How the calculation works
The budget follows the law of propagation of uncertainty from the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008), for input quantities that are not correlated.
- Every source is turned into a standard uncertainty u(x): one standard deviation.
- Each is multiplied by the size of its sensitivity coefficient c, giving its contribution ui(y) in the unit of the result.
- The contributions are combined as the root sum of squares.
- The combined uncertainty is multiplied by a coverage factor k to give the expanded uncertainty U.
uc(y) = √( Σ (ci · u(xi))² ) U = k · uc(y)
The share bar next to each source is its fraction of the total variance, (ui/uc)². Because contributions add in squares, the largest one usually dominates: halving a source that holds 5 % of the variance changes almost nothing, while halving the dominant one can cut U nearly in half. Improve the amber source first.
Type A and Type B evaluations
Type A uses your own repeated readings. The standard deviation s of n readings is divided by √m, where m is the number of readings averaged in the reported result. If you report the mean of all n readings, m = n. If you report a single reading but estimated s from a separate study, set m = 1. The degrees of freedom are n − 1.
Type B uses any other information: calibration certificates, manufacturer specifications, resolution, reference data, or judgement. The value is converted to a standard uncertainty by dividing by the divisor of its assumed distribution.
| Information you have | Distribution | Divisor |
|---|---|---|
| Certificate states U and k | Normal | k |
| Specification or tolerance ±a, any value inside equally likely | Rectangular | √3 ≈ 1.732 |
| Digital display with resolution step d (half-width d/2) | Rectangular | 2√3 ≈ 3.464 |
| Limits ±a, values near the centre more likely | Triangular | √6 ≈ 2.449 |
| Quantity cycling between ±a, such as temperature control or cosine-shaped effects | U-shaped (arcsine) | √2 ≈ 1.414 |
| Standard uncertainty already known | Normal | 1 |
Type B sources normally have infinite degrees of freedom. If you are unsure of a Type B estimate itself, you can enter finite degrees of freedom: GUM G.4.2 gives ν ≈ ½ · (relative uncertainty of the estimate)⁻². An estimate you trust to about 25 % corresponds to ν ≈ 8.
Coverage factor and effective degrees of freedom
With plenty of data, k = 2 gives a coverage probability of about 95 % for a normal distribution. When Type A sources rest on few readings, the combined uncertainty is itself uncertain and k = 2 is too small. The effective degrees of freedom are then estimated with the Welch–Satterthwaite formula:
νeff = uc⁴ / Σ ( ui⁴ / νi )
In automatic mode the calculator truncates νeff to the next lower integer and takes k from the Student t-distribution at 95.45 %, exactly as in EA-4/02 M:2013 Annex E. For example, νeff = 10 gives k = 2.28 and νeff = 4 gives k = 2.87. EA-4/02 accepts k = 2 when no Type A source is based on fewer than ten readings.
Worked example: gauge block measured with a micrometer
A 25 mm gauge block is measured ten times with an outside micrometer with 0.001 mm resolution. The mean is 25.0012 mm and the standard deviation of the readings is 0.0012 mm. The micrometer certificate states U = 0.002 mm with k = 2, and the temperature difference between part and micrometer can shift the reading by up to ±0.0008 mm.
| Source | Information | u(x), mm | Share |
|---|---|---|---|
| Repeatability | s = 0.0012, n = 10, mean reported | 0.000379 | 10.0 % |
| Calibration | U = 0.002, k = 2 | 0.001 | 69.4 % |
| Resolution | d = 0.001, rectangular | 0.000289 | 5.8 % |
| Temperature | ±0.0008, rectangular | 0.000462 | 14.8 % |
The combined standard uncertainty is 0.00120 mm with νeff ≈ 900, so k = 2.00 and U = 0.0024 mm, reported as 0.0025 mm after rounding up. The result is (25.0012 ± 0.0025) mm. The calibration of the micrometer holds almost 70 % of the variance, so a better calibration would help far more than more readings. Press Load worked example to see it in the calculator.
Reporting the result
State the result as y ± U with the unit, the coverage factor, and the coverage probability, for example: “(25.0012 ± 0.0025) mm, k = 2, coverage probability approximately 95 %.” The expanded uncertainty is given to at most two significant figures (ILAC P14), and the measured value is rounded to the same decimal place. By default the calculator rounds U upward, so the reported uncertainty is never smaller than the calculated one; GUM 7.2.6 notes that rounding up is sometimes appropriate. Switch to nearest rounding if your procedure requires it.
Common mistakes
Counting the same effect twice
Repeatability readings already contain resolution and some operator effects. Adding resolution again is common and usually harmless when it is small, but if resolution is comparable to the observed spread, include only the larger of the two.
Dividing a certificate value by √3
A calibration certificate gives a normal expanded uncertainty: divide by the stated k, not by √3. The rectangular divisor is for limits, such as a maximum permissible error or a tolerance.
Mixing units
A temperature uncertainty in °C cannot be added to a length in mm. Convert it with a sensitivity coefficient, for steel roughly c = 11.5 × 10⁻⁶ /°C × L. For L = 25 mm that is 0.00029 mm/°C.
Trusting k = 2 when one rectangular source dominates
If a single rectangular or U-shaped source makes up almost all of the variance, the result is not normally distributed and k = 2 does not mean 95 %. The calculator warns you; the rigorous check is a Monte Carlo evaluation following GUM Supplement 1.
Questions
Does this replace an accredited uncertainty evaluation?
No. It applies the GUM formulas correctly to the sources you enter, but choosing the sources, the measurement model and the distributions is your responsibility and part of your laboratory procedure. Check the result against your procedure before using it on a certificate.
How do I know the calculations are right?
The calculator is tested against published values: the coverage factors of GUM Table G.2 and EA-4/02 Table E.1, and the full budget of EA-4/02 example S3 (a 10 kΩ standard resistor, u = 8.33 mΩ, U = 17 mΩ), which it reproduces exactly. The Monte Carlo check is tested against the exact intervals of the normal, uniform and t-distributions.
Are correlated inputs supported?
Not yet. All inputs are treated as independent. If two sources are strongly correlated, for example two thermometers calibrated against the same reference, combine them into one source or add their covariance term by hand.
Is my data stored anywhere?
No. The calculation runs entirely in your browser. Your budget is kept only in this browser’s local storage, and the share link carries the budget in the link itself.
Why does the automatic k differ from 2?
Because one or more Type A sources rest on few readings, which lowers νeff. Take more readings, or check whether the Type A source is significant at all; a small source with few readings barely changes νeff.
References
- JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM).
- EA-4/02 M:2013, Evaluation of the Uncertainty of Measurement in Calibration.
- ILAC P14, ILAC Policy for Measurement Uncertainty in Calibration.