Measurement Uncertainty

In plain English

Every measurement is a little uncertain. If you measure the same shaft several times you get slightly different numbers, the instrument has its own small errors, and temperature changes the part. Measurement uncertainty puts a number on that doubt, so a result can be written as, for example, 25.012 mm ± 0.003 mm.

Technical definition

Measurement uncertainty is a non-negative parameter characterising the dispersion of the values attributed to a measured quantity, based on the information used (VIM). A standard uncertainty is expressed as a standard deviation. The combined standard uncertainty u_c combines all the contributions. The expanded uncertainty U = k × u_c defines an interval expected to contain a large fraction of the values, where k is the coverage factor.

How it works and where it is used

The GUM method builds an uncertainty budget:

1. List every significant source — calibration of the instrument, resolution, repeatability, temperature, the operator, the part’s form.
2. Evaluate each as a standard uncertainty: Type A from statistics of repeated readings, Type B from other information such as certificates, specifications or resolution.
3. Combine them (by root-sum-square when independent), then multiply by k.

Formula

u_c = √[ Σ (c_i × u_i)² ]    U = k × u_c

Rectangular limits ±a: u = a ÷ √3    From a certificate: u = U ÷ k

u_i = standard uncertainty of each independent component, c_i = its sensitivity coefficient. k = 2 gives roughly 95% coverage when the result is close to normally distributed with enough degrees of freedom.

Worked example

Measuring a length: certificate U = 1.5 µm (k = 2) → u = 0.75 µm; display resolution 1 µm → ±0.5 µm rectangular → u = 0.29 µm; repeatability u = 0.8 µm.

u_c = √(0.75² + 0.29² + 0.8²) = 1.13 µm; U = 2 × 1.13 = 2.3 µm. The result is reported as, for example, 25.0120 mm ± 0.0023 mm (k = 2).

In the workshop and in practice

Uncertainty decides whether a measurement can prove a part is in tolerance. ISO 14253-1 sets the default rule: to prove conformance, the measured value must lie inside the tolerance zone reduced by the expanded uncertainty at each end. If U is a large share of the tolerance, good parts get rejected or bad parts accepted — the instrument or method has to improve. Calibration laboratories state uncertainty on every certificate.

Common mistakes and misconceptions

• Confusing uncertainty with error. Error is the (unknown) difference from the true value; uncertainty describes the doubt about the result.
• Using resolution alone. Resolution is only one component; repeatability and calibration are usually larger.
• Adding uncertainties directly. Independent standard uncertainties combine by root-sum-square, not by simple addition.
• Not stating k. “± 2.3 µm” without the coverage factor is incomplete.

Related calculators

• Measurement Uncertainty Calculator (GUM) — Combined standard and expanded uncertainty from independent components (root-sum-square).
• Vernier Caliper Least Count & Reading Calculator — Least count of a vernier scale, and the reading with zero-error correction.
• Micrometer Least Count & Reading Calculator — Least count from pitch and thimble divisions, and a micrometer reading with zero error.

Related topics

• Least Count — The smallest change in size that a measuring instrument can show.
• Limits of Size and Tolerance — The largest and smallest acceptable sizes of a feature, and the tolerance between them.

References

• JCGM 100:2008 Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM)
• JCGM 200:2012 International vocabulary of metrology (VIM)
• ISO 14253-1: GPS — Inspection by measurement of workpieces and measuring equipment — Decision rules for verifying conformity or nonconformity with specifications