, so the page opens solved at 30.10% and 301.0.
The shape comes from logarithms. When the logs of values spread evenly, the mantissa lands in low digit territory more often, which yields P(d) = log10(1 + 1/d). Scale does not matter: whether figures sit in dollars or thousands of dollars, the first digits keep the same profile as long as the data span orders of magnitude.
Scope matters as much as math. Prices pinned by psychology, heights of adults, and ID numbers with fixed prefixes break the pattern for honest reasons. Clean data can also miss the mark in small samples. Treat a gap as a pointer toward records to inspect, then confirm with source documents and a real goodness of fit test.
The table lists all nine shares for reference, each checked against the log formula. Run your digit through the tool for exact counts at your N, then move digit by digit until the full observed profile stands next to the expected one.
How to use
- Pick a first digit from 1 to 9.
- Enter the sample size N.
- Read the expected percent and the expected count.
- Read the ratio versus a uniform split.
- Repeat for the other digits to build the full profile.
- Compare observed counts against these expectations with a proper test.
Formula
- d
- First significant digit, 1 to 9.
- N
- Number of records examined.
- P(d)
- Expected share for digit d.
- Uniform
- Share if all nine digits were equal, 11.11%.
The law covers the first significant digit in base 10. N is the sample size and d runs from 1 to 9. The ratio column compares each share with the uniform 11.11% baseline so the skew reads at a glance. Fit quality still needs a proper statistical test outside this page.
Worked example
Example: leading 1s in 1000 records
In 1000 accounting entries that follow Benford law, how many should start with digit 1?
- P(1) = log10(1 + 1/1) = log10(2) = 0.3010, shown as 30.10 %.
- Expected count in 1000 records = 0.3010 x 1000 = 301.0.
- Uniform share would be 100/9 = 11.11 %, so the ratio is 30.10/11.11 = 2.71 x.
- Digits 2 through 9 split the rest, with 9 near 4.58 %.
- A clean ledger with far fewer 1s than 30% deserves a second look.
- Fit alone never proves fraud; it only directs the audit.
Reference table
| Digit | Share | In 1000 records |
|---|---|---|
| 1 | 30.10 % | 301.0 |
| 2 | 17.61 % | 176.1 |
| 3 | 12.49 % | 124.9 |
| 4 | 9.69 % | 96.9 |
| 5 | 7.92 % | 79.2 |
| 6 | 6.69 % | 66.9 |
| 7 | 5.80 % | 58.0 |
| 8 | 5.12 % | 51.2 |
| 9 | 4.58 % | 45.8 |
Limits
- First digit only; later digits need wider forms.
- Needs data spanning orders of magnitude.
- Small samples swing widely around the shares.
- Fit alone never proves or clears fraud.
Common mistakes
- Testing assigned numbers like ZIP codes against it.
- Expecting exact counts instead of a band.
- Including 0 as a possible leading digit.
- Stopping at one digit instead of profiling all nine.
- Calling a gap proof of wrongdoing without documents.
Try it yourself
Quick checks. Answers sit tucked under each question.
Digit 9 in 5000 records?
Share is 4.58 %. Count is 228.8. Ratio is 0.41 x.
Digit 2 in 1000 records?
Share is 17.61 %. Count is 176.1. Ratio is 1.58 x.
Digit 3 in 2000 records? Work it by hand.
P(3) = log10(4/3) = 12.49 %. In 2000 records expect 249.9.
Questions
What is Benford law?
The claim that in many natural tables the first digit d appears with probability log10(1 + 1/d). Digit 1 leads about 30.1% of the time, digit 9 about 4.6%.
Does every dataset fit it?
No. It fits data that span several orders of magnitude with no built in cutoffs, such as populations, river lengths, and many ledgers. Assigned numbers like phone codes do not fit.
How do auditors use it?
Compare observed shares against the expected ones, often with a chi square or a mean absolute deviation check. Large gaps flag the records to inspect first.
How close should real data land?
Small samples swing widely. With 1000 records the 1 count should sit near 301, but random sets can still land 30 or more away.
Why is there no digit 0 input?
Use digits 1 to 9 only. Zero never leads a number, so it has no Benford share.
Which logarithm base does the tool use?
It applies to base 10 here with the log10 form. Other bases use log base b of (1 + 1/d).