Test whether your dataset follows Benford's Law. Analyze the distribution of first digits and detect anomalies or fraud.
Last updated: March 2026
Only positive numbers are used. Zeros and negative numbers are ignored.
| Digit | Observed | Expected % | Expected Count |
|---|---|---|---|
| 1 | 3 | 30.10% | 3.61 |
| 2 | 1 | 17.61% | 2.11 |
| 3 | 1 | 12.49% | 1.50 |
| 4 | 1 | 9.69% | 1.16 |
| 5 | 1 | 7.92% | 0.95 |
| 6 | 1 | 6.69% | 0.80 |
| 7 | 1 | 5.80% | 0.70 |
| 8 | 1 | 5.12% | 0.61 |
| 9 | 2 | 4.58% | 0.55 |
| Digit | Expected % | Applications Where Benford's Applies |
|---|---|---|
| 1 | 30.1% | Financial records, tax returns, election data, population sizes |
| 2 | 17.6% | Stock prices, scientific constants, street addresses |
| 3-9 | Decreasing (12-5%) | Digit 9 appears least frequently (~4.6%) |
Significant deviations (χ² test p < 0.05) may indicate data fabrication, rounding, or quality issues. Not all datasets follow Benford's—typically applies to naturally occurring numbers spanning multiple orders of magnitude.
Benford's Law is a fascinating statistical principle stating that in many real-world datasets, the leading digit is not uniformly distributed (1/9 each). Instead, smaller digits (especially 1) appear more frequently than larger digits. Specifically, the first digit $d$ follows the distribution: $P(d) = \log_10(1 + 1/d)$.
This means digit 1 appears ~30% of the time, digit 2 appears ~18%, and digit 9 appears only ~5%. The pattern emerges naturally in many datasets: financial records, tax returns, city populations, stock prices, and more. Deviations from Benford's Law can indicate data anomalies, rounding errors, or even fraud.
The law was first noted by Frank Benford in 1938, though it applies to any exponentially growing or multiplicatively scaled data. Fraudsters often don't know about Benford's Law and misrepresent digits uniformly, allowing auditors to detect suspicious patterns.
Enter a list of positive numbers separated by commas, spaces, or newlines. Only the first digit of each number is analyzed.
The calculator computes a χ² statistic comparing your observed distribution to Benford's expected distribution.
If p-value < 0.05, your data significantly deviates from Benford's Law—possible indicators of data quality issues or fraud.
Testing a dataset of 100 invoices from a company's accounting records:
The invoices follow Benford's Law well. The observed distribution matches the expected distribution within normal statistical variation. No evidence of fraud or anomalies.
Interpretation: If the p-value were < 0.05 (e.g., χ² > 15.5), we'd suspect data manipulation.
No. It applies to data that spans multiple orders of magnitude (exponential growth or multiplication): financial data, populations, stock prices, gene lengths. Small datasets or those with constrained ranges (e.g., grades 0–100) don't follow it.
A p-value < 0.05 suggests your data significantly deviates from Benford's Law. This could indicate data entry errors, rounding, artificial constraints, or fraud. Larger p-values mean your data fits well.
Yes, as a screening tool. Fraudsters rarely account for first-digit distributions, so fabricated financial records often deviate. However, it's not proof alone—combine with other audits.
When numbers are distributed across multiple orders of magnitude and scales (multiplied by random factors), the logarithmic distribution of first digits emerges naturally. It's scale-invariant.
Results are unreliable. Chi-square tests require sufficient sample size (typically ≥ 20–30). With few numbers, random variation dominates.
No. Benford's Law applies only to positive numbers' first significant digit. Leading zeros (e.g., 0.123) have first digit 1 (from 1.23×10⁻¹). Negatives: use absolute value.
That's a huge red flag (χ² would be enormous, p ≈ 0). Suggests data is either fabricated, heavily rounded, or bounded (e.g., prices in a fixed range than expected by Benford's Law).
Don't 'fix' it artificially. Investigate root causes: rounding, data entry errors, measurement constraints. If it's high-quality real data with constrained ranges, Benford's Law simply doesn't apply.
Related Tools