"algo-risk-benford"

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

1. Extract first digit of each number
2. Count frequency of each digit (1-9)
3. Compare observed frequencies against Benford's expected: P(d) = log₁₀(1 + 1/d)
4. 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.