Benford's Law Calculator
Benford's Law calculator. Test data against first-digit law. Chi-square, MAD, fraud detection. Obser
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Benford's Law โ Test Data Against First-Digit Law
P(d) = logโโ(1 + 1/d). Chi-square fraud detection. Natural data follows; fabricated data often doesn't. MAD conformity.
Real-World Scenarios โ Click to Load
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For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- Benford's Law: P(d) = logโโ(1 + 1/d) for first digit d = 1..9. Digit 1 appears ~30.1%, 2 ~17.6%, 9 ~4.6%.
- Natural data: Many real-world datasets (populations, stock prices, river lengths) follow Benford's Law due to scale invariance.
- Fraud detection: Fabricated or manipulated data often deviate from Benford. MAD and chi-square tests flag suspicious datasets.
- MAD conformity: MAD < 0.006 = close, 0.006โ0.012 = acceptable, > 0.015 = nonconforming.
Did You Know?
How It Works
1. First-Digit Law
P(d) = logโโ(1 + 1/d). P(1) = 30.1%, P(2) = 17.6%, P(3) = 12.5%, โฆ P(9) = 4.6%.
2. First-Two Digits
P(dโdโ) = logโโ(1 + 1/(10dโ+dโ)) for 10..99. More sensitive for fraud detection.
3. Chi-Square Test
ฯยฒ = ฮฃ (observed โ expected)ยฒ / expected. df = 8 for first digit. Critical ฯยฒ(0.05) = 15.51.
4. MAD (Mean Absolute Deviation)
MAD = (1/9) ร ฮฃ|observed% โ expected%|. Nigrini (2012) conformity thresholds.
Expert Tips
Use sample datasets
Try Country Populations, Fibonacci โ see Benford in action
Fraud detection
Suspicious invoices โ compare to natural data
First-two digits
More granular โ 90 bins, better for forensic accounting
Scale invariance
Why natural data follows Benford โ multiplicative processes
First-Digit Expected Frequencies
| Digit | Expected % |
|---|---|
| 1 | 30.10% |
| 2 | 17.61% |
| 3 | 12.49% |
| 4 | 9.69% |
| 5 | 7.92% |
| 6 | 6.69% |
| 7 | 5.80% |
| 8 | 5.12% |
| 9 | 4.58% |
Frequently Asked Questions
Why does natural data follow Benford's Law?
Scale invariance: multiplicative processes (e.g., growth rates) produce numbers spanning many orders of magnitude. The distribution of mantissas in scientific notation is uniform, leading to Benford.
How is Benford's Law used in fraud detection?
Fabricated numbers often have human bias (e.g., round numbers, excess 5s and 6s). Real data follows Benford. Auditors compare observed first-digit distribution to expected.
What is MAD conformity?
MAD = mean absolute deviation between observed and expected percentages. Nigrini: < 0.006 = close, 0.006โ0.012 = acceptable, 0.012โ0.015 = marginal, > 0.015 = nonconforming.
Do Fibonacci numbers follow Benford?
Yes. The ratio of consecutive Fibonaccis tends to ฯ (golden ratio), and such sequences follow Benford exactly.
What datasets do NOT follow Benford?
Assigned numbers (IDs, ZIP codes), numbers with min/max constraints, or human-made data with bias.
By the Numbers
Official Sources
Scale Invariance Explained
Benford's Law is scale-invariant: if you multiply all numbers in a Benford-compliant dataset by a constant, the first-digit distribution remains unchanged. This is because the mantissa (fractional part of logโโ) of the product equals the mantissa of the original times the mantissa of the constant. For data spanning many orders of magnitude (populations, prices, physical constants), the uniform distribution of mantissas in scientific notation leads directly to Benford's Law.
Disclaimer: This calculator is for educational and exploratory analysis. Benford's Law is a useful heuristic for fraud detection but is not definitive proof. Consult professional auditors for forensic analysis.
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