How to calculate measurement uncertainty
The method is the GUM (JCGM 100:2008, Evaluation of measurement data: Guide to the expression of uncertainty in measurement) in the simplified form calibration labs and plants use every day, the same one as EA-4/02. Five steps:
Type A: uA = s ÷ √n, with ν = n − 1
Type B: ui = value ÷ divisor
Combined: uc = √(uA² + u1² + u2² + …)
Effective degrees of freedom: νeff = uc⁴ ÷ Σ (ui⁴ ÷ νi)
Expanded: U = k × uc, with k from the t distribution at νeff and 95.45%
The divisor depends on how the source behaves:
| Source | Distribution | Divisor |
|---|---|---|
| Calibration certificate U | normal | the certificate’s k (usually 2) |
| Digital instrument resolution | rectangular | 2√3 ≈ 3.46 |
| A ±a limit (drift, specification, temperature) | rectangular | √3 ≈ 1.73 |
| A ±a limit with values more likely near the center | triangular | √6 ≈ 2.45 |
The calculator uses a sensitivity coefficient of 1 for every source. That holds when each source is already in the unit of the result, as in a direct measurement with a caliper, micrometer, scale or pressure gauge. If a source needs converting (a temperature effect in °C on a result in mm, for example), convert it first and enter the value in mm.
Effective degrees of freedom come from the Welch–Satterthwaite formula. Type B sources count as infinite degrees of freedom unless you enter another value in the fifth column. A fractional value is truncated to the integer below before looking up k, as the GUM says.
Worked example
The values preloaded in the calculator are a 25 mm shaft measured 10 times with a digital caliper with 0.01 mm resolution:
25.02, 25.01, 25.03, 25.02, 25.00, 25.02, 25.01, 25.03, 25.02, 25.01 mm
- Mean = 25.017 mm. Standard deviation s = 0.0095 mm
- Type A: uA = 0.0095 ÷ √10 = 0.0030 mm, with 9 degrees of freedom
- Caliper certificate: U = 0.02 mm at k = 2, so u = 0.02 ÷ 2 = 0.0100 mm
- Resolution: u = 0.01 ÷ (2√3) = 0.0029 mm
- Drift between calibrations of ±0.005 mm, rectangular: u = 0.005 ÷ √3 = 0.0029 mm
- Combined: uc = √(0.0030² + 0.0100² + 0.0029² + 0.0029²) = 0.0112 mm
- Effective degrees of freedom: only Type A has a finite ν, so νeff = 0.0112⁴ ÷ (0.0030⁴ ÷ 9) ≈ 1,754
- k at 1,754 degrees of freedom and 95.45% = 2.00
- Expanded: U = 2.00 × 0.0112 = 0.022 mm
Report it as: 25.017 ± 0.022 mm, k = 2.00, coverage probability of approximately 95%.
How to read the result
- The expanded uncertainty U is the band around the mean where the true value should lie, with about 95% confidence. Give U with at most two significant digits and round the mean to the same decimal place.
- Each source’s contribution shows where to act. In the example, the caliper certificate accounts for 80% of the variance. Taking more readings barely changes the result; a better calibrated instrument does.
- Low degrees of freedom show up when Type A dominates and there are few readings. Then k grows (2.87 at ν = 4, 2.32 at ν = 9) and so does U. More readings fix it.
- Compare U with the tolerance. An uncertainty that takes up a big part of the tolerance leaves parts near the limit without a safe decision. To check whether the instrument itself passed calibration, use the calibration acceptance check. To see whether the process fits the tolerance, try the Cp and Cpk calculator and the control chart.
Common mistakes
- Using the certificate’s U directly as a standard uncertainty. The certificate gives the expanded uncertainty. Divide by its k before combining.
- Adding uncertainties instead of their squares. Independent sources combine as the root sum of squares. Adding them straight overstates the uncertainty.
- Dividing resolution by √3 instead of 2√3. The half-width of one digit is half the resolution. On an analog instrument, judge the smallest fraction you can read and use that as the limit.
- Using the standard deviation s as Type A. For the mean of n readings the uncertainty is s ÷ √n. If you report a single reading, then use s.
- Always fixing k = 2. With few readings and a heavy Type A, k = 2 gives less than 95% coverage. Work out the effective degrees of freedom.
- Leaving sources out. Temperature, instrument drift, the reference standard and the operator also belong in the budget when they matter for the measurement.