## Error models and measurement uncertainty

#### Error models and measurement uncertainty

Normally it is not sufficient to divide up a measurement error in a systematic and a random component, but it can also be necessary to describe the error structure with a more complex model.

Measurement system

In the interpretation and estimation of measurement errors and uncertainty it can be appropriate to make a division of error according to various partial populations of measurement defined by having one or several of the following factors held constant:

• equipment
• operator/laboratory
• environment
• method
• time (short time interval)

All of these factors – which together constitute the Measurement System (figure) – are constant under conditions of repeatability (r) while all vary under reproducibility (R) conditions.

In comparing measurement results, it is important to estimate the distribution of errors under relevant conditions. Besides error components, as arranged as above, there can even be test specific errors, that is errors caused by characeristics of the actual test.

Reproducibility & repeatability

Measurement error may also be decomposed according various error sources, e.g. belonging to different stages of the measurement procedure, measurement equipment, performance, conditions of measurement or the measurement object itself.

### Measurement model – Ishikawa diagram

A useful tool to visualise how different sources of measurement error and uncertainty contribute to the overall measurement result is the Ishikawa diagram also known as a ‘fishbone’ diagram and illustrating a hierarchy of cause and effect.

Ishikawa diagram

Such as diagram is drawn with one principal ‘bone’ for each element of the measurement system and bones for each source of measurement error and uncertainty.

### Measurement model – errors & uncertainties

Model: y = m + ε = m +  f(x1, …, xn) + g(z1, …, zn)

where the observed measurement value, y, is the sum of a (‘correct’) measurement value, m, plus a measurement error, ε.

The measurement error is generally a sum of functions of the input measurement quantities, f(x1, …, xn) and the influence quantities g(z1, …, zn).

Examples:

If the measurement error is calculated as a function of other quantities (including corrections for possible influence quantities) then it can be easier to judge the error structure for these (inputs) and thereby derive how the errors propagate to the final result. For at least some of the inputs it can be appropriate to decompose and describe the error structure as above. The input quantities are often themselves measurement values, determined by other input quantities, and an hierarchical structure can be set up of successive decompositions where the actual measurement value is at the uppermost level.

Through calibration and correction it is possible to eliminate some of the error components to be replaced instead by calibration and correction errors. This can make an error model complex and sometimes it can be difficult to judge which error components should be included. It is important however that all relevant error components are accounted for, even if their contributions can be difficult to estimate. To neglect or ignore error components can lead to a systematic underestimate of the uncertainty.

There is nothing wrong in using different error models in describing the error structure and as a basis for calculating measurement uncertainty. Which kind of model which is most relevant depends amongst others on use, available information and to what extent the total error can be explained by errors in input quantities in a formula for calculation of the measurement result.

### Variance propagation

Variance propagation

Independent of which kind of error model used, and after correction for known systematic errors, one can assume that error consists of a number of randomly varying components with the best estimate having an expectation value = 0 (otherwise one would have corrected for deviations from zero) and a measure of the scatter, such as the standard deviation. Uncertainty may be associated with the standard deviation; the so-called standard uncertainty.