Measurement uncertainty budget calculator
List what contributes to the uncertainty of your measurement, with the value and distribution of each. The calculator converts them to standard uncertainties, combines them and gives the expanded uncertainty.
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New here? Load the example to see a finished job, then replace it with your own.
Contributors
One row for each source of uncertainty. Use the same unit throughout.
2 for about 95 % coverage.
| Source | Value | Distribution | Sensitivity | Standard uncertainty | Remove |
|---|---|---|---|---|---|
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Uncertainty budget
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Next: Uncertainty ratio (TUR)
The expanded uncertainty carries over, so the ratio is worked out as soon as there is a tolerance.
How the budget is calculated
- Standard uncertainty. Each value is divided by the divisor for its distribution and multiplied by its sensitivity coefficient.
- Combined standard uncertainty. The standard uncertainties are squared, added and the square root taken. This is the root sum of squares, and it assumes the contributors are independent of each other.
- Expanded uncertainty. The combined figure is multiplied by the coverage factor, normally k = 2 for about 95 % coverage.
This is the method of the Guide to the Expression of Uncertainty in Measurement (GUM). The share column shows how much of the combined variance each contributor accounts for, which tells you where an improvement would actually reduce the result.
Typical contributors for a field calibration
- The reference standard. The expanded uncertainty on its calibration certificate. Normal, k = 2.
- Drift of the standard. How far it may have moved since that calibration, from its history or the manufacturer's specification. Rectangular.
- Resolution. The smallest step of the instrument under test, and of the standard if it is coarse. Use the resolution option and enter the full step.
- Repeatability. The standard deviation of repeated readings, divided by the square root of the number of readings. Normal, k = 1.
- Environment and setup. Temperature effects, head height on a pressure line, lead resistance, loading. Usually a ± limit, so rectangular.
The example budget gives a combined standard uncertainty of 0.050 psi and an expanded uncertainty of ±0.10 psi. Against a tolerance of ±0.5 psi that is a test uncertainty ratio of 5:1.
Sensitivity coefficients
Leave the sensitivity at 1 when the contributor is already in the unit of the result. Use it when it is not: a temperature limit of ±2 °C on a gauge that changes by 0.01 psi per °C has a sensitivity of 0.01, which turns the limit into psi.
What this calculator does not do
It treats every contributor as independent and does not calculate effective degrees of freedom. For most field calibrations that is the usual simplification. If two contributors are correlated, or one contributor from a handful of readings dominates the budget, the result needs a closer look than this tool gives it.
Questions
- What is an uncertainty budget?
- A table of everything that makes a measurement result uncertain: the reference standard, resolution, repeatability, environment and so on. Each contributor is converted to a standard uncertainty, the standard uncertainties are combined by root sum of squares, and the result is multiplied by a coverage factor to give the expanded uncertainty you report.
- Which distribution should I choose?
- Use normal with k = 2 for an expanded uncertainty taken from a calibration certificate. Use normal with k = 1 for a standard deviation you worked out yourself, such as repeatability. Use rectangular when you only know limits, such as a manufacturer accuracy specification or drift between calibrations. Use the resolution option for the smallest step of a digital display. Triangular and U-shaped cover the less common cases where values cluster in the middle or at the limits.
- What are the divisors?
- The divisor turns the value you enter into a standard uncertainty. Normal with k = 2 divides by 2, normal with k = 1 by 1, rectangular by the square root of 3, triangular by the square root of 6 and U-shaped by the square root of 2. For resolution the full step is divided by 2 times the square root of 3, because the half-step is treated as a rectangular limit.
- Why is the coverage factor 2?
- For a combined uncertainty that is approximately normal, k = 2 gives a coverage probability of about 95 %, which is what most calibration certificates state. If one contributor with few repeated readings dominates the budget, the effective degrees of freedom are low and a larger factor may be needed. This calculator does not work out effective degrees of freedom, so check that case yourself.
Other free calibration tools
As-found / as-left calibration sheet calculator
Enter range, tolerance and test points. Get the error and pass or fail for every point.
Test uncertainty ratio (TUR) calculator
Enter the tolerance and your expanded uncertainty. Get the ratio and a check against 4:1.
Calibration certificate generator
Add lab, customer and reference standard details. Save the certificate as a PDF.
ISO/IEC 17025 calibration certificate checklist
Tick what your certificate includes. Get the list of gaps against ISO/IEC 17025 clause 7.8.
Doing this for every instrument, every month?
Probatus keeps the uncertainty with each reference standard and procedure, puts it on every certificate and applies your decision rule, so a budget is worked out once and reused.