• Table of Content
• Foreword
• Scope
• Uncertainty
• Analytical Measurement
• The Process
• Step 1: Specification
• Step 2: Identification
• Step 3: Quantification
• Step 4: Calculation
• Reporting Uncertainty
• Appendix A: Examples
  • Example 1: Calibration Standard
  • Example 2: Standardising NaOH
  • Example 3: An Acid/Base Titration
  • Example 4: In-house Validation Studies
  • Example 5: Determ. Cadmium Release
  • Example 6: Crude Fibre in Animal Feed
  • Example 7: Lead in Water - double IDMS
• Appendix B: Definitions
• Appendix C: Analytical Proc.
• Appendix D: Analyse U Sources
• Appendix E: Statistics
• Appendix F: MU at the Limit
• Appendix G: Common Sources
• Appendix H: Bibliography

Appendix F: Measurement Uncertainty at the Limit of Detection/ Determination

F.1 Introduction

F1.1. At low concentrations, an increasing variety of effects becomes important, including, for example,

• the presence of noise or unstable baseline,
• the contribution of interferences to the (gross) signal,
• the influence of any analytical blank used, and
• losses during extraction, isolation or clean-up.

Because of such effects, as analyte concentrations drop, the relative uncertainty associated with the result tends to increase, first to a substantial fraction of the result and finally to the point where the (symmetric) uncertainty interval includes zero. This region is typically associated with the practical limit of detection for a given method.

NOTE: The terminology and conventions associated with measuring and reporting low levels of analyte have been widely discussed elsewhere (See bibliography [h.15, h.16, h.17] for examples and definitions). Here, the term 'limit of detection' only implies a level at which detection becomes problematic, and is not associated with any specific definition.

F1.2. It is widely accepted that the most important use of the 'limit of detection' is to show where method performance becomes insufficient for acceptable quantitation, so that improvements can be made. Ideally, therefore, quantitative measurements should not be made in this region. Nonetheless, so many materials are important at very low levels that it is inevitable that measurements must be made, and results reported, in this region.

F1.3. The ISO Guide on Measurement Uncertainty [h.2] does not give explicit instructions for the estimation of uncertainty when the results are small and the uncertainties large compared to the results. Indeed, the basic form of the 'law of propagation of uncertainties', described in chapter 8 of this guide, may cease to apply accurately in this region; one assumption on which the calculation is based is that the uncertainty is small relative to the value of the measurand. An additional, if philosophical, difficulty follows from the definition of uncertainty given by the ISO Guide: though negative observations are quite possible, and even common in this region, an implied dispersion including values below zero cannot be "... reasonably ascribed to the value of the measurand" when the measurand is a concentration, because concentrations themselves cannot be negative.

F1.4. These difficulties do not preclude the application of the methods outlined in this guide, but some caution is required in interpretation and reporting the results of measurement uncertainty estimation in this region. The purpose of the present Appendix is to provide limited guidance to supplement that already available from other sources.

NOTE: Similar considerations may apply to other regions; for example, mole or mass fractions close to 100% may lead to similar difficulties.

F.2 Observations and Estimates

F2.1. A fundamental principle of measurement science is that results are estimates of true values. Analytical results, for example, are available initially in units of the observed signal, e.g. mV, absorbance units etc. For communication to a wider audience, particularly to the customers of a laboratory or to other authorities, the raw data need to be converted to a chemical quantity, such as concentration or amount of substance. This conversion typically requires a calibration procedure (which may include, for example, corrections for observed and well characterised losses). Whatever the conversion, however, the figure generated remains an observation, or signal. If the experiment is properly carried out, this observation remains the 'best estimate' of the value of the measurand.

F2.2. Observations are not often constrained by the same fundamental limits that apply to real concentrations. For example, it is perfectly sensible to report an 'observed concentration', that is, an estimate, below zero. It is equally sensible to speak of a dispersion of possible observations which extends into the same region. For example, when performing an unbiased measurement on a sample with no analyte present, one should see about half of the observations falling below zero. In other words, reports like

observed concentration = 2.4 ± 8 mg l^-1

observed concentration = -4.2 ± 8 mg l^-1

are not only possible; they should be seen as valid statements.

F2.3. The methods of uncertainty estimation described in this guide apply well to the estimation of uncertainties on observations. It follows that while reporting observations and their associated uncertainties to an informed audience, there is no barrier to, or contradiction in, reporting the best estimate and its associated uncertainty even where the result implies an impossible physical situation. Indeed, in some circumstances (for example, when reporting a value for an analytical blank which will subsequently be used to correct other results) it is absolutely essential to report the observation and its uncertainty (however large).

F2.4. This remains true wherever the end use of the result is in doubt. Since only the observation and its associated uncertainty can be used directly (for example, in further calculations, in trend analysis or for re-interpretation), the uncensored observation should always be available.

F2.5. The ideal is accordingly to report valid observations and their associated uncertainty regardless of the values.

F.3 Interpreted Results and Compliance Statements

F3.1. Despite the foregoing, it must be accepted that many reports of analysis and statements of compliance include some interpretation for the end user's benefit. Typically, such an interpretation would include any relevant inference about the levels of analyte which could reasonably be present in a material. Such an interpretation is an inference about the real world, and consequently would be expected (by the end user) to conform to real limits. So, too, would any associated estimate of uncertainty in 'real' values.

F3.2. Under such circumstances, where the end use is well understood, and where the end user cannot realistically be informed of the nature of measurement observations, the general guidance provided elsewhere (for example in references h.15, h.16, h.17) on the reporting of low level results may reasonably apply.

F3.3. One further caution is, however, pertinent. Much of the literature on capabilities of detection relies heavily on the statistics of repeated observations. It should be clear to readers of the current guide that observed variation is only rarely a good guide to the full uncertainty of results. Just as with results in any other region, careful consideration should accordingly be given to all the uncertainties affecting a given result before reporting the values.