Benford's Law says that in most naturally occurring, multi-order-of-magnitude numeric datasets, the leading (first) digit is not uniformly distributed — small digits dominate. The probability that a record's leading digit is d follows a logarithmic curve:
P(d) = log10(1 + 1/d), d = 1..9
P(1) ≈ 30.1% P(5) ≈ 7.9% P(9) ≈ 4.6%
This audit treats each simulated UFO-report reference number / timestamp as one draw. Genuine records are sampled directly from the Benford distribution via inverse-CDF sampling (real scale-invariant data — report IDs, timestamps, distances — behaves this way because it spans many orders of magnitude). Fabricated or manually altered records are drawn from a uniform distribution over digits 1–9 — people inventing "random-looking" numbers avoid low leading digits and spread their guesses roughly evenly, which is exactly the tell forensic accountants look for.
The audit compares the observed digit histogram to the theoretical curve with a chi-squared goodness-of-fit statistic:
χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ, Eᵢ = N·P(i)
- Records in database — how many reference numbers the audit draws (N).
- Fabricated-record fraction — the share of records generated by the uniform (non-Benford) process, simulating altered or invented entries mixed into the real database.
- χ² statistic — larger values mean the observed histogram is farther from the Benford curve than chance alone would produce.
- Verdict — flagged when χ² exceeds the 8-degrees-of-freedom, 95%-confidence critical value of 15.51, the standard threshold forensic auditors use for this exact test.
Real-world relevance: Benford's Law audits are a genuine forensic-accounting and data-integrity technique, used to flag tax fraud, election-count tampering and falsified scientific data — any dataset where a real generative process should span orders of magnitude but the numbers were instead typed by a person.