Chapter 04· 4 min read

Mathematics & Statistics

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Forensic statistics is the discipline that turns observations into defensible probabilistic statements. The court asks "did the suspect leave this trace?"; the analyst answers in the language of likelihood ratios, prior odds, posterior probabilities, and confidence intervals. This chapter covers the small set of statistical tools that appear in casework.

4.1The Likelihood Ratio

Likelihood Ratio

LR = P(E | H_p) / P(E | H_d)

E = the observed evidence · H_p = prosecution hypothesis (suspect is the source) · H_d = defence hypothesis (some other person is the source)

The LR answers: how much more likely is the evidence under H_p than under H_d? Critically, the LR is not a probability of guilt. Even an LR of 1,000,000 doesn't establish guilt by itself — it must be combined with the prior odds (set by the rest of the case-record) to produce posterior odds. The court holds the prior; the forensic scientist supplies the LR.

Verbal scale (RSS / ENFSI)

LR	Verbal interpretation
1–10	Limited support
10–100	Moderate support
100–1,000	Strong support
1,000–10,000	Very strong support
> 10,000	Extremely strong support

4.2Bayesian Inference

Bayesian Update (Odds Form)

Posterior odds = Prior odds × LR

Multiply, never divide. Worked example: prior 1:1000 × LR 1,000,000 = 1000:1 ≈ 99.9% posterior probability of guilt.

Fig 4.1 — Bayesian update in odds form. The court owns the prior; the analyst owns the LR.

Why the prior matters: with the same LR 10⁶ but a prior of 1 in 10⁹ (suspect chosen from world population without other evidence), the posterior is 10⁶ / 10⁹ = 1/1000 — the suspect is probably not the source despite the strong forensic evidence.

4.3The Prosecutor's Fallacy

The most common statistical error in courtroom forensic testimony: confusing P(E | H_d) with P(H_d | E). A "1 in a million random match probability" is not "1 in a million chance the suspect is innocent". The first is a property of the evidence; the second is the posterior, which depends on the prior.

4.4Descriptive Statistics

Statistic	Formula	Meaning
Mean	μ = Σx_i / n	Arithmetic average
Median	middle value (sorted)	50th percentile
Mode	most frequent value	Peak of distribution
Standard deviation	σ = √variance	Spread (data variability)
SEM	SEM = σ / √n	Precision of the sample mean
Coefficient of variation	CV = (σ / μ) × 100%	Dimensionless precision

SD vs SEM — frequently confused: SD describes the spread of individual data and doesn't decrease with sample size; SEM describes the precision of the sample mean and decreases as √n. For 100 measurements with SD 5, SEM = 5 / √100 = 0.5.

4.5The Normal Distribution

Fig 4.2 — The empirical 68-95-99.7 rule for the normal distribution.

68% of values within 1 SD of the mean (μ ± σ)
95% within 2 SD (μ ± 2σ) — the standard 95% CI (more precisely 1.96σ)
99.7% within 3 SD (μ ± 3σ)

QC application: a measurement within mean ± 2 SD is accepted; outside that interval (5% by chance) prompts investigation; outside mean ± 3 SD (0.3% by chance) typically triggers root-cause analysis.

4.6Type I and Type II Errors

Decision \ Reality	H₀ true	H₀ false
Reject H₀	Type I error (α, false positive)	Correct rejection
Fail to reject H₀	Correct retention	Type II error (β, false negative)

Statistical power = 1 − β = probability of correctly rejecting a false null. Forensic interpretation in match testing: Type I = declaring a match when scene and suspect are different sources; Type II = missing a match when they are the same source.

4.7DNA Random Match Probability

For a multi-locus STR profile under HWE + linkage independence:

Random Match Probability

RMP = ∏ (per-locus match probabilities)

With per-locus heterozygosity ~0.75–0.85 → per-locus match probability ~0.05–0.15. For 13 CODIS loci: RMP ≈ 0.10¹³ = 10⁻¹³. Modern 24-locus kits achieve 10⁻²⁰.

Theta (θ) correction for population substructure

θ	Population structure	Standard usage
0.01	Homogeneous (single ethnic group)	Default for narrow population
0.02	Mildly subdivided	Multi-state populations
0.03	Substantially subdivided	Conservative default; protects against undisclosed substructure

Identical (monozygotic) twins share their entire nuclear genome, so STR profiles match completely regardless of locus count. Distinguishing twins requires high-coverage whole-genome sequencing or epigenetic methylation profiling.

4.8ROC Analysis

The Receiver Operating Characteristic curve evaluates binary classifiers across all possible decision thresholds.

Fig 4.3 — ROC curve and AUC. Higher AUC = better classifier discrimination.

AUC ranges: 1.0 = perfect, 0.9–1.0 = excellent, 0.8–0.9 = good, 0.7–0.8 = fair, 0.5 = random. ML classifiers in forensic settings (signature, voice, biometric) report AUC alongside explicit error rates so the court can weigh the evidence properly.

Memory hooks · Chapter 4

LR = P(E | H_p) / P(E | H_d). Strength of evidence, not probability of guilt. Bayesian update: posterior odds = prior odds × LR. Multiply, never divide. Prosecutor's fallacy: P(E | H_d) ≠ P(H_d | E). Empirical rule: 68 / 95 / 99.7 within 1 / 2 / 3 SD. SD vs SEM: SD = data spread (fixed); SEM = SD / √n. RMP = ∏ per-locus probabilities; 13 loci → 10⁻¹³. Theta: 0.01–0.03; conservative default 0.03. Type I (α) = false positive; Type II (β) = false negative. AUC: 1 = perfect, 0.5 = random, 0.9+ = excellent.

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