TAHA Engineering Toolsby TAHA Knowledge Lab

Metrology & Quality

Measurement Uncertainty Calculator (GUM)

Combined standard and expanded uncertainty from independent components (root-sum-square).

Inputs

For example µm, mm, N or °C. All components must use the same unit.

k = 2 is commonly used and gives a coverage probability of about 95% when the result is approximately normally distributed with enough degrees of freedom.

Result

Expanded uncertainty U = k × uc
2.26789
Combined standard uncertainty uc
1.13395
Uncertainty budget
#1: u = 0.75 µm → 43.7% of uc² #2: u = 0.2887 µm → 6.48% of uc² #3: u = 0.8 µm → 49.8% of uc²
Working
uc = √(0.75² + 0.2887² + 0.8²) = 1.134 µm U = 2 × 1.134 = 2.268 µm

Combined standard uncertainty uc = 1.13395; expanded uncertainty U = 2.26789. Report the result as “value ± U” and state the coverage factor k.

Educational estimate — check the assumptions below.

About this calculator

Every measurement has some doubt. The internationally agreed method (the GUM, JCGM 100) lists each source of uncertainty — calibration, resolution, repeatability, temperature and so on — converts each to a standard uncertainty, and combines them. When the sources are independent, they combine by root-sum-square. Multiplying by a coverage factor k gives the expanded uncertainty U that is reported with the result.

This calculator does the arithmetic of an uncertainty budget for independent inputs. You still have to decide which components exist and estimate them honestly.

Formula

uc = √[ Σ (ci × ui)² ]    U = k × uc

Rectangular: u = a ÷ √3    Triangular: u = a ÷ √6    From a certificate: u = U ÷ k

ui = standard uncertainty of component i, ci = sensitivity coefficient (how much the result changes per unit change of that input; 1 if the component is already in the unit of the result), a = half-width of the limits, k = coverage factor.

Worked example

Measuring a length in µm:

  • Calibration certificate: U = 1.5 µm at k = 2 → u = 0.75 µm
  • Display resolution 1 µm → ±0.5 µm rectangular → u = 0.5 ÷ √3 = 0.289 µm
  • Repeatability (standard deviation of the mean) u = 0.8 µm
  • uc = √(0.75² + 0.289² + 0.8²) = 1.134 µm; U = 2 × 1.134 = 2.27 µm (k = 2)

Assumptions and limitations

  • All components are treated as <strong>independent</strong> (uncorrelated). Correlated inputs need covariance terms, which are not included.
  • k = 2 corresponds to about 95% coverage only for an approximately normal result with enough effective degrees of freedom; with few repeat readings a larger k may be needed (Welch–Satterthwaite, GUM Annex G).
  • The calculator does not estimate the components for you. Missing a significant source makes the result too small.

References

  • JCGM 100:2008 (GUM): Evaluation of measurement data — Guide to the expression of uncertainty in measurement.
  • JCGM 200:2012 (VIM): International vocabulary of metrology.