GUM Uncertainty Budget
Combine Type A and Type B uncertainty components into a combined and expanded uncertainty (k=2).
Calculator
Uncertainty components
Theory & method
The GUM (JCGM 100:2008, "Guide to the Expression of Uncertainty in Measurement", also redesignated ISO/IEC Guide 98-3) is the internationally accepted framework for stating how confident a measurement result really is. Every measurement is affected by multiple sources of error — sampling, instrument resolution, calibration uncertainty, operator technique — and the GUM gives a disciplined way to combine them into a single number.
Sources are classified by how their uncertainty is evaluated, not by whether they are "random" or "systematic": Type A components come from the statistical analysis of repeated observations (e.g. the standard deviation of the mean of several sieving runs); Type B components come from any other evidence — calibration certificates, manufacturer specifications, resolution limits, or professional judgement about a known bound.
When every component is already expressed in the same unit as the quantity of interest (this calculator's simplifying assumption for an educational v1), the combined standard uncertainty is the root-sum-square of the individual standard uncertainties: u_c = √(Σ u_i²) — JCGM 100:2008 §5.1.2, equation 11. Each component's variance (u_i²) divided by the total variance gives its share of the combined uncertainty, which is what the contribution breakdown shows.
The expanded uncertainty U = k·u_c reports a wider interval intended to contain the true value with a stated level of confidence. This calculator fixes k = 2, which corresponds to roughly 95% confidence under the common assumption that the combined distribution is approximately normal with many effective degrees of freedom (JCGM 100:2008 §6.3.3 and Annex G.2) — a documented approximation, since it does not compute the effective degrees of freedom via the Welch–Satterthwaite formula.
How to use
- 01Choose the unit shared by every component in this budget (% mass or µm) — for example, sampling and weighing uncertainties expressed as % of test mass, or aperture-measurement uncertainties expressed in µm.
- 02For each source of uncertainty (sampling, sieve overload, weighing resolution, aperture calibration, etc.), add a row with a short description, mark it Type A or Type B, and enter its standard uncertainty u_i in the chosen unit.
- 03If you only have a bound or tolerance (a half-width) rather than a standard uncertainty, convert it first assuming the appropriate distribution: divide by √3 for a rectangular (equal-probability) distribution, or by √6 for a triangular distribution.
- 04Read the combined standard uncertainty u_c, the expanded uncertainty U (k = 2), and the contribution of each component — the largest bars point to where improving the measurement procedure would have the most impact.
Frequently asked questions
Why does this tool require a single unit for the whole budget?
Combining components expressed in different units (e.g. % mass and µm) rigorously requires sensitivity coefficients — partial derivatives of the measurement model with respect to each input quantity. That is beyond the scope of this simplified educational tool; keeping every component in one unit corresponds to the GUM's basic case where the measurand is a direct sum of the inputs (sensitivity coefficients equal to 1).
What are the main sources of uncertainty in sieve analysis?
Sampling and sample division (quartering, riffling) are typically the largest contributor. Others include sieve overload or blinding, agitation technique, sieve wear, balance resolution during weighing (mass losses above 0.5–1% invalidate the test per NBR NM 248/ASTM C136), and the uncertainty of the sieve's own aperture calibration.
Why is k = 2 called an approximation here?
The GUM's rigorous expanded-uncertainty calculation determines an effective number of degrees of freedom for the combined uncertainty (Welch–Satterthwaite formula, Annex G) and looks up the corresponding coverage factor for the desired confidence level. This calculator instead fixes k = 2 as a standard, widely used approximation of ~95% confidence, valid when the combined distribution is approximately normal and based on enough data — it does not perform the full degrees-of-freedom calculation.
Does a Type B component mean the uncertainty is less rigorous?
No — Type A and Type B refer only to how the uncertainty was evaluated (statistically from repeated measurements, versus from other evidence such as a calibration certificate), not to its reliability or magnitude. A well-documented Type B uncertainty from a certificate can be more trustworthy than a Type A uncertainty from very few measurements.
Normative references
- JCGM 100:2008 — Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM), BIPM/JCGM.
- ISO/IEC Guide 98-3:2008 — Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995).
- NBR NM 248 / ASTM C136 — Sieve analysis of aggregates (mass-loss validity limit referenced in the sampling/weighing discussion).