An acoustic inspection is only as good as its limits. Working intuitively (or worse: copy-paste from a similar project) creates either too many false rejects or lets real defects through. Statistics helps.

The base assumption: normal distribution

Most inspection metrics (natural frequency, level per order, loudness) follow a normal distribution in series production. Practical finding from thousands of projects.

p(x) = (1 / (σ · √(2π))) · exp(−(x − μ)² / (2σ²))

Step 1: collect reference data

  1. 30–100 OK parts, spread across shifts and days.
  2. Compute μ and σ per metric.
  3. Test for normality (Q-Q plot, Shapiro-Wilk).

Step 2: set limits

LSL = μ − 3σ USL = μ + 3σ

3σ → 99.73 % within limits, ~2,700 ppm false rejects. 4σ → 63 ppm, 6σ → 3.4 ppb.

Step 3: process capability

Cp = (USL − LSL) / (6σ)
Cpk = min((μ − LSL) / (3σ), (USL − μ) / (3σ))
  • Cpk < 1.0: not capable.
  • Cpk = 1.33: standard.
  • Cpk ≥ 1.67: high capability.
  • Cpk ≥ 2.0: Six Sigma.

Step 4: tolerance intervals for finite samples

With only 50 parts σ is estimated, not known. Tolerance intervals account for this with k from DIN ISO 16269. Example: n = 30, 95 % confidence, 99 % coverage → k ≈ 3.35 instead of 3.0.

Step 5: handle drift

  • Static: limits cover all known variation.
  • Adaptive: limits track sliding mean over last 1,000 parts.
  • Hierarchical: hard safety limits + adaptive warning thresholds.

Practical example: brake disc mode 4

50 OK discs: μ = 4,318 Hz, σ = 6.2 Hz. 3σ rule: 4,299–4,337 Hz. Tolerance interval (n = 50, 95 %/99 %): 4,297–4,339 Hz. False reject rate ~250 ppm. Real cracks all < 4,280 Hz – safely outside.