Benford's Law Analysis

Overview

Benford's Law predicts that in naturally occurring datasets, the leading digit d appears with probability P(d) = log₁₀(1 + 1/d). Digit 1 appears ~30.1% of the time, digit 9 only ~4.6%. Deviations from this distribution may indicate data fabrication or manipulation. Analysis runs in O(n).

When to Use

Trigger conditions:

Auditing financial data (expenses, invoices, tax returns) for manipulation
Screening large datasets for data integrity issues
Detecting fabricated or artificially rounded numbers

When NOT to use:

For assigned/sequential numbers (zip codes, phone numbers, IDs)
For datasets with constrained ranges (e.g., human ages, percentages)
For small datasets (< 500 records — insufficient statistical power)

Algorithm

IRON LAW: Benford's Law Applies to NATURALLY OCCURRING Data Spanning Orders of Magnitude
Data that doesn't span multiple orders of magnitude (e.g., temperatures
in Celsius, human heights) will NOT follow Benford's Law. Deviation from
Benford's in such data is EXPECTED, not suspicious. Always verify the
data type is appropriate before concluding fraud.

Phase 1: Input Validation

Extract leading digits from dataset. Filter: remove zeros, negatives (take absolute value), values < 10. Verify dataset spans multiple orders of magnitude. Gate: 500+ records, data spans at least 2 orders of magnitude.

Phase 2: Core Algorithm

Extract first digit of each number
Count frequency of each digit (1-9)
Compare observed frequencies against Benford's expected: P(d) = log₁₀(1 + 1/d)
Statistical tests: chi-squared test, MAD (Mean Absolute Deviation), KS test

Phase 3: Verification

MAD thresholds: < 0.006 (close conformity), 0.006-0.012 (acceptable), 0.012-0.015 (marginal), > 0.015 (non-conforming). Flag specific digits with large deviations. Gate: MAD computed, non-conforming digits identified.

Phase 4: Output

Return conformity assessment with digit-level analysis.

Output Format

{
  "conformity": "marginal",
  "mad": 0.013,
  "chi_squared": {"statistic": 18.5, "p_value": 0.018, "df": 8},
  "digit_analysis": [{"digit": 1, "observed_pct": 25.1, "expected_pct": 30.1, "deviation": -5.0}],
  "metadata": {"records": 5000, "dataset": "Q4 expense reports"}
}

Examples

Sample I/O

Input: 1000 invoice amounts from a company's AP ledger Expected: First digits should approximate 30.1%, 17.6%, 12.5%, 9.7%, 7.9%, 6.7%, 5.8%, 5.1%, 4.6%. MAD < 0.012 for legitimate data.

Edge Cases

Input	Expected	Why
All amounts $90-$99	Digit 9 dominates	Constrained range — Benford's doesn't apply
Round number spike (digit 1, 5)	Flag for review	May indicate round-number estimation or threshold manipulation
Government budget data	Typically conforms well	Large naturally-occurring financial datasets fit Benford's

Gotchas

Not proof of fraud: Non-conformity is a RED FLAG, not evidence. Many legitimate processes produce non-Benford distributions. Always investigate further.
Second-digit test: First digit test catches gross fabrication. Second-digit analysis catches more subtle manipulation (e.g., rounding to approval thresholds).
Combining datasets: Mixing datasets from different processes may artificially create or destroy Benford conformity. Analyze homogeneous datasets.
Approval thresholds: If expenses over $5,000 require VP approval, expect a spike of amounts just below $5,000 (digit 4 in the $4,9xx range). This is a behavioral pattern, flagged by second-digit analysis.
Sample size matters: Chi-squared test is sensitive to sample size. With 100K+ records, even trivial deviations become statistically significant. Use MAD as primary metric.

References

For second and third digit extensions, see references/higher-digit-tests.md
For case studies in fraud detection, see references/fraud-case-studies.md

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