Measurement uncertainty budget calculator
List your uncertainty contributors, pick a distribution for each, and get the combined standard uncertainty and expanded uncertainty U, with each contributor’s share of the variance. Uses the GUM (JCGM 100:2008) root-sum-square method. Everything runs in your browser.
Uncertainty contributors
Each value is reduced to a standard uncertainty using the divisor for its distribution, multiplied by its sensitivity coefficient, then combined by root-sum-square.
| Contributor | Value (half-width or u) | Distribution | Sensitivity c | Remove |
|---|
Value is the number you are reducing: a certificate expanded uncertainty, a plus or minus limit, or a standard uncertainty. The distribution sets the divisor. The sensitivity coefficient defaults to 1; enter another value when the contributor does not map one to one onto the result.
k = 2 gives about 95% coverage under an approximately normal combined distribution.
All contributors must share the same units after their sensitivity coefficients are applied.
Result
Method: GUM (JCGM 100:2008). Standard uncertainty u_i = (value / divisor) × |c|; combined u_c = √(Σ u_i²) for independent contributors; expanded U = k · u_c. Correlations between contributors and effective degrees of freedom are not modeled here.
Uncertainty budgets, in plain terms
The GUM method
The Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008) is the standard recipe for putting a number on how much a measurement could be off. You list every source that matters, put each on the same footing as a standard uncertainty (one standard deviation), combine them, and expand the result to a stated level of confidence. Sources you evaluate from repeated readings are called Type A; sources you evaluate from a certificate, a specification, a handbook, or judgement are called Type B. The budget is where all of that gets written down so it can be checked.
Distributions and their divisors
A contributor is rarely quoted as a standard deviation, so you convert it by dividing by a divisor set by the distribution you assume for it:
- Normal, k = 2: divide by 2. A calibration certificate value given at 95% (k = 2).
- Normal, k = 1: divide by 1. A value already reported as one standard deviation, such as a Type A repeatability.
- Rectangular: divide by the square root of 3 (about 1.732). A plus or minus limit where any value inside is equally likely, such as resolution or a digital step.
- Triangular: divide by the square root of 6 (about 2.449). A plus or minus limit where values near the center are more likely.
- U-shaped: divide by the square root of 2 (about 1.414). A plus or minus limit where the extremes are more likely, typical of some cyclic effects.
Sensitivity coefficients and combining
Not every input moves the result by the same amount. A sensitivity coefficient c is how much the result changes per unit change in that input, taken from your measurement model. Fold each coefficient into its contributor so that each u_i is the input's standard uncertainty times the size of its coefficient, then combine by root-sum-square: u_c = √(Σ u_i²). Squaring and adding is why the largest one or two contributors usually dominate the budget, and why the share of variance is worth reading: it points you at where a better reference standard or a tighter method would actually help.
Coverage factor, and what this tool does not do
The expanded uncertainty is U = k times the combined standard uncertainty. With k = 2 and a combined distribution that is approximately normal, U spans an interval of roughly 95% confidence, and the central limit theorem makes normality reasonable once several comparable contributors are added together. Two things are deliberately left to you. This calculator treats the contributors as independent, so it does not add covariance terms for sources that are correlated. And it does not compute a coverage factor from the effective degrees of freedom (the Welch-Satterthwaite formula), because that needs the degrees of freedom behind each contributor, which is a judgement you own. It reports the k you enter rather than inventing one. References: JCGM 100:2008 (GUM); ISO/IEC 17025:2017.
Common questions
What is a measurement uncertainty budget?
A measurement uncertainty budget is a structured list of every source of uncertainty that affects a measurement, each expressed as a standard uncertainty, combined into a single combined standard uncertainty and then an expanded uncertainty. It follows the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008). The budget makes the calculation transparent: an assessor can see which contributors you included, how you treated each one, and which source dominates.
How do you turn a specification or certificate value into a standard uncertainty?
Each contributor is quoted in its own way, so you first reduce it to a standard uncertainty (one standard deviation) by dividing by a divisor set by its assumed distribution. A calibration certificate value stated at k = 2 is divided by 2. A one sigma value is used as is (divide by 1). A plus or minus limit with a uniform (rectangular) distribution is divided by the square root of 3, about 1.732. A triangular distribution is divided by the square root of 6, about 2.449. A U-shaped distribution, common for some cyclic effects, is divided by the square root of 2, about 1.414.
What is a sensitivity coefficient?
A sensitivity coefficient converts a change in an input quantity into the change it causes in the measurement result. If an input contributes directly and in the same units, its coefficient is 1. If a 1 unit change in the input causes a 0.2 unit change in the result, the coefficient is 0.2. Each contributor standard uncertainty is multiplied by the absolute value of its sensitivity coefficient before the contributions are combined. Working the coefficients out from your measurement model is your responsibility; this tool applies the value you enter.
Why combine by root-sum-square, and what does it assume?
Root-sum-square (adding the squares of the standard uncertainties and taking the square root) is the GUM law of propagation of uncertainty for contributors that are independent and uncorrelated. It assumes the sources do not move together. If two contributors are correlated, plain root-sum-square understates or overstates the combined uncertainty and you need the correlation (covariance) term, which this tool does not include. Confirm your contributors are independent before you rely on the combined figure.
Does a coverage factor of k = 2 always mean 95%?
The expanded uncertainty is U = k times the combined standard uncertainty. A coverage factor of k = 2 gives an interval with an approximately 95% level of confidence when the combined distribution is approximately normal, which the central limit theorem makes reasonable when several comparable contributors combine. A rigorous coverage factor for a small number of contributors comes from the effective degrees of freedom (the Welch-Satterthwaite formula), which needs the degrees of freedom of each contributor. This tool does not ask for degrees of freedom, so it reports the k you choose and does not invent one.
Does ISO/IEC 17025 require an uncertainty budget?
ISO/IEC 17025:2017 clause 7.6 requires a laboratory to evaluate measurement uncertainty and to account for all significant contributions, and a documented budget is the usual way to show that work. This calculator helps you assemble and combine the budget using transparent GUM methods. It does not set your uncertainty policy, choose your contributors, or replace JCGM 100:2008. The evaluation and its defensibility remain your laboratory responsibility.
This calculator combines the contributors you enter using the transparent GUM root-sum-square method and reports u_c, U, and each contributor’s share of the variance. It assumes the contributors are independent, does not add correlation terms, and does not derive a coverage factor from effective degrees of freedom. It does not choose your contributors, set your uncertainty policy, or replace JCGM 100:2008 or ISO/IEC 17025. Treat the result as a working estimate and validate it against your documented uncertainty procedure before you use it.
Keep the budget with the record
Axiospec keeps every calibration, uncertainty budget, and certificate in one tamper-evident ledger built for ISO/IEC 17025, with per-instrument uncertainty budgets and test uncertainty ratios. See it on real calibration data, no signup and no credit card.