How to build a measurement uncertainty budget: steps and a worked example

What an uncertainty budget is

The method comes from the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008), published by the Joint Committee for Guides in Metrology and free from the BIPM website. EA-4/02 M:2022, the European accreditation guidance on uncertainty in calibration, applies it to calibration work. It describes the budget as a list of all sources of uncertainty with their standard uncertainties and the way each was evaluated, and recommends presenting it as a table.

Each row of that table carries the same items: the input quantity, its estimate, its standard uncertainty, the probability distribution assumed, the sensitivity coefficient and the resulting contribution to the uncertainty of the result. The bottom line is the combined standard uncertainty. One more line multiplies it by the coverage factor to give the expanded uncertainty.

A budget belongs to one measurement process: one kind of instrument, one method, one range or point. A micrometer checked against gauge blocks and a pressure gauge checked against a deadweight tester need separate budgets, even when the same person does both.

• Free measurement uncertainty budget calculator
• Uncertainty budget template (Excel)

When you need one

A calibration laboratory working to ISO/IEC 17025:2017 has to evaluate uncertainty. Clause 7.6.1 requires the lab to identify the contributions to measurement uncertainty and to take all significant ones into account. Clause 7.6.2 requires a lab that performs calibrations, including calibrations of its own equipment, to evaluate the measurement uncertainty for all of them. A documented budget is the usual way to show that work to an assessor.

The budget also feeds decisions. When a lab states conformity to a specification, ISO/IEC 17025 asks for the decision rule to be defined (clause 7.1.3) and reported with the statement (clause 7.8.6). Decision rules that use a guard band take its width from the expanded uncertainty, so the budget is the input.

Shops that are not laboratories need the number too. Without an expanded uncertainty you cannot calculate a test uncertainty ratio (TUR), set a guard band or show that an in-house calibration is good enough for the tolerance it checks. Customers sometimes ask for it directly.

• Guard banding in calibration

Start with the measurement model

Write down what you are measuring and how the result is calculated. The GUM calls this the model, Y = f(X1, X2, ..., XN): the output Y depends on the input quantities X1 to XN. Each input that can vary is a candidate row in the budget.

For most calibrations the model is short. When you calibrate a micrometer against a gauge block, the result is the error of indication: the micrometer reading minus the length of the gauge block, plus small corrections such as one for temperature. Apply the corrections you can estimate. Their uncertainty stays in the budget.

List the sources of uncertainty

Work through the measurement from the reference to the reading. These are the usual sources in a calibration.

• The reference standard. Its calibration certificate states an expanded uncertainty and a coverage factor. This term is easy to forget because the standard feels like the truth.
• Drift of the reference standard since its last calibration, estimated from its calibration history.
• Resolution of the instrument being calibrated, and of the reference if it has a display. A digital reading is rounded to its last digit.
• Repeatability, from repeated readings under the same conditions.
• Temperature. The deviation from the reference temperature, and any difference between the reference and the item, act through thermal expansion.
• The method and setup, such as alignment, measuring force, zero setting or the fixture.
• The operator, when different people get different results. A study with several operators measures this.
• The item's own behavior during the measurement, such as hysteresis or an unstable zero.

Type A and Type B evaluation

The GUM sorts evaluations by method, not by the kind of effect. A Type A evaluation uses statistical analysis of a series of observations. A Type B evaluation uses other information: a calibration certificate, a manufacturer's specification, a handbook value or experience with the instrument. Both produce a standard uncertainty, which is one standard deviation, and the budget combines them the same way.

For Type A, take n repeated readings and calculate their experimental standard deviation s. If the result you report is the mean of those n readings, its standard uncertainty is s / √n (GUM 4.2.3). If the result is a single reading, use s itself. The degrees of freedom are n − 1.

For Type B, convert what the source gives you into a standard deviation. A certificate that states U = 0.10 µm at k = 2 gives a standard uncertainty of 0.10 / 2 = 0.05 µm. NIST Technical Note 1297 gives the same rule: divide a quoted uncertainty by the multiplier it was stated with. A source that gives only limits of ±a needs a divisor that depends on the distribution you assume.

• Type A uncertainty calculator

Distributions and divisors

When a source gives limits of ±a instead of a standard deviation, the standard uncertainty is a divided by a factor. The factor depends on how the possible values are likely to spread between the limits.

• Resolution uncertainty calculator

Sensitivity coefficients

Inputs often come in different units from the result. A temperature arrives in °C while the result is in µm. The sensitivity coefficient c says how much the result changes per unit change in an input, so each contribution is c × u. In the GUM it is the partial derivative of the model with respect to that input (clause 5.1.3).

When the model is a plain sum or difference, every coefficient is 1 or −1 and the units already match. For a temperature effect on length, the coefficient is the length times the expansion coefficient. A 25 mm steel part with an expansion coefficient of 11.5 × 10⁻⁶ per °C changes by 25 mm × 11.5 × 10⁻⁶ = 0.2875 µm per °C.

Combining the contributions: root sum of squares

For independent inputs, the combined standard uncertainty is the square root of the sum of the squared contributions: u_c = √((c1 × u1)² + (c2 × u2)² + ... + (cN × uN)²). This is the GUM's law of propagation of uncertainty for uncorrelated inputs (clause 5.1.2).

Squaring makes the large terms dominate. A contribution one third the size of the largest raises the total by only about 5 percent. Effort spent shrinking the largest term pays far more than effort on the small ones.

The formula assumes the inputs are uncorrelated. When two inputs move together, for example because both were measured with the same reference standard or the same thermometer, the GUM adds covariance terms (clause 5.2). EA-4/02 notes that a different choice of model can sometimes remove the correlation.

Effective degrees of freedom and the coverage factor k

The expanded uncertainty is U = k × u_c (GUM 6.2.1). The usual k = 2 gives about 95 percent coverage only under two conditions. The combined result has to be close to normal, which holds when several contributions of comparable size are combined. And u_c has to be reliable, which means its effective degrees of freedom are large. The GUM's simplified approach asks for more than about 10 (G.6.6). EA-4/02 treats the reliability condition as met when no Type A contribution comes from fewer than ten repeated readings.

When a Type A term from a few readings carries a large share of the budget, estimate the effective degrees of freedom with the Welch-Satterthwaite formula (GUM G.4.1): ν_eff = u_c⁴ / Σ(u_i⁴ / ν_i). Each Type A term has ν = n − 1. A Type B term whose limits were set so that values outside them are very unlikely is usually given infinite degrees of freedom, so it drops out of the sum (GUM G.4.3).

Then read k from the t-distribution at 95.45 percent, the coverage that k = 2 gives for a normal distribution. If ν_eff is not a whole number, truncate it to the next lower one.

• Welch-Satterthwaite effective degrees of freedom calculator

Worked example: a micrometer at 25 mm

A 0 to 25 mm digital micrometer that reads to 0.001 mm is calibrated at 25 mm against a steel gauge block. The block's certificate gives its length as 25.0000 mm with an expanded uncertainty of 0.10 µm at k = 2. The technician takes three readings: 25.001, 25.002 and 25.001 mm. Their mean is 25.0013 mm, so the error of indication is +1.3 µm. After handling, the block and the micrometer can differ in temperature by up to ±1 °C.

• Combined standard uncertainty: u_c = √(0.050² + 0.289² + 0.333² + 0.166²) = √0.2245 = 0.474 µm.
• Repeatability carries about half of the variance, resolution 37 percent, temperature 12 percent and the gauge block about 1 percent.
• Effective degrees of freedom: ν_eff = 0.474⁴ / (0.333⁴ / 2) = 8.2. The other terms have infinite degrees of freedom and drop out.
• Coverage factor: truncate 8.2 to 8, so k = 2.37 from the table above.
• Expanded uncertainty: U = 2.37 × 0.474 = 1.12 µm. Rounded up to two significant digits, as the GUM allows, that is 1.2 µm.
• Result: the error of indication at 25 mm is +1.3 µm with U = 1.2 µm (k = 2.37, about 95 percent coverage).

How the budget feeds TUR and guard banding

The test uncertainty ratio compares the tolerance with the expanded uncertainty of the calibration: TUR = tolerance half-width ÷ U. If the tolerance at this point is ±3 µm, TUR = 3 ÷ 1.2 = 2.5:1. The test accuracy ratio (TAR) uses only the accuracy of the reference standard, so it misses resolution, repeatability and temperature. In this example those three make up almost the whole budget.

A guard band uses the same U. Under the simple rule of subtracting the full expanded uncertainty from each tolerance limit, the acceptance limits become ±(3 − 1.2) = ±1.8 µm. The measured error of +1.3 µm is inside them, so the micrometer passes under that rule.

The coverage factor matters here too. With k = 2 the budget would give U = 0.95 µm and a TUR of 3.2:1, which makes the measurement look better than it is.

• TUR and TAR calculator
• Guard banding and decision rules

Common mistakes

• Leaving out the reference standard's own uncertainty. The value on its certificate is uncertain too, and that uncertainty passes into every calibration made with it. It is often small, as in the example above, but it has to be in the budget.
• Using k = 2 with only a few readings. When a Type A term from three or four readings dominates, k = 2 gives well under 95 percent coverage. Work out the effective degrees of freedom first.
• Double counting resolution and repeatability, or dropping resolution. Readings taken from a display already carry its rounding, so the two terms partly describe the same thing. The GUM approach adds both. A NIST study (Phillips, Toman and Estler, 2008) compared it with the ISO 14253-2 rule of keeping only the larger, and found that keeping only the larger can understate the uncertainty. Pick one rule, write it into your procedure and apply it every time. Never drop resolution when the repeated readings are all identical, because the Type A term is then zero and nothing else represents the rounding (GUM F.2.2.1).
• Dividing by √n when you report a single reading. s / √n is the uncertainty of a mean. If the result you report is one reading, the repeatability term is s.
• Mixing units or confidence levels. Convert every certificate value to a standard uncertainty before you combine, and express every contribution in the units of the result.

Where Axiospec fits

Axiospec is calibration management software. It records each calibration with its as-found and as-left readings, the certificate and the due date. It can also store an uncertainty budget for each instrument and use it for the test uncertainty ratio when a calibration states no uncertainty of its own. Building the budget and judging it stay with you.