What this calculator does
Benford law says that in many naturally occurring data sets, the leading digit is 1 about 30.1% of the time and 9 only 4.6%. The distribution is logarithmic, not uniform, and the pattern is strong enough that departures from it are used to flag possible fabrication.
It works because data spanning several orders of magnitude spends more of its range with a leading 1 than a leading 9. Going from 100 to 200 doubles a value; going from 900 to 1000 adds only 11%. Numbers therefore linger longer in the low leading digits as they grow.
The formula
The leading digit of each value is extracted and the observed shares are compared with the Benford expectation, log₁₀(1 + 1/d) for each digit. The mean absolute deviation averages the gaps in percentage points. It is a descriptive comparison rather than a formal significance test, so interpret the result as a prompt to look closer.
| Term | Meaning |
|---|---|
| Leading digit | The first non-zero digit, so 0.0034 and 340 both lead with 3. |
| Benford distribution | P(d) = log₁₀(1 + 1/d), giving 30.1% for 1 down to 4.6% for 9. |
| Mean absolute deviation | The average gap between observed and expected shares, in percentage points. |
| Scale invariance | Benford distribution is unchanged by a change of units, which is part of why it appears so widely. |
The inputs explained
| Field | What to enter |
|---|---|
| Data (comma or space separated) | Your data, comma or space separated. It should span several orders of magnitude and be free of imposed minimums or maximums for the law to apply. |
When to use it
Screening financial data
Fabricated figures tend to have too-uniform leading digits, because people inventing numbers do not naturally produce a logarithmic distribution.
Checking data integrity
A large deviation can indicate rounding, truncation or a hidden constraint rather than fraud.
Auditing election or census counts
The law has been applied to both, though its reliability in these settings is genuinely contested.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the deviation respond to different data?
Roughly Benford-like data against two artificially constrained sets.
| Data set | Mean absolute deviation from Benford | Digit 1 observed vs expected | Interpretation |
|---|---|---|---|
| 234, 45, 891, 102, 33, 567, 12, 789, 456, 78… | 5.98 percentage points | 20.0% vs 30.1% | Some deviation: worth a closer look |
| 1,10,100,1000,12,15,19,11,13,17,120,150,18,1… | 15.53 percentage points | 100.0% vs 30.1% | Large deviation from Benford |
| 91,92,93,94,95,96,97,98,99,90,89,88,87,86,85 | 20.07 percentage points | 0.000% vs 30.1% | Large deviation from Benford |
Questions
Which data sets follow Benford law?
Data spanning several orders of magnitude that arises from multiplicative or growth processes: populations, river lengths, financial transactions, physical constants. It does not apply to data with imposed bounds, such as heights, IQ scores or anything assigned sequentially.
Can Benford law prove fraud?
No. It flags a data set as worth examining, nothing more. Deviations arise from rounding, truncation, price thresholds, minimums and many other innocent causes. It is a screening tool used to direct attention, and it has been rejected as standalone evidence in court.
Why is 1 the most common leading digit?
Because on a logarithmic scale the span from 1 to 2 is much wider than from 9 to 10. A growing quantity spends about 30% of its time with a leading 1 and about 5% with a leading 9, simply because reaching the next digit from 1 requires a 100% increase and from 9 requires only 11%.
How much data do I need?
Several hundred values at minimum for a meaningful comparison, and preferably more. With only fifteen values, as in the table above, individual digits swing the shares enormously and the deviation figure is dominated by noise.
For general summary statistics, see the descriptive statistics calculator. For a formal fit test, see the chi-square goodness of fit calculator.