How to make a measurement uncertainty budget, step by step

By Anna Badanina, MSc in Metrology and Information-Measuring Technology · 8 November 2026

An uncertainty budget is a table that lists every significant source of uncertainty in a measurement, converts each one to a standard uncertainty, and combines them into one number: the expanded uncertainty U you put on the certificate. The method is defined in JCGM 100:2008, the Guide to the expression of uncertainty in measurement (GUM). EA-4/02 M applies it to calibration.

This guide walks through the seven steps with one running example: the calibration of a digital thermometer at 50 °C in a liquid bath against a reference thermometer.

Step 1. Define the measurand and the model

Write down exactly what you are reporting and how it is calculated from the quantities you measure or know. For the thermometer, the measurand is the error of the indication:

where t_ix is the mean indication of the thermometer under test, t_S is the temperature of the reference thermometer, and the δ terms are corrections whose best estimate is zero but whose value is not exactly known: drift of the reference since its calibration, inhomogeneity of the bath and the resolution of the display. Writing the model first prevents the two most common errors in budgets: forgetting a source, and counting one twice.

Step 2. List the sources of uncertainty

Each input quantity of the model is a row of the budget. Typical sources in calibration are the reference standard (from its certificate), its drift, environmental conditions, the resolution of the device under test and repeatability. In the example:

• Reference thermometer: certificate states U = 0.10 °C with k = 2.
• Drift of the reference between calibrations: within ±0.05 °C, from its calibration history.
• Bath inhomogeneity: within ±0.08 °C, from a bath characterisation.
• Resolution of the thermometer under test: 0.1 °C, so the reading is within ±0.05 °C of the indicated value.
• Repeatability: five readings, 50.31, 50.28, 50.33, 50.30 and 50.27 °C.

Step 3. Convert each source to a standard uncertainty u(x_i)

Every row must be expressed as a standard uncertainty, the equivalent of one standard deviation. How you get there depends on what you know about the source. This is the difference between Type A and Type B evaluation, explained in detail in Type A and Type B uncertainty:

• Certificate value (normal distribution): divide U by its k. 0.10 / 2 = 0.050 °C.
• Limits ±a with no further knowledge (rectangular distribution): divide a by √3. Drift: 0.05 / √3 = 0.029 °C; bath: 0.08 / √3 = 0.046 °C; resolution: 0.05 / √3 = 0.029 °C.
• Repeated readings (Type A): the experimental standard deviation s = 0.0239 °C. Because the result is the mean of n = 5 readings, the standard uncertainty is s / √n = 0.0107 °C, with ν = n − 1 = 4 degrees of freedom.

Step 4. Find the sensitivity coefficients c_i

A sensitivity coefficient tells you how much the result changes when an input changes by one unit. It is the partial derivative of the model with respect to that input. It also converts units: if an input is in ohms and the result in volts, c_i is in volts per ohm. In the thermometer model every term enters with a factor of 1 (or −1), so all c_i have magnitude 1. When an input enters a product or a power, c_i is no longer 1, and forgetting it is a frequent source of wrong budgets.

Step 5. Combine: the combined standard uncertainty u_c

For uncorrelated inputs, the contributions c_i·u(x_i) add as a root sum of squares:

The full budget for the example:

The share column is where the budget earns its keep. Here the reference thermometer and the bath make up 72 % of the variance. Improving the repeatability, which is 1.8 %, would change almost nothing. Buying a better reference or characterising the bath more carefully would.

Step 6. Choose the coverage factor k

The expanded uncertainty is U = k · u_c. For a coverage probability of about 95 %, k = 2 is correct only when the combined uncertainty has many effective degrees of freedom. You check this with the Welch–Satterthwaite formula. In the example, the only source with finite degrees of freedom is repeatability, and it is small, so ν_eff is very large (above 10 000) and k = 2.00. When a Type A component dominates and n is small, k can be noticeably larger than 2: see Coverage factor k and Welch–Satterthwaite.

Step 7. Calculate and report the expanded uncertainty

EA-4/02 recommends giving the uncertainty with at most two significant digits, and the result rounded to the same decimal place. The certificate line would read: error of indication E = 0.30 °C, U = 0.16 °C (k = 2, coverage probability approximately 95 %). The statement of k and the coverage probability is part of the result, not decoration.

Common mistakes

• Using the certificate U as a standard uncertainty. U from a certificate must be divided by its k before it goes into the budget.
• s or s / √n? Divide by √n only when the reported result is the mean of those n readings. If the customer will use a single reading in practice, the repeatability contribution is s itself.
• Ignoring sensitivity coefficients when inputs are in different units or enter the model non-linearly.
• Counting the same effect twice, for example the resolution of the device and a repeatability study that already includes the scatter caused by that resolution. Whether to keep both is a judgement you should document.
• Assuming k = 2 without checking the effective degrees of freedom.

Once you have U, the next question is usually whether the instrument meets its specification. That is a separate decision with its own rules: see ILAC G8 decision rules.