Engineering Measurements: Tolerance and Uncertainty

A symmetric dimensional tolerance states lower and upper limits around a nominal value. A unit conversion changes the numerical scale, not the permitted physical interval. Decimal places alone do not establish measurement capability. The example is arithmetic, not a recommendation for inspection equipment. [1] 20.00 ± 0.02 mm: 19.98–20.02 mm. 0.001 in = 25.4 µm.

For a measurement model y = f(x1, …, xn), first-order propagation uses sensitivity coefficients ci = ∂f/∂xi and the estimated covariances u(xi,xj). The local linear approximation must be adequate. Only for uncorrelated inputs does the covariance sum vanish. The plain RSS formula applies to independent contributions already expressed in the output unit, including their sensitivity factors; independence alone does not justify dropping those factors. [2] uc²(y) = Σ ci² u²(xi) + 2 Σ(i<j) ci cj u(xi,xj). y = 2x; u(x) = 1 mm; u(y) = 2 mm. uc = √((3 µm)² + (4 µm)²) = 5 µm.

Expanded uncertainty is U = k uc. A coverage factor k = 2 gives approximately 95% coverage when the output distribution is approximately normal and uc is reliably estimated with sufficient effective degrees of freedom. It is not a universal probability guarantee; other distributions or limited degrees of freedom require an appropriate coverage evaluation. The example below assumes these conditions. [3] U = k uc = 2 × 5 = 10 µm.