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Measurement uncertainty assessment is an essential requirement of ISO 17025 (§7.6). This guide presents the 5 GUM equations applied to digital micrometers, with numerical simulation and ready-to-copy Excel formulas.
The 5 GUM Equations
Equation A — Repeatability Uncertainty (u_rep)
u_rep = σ / √n = standard deviation of measurements / √3
Equation B — Electronic Resolution Uncertainty (u_res)
u_res = Resolution / √12 = 0.001 mm / √12 ≈ 0.00029 mm
Equation C — Gauge Block Uncertainty (u_cale)
u_cale = U_certificate / 2
Equation D — Combined Uncertainty (u_c)
u_c = √(u_rep² + u_res² + u_cale²)
Equation E — Expanded Uncertainty (U) at 95%
U = 2 × u_c (coverage factor k=2)
Numerical Simulation — 21.500 mm Point
| Source | Calculation | Result |
|---|---|---|
| u_rep | 0.00058 / √3 | 0.00033 mm |
| u_res | 0.001 / √12 | 0.00029 mm |
| u_cale | 0.00050 / 2 | 0.00025 mm |
| u_c | √(0.00033² + 0.00029² + 0.00025²) | 0.00051 mm |
| U (k=2) | 2 × 0.00051 | 0.0010 mm |
Validation: |Error| + U = 0.0017 + 0.0010 = 0.0027 mm ≤ 0.0030 mm (MPE) → ✅ COMPLIANT
Excel Formulas
| Column | Excel Formula |
|---|---|
| F2 — Average | =AVERAGE(B2:D2) |
| G2 — Error | =F2-A2 |
| H2 — u_rep | =STDEV.S(B2:D2)/SQRT(3) |
| I2 — Expanded U | =2*SQRT(H2^2+(0.001/SQRT(12))^2+E2^2) |
| J2 — Total error | =ABS(G2)+I2 |
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