Measurement uncertainty: how certain is the measured value?

A value just inside the limit

A shaft must have a diameter of 20.00 ± 0.02 mm. A part is measured at 20.018 mm. It is within tolerance, so it is approved. Or is it? The gauge has a resolution of 0.001 mm, but the true value may differ from the displayed one by several thousandths, depending on the gauge’s calibration, the temperature and how the measurement was made. If the true value is 20.023 mm, the part does not meet the requirement.

The measurement uncertainty is the answer to the question of how far from the displayed value the true value may lie.

What measurement uncertainty is

The international guide to measurement uncertainty, the GUM (JCGM 100:2008), describes measurement uncertainty as a measure of the dispersion of the values that can reasonably be attributed to what is being measured. A complete measurement result is therefore written as a value and an uncertainty, for example 20.018 ± 0.004 mm.

The uncertainty is determined by listing everything that can affect the result and estimating how much each part contributes. The list is called an uncertainty budget. Typical contributions for a length measurement are:

• the uncertainty of the reference, from the gauge’s calibration certificate
• the resolution of the gauge
• the temperature, because part and gauge expand by different amounts
• the repeatability, that is, the spread when the same measurement is repeated
• differences between operators, fixtures and measuring stations

The GUM groups the contributions by how they are estimated. Type A is calculated statistically from repeated measurements. Type B is estimated in another way, for example from a certificate, a specification or experience. The contributions are expressed as standard uncertainties and combined into a combined standard uncertainty. This is then multiplied by a coverage factor k, often k = 2, to give an expanded uncertainty U. With k = 2 and normally distributed contributions, the interval corresponds to a probability of about 95%.

Measurement uncertainty and MSA

Measurement uncertainty and measurement system analysis (MSA) are often mixed up, but they answer different questions.

• An uncertainty budget according to the GUM describes how certain a single measurement result is. It builds the uncertainty from the bottom up, contribution by contribution, and also includes systematic contributions such as the uncertainty of the reference.
• An MSA, for example a Gage R&R study, describes how much the measurement system varies when it is used as in production, with real parts and operators. The result is compared with the process variation or with the tolerance to decide whether the measurement system is good enough for its task.

They complement each other. The repeatability and reproducibility from a Gage R&R study can be contributions in the uncertainty budget. And a budget can show which contribution is largest, and therefore what needs improving, when an MSA study shows that the measurement system varies too much.

Uncertainty near the tolerance limit

In the middle of the tolerance the uncertainty rarely matters: even if the true value is a few thousandths away, the part is approved. Near a limit it is different. There the interval around the measured value can reach beyond the limit, and then you cannot say for certain whether the part meets the requirement.

ISO 14253-1:2017 describes decision rules for exactly this situation. The basic rule is that the uncertainty counts against whoever has to prove something:

• To show that a part meets the requirement, the measured value must lie within the tolerance reduced by the expanded uncertainty U on each side.
• To show that a part does not meet the requirement, the measured value must lie outside the tolerance extended by U.
• In between there is an uncertainty range, where neither conformity nor nonconformity can be proved.

In the example above, with U = 0.004 mm, the supplier would need a measured value between 19.984 and 20.016 mm to show that the shaft is approved. The value 20.018 mm lies in the uncertainty range. (The figures are a worked example.)

Such a rule means that the approved range is in practice narrower than the tolerance. The larger the uncertainty is in relation to the tolerance, the more of the tolerance is lost. That is a strong reason to choose a gauge and a method with small uncertainty for tight tolerances.

Common misunderstandings

• That the resolution is the uncertainty. A gauge that displays thousandths does not necessarily measure to a thousandth. The resolution is only one of the contributions.
• That a calibrated gauge has no uncertainty. Calibration makes the uncertainty known, it does not remove it.
• That the uncertainty is the same everywhere. It depends on gauge, method, environment and part, and can be different in the measuring room than at the machine.

What you do in practice

Draw up an uncertainty budget for the measurements where the tolerance is tight or where decisions often end up close to the limit. Decide which decision rule applies, and agree it with the customer before a value in the uncertainty range turns up. And make sure the gauges’ calibrations, where the uncertainty of the reference is stated, are up to date, because they are the basis for the whole budget.