How to Write an Uncertainty Budget: Step-by-Step with Examples

Rule Statement: What an Uncertainty Budget Must Do

An uncertainty budget is a structured accounting of every input quantity that affects a measurement result. It follows the Guide to the Expression of Uncertainty in Measurement (GUM), JCGM 100:2008, and related standards such as NIST TN 1297 and ISO/IEC Guide 98-3:2008. The budget must identify the measurand, define the measurement model, list uncertainty sources, classify each as Type A or Type B, convert each to a standard uncertainty u_i, apply sensitivity coefficients c_i, combine independent components, expand with a coverage factor k, and report the result with defensible significant figures.

GUM principle: The combined standard uncertainty u_c(y) is the positive square root of the total variance obtained from all input uncertainty components, weighted by their sensitivity coefficients. Expanded uncertainty is U = k u_c(y).

This site is a precision and rounding reference, not just a calculator. The significant figures rules and rounding rules pages explain the conventions needed to report an uncertainty budget correctly. The significant figures calculator is a fast accuracy check, but the budget itself remains the authoritative record.

Worked Example: Caliper Measurement of a Steel Rod

A steel rod is measured with a digital caliper. The mean of repeated readings is 50.012 mm. The measurement model is L = L_ind + δ_cal + δ_res + δ_rep + δ_temp, where each delta is a correction or error contribution.

Step 1: Quantify Sources and Standard Uncertainties

For resolution, the half-interval is 0.005 mm; the rectangular standard uncertainty is 0.005/√3 = 0.0029 mm, equivalently 0.01/√12. For temperature, the contribution is L α ΔT = 50 mm × 11.5 × 10^-6/°C × 1 °C = 0.000575 mm; dividing by √3 gives 0.00033 mm.

Step 2: Combine and Expand

For independent inputs, u_c = √(0.0100² + 0.0029² + 0.0150² + 0.00033²) = 0.01826 mm. Using k = 2 for approximately 95% confidence, U = 2 × 0.01826 = 0.0365 mm.

Step 3: Round and Report

Expanded uncertainty is conventionally reported to two significant digits: U ≈ 0.037 mm. The result is rounded to the same decimal place as the uncertainty: L = (50.012 ± 0.037) mm, k = 2, approximately 95% coverage. The intermediate value 0.01826 mm should not be rounded before expansion.

Counter-Examples: How Not to Build a Budget

• Adding uncertainties directly: 0.0100 + 0.0029 + 0.0150 = 0.0279 mm overstates the combined uncertainty. Independent standard uncertainties combine by root-sum-square, not by linear addition.
• Using expanded uncertainty as standard uncertainty: treating calibration U = 0.02 mm as u = 0.02 mm instead of dividing by k = 2 doubles that component.
• Ignoring the rectangular divisor: using 0.005 mm for resolution instead of 0.0029 mm inflates the budget and breaks the distribution assumption.
• Rounding too early: rounding 0.0029 to 0.003 mm or 0.01826 to 0.018 mm before expansion introduces avoidable rounding error.
• False precision: reporting 50.01234 ± 0.03654 mm implies more resolution than the measurement supports.

Convention Comparison Table: GUM, NIST, ASTM, ISO

Standards Citation and Rounding Rules

The GUM explicitly separates Type A evaluations, based on statistical analysis of repeated observations, from Type B evaluations, based on certificates, manufacturer specifications, calibration reports, or scientific judgment. GUM 4.2 and 4.3 define these evaluations. GUM 5.1.2 defines standard uncertainty, and GUM 6.2.1 gives the combined standard uncertainty for independent inputs. GUM 7.2.2 addresses expanded uncertainty.

NIST TN 1297 recommends that expanded uncertainty be reported with a coverage factor, commonly k = 2 for approximately 95% confidence when the effective degrees of freedom are large. ASTM E29-22 provides significant-digit and rounding practice for test data and conformance decisions. ISO 80000-1:2022 clause 7.3.3 governs rounding of quantity values. In all cases, the numerical result and its uncertainty must be rounded to the same decimal place.

Common Mistakes

• Mixing standard and expanded uncertainty: never combine an expanded U with standard u_i values.
• Forgetting sensitivity coefficients: a source may enter the model multiplied by a factor other than 1, such as temperature coefficient times length.
• Omitting correlation: if inputs are correlated, include covariance terms rather than assuming independence.
• Using k = 2 blindly: state the distribution assumption and, for small degrees of freedom, use a Student t factor.
• Inconsistent rounding: report the result to the same decimal place as the expanded uncertainty.
• Dropping units: every component, combined uncertainty, and expanded uncertainty must carry the measurand unit.

Quick Reference Table: Uncertainty Budget Checklist

Related Rules and Internal Reference

Uncertainty budgets depend on the same rounding discipline as all measurement reporting. Review significant figures, rounding rules, error propagation, and measurement uncertainty. For quick checks of how a result should be rounded after expansion, the site’s significant figures calculator provides an independent verification of significant-digit behavior. The reference articles explain why the calculator gives its answer, making this site useful for students, engineers, scientists, and educators.

FAQ: Uncertainty Budgets and Significant Figures

How many significant figures should uncertainty have?

Most metrology guidance uses one or two significant digits for expanded uncertainty; two is common for final reporting. The result should then be rounded to the same decimal place as the uncertainty. For example, U = 0.0365 mm becomes 0.037 mm, so report 50.012 ± 0.037 mm.

When can I use k = 2?

Use k = 2 for approximately 95% coverage when the output is approximately normal and the effective degrees of freedom are large. For small samples or dominant Type A components with few degrees of freedom, use a Student t coverage factor from GUM 6.3.

Can I ignore a very small uncertainty component?

A component contributing less than about one-tenth of the largest standard uncertainty often has negligible effect on the combined result, but the decision should be documented. In the worked example, temperature contributes much less than calibration, resolution, and repeatability.

What if my uncertainty is zero?

Zero standard uncertainty is rarely physical. A resolution of zero, for example, may mean the display is finer than the process capability, not that the component is truly absent. Re-evaluate the model and available evidence.

Sources & Further Reading

• JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM), BIPM/IEC/ISO/IUPAC/IUPAP/OIML.
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, 1994.
• ISO/IEC Guide 98-3:2008, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement.
• ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
• ISO 80000-1:2022, Quantities and units — Part 1: General, clause 7.3.3.