Measurement uncertainty in practice: The general approach in seven steps

Measurement uncertainty describes the accuracy of a measurement. But what does accuracy actually mean? A measurement result is considered accurate if it exhibits both trueness and precision.

Trueness describes how closely the mean of several measurements matches a target value, such as the “correct value” of a reference sample. Precision, by contrast, describes the scatter of individual measured values around this mean when measurements are repeated under identical conditions.

Determining measurement uncertainty first requires the measurement process to have the highest possible trueness. On this basis, it then describes the remaining deviations of individual measured values or means from the true value (see Part 1). Measurement uncertainty therefore does not describe a faulty measurement process, but rather the degree of confidence – ideally as high as possible – with which a measurement result can be stated.

The general approach to determining measurement uncertainty can be summarised in seven steps.

Step 1: Defining the measurement method

First, the measurement method used to determine a physical quantity is defined. For example, there are several ways to determine the mass m of an object:

• determining density ρ and volume V (m=ρ⋅V)
• measuring the gravitational force F and using gravitational acceleration g (m=F/g)

Different measurement methods involve different influencing quantities and therefore different contributions to measurement uncertainty.

Step 2: Correctly setting up the measuring instruments

Determining meaningful measurement uncertainty requires a properly functioning measurement process. Measuring instruments must therefore be correctly set up or checked. This can be done using reference samples or reference methods, or by calibrating the instruments, for example through an accredited calibration laboratory that in turn uses reference samples or reference methods.

Step 3: Identifying all influencing quantities

The next step is to identify all quantities that may influence the measurement result. If we measure the gravitational force F using a balance, relevant factors may include:

• the accuracy of the balance’s mechanics or electronics
• the position of the object on the balance
• the accuracy of the reference mass used to check the balance
• the buoyancy of the object

Each of these influencing quantities contributes a certain share to the overall measurement uncertainty.

Step 4: Determining distributions and standard deviations

The distribution function of each influencing quantity is then determined, for example a normal, rectangular or triangular distribution. The standard deviation, which indicates the scatter of the respective influencing quantity, is also determined. For a normally distributed quantity, this means working with a confidence or significance level of approximately 68% (±1⋅σ).

Step 5: Checking for correlations

The next step is to examine whether the influencing quantities are correlated. Two influencing quantities are described as correlated if they depend on one another. For example, if two influencing quantities are determined using the same reference sample, they are no longer independent. Such correlations make measurement uncertainty more difficult to calculate and should be avoided where possible.

Step 6: Calculating standard measurement uncertainty

Standard measurement uncertainty is then calculated from the standard uncertainties of the individual influencing quantities. For independent influencing quantities, the following applies:

• The mean measurement uncertainty is obtained from the square root of the sum of the squares of the individual standard deviations (the usual method).
• A maximum measurement uncertainty can be estimated approximately by adding the standard deviations.

Step 7: Determining expanded measurement uncertainty

Finally, expanded measurement uncertainty is determined, defining an expanded confidence interval associated with a higher confidence level (increased significance). In laboratory practice, a coverage factor of 2 is commonly used to multiply the standard measurement uncertainty.

A factor of 2 (or ±2σ) corresponds to a confidence level of approximately 95%. This means that the true value of the measurand lies within the stated confidence interval with a probability (confidence level) of approximately 95%.

The Higgs boson – A different perspective on measurement uncertainty

Measurement uncertainty matters not only in materials testing, but also in particle physics. A well-known example is the detection of the Higgs boson. Two international research teams, each comprising around 2,500 scientists, worked on the ATLAS and CMS detectors at the Large Hadron Collider (LHC). Both teams observed a particle with an energy of approximately 125 GeV, as had been predicted for the Higgs boson.

One team achieved a significance of 5σ, the other 4.9σ.

Only when both measurements achieve a significance of at least 5σ is the result regarded as an established discovery in physics. A significance of 5σ corresponds to a probability of approximately 99.999943% that the result is not due to random fluctuations.

In other words, only with this extremely high level of confidence does science actually speak of a discovery.

W.S. Werkstoff Service – Expertise in materials engineering

Werkstoff Service offers practical seminars on measurement uncertainty in materials testing. Further information is available at: Measurement uncertainty online.