What Is Measurement Uncertainty and How Is It Calculated?

Measurement uncertainty is a parameter, stated with a measurement result, that characterises the range within which the true value of the measured quantity can reasonably be expected to lie. No measurement is exact: the instrument's resolution and repeatability, the uncertainty of the reference and the environmental conditions all affect the result. Uncertainty expresses, as a number, how much doubt these effects create together.

This guide explains what measurement uncertainty is, how it differs from error, and how it is calculated step by step, with a numerical uncertainty budget for a caliper calibration. The method follows the internationally accepted GUM (JCGM 100:2008, Guide to the Expression of Uncertainty in Measurement).

Measurement error versus measurement uncertainty

A complete statement of a result contains both: "Error +0.014 mm, expanded uncertainty 0.008 mm (k=2, about 95% confidence)."

How to calculate measurement uncertainty in 7 steps

1. Write the measurement model: express in an equation how the result is calculated from the input quantities, for example error = reading − reference + temperature effect.
2. List the sources of uncertainty: repeatability, resolution, uncertainty of the reference, temperature, operator, fixturing, drift and so on.
3. Find the standard uncertainty of each source: by a Type A or Type B evaluation (see below).
4. Determine sensitivity coefficients: they show how strongly each input affects the result and are the partial derivatives of the model. When units are the same they are often 1.
5. Calculate the combined standard uncertainty: for independent contributions, take the square root of the sum of squares: u_c = √(Σ (c_i·u_i)²).
6. Find the expanded uncertainty: U = k · u_c; k=2 corresponds to about 95% confidence.
7. Report the result: give the uncertainty to at most two significant digits, round the result to the same decimal place and state k.

Type A and Type B uncertainty

A Type A evaluation uses statistical analysis of repeated measurements. If the standard deviation of n readings is s, the standard uncertainty of the mean is u = s / √n.

A Type B evaluation uses information other than statistics: the reference's calibration certificate, manufacturer specifications, instrument resolution, handbook values and experience. This information is usually given as a limit (±a) and converted to a standard uncertainty according to the assumed distribution:

Worked example: uncertainty budget for a digital caliper

A digital caliper with 0.01 mm resolution is calibrated with a 100 mm nominal gauge block. The numbers are illustrative; in a real calculation, the sources must be defined according to your method and instrument.

• Five readings: 100.01 / 100.02 / 100.01 / 100.02 / 100.01 mm. Mean 100.014 mm, standard deviation s = 0.0055 mm.
• Uncertainty on the gauge block certificate: 0.0006 mm (k=2).
• Temperature difference between caliper and gauge block at most ±1 °C; thermal expansion coefficient of steel about 11.5 × 10⁻⁶ /°C.

Error: 100.014 − 100.000 = +0.014 mm.

Result: error +0.014 mm, U = 0.008 mm (k=2). The table shows the largest contributions come from resolution and repeatability; the gauge block's contribution is negligible. To reduce the uncertainty, a better reference would not help, but more readings or a higher-resolution instrument would. That is the real value of an uncertainty budget: it shows where to invest.

If the caliper's maximum permissible error is ±0.03 mm, then |+0.014| + 0.008 = 0.022 mm is within the limit and the instrument conforms at this point. We explain this decision in our article on evaluating a calibration certificate.

Is the coverage factor always 2?

k=2 gives about 95% confidence when the combined uncertainty has sufficient degrees of freedom. If the dominant contribution comes from few repeated readings, the effective degrees of freedom are calculated with the Welch-Satterthwaite formula and k is taken from the Student t distribution. In our example the effective degrees of freedom are about 25; using the corresponding k ≈ 2.1, U still rounds to 0.008 mm.

Uncertainty, CMC and decision rules

• CMC (Calibration and Measurement Capability): the smallest uncertainty an accredited laboratory can achieve for a measurement in its scope under normal conditions, published in its accreditation scope. Under the ILAC P14 policy, a laboratory may not state an uncertainty smaller than its CMC on a certificate.
• Decision rule: if a statement of conformity is given, how uncertainty is taken into account must be defined. ISO/IEC 17025 and ILAC-G8 address this.
• References: EA-4/02 for calibration and the GUM as the general framework.

Common mistakes in uncertainty calculations

1. Adding the expanded uncertainty from a certificate to the budget without dividing by k.
2. Counting resolution twice, or adding it separately when repeatability already includes it (a common approach is to take the larger; define this in your method).
3. Adding contributions with different units without sensitivity coefficients.
4. Adding standard uncertainties directly instead of in quadrature.
5. Reporting the uncertainty with too many digits.
6. Not updating a calculation when the method, instrument or conditions change.

Moving uncertainty calculations from Excel to software

Most laboratories keep uncertainty budgets in Excel. The problem is not the calculation but control: nobody knows who changed a formula and when, or which version was used for which certificate. KALDATA Calibration Laboratory Software uses the laboratory's own calculation templates in an online spreadsheet that runs in the browser; raw data is entered once and carried to the certificate. We describe a smooth transition in moving from Excel to calibration software.

Frequently Asked Questions

What is measurement uncertainty?
A parameter stated with a measurement result that shows the range in which the true value can reasonably be expected to lie. It is usually given as ±U with k=2, about 95% confidence.

How is measurement uncertainty calculated?
Write the measurement model, list the sources, find each standard uncertainty by Type A or Type B evaluation, combine them with sensitivity coefficients and expand with a coverage factor.

What is the difference between Type A and Type B uncertainty?
Type A is calculated from the statistics of repeated measurements; Type B from other information such as certificates, specifications and resolution. Both enter the budget as standard uncertainties.

What does k=2 mean?
The combined standard uncertainty has been multiplied by 2. With sufficient degrees of freedom this corresponds to about 95% confidence.

Are uncertainty and error the same thing?
No. Error is the difference between the measured value and the reference and can be corrected. Uncertainty is a measure of how far the result can be trusted and cannot be eliminated.

How many digits should uncertainty have?
Usually at most two significant digits, with the result rounded to the same decimal place as the uncertainty.

What is CMC?
The smallest uncertainty an accredited laboratory can achieve for a measurement in its scope under normal conditions. A laboratory may not state a smaller uncertainty on a certificate.

Can measurement uncertainty be calculated in Excel?
Yes, many laboratories use Excel. What matters is keeping templates under control, tracking changes and knowing which template version produced which result.