## Measurement error and measurement uncertainty

### Measurement error analysis

Aims:

• describe how measurement result or test has been made, and
• make up a so complete list (“budget”) as possible of all potential error sources which might affect result.

Some measurement errors can be reduced by:

• repeated measurement (reduction of random errors) – more exactly, errors are not reduced but mean value will be better representation of reality.
• application of corrections (reduction of systematic errors) through calibration

#### Random and systematic measurement errors

Regarding repeated measurements of a particular measurement object under given conditions, then the error can normally be divided into three components:

• A component of measurement error which varies randomly between measurements and is assumed to have a mean value = 0.
• A component which is constant during the actual measurements – a (locally) systematic error
• A component which varies systematically during the actual measurement

Normally these components are only partially known and give contributions to the uncertainty of the measurement value.

The random component has a real distribution with repeated measurement and, since these errors cannot be predicted and the expectation value = 0, then no correction for them can be made. The uncertainty can appropriately be expressed as an interval (about 0) which covers a given proportion of the distribution.

2. Correct for known errors

If the systematic errors are known – both the constant and systematically varying components – , then one of course can correct for them and they thus do not contribute to the uncertainty. If the systematic error has been estimated in some way, then it can also be corrected for, but residual uncertainties in these estimates must be included in the total uncertainty.

In cases where it is impractical to make repeated estimates of systematic errors, it may still be possible to imagine doing it and thereby think of a plausible distribution of possible correction errors which may be treated mathematically as above. In these cases, if a ‘guessestimate’ of the systematic error has been made, then no correction is usually made. However, allowance must still be made for systematic errors and also contributions to the uncertainty. For instance, a correction of zero can be said to have been made, but this is an uncertain zero, which as earlier may have an associated correction error for which an uncertainty should be given.

#### Uncertainty in a measurement value – “Unknown measurement errors”

is an interval which expresses our lack of knowledge of the real value of the measurement error. For practical use, the measurement uncertainty should be interpreted as follows: With our present knowledge of the measurement error structure, one expects that the measurement error is less than the measurement uncertainty with at least an approximate probability.

### Measurement uncertainty and knowledge

In most cases there is seldom time or resources to investigate all possible sources of measurement error. Where knowledge about measurement is limited – as it always is – then measurement result will have an uncertainty.

__________________________________________________________________________________

### ‘Unknown’ measurement errors, examples:

__________________________________________________________________________________