• 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

8. Step 4: Calculating the Combined Uncertainty

8.1. Standard Uncertainties

8.1.1. Before combination, all uncertainty contributions must be expressed as standard uncertainties, that is, as standard deviations. This may involve conversion from some other measure of dispersion. The following rules give some guidance for converting an uncertainty component to a standard deviation.

8.1.2. Where the uncertainty component was evaluated experimentally from the dispersion of repeated measurements, then it can readily be expressed as a standard deviation. For the contribution to uncertainty in single measurements, the standard uncertainty is simply the observed standard deviation; for results subjected to averaging, the standard deviation of the mean [b.25] is used.

8.1.3. Where an uncertainty estimate is derived from previous results and data, it may already be expressed as a standard deviation. However where a confidence interval is given with a level of confidence, (in the form ± at p%) then divide the value by the appropriate percentage point of the Normal distribution for the level of confidence given to calculate the standard deviation.

EXAMPLE

A specification states that a balance reading is within ±0.2 mg with 95% confidence. From standard tables of percentage points on the normal distribution, a 95% confidence interval is calculated using a value of 1.96. Using this figure gives a standard uncertainty of (0.2/1.96) 0.1.

8.1.4. If limits of ± are given without a confidence level and there is reason to expect that extreme values are likely, it is normally appropriate to assume a rectangular distribution, with a standard deviation of / (see appendix e).

EXAMPLE

A 10 ml Grade A volumetric flask is certified to within ±0.2 ml. The standard uncertainty is 0.2/ 0.11 ml.

8.1.5. If limits of ± are given without a confidence level, but there is reason to expect that extreme values are unlikely, it is normally appropriate to assume a triangular distribution, with a standard deviation of / (see appendix e).

EXAMPLE

A 10 ml Grade A volumetric flask is certified to within ±0.2 ml, but routine in-house checks show that extreme values are rare. The standard uncertainty is 0.2/ 0.08 ml.

8.1.6. Where an estimate is to be made on the basis of judgement, then it may be possible to estimate the component directly as a standard deviation. If this is not possible then an estimate should be made of the maximum deviation which could reasonably occur in practice (excluding simple mistakes). If a smaller value is considered substantially more likely, this estimate should be treated as descriptive of a triangular distribution. If there are no grounds for believing that a small error is more likely than a large error, the estimate should be treated as characterising a rectangular distribution.

8.1.7. Conversion factors for the most commonly used distribution functions are given in appendix e.1.

8.2. Combined standard uncertainty

8.2.1. Following the estimation of individual or groups of components of uncertainty and expressing them as standard uncertainties, the next stage is to calculate the combined standard uncertainty using one of the procedures described below.

8.2.2. The general relationship between the uncertainty u(y) of a value y and the uncertainty of the independent parameters x₁, x₂, ...x_n on which it depends is

u(y(x₁,x₂,...)) =

where y(x₁,x₂,..) is a function of several parameters x₁,x₂,.., and c_i is a sensitivity coefficient evaluated as c_i=y/x_i, the partial differential of y with respect to x_i. Each variable's contribution is just the square of the associated uncertainty expressed as a standard deviation multiplied by the square of the relevant sensitivity coefficient. These sensitivity coefficients describe how the value of y varies with changes in the parameters x₁, x₂ etc.

NOTE: Sensitivity coefficients may also be evaluated directly by experiment; this is particularly valuable where no reliable mathematical description of the relationship exists.

8.2.3. Where variables are not independent, the relationship is more complex:

u(y(x_i,_j...)) =

where s(x_i,x_k) is the covariance between x_i and x_i and c_i and c_k are the sensitivity coefficients as described and evaluated in 8.2.2. The covariance is related to the correlation coefficient r_ik by

s(x_i,x_k) =

where

8.2.4. These general procedures apply whether the uncertainties are related to single parameters, grouped parameters or to the method as a whole. However, when an uncertainty contribution is associated with the whole procedure, it is usually expressed as an effect on the final result. In such cases, or when the uncertainty on a parameter is expressed directly in terms of its effect on y, the sensitivity coefficient y/x_i is equal to 1.0.

EXAMPLE

8.2.5. Except for the case above, when the sensitivity coefficient is equal to one, and for the special cases given in Rule 1 and Rule 2 below, the general procedure, requiring the generation of partial differentials or the numerical equivalent must be employed. Appendix e gives details of a numerical method, suggested by Kragten, which makes effective use of standard spreadsheet software to provide a combined standard uncertainty from input standard uncertainties and a known measurement model [h.5]. It is recommended that this method, or another appropriate computer-based method, be used for all but the simplest cases.

8.2.6. In some cases, the expressions for combining uncertainties reduce to much simpler forms. Two simple rules for combining standard uncertainties are given here.

Rule 1

For models involving only a sum or difference of quantities, e.g. y=(p+q+r+...), the combined standard uncertainty u_c(y) is given by

Rule 2

For models involving only a product or quotient, e.g. y=(p x q x r...) or y= p/(q x r...), the combined standard uncertainty u_c(y) is given by

where (u(p)/) etc. are the uncertainties in the parameters, expressed as relative standard deviations.

NOTE: Subtraction is treated in the same manner as addition, and division in the same way as multiplication.

8.2.7. For the purposes of combining uncertainty components, it is most convenient to break the original mathematical model down to expressions which consist solely of operations covered by one of the rules above. For example, the expression

should be broken down to the two elements (o+p) and (q+r). The interim uncertainties for each of these can then be calculated using rule 1 above; these interim uncertainties can then be combined using rule 2 to give the combined standard uncertainty.

8.2.8. The following examples illustrate the use of the above rules:

EXAMPLE 1

The values are m=1, p=5.02, q=6.45 and r=9.04 with standard uncertainties deviations u(p)=0.13, u(q)=0.05 and u(r)= 0.22.

y = 5.02 - 6.45 + 9.04 = 7.61

EXAMPLE 2

y = (op/qr). The values are o=2.46, p=4.32, q=6.38 and r=2.99, with standard uncertainties of u(o)=0.02, u(p)=0.13, u(q)=0.11 and u(r)=0.07.

y=(2.46 x 4.32 ) / (6.38 x 2.99 ) = 0.56

8.2.9. There are many instances in which the magnitudes of components of uncertainty vary with the level of analyte. For example, uncertainties in recovery may be smaller for high levels of material, or spectroscopic signals may vary randomly on a scale approximately proportional to intensity (constant coefficient of variance). In such cases, it is important to take account of the changes in the combined standard uncertainty with level of analyte. Approaches include:

• Restricting the specified procedure or uncertainty estimate to a small range of analyte concentrations.
• Providing an uncertainty estimate in the form of a relative standard deviation.
• Explicitly calculating the dependence and recalculating the uncertainty for a given result.

Appendix e4 gives additional information on these approaches.

8.3. Expanded Uncertainty

8.3.1. The final stage is to multiply the combined standard uncertainty by the chosen coverage factor in order to obtain an expanded uncertainty. The expanded uncertainty is required to provide an interval which may be expected to encompass a large fraction of the distribution of values which could reasonably be attributed to the measurand.

8.3.2. In choosing a value for the coverage factor k, a number of issues should be considered. These include:

• The level of confidence required
• Any knowledge of the underlying distributions
• Any knowledge of the number of values used to estimate random effects (see 8.3.3 below).

8.3.3. For most purposes it is recommended that k is set to 2. However, this value of k may be insufficient where the combined uncertainty is based on statistical observations with relatively few degrees of freedom (less than about six). The choice of k then depends on the effective number of degrees of freedom.

8.3.4. Where the combined standard uncertainty is dominated by a single contribution with fewer than six degrees of freedom, it is recommended that k be set equal to the two-tailed value of Student's t for the number of degrees of freedom associated with that contribution, and for the level of confidence required (normally 95%). Table 1 gives a short list of values for t.

EXAMPLE:

A combined standard uncertainty for a weighing operation is formed from contributions u_cal=0.01 mg arising from calibration uncertainty and s_obs=0.08 mg based on the standard deviation of five repeated observations. The combined standard uncertainty u_c is equal to = 0.081 mg. This is clearly dominated by the repeatability contribution s_obs, which is based on five observations, giving 5- 1=4 degrees of freedom. k is accordingly based on Student's t. The two-tailed value of t for four degrees of freedom and 95% confidence is, from tables, 2.8; k is accordingly set to 2.8 and the combined expanded uncertainty U_c = 2.8 x 0.081 = 0.23 mg.

8.3.5. The Guide [h.2] gives additional guidance on choosing k where a small number of measurements is used to estimate large random effects, and should be referred to when estimating degrees of freedom where several contributions are significant.

8.3.6. Where the distributions concerned are normal, a coverage factor of 2 (or chosen according to paragraphs 8.3.3. - 8.3.5. using a level of confidence of 95%) gives an interval containing approximately 95% of the distribution of values. It is not recommended that this interval is taken to imply a 95% confidence interval without a knowledge of the distribution concerned.