Key Topics

Introduction to evaluating uncertainty

There will nearly always be more than one source of error that will influence measurement results. The aim when evaluating measurement uncertainty is to combine the effects of all of the errors into a single value which can be quoted with the measurement result. The uncertainty represents the range within which the analyst believes the true value lies, with a specified level of confidence.

There are four main stages in the uncertainty evaluation process:

• Specify the measurand

• Identify the sources of uncertainty

• Quantify the uncertainty components

• Combine the uncertainty estimates

Specify the measurand

At the beginning of the process of evaluating uncertainty it is important to be clear about what is being measured (e.g. the total amount of lead in a sample of paint, or the amount extracted under specified test conditions). It should be possible to write down the equation that will be used to calculate the result. The uncertainties associated with the parameters in the equation (masses, volumes, instrument response, etc) will contribute to the uncertainty in the measurement result.

Identify the sources of uncertainty

A critical step in evaluating uncertainty is compiling a list of all the possible sources of uncertainty. This will include the uncertainties in the parameters identified in the specification stage, but there are likely to be other sources of uncertainty. The analyst should consider each stage of the test method in turn and think about the things which could cause results to vary. These will be sources of uncertainty.

Quantify the uncertainty components

The next step is to estimate the size of each uncertainty component identified in previous stage. This can be done by using published data, or method performance data from validation studies, or ongoing quality control. Where such data are not available additional experimental studies will usually be required. It is often possible to plan experiments that can account for a number of different sources of uncertainty.

Combine the uncertainty estimates

Stage 3 will result in a list of a number of sources of uncertainty together with an estimate of their magnitude. This is often referred to as the ‘uncertainty budget’. The individual contributions must be expressed as standard deviations. An uncertainty expressed as a standard deviation is described as a ‘standard uncertainty’, and is denoted by the symbol u.

Standard uncertainties are combined using the mathematical rule for combining variances. This is sometimes referred to as the ‘root sum of squares’ rule and is illustrated in the figure. u₁ and u₂ are independent uncertainty components, expressed as standard deviations and u_c is the combined uncertainty. Due to the way the standard uncertainties are combined, if u₁ is much greater than u₂ then u_c will be approximately equal to u₁.

To increase the level of confidence that the uncertainty quoted includes the true value, the combined standard uncertainty is usually expanded to include 95% of values. This is achieved by multiplying the combined standard uncertainty by a coverage factor, k. To obtain a confidence level of approximately 95%, a coverage factor of 2 is generally used. Expanded uncertainties are denoted by the symbol U.