How to build a measurement uncertainty budget for a calibration

Why is it a budget per point and not per instrument?

Because the contributions change across the range. The uncertainty at 10 MPa on a 68 MPa pre-stress jack is not the uncertainty at 60 MPa: the repeatability differs, the share of the resolution differs, and the reference standard's own certificate may quote a different figure at each force. A budget is therefore computed at every calibration point, and a certificate that reports one figure for the whole instrument is either reporting the worst point or hiding the others.

What does one contribution contain?

A budget is a declared list. Each line in it carries six things:

A list rather than a fixed set of columns matters more than it looks. The legacy spreadsheets this was written against hard-coded four columns, so adding a temperature coefficient or a drift term meant forking the workbook. That is how one laboratory came to own some thirty incompatible variants of the same budget.

Which divisor goes with which distribution?

The standard uncertainty of a contribution is its value divided by its divisor, multiplied by the absolute value of its sensitivity coefficient.

What are the four contributions almost every budget carries?

Repeatability, Type A. The sample standard deviation of the n readings at the point, divided by √n, with ν = n − 1 (GUM 4.2.3). Three readings give ν = 2, which is why this term, small as it often is, usually decides the coverage factor. A point with a single reading has no Type A term and the certificate should say so rather than assume one.

The reference standard, Type B. The expanded uncertainty on the master's own certificate, divided by the coverage factor that certificate states, expressed in the unit of the result. If the master's certificate quotes a relative figure, it is applied to the value at this point, not to full scale.

Resolution, Type B. Half the smallest division of the unit under test, rectangular, so divided by √3. For a reading somebody interpolates between marks, the division is the interpolated one, and the budget should say which. The resolution used must be the one actually recorded for this instrument, never a default.

Zero or drift, Type B. The observed zero return or the master's drift between calibrations, rectangular. For a load cell each run subtracts its own preceding zero, and what is left is this term.

Where a contribution is in a different unit from the result, the sensitivity coefficient does the conversion. A load cell's repeatability is observed in its output and carried into force through the fitted slope; that slope is c, and leaving it at 1 would add mV to kN.

How are the contributions combined?

In quadrature, treating them as uncorrelated (GUM 5.1.2): the combined standard uncertainty u_c is the square root of the sum of the squared standard uncertainties. Where a correlation is known to exist it is entered as an explicit contribution with its own sensitivity; a budget does not infer one.

Report the share of the combined variance for each term as well, u_i² over u_c². It is the most useful line in the budget for anybody trying to reduce an uncertainty: if the reference standard contributes 85 % of the variance, a better gauge on the bench changes nothing, and the next purchase should be a better master.

Where does the coverage factor come from?

Two defensible policies, and the method must declare which. The first computes the effective degrees of freedom with the Welch–Satterthwaite formula (GUM G.4.1): u_c⁴ divided by the sum over contributions of u_i⁴ / ν_i. Terms with infinite ν add nothing to the denominator, so a budget dominated by Type B terms has a large ν_eff and k converges on 1.96. The coverage factor is then the inverse of the t-distribution at the stated confidence, usually 95 %, and that ν_eff.

The second declares a fixed k, conventionally 2. Either is acceptable. What is not acceptable is applying one and printing the other: one set of real laboratory spreadsheets expanded with a hard-coded 2 while the certificate printed the computed 1.99. Whichever policy is chosen has to be both applied and printed, and the way to make that impossible to get wrong is to record the policy on the method and let the software print what it used.

A worked example, in kN

A force point at nominally 180 kN. The reference load cell's certificate quotes U = 0.10 kN at k = 2. The unit under test reads to 0.1 kN. Three readings have a sample standard deviation of 0.06 kN.

u_c = √(0.0500² + 0.0289² + 0.0346²) = 0.067 kN. Only the repeatability term has finite ν, so ν_eff = 0.067⁴ / (0.0346⁴ / 2) ≈ 29, and the t-distribution at 95 % gives k ≈ 2.05. The expanded uncertainty is U = 2.05 × 0.067 ≈ 0.14 kN, which is what the certificate prints at this point, in kN, with k and the confidence level beside it. The reference standard carries 55 % of the variance; the resolution 18 %.

The uncertainty budget calculator runs this arithmetic in the browser, with the same engine a certificate uses.

What mistakes do real laboratory spreadsheets make?

These were all found in the certificate workbooks of one working laboratory, and each one had been issuing certificates for years.

• The range as a Type A estimate. The spread of three readings as a percentage, divided by √3 as if rectangular, with ν set to 200. A Type A term is statistical, the range of three readings is not the standard deviation of their mean, and ν = 200 drives k to 1.97 where the GUM gives about 4.3 for ν = 2. The result was an uncertainty understated at every point.
• A term multiplied by itself. One fork computed the repeatability percentage times the repeatability percentage instead of times the applied force, in fourteen cells. A 1 % repeatability produced a contribution of 0.01 %.
• The budget in one unit, the result in another. A jack is read in MPa and calibrated in kN. A budget assembled in the gauge's unit and printed beside a force gives a certificate that says ±1.0 MPa about 1.0 kN.
• A divisor that does not match the data. Dividing by a hard-coded 3 runs when the sheet holds two, or four.
• The master's k forgotten. Entering the master's expanded uncertainty as if it were a standard uncertainty doubles its weight.

None of these produces an obviously wrong number, which is why they survive. The protection is a budget that is recomputed from its inputs every time rather than copied, and compared with what was issued. ISO/IEC 17025 §7.11.2 asks for exactly that of any software doing the arithmetic.

What has to appear on the certificate?

ISO/IEC 17025 §7.8.4 requires the measurement uncertainty of the result, in the same unit as the result or relative to it, together with the coverage factor and the confidence level. The certificate requirements guide lists the rest. The GUM itself is JCGM 100:2008, free from the BIPM.