Error and uncertainty
The terms ‘error’ and ‘uncertainty’ are not the same, although they are often confused. To understand the difference between these two terms, let’s consider the result obtained from a measurement. If we could make a ‘perfect’ measurement, we would obtain the ‘true value’ of the measurand. However, in reality, the measurements that we make are never perfect, and the values obtained from measurement (measured values) are therefore estimates of this ‘true value’. A measured value will differ from the hypothetical ‘true value’ of the measurand for a number of reasons, some of which we may know about, others of which we may not.
To account for known differences, we can apply a correction to the measured value. For instance, a measured value may be multiplied by a gain determined during the instrument’s calibration, or a measured optical signal may have a dark reading subtracted from it. It is, however, important to realise that the correction will never be perfectly known. In other words, there will always be some difference between the corrected measured value and the (unknown and, in fact, unknowable) ‘true value’ of the measurand. We call the difference between a measured (or corrected) value and the ‘true value’ of the measurand the error, as shown in the diagram below.
Since we can never know the ‘true value’ of the measurand, we can never know the difference between it and the measured values that we obtain. This means that we can never know the value of an error.
We can, however, describe the error in the result of a particular measurement as a draw from a probability distribution. The uncertainty associated with a measured value is a measure of that probability distribution. In particular, the standard uncertainty is the standard deviation of this probability distribution.
although errors are unknown and unknowable, we can evaluate the uncertainty associated with a measured value. As we have seen, any measurement is sensitive to several different ‘effects’, or ‘sources of uncertainty’ such as noise, non-linearity, non-uniformity, etc. Each of these effects gives rise to an unknown error drawn from a probability distribution described by the uncertainty.
There are two approaches to evaluating this uncertainty:
- Type A evaluations, which rely on a statistical approach, and generally deal with variation in values obtained when a measurement is repeated
- Type B evaluations, which are based not on statistical analysis of data, but on other forms of information and knowledge of the individual effect in the measurement situation. Common sources of such information are past experience, calibration certificates, manufacturers’ specifications, published information and common sense
We won’t discuss these two methods of uncertainty evaluation in detail in this recipe, but detailed information on both can be found in ‘The Guide to the Expression of Uncertainty in Measurement’ (The GUM) and in NPL’s ‘Uncertainty Analysis for Earth Observation’ e-Learning course.
Now that we’ve examined the difference between error and uncertainty, and seen the two ways in which measurement uncertainty can be evaluated, let’s move on to examine a key component of uncertainty evaluation: sensitivity coefficients.