• 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 D: Analysing Uncertainty Sources

D.1 Introduction

It is commonly necessary to develop and record a list of sources of uncertainty relevant to an analytical method. It is often useful to structure this process, both to ensure comprehensive coverage and to avoid over-counting. The following procedure (based on a previously published method [h.13]), provides one possible means of developing a suitable, structured analysis of uncertainty contributions.

D.2 Principles of Approach

D.2.1 The strategy has two stages:

• Identifying the effects on a result
  In practice, the necessary structured analysis is effected using a "cause and effect diagram" (sometimes known as an Ishikawa or 'fishbone' diagram) [h.14].
• Simplifying and resolving duplication
  The initial list is refined to simplify presentation and ensure that effects are not unnecessarily duplicated.
  
  

D.3 Cause and Effect Analysis

D.3.1 The principles of constructing a cause and effect diagram are described fully elsewhere. The procedure employed is as follows:

1. Write the complete equation for the result. The parameters in the equation form the main branches of the diagram. It is almost always necessary to add a main branch representing a nominal correction for overall bias, usually as recovery, and this is accordingly recommended at this stage if appropriate.
2. Consider each step of the method and add any further factors to the diagram, working outwards from the main effects. Examples include environmental and matrix effects.
3. For each branch, add contributory factors until effects become sufficiently remote, that is, until effects on the result are negligible.
4. Resolve duplications and re-arrange to clarify contributions and group related causes. It is convenient to group precision terms at this stage on a separate precision branch.

D.3.2 The final stage of the cause and effect analysis requires further elucidation. Duplications arise naturally in detailing contributions separately for every input parameter. For example, a run-to-run variability element is always present, at least nominally, for any influence factor; these effects contribute to any overall variance observed for the method as a whole and should not be added in separately if already so accounted for. Similarly, it is common to find the same instrument used to weigh materials, leading to over-counting of its calibration uncertainties. These considerations lead to the following additional rules for refinement of the diagram (though they apply equally well to any structured list of effects):

• Cancelling effects: remove both. For example, in a weight by difference, two weights are determined, both subject to the balance 'zero bias'. The zero bias will cancel out of the weight by difference, and can be removed from the branches corresponding to the separate weighings.
• Similar effect, same time: combine into a single input. For example, run-to-run variation on many inputs can be combined into an overall run-to-run precision 'branch'. Some caution is required; specifically, variability in operations carried out individually for every determination can be combined, whereas variability in operations carried out on complete batches (such as instrument calibration) will only be observable in between-batch measures of precision.
• Different instances: re-label. It is common to find similarly named effects which actually refer to different instances of similar measurements. These must be clearly distinguished before proceeding.

D.3.3 This form of analysis does not lead to uniquely structured lists. In the present example, temperature may be seen as either a direct effect on the density to be measured, or as an effect on the measured mass of material contained in a density bottle; either could form the initial structure. In practice this does not affect the utility of the method. Provided that all significant effects appear once, somewhere in the list, the overall methodology remains effective.

D.3.4 Once the cause-and-effect analysis is complete, it may be appropriate to return to the original equation for the result and add any new terms (such as temperature) to the equation.

D.4 Example

D.4.1 The procedure is illustrated by reference to a simplified direct density measurement. Consider the case of direct determination of the density d(EtOH) of ethanol by weighing a known volume V in a suitable volumetric vessel of tare weight m_tare and gross weight including ethanol m_gross. The density is calculated from

d(EtOH)=(m_gross - m_tare)/V

For clarity, only three effects will be considered: Equipment calibration, Temperature, and the precision of each determination. Figures D1, D2 and D3 illustrate the process graphically.

D.4.2 A cause and effect diagram consists of a hierarchical structure culminating in a single outcome. For the present purpose, this outcome is a particular analytical result ('d(EtOH)' in figure d1 more...). The 'branches' leading to the outcome are the contributory effects, which include both the results of particular intermediate measurements and other factors, such as environmental or matrix effects. Each branch may in turn have further contributory effects. These 'effects' comprise all factors affecting the result, whether variable or constant; uncertainties in any of these effects will clearly contribute to uncertainty in the result.

D.4.3 Figure d1 shows a possible diagram obtained directly from application of steps 1-3. The main branches are the parameters in the equation, and effects on each are represented by subsidiary branches. Note that there are two 'temperature' effects, three 'precision' effects and three 'calibration' effects. more...

D.4.4 Figure d2 shows precision and temperature effects each grouped together following the second rule (same effect/time); temperature may be treated as a single effect on density, while the individual variations in each determination contribute to variation observed in replication of the entire method. more...

D.4.5 The calibration bias on the two weighings cancels, and can be removed (figure d3) following the first refinement rule (cancellation). more...

D.4.6 Finally, the remaining 'calibration' branches would need to be distinguished as two (different) contributions owing to possible non-linearity of balance response, together with the calibration uncertainty associated with the volumetric determination.