Uncertainty on Kbias This bias factor is introduced to account for possible deviations in the value of the mass discrimination factor. As can be seen in the cause and effect diagram above, and in equation ( 2), there is a bias associated with every K factor. The values of these biases are in our case not known, and a value of 0 is applied. An uncertainty is, of Table A7.3 Blend b b'

Solutions used Spike Sample Spike Assay 2 Parameter my mx m'y mz Mass ( g) 1.1360 1.0440 1.0654 1.1029 Table A7.4 cx ( µ mol g 1)

Replicate 1 ( our example) 0.053738 Replicate 2 0.053621 Replicate 3 0.053610 Replicate 4 0.053822 Average 0.05370 Experimental standard deviation ( s) 0.0001 Quantifying Uncertainty Example A7 QUAM: 2000. P1 Page 91 course, associated with every bias and this has to be taken into consideration when calculating the final uncertainty. In principle, a bias would be applied as in equation ( 11), using an excerpt from equation ( 5) and the parameters Ky1 and Ry1 to demonstrate this principle.

( ) . .. . . . . . . . . . ) 1 ( ) 1 ( .. . . y1 bias 0 x × - × + × = R y K y K c ( 11)

The values of all biases, Kbias( yi, xi, zi), are ( 0 ± 0.001). This estimation is based on a long experience of lead IDMS measurements. All Kbias( yi, xi, zi) parameters are not included in detail in Table A7.5, Table A7.8 or in equation 5, but they are used in all uncertainty calculations.

Uncertainty of the weighed masses In this case, a dedicated mass metrology lab performed the weighings. The procedure applied was a bracketing technique using calibrated weights and a comparator. The bracketing technique was repeated at least six times for every sample mass determination. Buoyancy correction was applied. Stoichiometry and impurity corrections were not applied in this case. The uncertainties from the weighing certificates were treated as standard uncertainties and are given in Table A7.8.

Uncertainty in the amount content of the Standard Assay Solution, cz i) Uncertainty in the atomic weight of Pb First, the combined uncertainty of the molar mass of the assay solution, Assay 1, will be calculated. The values in Table A7.5 are known or have been measured:

According to equation ( 7), the calculation of the molar mass takes this form:

z4 z4 z3 z3 z2 2 z1 z1 4 z4 z4 3 z3 z3 2 z2 1 z1 z1 ) 1 , Pb (

R K R K R K R K M R K M R K M R M R K Assay M z × + × + × + × × × + × × + × + × × = ( 12)

To calculate the combined standard uncertainty of the molar mass of Pb in the standard assay solution, the spreadsheet model described in Appendix E was used. There were eight measurements of every ratio and K0. This gave a molar mass M( Pb, Assay 1)= 207.2103 g mol 1, with uncertainty 0.0010 g mol 1 calculated using the spreadsheet method.

ii) Calculation of the combined standard uncertainty in determining cz To calculate the uncertainty on the amount content of Pb in the standard assay solution, cz the data from A7.2 and equation ( 8) are used. The uncertainties were taken from the weighing certificates, see A7.3. All parameters used in equation ( 8) are given with their uncertainties in Table A7.6.

The amount content, cz, was calculated using equation ( 8). Following Appendix D. 5 the combined standard uncertainty in cz, is calculated to be uc( cz)= 0.000028. This gives cz= 0.092606 µ mol g 1 with a standard uncertainty of 0.000028 µ mol g 1 ( 0.03% as % rsd).

To calculate uc( cx), for replicate 1, the spreadsheet model was applied ( Appendix E). The uncertainty budget for replicate 1 will be representative for the measurement. Due to the number of parameters in equation ( 5), the spreadsheet will not be displayed. The value of the parameters and their uncertainties as well as the combined uncertainty of cx can be seen in Table A7.8. Table A7.5 Value Standard Uncertainty TypeNote 1 Kbias( zi) 0 0.001 B Rz1 2.1429 0.0054 A K0( z1) 0.9989 0.0025 A K0( z3) 0.9993 0.0035 A K0( z4) 1.0002 0.0060 A Rz2 1 0 A Rz3 0.9147 0.0032 A Rz4 0.05870 0.00035 A M1 207.976636 0.000003 B M2 205.974449 0.000003 B M3 206.975880 0.000003 B M4 203.973028 0.000003 B Note 1. Type A ( statistical evaluation) or Type B ( other) Quantifying Uncertainty Example A7 QUAM: 2000. P1 Page 92 A7.4 Step 4: Calculating the combined standard uncertainty The average and the experimental standard deviation of the four replicates are displayed in Table A7.7. The numbers are taken from Table A7.4 and Table A7.8.

Table A7.7 Replicate 1 Mean of replicates 1 4 cx= 0.05374 cx= 0.05370 µ mol g 1 uc( cx)= 0.00018 s = 0.00010 Note 1 µ mol g 1 Note 1. This is the experimental standard uncertainty and not the standard deviation of the mean.

In IDMS, and in many non routine analyses, a complete statistical control of the measurement procedure would require limitless resources and time. A good way then to check if some source of uncertainty has been forgotten is to compare the uncertainties from the type A evaluations with the experimental standard deviation of the four replicates. If the experimental standard deviation is higher than the contributions from the uncertainty sources evaluated as type A, it could indicate that the measurement process is not fully understood. As an approximation, using data from Table 8, the sum of the type A evaluated experimental uncertainties can be calculated by taking 92.2% of the total experimental uncertainty, which is 0.00041 µ mol g 1. This value is then clearly higher than the experimental standard deviation of 0.00010 µ mol g 1, see Table A7.7. This indicates that the experimental standard deviation is covered by the contributions from the type A evaluated uncertainties and that no further type A evaluated uncertainty contribution, due to the preparation of the blends, needs to be considered. There could however be a bias associated with the preparations of the blends. In this example, a possible bias in the preparation of the blends is judged to be insignificant in comparison to the major sources of uncertainty.

The amount content of lead in the water sample is then:

cx=( 0.05370 ± 0.00036) µ mol g 1 The result is presented with an expanded uncertainty using a coverage factor of 2.

References for Example 7 1. T. Cvitas, Metrologia, 1996, 33, 35 39 2 G. Audi and A. H. Wapstra, Nuclear Physics, A565 ( 1993) Table A7.6 Value Uncertainty Mass of lead piece, m1 ( g) 0.36544 0.00005 Total mass first dilution, d1 ( g) 196.14 0.03 Aliquot of first dilution, m2 ( g) 1.0292 0.0002 Total mass of second dilution, d2 ( g) 99.931 0.01 Purity of the metallic lead piece, w ( mass fraction) 0.99999 0.000005 Molar mass of Pb in the Assay Material, M ( g mol 1) 207.2104 0.0010 Quantifying Uncertainty Example A7 QUAM: 2000. P1 Page 93 Table A7.8 parameter uncertainty evaluation value experimental uncertainty ( Note 1) contribution to total uc(%) final uncertainty ( Note 2) contribution to total uc(%)

S Kbias B 0 0.001Note 3 7.2 0.001Note 3 37.6 cz B 0.092605 0.000028 0.2 0.000028 0.8 K0( b) A 0.9987 0.0025 14.4 0.00088 9.5 K0( b') A 0.9983 0.0025 18.3 0.00088 11.9 K0( x1) A 0.9992 0.0025 4.3 0.00088 2.8 K0( x3) A 1.0004 0.0035 1 0.0012 0.6 K0( x4) A 1.001 0.006 0 0.0021 0 K0( y1) A 0.9999 0.0025 0 0.00088 0 K0( z1) A 0.9989 0.0025 6.6 0.00088 4.3 K0( z3) A 0.9993 0.0035 1 0.0012 0.6 K0( z4) A 1.0002 0.006 0 0.0021 0 mx B 1.0440 0.0002 0.1 0.0002 0.3 my1 B 1.1360 0.0002 0.1 0.0002 0.3 my2 B 1.0654 0.0002 0.1 0.0002 0.3 mz B 1.1029 0.0002 0.1 0.0002 0.3 Rb A 0.29360 0.00073 14.2 0.00026Note 4 9.5 R'b A 0.5050 0.0013 19.3 0.00046 12.7 Rx1 A 2.1402 0.0054 4.4 0.0019 2.9 Rx2 Cons. 1 0 0 Rx3 A 0.9142 0.0032 1 0.0011 0.6 Rx4 A 0.05901 0.00035 0 0.00012 0 Ry1 A 0.00064 0.00004 0 0.000014 0 Rz1 A 2.1429 0.0054 6.7 0.0019 4.4 Rz2 Cons. 1 0 0 Rz3 A 0.9147 0.0032 1 0.0011 0.6 Rz4 A 0.05870 0.00035 0 0.00012 0 cBlank A 4.5 ´ 10 7 4.0 ´ 10 7 0 2.0 ´ 10 7 0 cx 0.05374 0.00041 0.00018 S Acontrib.= 92.2 S Acontrib.= 60.4 S Bcontrib.= 7.8 S Bcontrib.= 39.6 Notes overleaf Quantifying Uncertainty Example A7 QUAM: 2000. P1 Page 94 Notes to Table A7.8 Note 1. The experimental uncertainty is calculated without taking the number of measurements on each parameter into account.

Note 2. In the final uncertainty the number of measurements has been taken into account. In this case all type A evaluated parameters have been measured 8 times. Their standard uncertainties have been divided by Ö 8.

Note 3. This value is for one single Kbias. The parameter S Kbias is used instead of listing all Kbias( zi, xi, yi), which all have the same value ( 0 ± 0.001).

Note 4. Rb has been measured 8 times per blend giving a total of 32 observations. When there is no blend to blend variation, as in this example, all these 32 observations could be accounted for by implementing all four blend replicates in the model. This can be very time consuming and since, in this case, it does not affect the uncertainty noticeably, it is not done. Quantifying Uncertainty Appendix B  Definitions QUAM: 2000. P1 Page 95 Appendix B. Definitions General B. 1 Accuracy of measurement The closeness of the agreement between the result of a measurement and a true value of the measurand [ H. 4].

NOTE 1 " Accuracy" is a qualitative concept.

NOTE 2 The term " precision" should not be used for " accuracy".

B. 2 Precision The closeness of agreement between independent test results obtained under stipulated conditions [ H. 5].

NOTE 1 Precision depends only on the distribution of random errors and does not relate to the true value or the specified value.

NOTE 2 The measure of precision is usually expressed in terms of imprecision and computed as a standard deviation of the test results. Less precision is reflected by a larger standard deviation.

NOTE 3 " Independent test results" means results obtained in a manner not influenced by any previous result on the same or similar test object. Quantitative measures of precision depend critically on the stipulated conditions. Repeatability and reproducibility conditions are particular sets of extreme stipulated conditions.

B. 3 True value Value consistent with the definition of a given particular quantity [ H. 4].

NOTE 1 This is a value that would be obtained by a perfect measurement.

NOTE 2 True values are by nature indeterminate.

NOTE 3 The indefinite article " a" rather than the definite article " the" is used in conjunction with " true value" because there may be many values consistent with the definition of a given particular quantity.

B. 4 Conventional true value Value attributed to a particular quantity and accepted, sometimes by convention, as having an uncertainty appropriate for a given purpose [ H. 4].

EXAMPLES a) At a given location, the value assigned to the quantity realised by a reference standard may be taken as a conventional true value.

b) The CODATA ( 1986) recommended value for the Avogadro constant, NA: 6.0221367 ´ 1023 mol 1 NOTE 1 " Conventional true value" is sometimes called assigned value, best estimate of the value, conventional value or reference value.

NOTE 2 Frequently, a number of results of measurements of a quantity is used to establish a conventional true value.

B. 5 Influence quantity A quantity that is not the measurand but that affects the result of the measurement [ H. 4].

EXAMPLES 1. Temperature of a micrometer used to measure length;

2. Frequency in the measurement of an alternating electric potential difference;

3. Bilirubin concentration in the measurement of haemoglobin concentration in human blood plasma. Quantifying Uncertainty Appendix B  Definitions QUAM: 2000. P1 Page 96 Measurement B. 6 Measurand Particular quantity subject to measurement [ H. 4].

NOTE The specification of a measurand may require statements about quantities such as time, temperature and pressure..

B. 7 Measurement Set of operations having the object of determining a value of a quantity [ H. 4].

B. 8 Measurement procedure Set of operations, described specifically, used in the performance of measurements according to a given method [ H. 4].

NOTE A measurement procedure is usually recorded in a document that is sometimes itself called a " measurement procedure" ( or a measurement method) and is usually in sufficient detail to enable an operator to carry out a measurement without additional information.

B. 9 Method of measurement A logical sequence of operations, described generically, used in the performance of measurements [ H. 4].

NOTE Methods of measurement may be qualified in various ways such as:

 substitution method  differential method  null method B. 10 Result of a measurement Value attributed to a measurand, obtained by measurement [ H. 4].

NOTE 1 When the term " result of a measurement" is used, it should be made clear whether it refers to:

 The indication.

 The uncorrected result.

 The corrected result.

and whether several values are averaged.

NOTE 2 A complete statement of the result of a measurement includes information about the uncertainty of measurement. Uncertainty B. 11 Uncertainty ( of measurement)

Parameter associated with the result of a measurement, that characterises the dispersion of the values that could reasonably be attributed to the measurand [ H. 4].

NOTE 1 The parameter may be, for example, a standard deviation ( or a given multiple of it), or the width of a confidence interval.

NOTE 2 Uncertainty of measurement comprises, in general, many components. Some of these components may be evaluated from the statistical distribution of the results of a series of measurements and can be characterised by experimental standard deviations. The other components, which can also be characterised by standard deviations, are evaluated from assumed probability distributions based on experience or other information.

NOTE 3 It is understood that the result of the measurement is the best estimate of the value of the measurand and that all components of uncertainty, including those arising from systematic effects, such as components associated with corrections and reference standards, contribute to the dispersion.

B. 12 Traceability The property of the result of a measurement or the value of a standard whereby it can be related to stated references, usually national or international standards, through an unbroken chain of comparisons all having stated uncertainties [ H. 4].

B. 13 Standard uncertainty u( xi) Uncertainty of the result xi of a measurement expressed as a standard deviation [ H. 2].

B. 14 Combined standard uncertainty uc( y) Standard uncertainty of the result y of a measurement when the result is obtained from the values of a number of other quantities, equal to the positive square root of a sum of terms, the terms being Quantifying Uncertainty Appendix B  Definitions QUAM: 2000. P1 Page 97 the variances or covariances of these other quantities weighted according to how the measurement result varies with these quantities [ H. 2].

B. 15 Expanded uncertainty U Quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand [ H. 2].

NOTE 1 The fraction may be regarded as the coverage probability or level of confidence of the interval.

NOTE 2 To associate a specific level of confidence with the interval defined by the expanded uncertainty requires explicit or implicit assumptions regarding the probability distribution characterised by the measurement result and its combined standard uncertainty. The level of confidence that may be attributed to this interval can be known only to the extent to which such assumptions can be justified.

NOTE 3 An expanded uncertainty U is calculated from a combined standard uncertainty uc and a coverage factor k using U = k ´ uc B. 16 Coverage factor k Numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded uncertainty [ H. 2].

NOTE A coverage factor is typically in the range 2 to 3.

B. 17 Type A evaluation ( of uncertainty)

Method of evaluation of uncertainty by the statistical analysis of series of observations [ H. 2].

B. 18 Type B evaluation ( of uncertainty)

Method of evaluation of uncertainty by means other than the statistical analysis of series of observations [ H. 2] Error B. 19 Error ( of measurement)

The result of a measurement minus a true value of the measurand [ H. 4].

NOTE 1 Since a true value cannot be determined, in practice a conventional true value is used.

B. 20 Random error Result of a measurement minus the mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions [ H. 4].

NOTE 1 Random error is equal to error minus systematic error.

NOTE 2 Because only a finite number of measurements can be made, it is possible to determine only an estimate of random error.

B. 21 Systematic error Mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions minus a true value of the measurand [ H. 4].

NOTE 1: Systematic error is equal to error minus random error.

NOTE 2: Like true value, systematic error and its causes cannot be known.

Statistical terms B. 22 Arithmetic mean x Arithmetic mean value of a sample of n results.

n x x n i i å = = ,

1 B. 23 Sample Standard Deviation s An estimate of the population standard deviation s from a sample of n results.

1 )

(

1 2 - - = å = n x x s n i i Quantifying Uncertainty Appendix B  Definitions QUAM: 2000. P1 Page 98 B. 24 Standard deviation of the mean x s The standard deviation of the mean x of n values taken from a population is given by n s s x = The terms " standard error" and " standard error of the mean" have also been used to describe the same quantity. B. 25 Relative Standard Deviation ( RSD)

RSD An estimate of the standard deviation of a population from a sample of n results divided by the mean of that sample. Often known as coefficient of variation ( CV). Also frequently stated as a percentage.

x s = RSD Quantifying Uncertainty Appendix C   Uncertainties in Analytical Processes QUAM: 2000. P1 Page 99 Appendix C. Uncertainties in Analytical Processes C. 1 In order to identify the possible sources of uncertainty in an analytical procedure it is helpful to break down the analysis into a set of generic steps:

1. Sampling 2. Sample preparation 3. Presentation of Certified Reference Materials to the measuring system 4. Calibration of Instrument 5. Analysis ( data acquisition)

6. Data processing 7. Presentation of results 8. Interpretation of results C. 2 These steps can be further broken down by contributions to the uncertainty for each. The following list, though not necessarily comprehensive, provides guidance on factors which should be considered.

1. Sampling

Homogeneity.

Effects of specific sampling strategy ( e. g. random, stratified random, proportional etc.)

Effects of movement of bulk medium ( particularly density selection)

Physical state of bulk ( solid, liquid, gas)

Temperature and pressure effects.

Does sampling process affect composition? E. g. differential adsorption in sampling system.

2. Sample preparation

Homogenisation and/ or sub sampling effects.

Drying.

Milling.

Dissolution.

Extraction.

Contamination.

Derivatisation ( chemical effects)

Dilution errors.

( Pre) Concentration.

Control of speciation effects.

3. Presentation of Certified Reference Materials to the measuring system

Uncertainty for CRM.

CRM match to sample 4. Calibration of instrument

Instrument calibration errors using a Certified Reference Material.

Reference material and its uncertainty.

Sample match to calibrant

Instrument precision 5. Analysis

Carry over in auto analysers.

Operator effects, e. g. colour blindness, parallax, other systematic errors.

Interferences from the matrix, reagents or other analytes.

Reagent purity.

Instrument parameter settings, e. g. integration parameters

Run to run precision 6. Data Processing

Averaging.

Control of rounding and truncating.

Statistics.

Processing algorithms ( model fitting, e. g. linear least squares).

7. Presentation of Results

Final result.

Estimate of uncertainty.

Confidence level.

8. Interpretation of Results

Against limits/ bounds.

Regulatory compliance.

Fitness for purpose. Quantifying Uncertainty Appendix D   Analysing Uncertainty Sources QUAM: 2000. P1 Page 100 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. 14]), 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. 15].

· 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, Quantifying Uncertainty Appendix D   Analysing Uncertainty Sources QUAM: 2000. P1 Page 101 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 mtare and gross weight including ethanol mgross. The density is calculated from d( EtOH)=( mgross  mtare)/ V For clarity, only three effects will be considered: Equipment calibration, Temperature, and the precision of each determination. Figures D1 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). 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.

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.

D. 4.5 The calibration bias on the two weighings cancels, and can be removed ( Figure D3) following the first refinement rule ( cancellation).

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

Figure D1: Initial list d( EtOH) m( gross) m( tare)

Volume Temperature Temperature Calibration Precision Calibration Calibration Lin*. Bias Lin*. Bias Precision Precision * Lin. = Linearity Figure D2: Combination of similar effects d( EtOH) m( gross) m( tare)

Volume Temperature Temperature Calibration Precision Calibration Calibration Lin. Bias Lin. Bias Precision Precision Precision Temperature Figure D3: Cancellation d( EtOH) m( gross) m( tare)

Volume Calibration Calibration Calibration Lin. Bias Lin. Bias Precision Temperature Same balance: bias cancels Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 102 Appendix E. Useful Statistical Procedures E. 1 Distribution functions The following table shows how to calculate a standard uncertainty from the parameters of the two most important distribution functions, and gives an indication of the circumstances in which each should be used.

EXAMPLE A chemist estimates a contributory factor as not less than 7 or more than 10, but feels that the value could be anywhere in between, with no idea of whether any part of the range is more likely than another. This is a description of a rectangular distribution function with a range 2a= 3 ( semi range of a= 1.5). Using the function below for a rectangular distribution, an estimate of the standard uncertainty can be calculated. Using the above range, a= 1.5, results in a standard uncertainty of ( 1.5/ Ö 3) = 0.87.

Rectangular distribution Form Use when: Uncertainty 2a ( = ± a )

1/ 2a x · A certificate or other specification gives limits without specifying a level of confidence ( e. g. 25ml ± 0.05ml)

· An estimate is made in the form of a maximum range ( ± a) with no knowledge of the shape of the distribution. 3 ) ( a x u = Triangular distribution Form Use when: Uncertainty 2a ( = ± a )

1/ a x · The available information concerning x is less limited than for a rectangular distribution. Values close to x are more likely than near the bounds.

· An estimate is made in the form of a maximum range ( ± a) described by a symmetric distribution. 6 ) ( a x u = Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 103 Normal distribution Form Use when: Uncertainty § An estimate is made from repeated observations of a randomly varying process. u( x) = s · An uncertainty is given in the form of a standard deviation s, a relative standard deviation x s / , or a coefficient of variance CV% without specifying the distribution. u( x) = s u( x)= x × ( s x / )

u( x)= x × 100 % CV 2 s x · An uncertainty is given in the form of a 95% ( or other) confidence interval x ± c without specifying the distribution. u( x) = c / 2 ( for c at 95%)

u( x) = c/ 3 ( for c at 99.7%) Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 104 E. 2 Spreadsheet method for uncertainty calculation E. 2.1 Spreadsheet software can be used to simplify the calculations shown in Section 8. The procedure takes advantage of an approximate numerical method of differentiation, and requires knowledge only of the calculation used to derive the final result ( including any necessary correction factors or influences) and of the numerical values of the parameters and their uncertainties. The description here follows that of Kragten [ H. 12].

E. 2.2 In the expression for u( y( x1, x2... xn))

å å = = ÷ ÷ ø ö ç ç è æ × ¶ ¶ × ¶ ¶ + ÷ ÷ ø ö ç ç è æ × ¶ ¶ n k i k i n i x x u k x y i x y i x u i x y , 1 , , 1 ) , ( 2 ) (

provided that either y( x1, x2... xn) is linear in xi or u( xi) is small compared to xi, the partial differentials ( ¶ y/ ¶ xi) can be approximated by:

) ( ) ( )) ( (

i i i i i x u x y x u x y x y - + » ¶ ¶ Multiplying by u( xi) to obtain the uncertainty u( y, xi) in y due to the uncertainty in xi gives u( y, xi) » y( x1, x2,..( xi+ u( xi)).. xn) y( x1, x2,.. xi.. xn)

Thus u( y, xi) is just the difference between the values of y calculated for [ xi+ u( xi)] and xi respectively.

E. 2.3 The assumption of linearity or small values of u( xi)/ xi will not be closely met in all cases. Nonetheless, the method does provide acceptable accuracy for practical purposes when considered against the necessary approximations made in estimating the values of u( xi). Reference H. 12 discusses the point more fully and suggests methods of checking the validity of the assumption.

E. 2.4 The basic spreadsheet is set up as follows, assuming that the result y is a function of the four parameters p, q, r, and s:

i) Enter the values of p, q, etc. and the formula for calculating y in column A of the spreadsheet. Copy column A across the following columns once for every variable in y ( see Figure E2.1). It is convenient to place the values of the uncertainties u( p), u( q) and so on in row 1 as shown.

ii) Add u( p) to p in cell B3, u( q) to q in cell C4 etc., as in Figure E2.2. On recalculating the spreadsheet, cell B8 then becomes f( p+ u( p), q , r..) ( denoted by f ( p', q, r, ..) in Figures E2.2 and E2.3), cell C8 becomes f( p, q+ u( q), r,..) etc.

iii) In row 9 enter row 8 minus A8 ( for example, cell B9 becomes B8 A8). This gives the values of u( y, p) as u( y, p)= f ( p+ u( p), q, r ..)  f ( p, q, r ..) etc.

iv) To obtain the standard uncertainty on y, these individual contributions are squared, added together and then the square root taken, by entering u( y, p) 2 in row 10 ( Figure E2.3) and putting the square root of their sum in A10. That is, cell A10 is set to the formula SQRT( SUM( B10+ C10+ D10+ E10))

which gives the standard uncertainty on y.

E. 2.5 The contents of the cells B10, C10 etc. show the squared contributions u( y, xi) 2=( ciu( xi)) 2 of the individual uncertainty components to the uncertainty on y and hence it is easy to see which components are significant.

E. 2.6 It is straightforward to allow updated calculations as individual parameter values change or uncertainties are refined. In step i) above, rather than copying column A directly to columns B E, copy the values p to s by reference, that is, cells B3 to E3 all reference A3, B4 to E4 reference A4 etc. The horizontal arrows in Figure E2.1 show the referencing for row 3. Note that cells B8 to E8 should still reference the values in columns B to E respectively, as shown for column B by the vertical arrows in Figure E2.1. In step ii) above, add the references to row 1 by reference ( as shown by the arrows in Figure E2.1). For example, cell B3 becomes A3+ B1, cell C4 becomes A4+ C1 etc. Changes to either parameters or uncertainties will then be reflected immediately in the overall result at A8 and the combined standard uncertainty at A10.

E. 2.7 If any of the variables are correlated, the necessary additional term is added to the SUM in A10. For example, if p and q are correlated, with a correlation coefficient r( p, q), then the extra term 2 ´ r( p, q) ´ u( y, p) ´ u( y, q) is added to the calculated sum before taking the square root. Correlation can therefore easily be included by adding suitable extra terms to the spreadsheet. Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 105 Figure E2.1 A B C D E 1 u( p) u( q) u( r) u( s) 2 3 p p p p p 4 q q q q q 5 r r r r r 6 s s s s s 7 8 y= f( p, q,..) y= f( p, q,..) y= f( p, q,..) y= f( p, q,..) y= f( p, q,..) 9 10 11 Figure E2.2 A B C D E 1 u( p) u( q) u( r) u( s) 2 3 p p+ u( p) p p p 4 q q q+ u( q) q q 5 r r r r+ u( r) r 6 s s s s s+ u( s) 7 8 y= f( p, q,..) y= f( p',...) y= f(.. q',..) y= f(.. r',..) y= f(.. s',..) 9 u( y, p) u( y, q) u( y, r) u( y, s) 10 11 Figure E2.3 A B C D E 1 u( p) u( q) u( r) u( s) 2 3 p p+ u( p) p p p 4 q q q+ u( q) q q 5 r r r r+ u( r) r 6 s s s s s+ u( s) 7 8 y= f( p, q,..) y= f( p',...) y= f(.. q',..) y= f(.. r',..) y= f(.. s',..) 9 u( y, p) u( y, q) u( y, r) u( y, s) 10 u( y) u( y, p) 2 u( y, q) 2 u( y, r) 2 u( y, s) 2 11 Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 106 E. 3 Uncertainties from linear least squares calibration E. 3.1 An analytical method or instrument is often calibrated by observing the responses, y, to different levels of the analyte, x. In most cases this relationship is taken to be linear viz:

y = b0 + b1x Eq. E3.1 This calibration line is then used to obtain the concentration xpred of the analyte from a sample which produces an observed response yobs from xpred = ( yobs   b0)/ b1 Eq. E3.2 It is usual to determine the constants b1 and b0 by weighted or un weighted least squares regression on a set of n pairs of values ( xi, yi).

E. 3.2 There are four main sources of uncertainty to consider in arriving at an uncertainty on the estimated concentration xpred:

· Random variations in measurement of y, affecting both the reference responses yi and the measured response yobs.

· Random effects resulting in errors in the assigned reference values xi.

· Values of xi and yi may be subject to a constant unknown offset, for example arising when the values of x are obtained from serial dilution of a stock solution · The assumption of linearity may not be valid Of these, the most significant for normal practice are random variations in y, and methods of estimating uncertainty for this source are detailed here. The remaining sources are also considered briefly to give an indication of methods available.

E. 3.3 The uncertainty u( xpred, y) in a predicted value xpred due to variability in y can be estimated in several ways:

From calculated variance and covariance.

If the values of b1 and b0, their variances var( b1), var( b0) and their covariance, covar( b1, b0), are determined by the method of least squares, the variance on x, var( x), obtained using the formula in Chapter 8. and differentiating the normal equations, is given by 2 1 0 1 0 1 2 ) var( ) , ( covar 2 ) var( ) var( ) var(

b b b b x b x y x pred pred obs pred + × × + × + = Eq. E3.3 and the corresponding uncertainty u( xpred, y) is Ö var( xpred).

From the calibration data.

The above formula for var( xpred) can be written in terms of the set of n data points, ( xi, yi), used to determine the calibration function:

÷ ÷ ø ö ç ç è æ å å - å × - + å × + = ) ) ( ) ( ( ) ( 1 / ) var( ) var(

2 2 2 2 1 2 2 1 i i i i i pred i obs pred w x w x w x x w b S b y x Eq. E3.4 where ) 2 ( ) ( 2 2 - - å = n i y y w S f i i , ) ( fi i y y - is the residual for the ith point, n is the number of data points in the calibration, b1 the calculated best fit gradient, wi the weight assigned to yi and ) ( x x pred - the difference between xpred and the mean x of the n values x1, x2....

For unweighted data and where var( yobs) is based on p measurements, equation E3.4 becomes ÷ ÷ ø ö ç ç è æ å - å - + + × = ) ) ( ) ( ( ) ( 1 1 ) var( 2 2 2 2 1 2 n x x x x n p b S x i i pred pred Eq. E3.5 This is the formula which is used in example 5 with Sxx = ( ) [ ] å å å - = - 2 2 2 ) ( ) ( x x n x x i i i .

From information given by software used to derive calibration curves.

Some software gives the value of S, variously described for example as RMS error or residual standard error. This can then be used in equation E3.4 or E3.5. However some software may also give the standard deviation s( yc) on a value of y calculated from the fitted line for some new value of x and this can be used to calculate var( xpred) since, for p= 1 ( ) ( ) å å - - + + = n x x x x n y s i i pred c 2 2 2 ) ( ) ( 1 1 ) ( Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 107 giving, on comparison with equation E3.5,

var( xpred) = [ s( yc) / b1] 2 Eq. E3.6 E. 3.4 The reference values xi may each have uncertainties which propagate through to the final result. In practice, uncertainties in these values are usually small compared to uncertainties in the system responses yi, and may be ignored. An approximate estimate of the uncertainty u( xpred, xi) in a predicted value xpred due to uncertainty in a particular reference value xi is u( xpred, xi) » u( xi)/ n Eq. E3.7 where n is the number of xi values used in the calibration. This expression can be used to check the significance of u( xpred, xi).

E. 3.5 The uncertainty arising from the assumption of a linear relationship between y and x is not normally large enough to require an additional estimate. Providing the residuals show that there is no significant systematic deviation from this assumed relationship, the uncertainty arising from this assumption ( in addition to that covered by the resulting increase in y variance) can be taken to be negligible. If the residuals show a systematic trend then it may be necessary to include higher terms in the calibration function. Methods of calculating var( x) in these cases are given in standard texts. It is also possible to make a judgement based on the size of the systematic trend.

E. 3.6 The values of x and y may be subject to a constant unknown offset ( e. g. arising when the values of x are obtained from serial dilution of a stock solution which has an uncertainty on its certified value). If the standard uncertainties on y and x from these effects are u( y, const) and u( x, const), then the uncertainty on the interpolated value xpred is given by:

u( xpred) 2 = u( x, const) 2 +

( u( y, const)/ b1) 2 + var( x) Eq. E3.8 E. 3.7 The four uncertainty components described in E. 3.2 can be calculated using equations Eq. E3.3 to Eq. E3.8. The overall uncertainty arising from calculation from a linear calibration can then be calculated by combining these four components in the normal way. Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 108 E. 4: Documenting uncertainty dependent on analyte level E. 4.1 Introduction E. 4.1.1 It is often observed in chemical measurement that, over a large range of analyte levels, dominant contributions to the overall uncertainty vary approximately proportionately to the level of analyte, that is u( x) µ x. In such cases it is often sensible to quote uncertainties as relative standard deviations or, for example, coefficient of variation (% CV).

E. 4.1.2 Where the uncertainty is unaffected by level, for example at low levels, or where a relatively narrow range of analyte level is involved, it is generally most sensible to quote an absolute value for the uncertainty.

E. 4.1.3 In some cases, both constant and proportional effects are important. This section sets out a general approach to recording uncertainty information where variation of uncertainty with analyte level is an issue and reporting as a simple coefficient of variation is inadequate.

E. 4.2 Basis of approach E. 4.2.1 To allow for both proportionality of uncertainty and the possibility of an essentially constant value with level, the following general expression is used:

2 1 2 0 ) ( ) ( s x s x u × + = [ 1]

where u( x) is the combined standard uncertainty in the result x ( that is, the uncertainty expressed as a standard deviation)

s0 represents a constant contribution to the overall uncertainty s1 is a proportionality constant.

The expression is based on the normal method of combining of two contributions to overall uncertainty, assuming one contribution ( s0) is constant and one ( xs1) proportional to the result. Figure E. 4.1 shows the form of this expression.

NOTE: The approach above is practical only where it is possible to calculate a large number of values. Where experimental study is employed, it will not often be possible to establish the relevant parabolic relationship. In such circumstances, an adequate approximation can be obtained by simple linear regression through four or more combined uncertainties obtained at different analyte concentrations. This procedure is consistent with that employed in studies of reproducibility and repeatability according to ISO 5725: 1994. The relevant expression is then u x s x s ( ) ' . ' » + 0 1 E. 4.2.2 The figure can be divided into approximate regions ( A to C on the figure):

A: The uncertainty is dominated by the term s0, and is approximately constant and close to s0.

B: Both terms contribute significantly; the resulting uncertainty is significantly higher than either s0 or xs1, and some curvature is visible.

C: The term xs1 dominates; the uncertainty rises approximately linearly with increasing x and is close to xs1.

E. 4.2.3 Note that in many experimental cases the complete form of the curve will not be apparent. Very often, the whole reporting range of analyte level permitted by the scope of the method falls within a single chart region; the result is a number of special cases dealt with in more detail below.

E. 4.3 Documenting level dependent uncertainty data E. 4.3.1 In general, uncertainties can be documented in the form of a value for each of s0 and s1. The values can be used to provide an uncertainty estimate across the scope of the method. This is particularly valuable when calculations for well characterised methods are implemented on computer systems, where the general form of the equation can be implemented independently of the values of the parameters ( one of which may be zero  see below). It is accordingly recommended that, except in the special cases outlined below or where the dependence is strong but not linear*, uncertainties * An important example of non linear dependence is the effect of instrument noise on absorbance measurement at high absorbances near the upper limit of the instrument capability. This is particularly pronounced where absorbance is calculated from transmittance ( as in infrared spectroscopy). Under these circumstances, baseline noise causes very large uncertainties in high absorbance figures, and the Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 109 are documented in the form of values for a constant term represented by s0 and a variable term represented by s1.

E. 4.4. Special cases E. 4.4.1. Uncertainty not dependent on level of analyte ( s0 dominant)

The uncertainty will generally be effectively independent of observed analyte concentration when:

· The result is close to zero ( for example, within the stated detection limit for the method). Region A in Figure E. 4.1 · The possible range of results ( stated in the method scope or in a statement of scope for the uncertainty estimate) is small compared to the observed level.

Under these circumstances, the value of s1 can be recorded as zero. s0 is normally the calculated standard uncertainty.

E. 4.4.2. Uncertainty entirely dependent on analyte ( s1 dominant)

Where the result is far from zero ( for example, above a ` limit of determination') and there is clear evidence that the uncertainty changes proportionally with the level of analyte permitted within the scope of the method, the term xs1 dominates ( see Region C in Figure E. 4.1). Under these circumstances, and where the method scope does not include levels of analyte near zero, s0 may reasonably be recorded as zero and s1 is simply the uncertainty expressed as a relative standard deviation.

E. 4.4.3. Intermediate dependence In intermediate cases, and in particular where the situation corresponds to region B in Figure E. 4.1, two approaches can be taken:

a) Applying variable dependence The more general approach is to determine, record and use both s0 and s1. Uncertainty uncertainty rises much faster than a simple linear estimate would predict. The usual approach is to reduce the absorbance, typically by dilution, to bring the absorbance figures well within the working range; the linear model used here will then normally be adequate. Other examples include the ` sigmoidal' response of some immunoassay methods. estimates, when required, can then be produced on the basis of the reported result. This remains the recommended approach where practical.

NOTE: See the note to section E. 4.2.

b) Applying a fixed approximation An alternative which may be used in general testing and where · the dependence is not strong ( that is, evidence for proportionality is weak) or · the range of results expected is moderate leading in either case to uncertainties which do not vary by more than about 15% from an average uncertainty estimate, it will often be reasonable to calculate and quote a fixed value of uncertainty for general use, based on the mean value of results expected. That is,

either a mean or typical value for x is used to calculate a fixed uncertainty estimate, and this is used in place of individually calculated estimates or a single standard deviation has been obtained, based on studies of materials covering the full range of analyte levels permitted ( within the scope of the uncertainty estimate), and there is little evidence to justify an assumption of proportionality. This should generally be treated as a case of zero dependence, and the relevant standard deviation recorded as s0.

E. 4.5. Determining s0 and s1 E. 4.5.1. In the special cases in which one term dominates, it will normally be sufficient to use the uncertainty as standard deviation or relative standard deviation respectively as values of s0 and s1. Where the dependence is less obvious, however, it may be necessary to determine s0 and s1 indirectly from a series of estimates of uncertainty at different analyte levels.

E. 4.5.2. Given a calculation of combined uncertainty from the various components, some of which depend on analyte level while others do not, it will normally be possible to investigate the dependence of overall uncertainty on analyte level by simulation. The procedure is as follows: Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 110 1: Calculate ( or obtain experimentally) uncertainties u( xi) for at least ten levels xi of analyte, covering the full range permitted.

2. Plot u( xi) 2 against xi 2 3. By linear regression, obtain estimates of m and c for the line u( x) 2 = mx2 + c 4. Calculate s0 and s1 from s0 = Ö c, s1 = Ö m 5. Record s0 and s1 E. 4.6. Reporting E. 4.6.1. The approach outlined here permits estimation of a standard uncertainty for any single result. In principle, where uncertainty information is to be reported, it will be in the form of [ result] ± [ uncertainty] where the uncertainty as standard deviation is calculated as above, and if necessary expanded ( usually by a factor of two) to give increased confidence. Where a number of results are reported together, however, it may be possible, and is perfectly acceptable, to give an estimate of uncertainty applicable to all results reported.

E. 4.6.2. Table E. 4.1 gives some examples. The uncertainty figures for a list of different analytes may usefully be tabulated following similar principles.

NOTE: Where a ` detection limit' or ` reporting limit' is used to give results in the form "< x" or " nd", it will normally be necessary to quote the limits used in addition to the uncertainties applicable to results above reporting limits.

Table E. 4.1: Summarising uncertainty for several samples Situation Dominant term Reporting example( s)

Uncertainty essentially constant across all results s0 or fixed approximation ( sections E. 4.4.1. or E. 4.4.3. a) Standard deviation: expanded uncertainty; 95% confidence interval Uncertainty generally proportional to level xs1 ( see section E. 4.4.2.) relative standard deviation; coefficient of variance (% CV)

Mixture of proportionality and lower limiting value for uncertainty Intermediate case ( section E. 4.4.3.) quote % CV or rsd together with lower limit as standard deviation. Quantifying Uncertainty Appendix E   Statistical Procedures QUAM: 2000. P1 Page 111 Figure E. 4.1: Variation of uncertainty with observed result C B A 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 1 2 3 4 5 6 7 8 Result x Uncertainty u( x)

s0 x. s1 u( x) Uncertainty approximately equal to s0 Uncertainty significantly greater than either s0 or x. s1 Uncertainty approximately equal to x. s1 Quantifying Uncertainty Appendix F   Detection limits QUAM: 2000. P1 Page 112 Appendix F. Measurement Uncertainty at the Limit of Detection/ Limit of Determination F. 1. Introduction F. 1.1. 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. 16, H. 17, H. 18] 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.

F. 1.2. 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.

F. 1.3. 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.

F. 1.4. 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 F. 2.1. 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.

F. 2.2. 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', Quantifying Uncertainty Appendix F   Detection limits QUAM: 2000. P1 Page 113 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.

F. 2.3. 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).

F. 2.4. 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.

F. 2.5. The ideal is accordingly to report valid observations and their associated uncertainty regardless of the values. F. 3. Interpreted results and compliance statements F. 3.1. 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.

F. 3.2. 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. 16, H. 17, H. 18) on the reporting of low level results may reasonably apply.

F. 3.3. 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. Quantifying Uncertainty Appendix G   Uncertainty Sources QUAM: 2000. P1 Page 114 Appendix G. Common Sources and Values of Uncertainty The following tables summarise some typical examples of uncertainty components. The tables give:

· The particular measurand or experimental procedure ( determining mass, volume etc)

· The main components and sources of uncertainty in each case · A suggested method of determining the uncertainty arising from each source.

· An example of a typical case The tables are intended only to summarise the examples and to indicate general methods of estimating uncertainties in analysis. They are not intended to be comprehensive, nor should the values given be used directly without independent justification. The values may, however, help in deciding whether a particular component is significant. Determination Uncertainty Cause Method of determination Typical values Components Example Value Mass Balance calibration uncertainty Limited accuracy in calibration Stated on calibration certificate,

converted to standard deviation 4 figure balance 0.5 mg Linearity i) Experiment, with range of certified weights ii) Manufacturer's specification ca. 0.5x last significant digit Readability Limited resolution on display or scale From last significant digit 0.5x last significant digit/ Ö 3 Daily drift Various, including temperature Standard deviation of long term check weighings. Calculate as RSD if necessary. ca. 0.5x last significant digit.

Run to run variation Various Standard deviation of successive sample or check weighings ca. 0.5x last significant digit.

Density effects ( conventional basis) Note 1 Calibration weight/ sample density mismatch causes a difference in the effect of atmospheric buoyancy Calculated from known or assumed densities and typical atmospheric conditions Steel, Nickel Aluminium Organic solids Water Hydrocarbons 1 ppm 20 ppm 50 100 ppm 65 ppm 90 ppm Density effects ( in vacuo basis) Note 1 As above. Calculate atmospheric buoyancy effect and subtract buoyancy effect on calibration weight. 100 g water 10 g Nickel + 0.1g ( effect)

< 1 mg ( effect)

Note 1. For fundamental constants or SI unit definitions, mass determinations by weighing are usually corrected to the weight in vacuum. In most other practical situations, weight is quoted on a conventional basis as defined by OIML [ H. 18]. The convention is to quote weights at an air density of 1.2 kg m 3 and a sample density of 8000 kg m 3, which corresponds to weighing steel at sea level in normal atmospheric conditions. The buoyancy correction to conventional mass is zero when the sample density is 8000 kg m 3 or the air density is 1.2 kg m 3. Since the air density is usually very close to the latter value, correction to conventional weight can normally be neglected. The standard uncertainty values given for density related effects on a conventional weight basis in the table above are sufficient for preliminary estimates for weighing on a conventional basis without buoyancy correction at sea level. Mass determined on the conventional basis may, however,

differ from the ` true mass' ( in vacuo) by 0.1% or more ( see the effects in the bottom line of the table above).

QUAM: 2000. P1 Page 115 Quantifying Uncertainty Appendix G   Uncertainty Sources * Assuming rectangular distribution Determination Uncertainty Cause Method of determination Typical values Components Example Value Volume ( liquid) Calibration uncertainty Limited accuracy in calibration Stated on manufacturer's specification, converted to standard deviation.

For ASTM class A glassware of volume V, the limit is approximately V0.6/ 200 10 ml ( Grade A) 0.02 / Ö 3 =

0.01 ml*

Temperature Temperature variation from the calibration temperature causes a difference in the volume at the standard temperature. D T × a / ( 2 Ö 3) gives the relative standard deviation, where D T is the possible temperature range and a the coefficient of volume expansion of the liquid. a is approximately 2 x10 4 K 1 for water and 1 x 10 3 K 1 for organic liquids. 100 ml water 0.03 ml for operating within 3 ° C of the stated operating temperature Run to run variation Various Standard deviation of successive check deliveries ( found by weighing) 25 ml pipette Replicate fill/ weigh:

s = 0.0092 ml QUAM: 2000. P1 Page 116 Quantifying Uncertainty Appendix G   Uncertainty Sources Determination Uncertainty Cause Method of determination Typical values Components Example Value Reference material concentration Purity Impurities reduce the amount of reference material present. Reactive impurities may interfere with the measurement. Stated on manufacturer's certificate. Reference certificates usually give unqualified limits;

these should accordingly be treated as rectangular distributions and divided by Ö 3.

Note: where the nature of the impurities is not stated,

additional allowance or checks may need to be made to establish limits for interference etc. Reference potassium hydrogen phthalate certified as 99.9 ± 0.1% 0.1/ Ö 3 =

0.06%

Concentration ( certified) Certified uncertainty in reference material concentration. Stated on manufacturer's certificate. Reference certificates usually give unqualified limits;

these should accordingly be treated as rectangular distributions and divided by Ö 3. Cadmium acetate in 4%

acetic acid.

Certified as ( 1000 ± 2) mg l 1. 2/ Ö 3 = 1.2 mg l 1 ( 0.0012 as RSD)*

Concentration ( made up from certified material) Combination of uncertainties in reference values and intermediate steps Combine values for prior steps as RSD throughout. Cadmium acetate after three dilutions from 1000 mg l 1 to 0.5 mg l 1 0034 .

0 0017 .

0 0021 .

0 0017 .

0 0012 .

0 2 2 2 2 = + + + as RSD * Assuming rectangular distribution QUAM: 2000. P1 Page 117 Quantifying Uncertainty Appendix G   Uncertainty Sources Determination Uncertainty Cause Method of determination Typical values Components Example Value Absorbance Instrument calibration Note: this component relates to absorbance reading versus reference absorbance,

not to the calibration of concentration against absorbance reading Limited accuracy in calibration. Stated on calibration certificate as limits, converted to standard deviation Run to run variation Various Standard deviation of replicate determinations, or QA performance. Mean of 7 absorbance readings with s= 1.63 1.63/ Ö 7 = 0.62 Sampling Homogeneity Sub sampling from inhomogeneous material will not generally represent the bulk exactly.

Note: random sampling will generally result in zero bias. It may be necessary to check that sampling is actually random. i) Standard deviation of separate sub sample results ( if the inhomogeneity is large relative to analytical accuracy).

ii) Standard deviation estimated from known or assumed population parameters. Sampling from bread of assumed twovalued inhomogeneity ( See Example A4) For 15 portions from 72 contaminated and 360 uncontaminated bulk portions:

RSD = 0.58 QUAM: 2000. P1 Page 118 Quantifying Uncertainty Appendix G   Uncertainty Sources Determination Uncertainty Cause Method of determination Typical values Components Example Value Extraction recovery Mean recovery Extraction is rarely complete and may add or include interferents. Recovery calculated as percentage recovery from comparable reference material or representative spiking.

Uncertainty obtained from standard deviation of mean of recovery experiments.

Note: recovery may also be calculated directly from previously measured partition coefficients. Recovery of pesticide from bread; 42 experiments,

mean 90%,

s= 28%

( See Example A4) 28/ Ö 42= 4.3%

( 0.048 as RSD)

Run to run variation in recovery Various Standard deviation of replicate experiments. Recovery of pesticides from bread from paired replicate data. ( See Example A4) 0.31 as RSD.

QUAM: 2000. P1 Page 119 Quantifying Uncertainty Appendix G   Uncertainty Sources Quantifying Uncertainty Appendix H  Bibliography QUAM: 2000. P1 Page 120 Appendix H. Bibliography H. 1. ISO/ IEC 17025: 1999. General Requirements for the Competence of Calibration and Testing Laboratories. ISO, Geneva ( 1999).

H. 2. Guide To The Expression Of Uncertainty In Measurement. ISO, Geneva ( 1993). ( ISBN 92 67 10188 9)

H. 3. EURACHEM, Quantifying Uncertainty in Analytical Measurement. Laboratory of the Government Chemist, London ( 1995). ISBN 0 948926 08 2 H. 4. International Vocabulary of basic and general terms in Metrology. ISO, Geneva, ( 1993). ( ISBN 92 67 10175 1)

H. 5. ISO 3534: 1993. Statistics  Vocabulary and Symbols. ISO, Geneva, Switzerland ( 1993).

H. 6. Analytical Methods Committee, Analyst ( London). 120 29 34 ( 1995).

H. 7. EURACHEM, The Fitness for Purpose of Analytical Methods. ( 1998) ( ISBN 0 948926 12 0)

H. 8. ISO/ IEC Guide 33: 1989. Uses of Certified Reference Materials. ISO, Geneva ( 1989).

H. 9. International Union of Pure and Applied Chemistry. Pure Appl. Chem., 67, 331 343, ( 1995).

H. 10 ISO 5725: 1994 ( Parts 1 4 and 6). Accuracy ( trueness and precision) of measurement methods and results. ISO, Geneva ( 1994). See also ISO 5725 5: 1998 for alternative methods of estimating precision.

H. 11. I. J. Good, " Degree of Belief", in Encyclopaedia of Statistical Sciences, Vol. 2, Wiley, New York ( 1982).

H. 12. J. Kragten, Analyst, 119, 2161 2166 ( 1994).

H. 13. British Standard BS 6748: 1986. Limits of metal release from ceramic ware, glassware, glass ceramic ware and vitreous enamel ware.

H. 14. S. L. R. Ellison, V. J. Barwick. Accred. Qual. Assur. 3 101 105 ( 1998).

H. 15. ISO 9004 4: 1993, Total Quality Management. Part 2. Guidelines for quality improvement. ISO, Geneva ( 1993).

H. 16. H. Kaiser, Anal. Chem. 42 24A ( 1970).

H. 17. L. A. Currie, Anal. Chem. 40 583 ( 1968).

H. 18. IUPAC, Limit of Detection, Spectrochim. Acta 33B 242 ( 1978).

H. 19. OIML Recommendations IR 33. Conventional value of the result of weighing in air.


[Federal Register Volume 88, Number 85 (Wednesday, May 3, 2023)]
[Notices]
[Pages 27947-27949]
[FR Doc No: 2023-09327]

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DEPARTMENT OF TRANSPORTATION

Federal Motor Carrier Safety Administration

[FMCSA Docket No. FMCSA-2023-0030]

Qualification of Drivers; Exemption Applications; Epilepsy and
Seizure Disorders

AGENCY: Federal Motor Carrier Safety Administration (FMCSA), Department
of Transportation (DOT).

ACTION: Notice of final disposition.

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SUMMARY: FMCSA announces its decision to exempt 11 individuals from the
requirement in the Federal Motor Carrier Safety Regulations (FMCSRs)
that interstate commercial motor vehicle (CMV) drivers have ``no
established medical history or clinical diagnosis of epilepsy or any
other condition which is likely to cause loss of consciousness or any
loss of ability to control a CMV.'' The exemptions enable these
individuals who have had one or more seizures and are taking anti-
seizure medication to operate CMVs in interstate commerce.

DATES: The exemptions were applicable on April 26, 2023. The exemptions
expire on April 26, 2025.

FOR FURTHER INFORMATION CONTACT: Ms. Christine A. Hydock, Chief,
Medical Programs Division, FMCSA, DOT, 1200 New Jersey Avenue SE, Room
W64-224, Washington, DC 20590-0001,<PHONE_NUMBER>,
[email protected]. Office hours are from 8:30 a.m. to 5 p.m. ET
Monday through Friday, except Federal holidays. If you have questions
regarding viewing or submitting material to the docket, contact Dockets
Operations,<PHONE_NUMBER>.

SUPPLEMENTARY INFORMATION:

I. Public Participation

A. Viewing Comments

    To view comments go to www.regulations.gov. Insert the docket
number, (FMCSA-2023-0030) in the keyword box and click ``Search.''
Next, sort the results by ``Posted (Newer-Older),'' choose the first
notice listed, and click ``Browse Comments.'' If you do not have access
to the internet, you may view the docket online by visiting Dockets
Operations in Room W12-140 on the ground floor of the DOT West
Building, 1200 New Jersey Avenue SE, Washington, DC 20590-0001, between
9 a.m. and 5 p.m. ET Monday through Friday, except Federal holidays. To
be sure someone is there to help you, please call<PHONE_NUMBER> or
(202) 366-9826 before visiting Dockets Operations.

B. Privacy Act

    In accordance with 49 U.S.C. 31315(b)(6), DOT solicits comments
from the public on the exemption requests. DOT posts these comments,
without edit, including any personal information the commenter
provides, to www.regulations.gov. As described in the system of records
notice DOT/ALL 14 (Federal Docket Management System), which can be
reviewed at https://www.transportation.gov/individuals/privacy/privacy-act-system-records-notices, the comments are searchable by the name of
the submitter.

II. Background

    On March 22, 2023, FMCSA published a notice announcing receipt of
applications from 11 individuals requesting an exemption from the
epilepsy and seizure disorders prohibition in 49 CFR 391.41(b)(8) and
requested comments from the public (88 FR 17287). The public comment
period ended on April 21, 2023, and no comments were received.
    FMCSA has evaluated the eligibility of these applicants and
determined that granting exemptions to these individuals would likely
achieve a level of safety that is equivalent to, or greater than, the
level that would be achieved by complying with Sec.  391.41(b)(8).
    The physical qualification standard for drivers regarding epilepsy
found in Sec.  391.41(b)(8) states that a person is physically
qualified to drive a CMV if that person has no established medical
history or clinical diagnosis of epilepsy or any other condition which
is likely to cause the loss of consciousness or any loss of ability to
control a CMV.
    In addition to the regulations, FMCSA has published advisory
criteria \1\ to assist medical examiners (MEs) in determining whether
drivers with certain medical conditions are qualified to operate a CMV
in interstate commerce.
---------------------------------------------------------------------------

    \1\ These criteria may be found in APPENDIX A TO PART 391--
MEDICAL ADVISORY CRITERIA, section H. Epilepsy: Sec.  391.41(b)(8),
paragraphs 3, 4, and 5, which is available on the internet at
https://www.gpo.gov/fdsys/pkg/CFR-2015-title49-vol5/pdf/CFR-2015-title49-vol5-part391-appA.pdf.
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III. Discussion of Comments

    FMCSA received no comments in this proceeding.

IV. Basis for Exemption Determination

    Under 49 U.S.C. 31136(e) and 31315(b), FMCSA may grant an exemption
from the FMCSRs for no longer than a 5-year period if it finds such
exemption would likely achieve a level of safety that is equivalent to,
or greater than, the level that would be achieved absent such
exemption. The statutes allow the Agency to renew exemptions at the end
of the 5-year period. However, FMCSA grants medical exemptions from the
FMCSRs for a 2-year period to align with the maximum duration of a
driver's medical certification.
    The Agency's decision regarding these exemption applications is
based on the 2007 recommendations of the Agency's Medical Expert Panel.
The Agency conducted an individualized assessment of each applicant's
medical information, including the root cause of the respective
seizure(s) and medical information about the applicant's seizure
history, the length of time that has elapsed since the individual's
last seizure, the stability of each individual's treatment regimen and
the duration of time on or off of anti-seizure medication. In addition,
the Agency reviewed the treating clinician's medical opinion related to
the ability of the driver to safely operate a CMV with a history of
seizure and each applicant's driving record found in the commercial
driver's license Information System for commercial driver's license
(CDL) holders, and interstate and intrastate inspections recorded in
the Motor Carrier Management Information System. For non-CDL holders,
the Agency reviewed the driving records from the State Driver's
Licensing Agency. A summary of each applicant's seizure history was
discussed in the March 22, 2023, Federal Register notice (88 FR 17287)
and will not be repeated in this notice.
    These 11 applicants have been seizure-free over a range of 31 years
while taking anti-seizure medication and maintained a stable medication
treatment regimen for the last 2 years. In each case, the applicant's
treating physician verified his or her seizure history and supports the
ability to drive commercially.
    The Agency acknowledges the potential consequences of a driver
experiencing a seizure while operating a CMV. However, the Agency
believes the drivers granted this exemption have demonstrated that they
are unlikely to have a seizure and their medical condition does not
pose a risk to public safety.
    Consequently, FMCSA finds further that in each case exempting these
applicants from the epilepsy and seizure disorder prohibition in Sec.
391.41(b)(8) would likely achieve a level of safety equal to that
existing without the exemption, consistent with the applicable standard
in 49 U.S.C. 31315(b)(1).

V. Conditions and Requirements

    The terms and conditions of the exemption are provided to the
applicants in the exemption document and include the following: (1)
each driver must remain seizure-free and maintain a stable treatment
during the 2-year exemption period; (2) each driver must submit annual
reports from their treating physicians attesting to the stability of
treatment and that the driver has remained seizure-free; (3) each
driver must undergo an annual medical examination by a certified ME, as
defined by Sec.  390.5T; and (4) each driver must provide a copy of the
annual medical certification to the employer for retention in the
driver's qualification file, or keep a copy of his/her driver's
qualification file if he/she is self-employed. The driver must also
have a copy of the exemption when driving, for presentation to a duly
authorized Federal, State, or local enforcement official.

VI. Preemption

    During the period the exemption is in effect, no State shall
enforce any law or regulation that conflicts with this exemption with
respect to a person operating under the exemption.

VII. Conclusion

    Based upon its evaluation of the 11 exemption applications, FMCSA
exempts the following drivers from the epilepsy and seizure disorder
prohibition in Sec.  391.41(b)(8), subject to the requirements cited
above:

Keith Dohrmann (MN)
Wallace Ferguson (CO)
Derek Jazdzewski (WI)

Charles E. Johnson (KS)
Michael Littleton (CO)
Robert Newhand (NY)
Kristopher Pettitt (CA)
Taylor Ramey (TX)
Herbert Spike (CT)
Scott Stone (WY)
Andrew Toler (VA)

    In accordance with 49 U.S.C. 31315(b), each exemption will be valid
for 2 years from the effective date unless revoked earlier by FMCSA.
The exemption will be revoked if the following occurs: (1) the person
fails to comply with the terms and conditions of the exemption; (2) the
exemption has resulted in a lower level of safety than was maintained
prior to being granted; or (3) continuation of the exemption would not
be consistent with the goals and objectives of 49 U.S.C. 31136, 49
U.S.C. chapter 313, or the FMCSRs.

Larry W. Minor,
Associate Administrator for Policy.
[FR Doc. 2023-09327 Filed 5-2-23; 8:45 am]
BILLING CODE 4910-EX-P


[Federal Register: February 14, 2007 (Volume 72, Number 30)]
[Notices]
[Page 7023]
[DOCID:fr14fe07-42]

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DEPARTMENT OF ENERGY

Federal Energy Regulatory Commission

[Docket No. RP98-18-028]

Iroquois Gas Transmission System, L.P.; Notice of Compliance
Filing

February 7, 2007.
    Take notice that on December 15, 2006, Iroquois Gas Transmission
System, L.P. (Iroquois) tendered for filing illustrative examples of
how a rate is calculated along with explanations and assumptions used
to calculate rates under certain negotiated rate formulas.
    Any person desiring to protest this filing must file in accordance
with Rule 211 of the Commission's Rules of Practice and Procedure (18
CFR 385.211). Protests to this filing will be considered by the
Commission in determining the appropriate action to be taken, but will
not serve to make protestants parties to the proceeding. Such protests
must be filed on or before the date as indicated below. Anyone filing a
protest must serve a copy of that document on all the parties to the
proceeding.
    The Commission encourages electronic submission of protests in lieu
of paper using the ``eFiling'' link at http://www.ferc.gov. Persons

unable to file electronically should submit an original and 14 copies
of the protest to the Federal Energy Regulatory Commission, 888 First
Street, NE., Washington, DC 20426.
    This filing is accessible on-line at http://www.ferc.gov, using the

``eLibrary'' link and is available for review in the Commission's
Public Reference Room in Washington, DC. There is an ``eSubscription''
link on the Web site that enables subscribers to receive e-mail
notification when a document is added to a subscribed docket(s). For
assistance with any FERC Online service, please e-mail
<EMAIL_ADDRESS>or call<PHONE_NUMBER> (toll free). For

TTY, call<PHONE_NUMBER>.
    Comment Date: 5 p.m. Eastern Time on February 14, 2007.

Magalie R. Salas,
Secretary.
[FR Doc. E7-2481 Filed 2-13-07; 8:45 am]

BILLING CODE 6717-01-P


[Federal Register Volume 87, Number 69 (Monday, April 11, 2022)]
[Notices]
[Page 21259]
[FR Doc No: 2022-07620]

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DEPARTMENT OF TRANSPORTATION

Federal Aviation Administration

[Docket No. FAA-2022-0345]

Airport Improvement Program (AIP) Grant Assurances; Errata Notice
Extending Comment Date

AGENCY: Federal Aviation Administration (FAA), Department of
Transportation.

    On April 4, 2022, the FAA issued a notice in the above-captioned
matter. The notice indicated that the FAA would accept comments
concerning the proposed modified grant assurances until ``April 12,
2022.'' This notice extends the comment due date to April 20, 2022.

    Issued in Washington, DC, on April 5, 2022.
Robert A. Hawks,
Deputy Director, Office of Airport Planning and Programming.
[FR Doc. 2022-07620 Filed 4-8-22; 8:45 am]
BILLING CODE 4910-13-P