Analytical Bias: the Neglected Component ofMeasurement Uncertainty†

Jean S. KaneRobert J. Kane Associates Inc., Brightwood, VA 22715, USA

All analysts are accustomed to reporting measurementresults accompanied by either the standard deviation ofindividual results from the mean or the standard error ofthe mean. These statements indicate repeatability ofmeasurement under unchanged conditions, orreproducibility, where the time period over whichmeasurements are taken is the source of changedconditions. Repeatability generally produces a smallerdeviation between replicates than does reproducibility.However, most of the variability of measurements madeby different laboratories, or using different methods in asingle laboratory, is not accounted for by eitherrepeatability or reproducibility attributable to timeperiod. This is evident in all interlaboratory data sets;these frequently contain between-laboratory andbetween-method discrepancies that are very large incomparison with the uncertainty of measurementstypically reported by individual laboratories. Recentlyissued ISO guidelines and related documents addressthese discrepancies as a legitimate component ofmeasurement uncertainty, and recommend expanding theconcept to include the deviation of a measurement fromthe true value of the measurand, so long as this deviationis small relative to fitness for purpose requirements.When the bias of measurement renders the resultunsuitable for purpose, however, that bias is a significanteffect and must be removed by use of a correction factor.To correct for significant bias, or to include smaller biasas an uncertainty component, laboratories must evaluateand quantify the bias in their measurements to the fullestextent possible. This paper presents uncertaintystatements developed in accordance with the ISOguidelines for several reference sample measurements.Some are very complex, drawn from published work ofmetrology laboratories. Others, drawn from the author’sdata for the Japanese Sedimentary rock referencematerials, are more suitable for routine laboratory use.

Keywords: Analytical bias; measurement uncertainty;reference materials

The Guide to the Expression of Uncertainty in Measurement1established general rules for evaluating and stating measure-ment uncertainty across a broad spectrum of measurements.This guide has particular relevance to laboratories whose datawill be used to develop reference values for geologicalreference materials. These laboratories will need to establishuncertainties for all contributed data in keeping with theprinciples of this important new ISO guide.

A detailed study of measurement uncertainty requiresextensive effort, and is typically undertaken only in nationalmetrology laboratories. The ISO guide1 provides guidance onthe process, and the Eurachem Working Group on Uncertaintyin Chemical Measurement2 has published a number of examples

to illustrate what is required. Two examples from the literaturedescribing NIST certifications3,4 provide similar illustration.Most routine laboratories will not be able to develop uncertaintystatements for their data in this manner; they will instead bedrawn largely on data from their routine quality control andquality assurance programs and the more in-depth knowledge ofmeasurement uncertainty developed by metrology laborato-ries.

Uncertainty in the broadest terms means the doubt about thevalidity of any measurement results. While doubt about validityarises from error, error and uncertainty are not the same. Thedoubt may exist about the exactness of the results, i.e., as aresult of repeatability or reproducibility attributable to timeperiod factors. Alternatively, it may exist about the validity of aresult attributable to inappropriate sample collection or han-dling, to heterogeneity leading to subsampling error or to thepresence of analytical bias.

In order to estimate uncertainty reliably for a measurement, itis essential that the measurement process be well defined andthat the process be under statistical control when the measure-ment is made. In defining the measurement process, thelaboratory will determine the degree of accuracy necessary for‘fitness of purpose’ and will identify those factors in the processwhich have greatest influence on the measurement result.

Influences may be either random, arising from unpredictablevariations, or systematic, arising from an influence quantitywhose effect is either constant or varies in a predictable way. Ifan influence quantity produces a systematic effect that issignificant from a fitness-for-purpose perspective, correction isessential, and the uncertainty associated with the correctionmust be included in the overall uncertainty of measure-ment.1,2,5

If, instead, the systematic effect is small relative to acceptableuncertainty for the purpose, the bias itself is the source ofuncertainty, and its magnitude may be included as an un-certainty component, as illustrated in example 4 of Appendix Ain the Eurachem document.2 A NIST document6 illustratingtests for bias in comparison with reference values is instructivein this regard. According to this reference,6 not all deviations ofa measurement result represent bias: only those which aresignificant in comparison with the uncertainty of the referencevalue. However, NIST includes any deviations of referencematerial (RM) results from reference values in its uncertaintyestimates,3 when an existing NIST Standard Reference Material(SRM) is analyzed concurrently with a new SRM undergoingcertification analysis. This uncertainty is termed ‘recovery ofstandard’ ( = measured/certified) in ref. 3 and in the text andTables 2 and 5 in this paper.

Many potential soruces of uncertainty are listed in theEurachem document.2 The list is intended to be illustrative,rather than definitive. These have been broadly categorized assampling, random variation and analytical bias, and are shownin Table 1. It should be noted that spurious error, also includedin Table 1, is an extreme case of random error; uncertaintyestimates do not allow for the possibility of spurious error.Sampling as a source of uncertainty is the subject of the otherpapers in this issue.7–9 The focus of all that follows will be onanalytical bias alone among these potential soruces of un-

† Presented at Geoanalysis 97: 3rd International Conference on the Analysis ofGeological and Environmental Materials, Vail, CO, USA, June 1–5, 1997.

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certainty. The goal is to illustrate the use of control data and thelike in estimating overall uncertainty. This process is includedas appropriate in the ISO guide and related documents,1,2,5 butillustrative examples are very limited in comparison with thoseshowing the more exhaustive metrological approaches.

Analytical Bias

All analysts are very familiar with instances of disagreementbetween results that are nominally for the same measurand.These discrepancies have been the topic of much discussionamong geoanalysts since Fairbairn et al.’s study10 first made itclear how common they are. Data ranges for all major rock-forming constituents other than SiO2 were completely un-

acceptable, and outlier rejection before deriving recommendedvalues was greater than 50% in some cases. This is seen in Figs.1 and 2, presenting the G-1 and W-1 data from Fairbairn et al.’sreport.10 Youden11 attributed most discrepancies of this type tosystematic error or analytical bias inherent in methods beingused.

It is these systematic errors leading to analytical bias, and thedoubt about the validity of the result due to them, that will beaddressed in all that follows. Some of these errors and resultinguncertainties can be evaluated from the statistical distribution ofa series of measurement results, and can therefore be charac-terized by a standard deviation, in just the same way thatrandom variations are. Other components are evaluated fromassumed probability distributions based on experience or otherinformation. These two categories for evaluating uncertaintiesare referred to as Type A and Type B in the ISO guide.1

Metrology Laboratory Approach

The isotope dilution mass spectrometric method is a definitivemethod, commonly used by metrology laboratories for certifica-tion analyses. When properly applied and fully under control,results will be obtained that are essentially free from analyticalbias. The uncertainty in result for the homogeneous sample willbe determined entirely by repeatability of measurement, withRSDs of about 0.2%. This small uncertainty of measurement is

Table 1 Potential sources of uncertainty in analytical measurements

Sampling uncertainty—Sample was inappropriate to the measured purposeSample was suitable, but has degraded in handling and/or storage

between collection and measurement (see ref. 7)Field sampling uncertainty was too large to be ‘fit for purpose’ (see ref.

8)Measurand is inhomogeneously distributed in the sample

Note: Sampling error due to heterogeneity of sample is more properlydescribed as sub-sampling error, leading to between-unit sub-samplingvariance. The uncertainty due to sub-sampling error may be quantifiedusing analysis of variance where the unit of issue is the classificationvariable (Type A evaluation) or by modeling mineralogy and using thetheoretical Poisson distribution for the count of mineralized grains in thesubsample (Type B evaluation) (see ref. 9)

Random error—Measurement repeatability for sample measured repetitivelyMeasurement repeatability for replicate subsamples of a single sample

Note: Random error can never be assigned a definitive value. Thestandard deviation of replicate measurements from the arithmetic mean isa measure of uncertainty due to some random effects, but is not therandom error2

Note: The standard deviation of the measured values from the arithmeticmean can be appropriately estimated from 15 or fewer measurements,unless very high precision is required, in which case a larger number ofreplicates is needed. This estimation of uncertainty is Type A

Spurious error—Transcription error in recording analytical mass and/or signalMisreading of analog signal outputUndetected instrument malfunction during measurementSample loss due to poor technique, spillage, etc., that did not result in

discarding sample

Note: Results affected by spurious error are invalid. They should bediscarded and there should be no attempt to incorporate the spurious errorinto statistical analysis of error sources1,2,5

Analytical bias—Incomplete knowledge of factors having critical impact on measurement

accuracyNo correction or incomplete correction for matrix effects (physical

interferences)No correction or incomplete correction for spectral interferencesIncomplete separation and/or preconcentration of measurand from

matrixCalibration error due to uncertainty in reference values used for

calibrationApproximations and assumptions incorporated in measurement proce-

dureUncertainty in masses, volumetric apparatus, etc.Uncertainties in values of parameters used in data reduction (e.g., atomic

mass and isotopic abundances of the elements for mass spectrome-try)

Instrument resolution or discrimination thresholdInadequate control of environmental conditions needed for measurement

accuracy

Fig. 1 Interlaboratory ranges relative to the recommended values formajor oxides in G-1. Data from Fairbairn et al.10 Open bar, full range; solidbar, 1s range for recommended value.

Fig. 2 Interlaboratory ranges relative to the recommended values formajor oxides in W-1. Data from Fairbairn et al.10 Open bar, full range; solidbar, 1s range for recommended value.

1284 Analyst, November 1997, Vol. 122

obtained at great effort and cost, however, and should not be theanalytical goal except when necessary for making validdecisions based on the measurement results.

Similarly, gravimetric analysis when carried out under fullcontrol can produce results that have random uncertaintiessimilar to the overall uncertainty of the isotope dilutionmeasurement. Freedom from systematic error in the gravimetricanalysis, however, is less assured than with isotope dilutionmeasurement. Two examples taken from NIST certifications areillustrative.3,4 The NaCl SRM 919a is the starting material forthe Na spectrometric solution SRM 3152 and the Cl anionchromatography solution SRM 3182. The solution SRM 3181 isthe sulfate anion chromatography solution standard.

For the certification of both, NIST employed gravimetry.Error evaluation examined incomplete precipitation, con-tamination of the precipitate and losses in sample handling atevery step in the process, among other things.

The method used for SRM 919a certification3 is a modifica-tion and improvement of the definitive method first developedas a National Committee for Clinical Laboratory Standardsmethod. The sample was first decomposed and then separatedon an AG 50W-X8 ion-exchange column. The sodium fractionwas collected and converted into the sulfate using H2SO4;evaporation produced the salt, which was then ignited andweighed to complete the gravimetric determination. The ion-exchange separation of sodium from any Li, K, Mg and Ca alsoin the sample avoided an error due to the co-existence of othersulfates in the ignited residue.

To summarize briefly the error sources examined, Moodyand Vetter3 considered the sample handling and transfer steps

(sample vial to digestion beaker, beaker to column, beaker afterevaporation to ignition crucible) to be very important. They alsoconsidered ignition losses due to evaporation and/or splattering.Ability to achieve the certified result for concurrent analysis ofpreviously certified materials (‘recovery of standard’) wasevaluated. Additionally, incompleteness of the chromato-graphic separation or failure to collect 100% of the sodiumfraction introduced uncertainty. Finally, the chemical blanksand any weighing or volumetric errors were evaluated.

Only measurement replication (random error) and ‘recoveryof standard’ were evaluated by statistical means (Type A). Allothers were evaluated in other ways (Type B). The individualuncertainties were first combined within each type followingthe law of propagation of errors, and then the uncertaintiesevaluated by Type A and Type B means were combined andexpanded to provide the final uncertainty for the certification. Acoverage factor of 2 for the expansion gives a level ofconfidence of approximately 95%, and a factor of 3 givesapproximately 99% confidence. The recommended coveragefactor1 for general use is 2. The detailed evaluation takendirectly from Moody and Vetter3 is given in Table 2.

In the case of the gravimetric analysis of the sulfate solutionSRM 3181,4 classical gravimetry was coupled with in-strumental techniques; the latter allowed corrections to be madefor any Ba that remained unprecipitated, for any BaCl2 orK2SO4 occluded in the BaSO4 precipitate and for any sulfatevolatilized during ignition. Uncertainties in the contribution ofthe blank and from measurement replication were determinedusing a Type A evaluation. Mechanical losses and all in-strumentally determined corrections were estimated using Type

Table 2 Uncertainty of measurement of sodium for the certification of SRM 919a3

Source of uncertainty Relative standard uncertainty (%)

Type A evaluation used—Measurement replication 0.007‘Recovery of standard’ 0.019Combined Type A ABBBBBBBB(0.007)2 + (0.019)2 = 0.020

Type B evaluation used—Sampling handling 0.036Sample volume 0.002Separation—loading/retention 0.005Separation—loss/contamination 0.001Weighing and ignition 0.015Combined Type B ABBBBBBBBBBBBBBBBBBBBBB(0.036)2 + (0.002)2 + (0.005)2 + (0.001)2 + (0.015)2 = 0.040Combined uncertainty ABBBBBBBB(0.020)2 + (0.040)2 = 0.0443Coverage factor 2Expanded uncertainty (relative 0.0089

Table 3 Measurement uncertainty for the certification of sulfate SRM 31814

Source of uncertainty Relative standard uncertainty (%)

Type A evaluation used—Measurement replication 0.018Blank 0.009Combined Type A ABBBBBBBB(0.018)2 + (0.009)2 = 0.020

Type B evaluation used—Mechanical loss 0.006BaCl2 correction 0.060K2SO4 correction 0.003Filtrate sulfate correction 0.033Volatile sulfate correction 0.035

Combined Type B ABBBBBBBBBBBBBBBBBBBBB(0.006)2 + (0.060)2 + (0.003)2 + (0.033)2 + (0.035)2 = 0.077Combined uncertainty ABBBBBBBB(0.020)2 + (0.077)2 = 0.080Coverage factor 2Expanded uncertainty (relative) 0.160

Analyst, November 1997, Vol. 122 1285

B methods. The results taken directly from Vetter et al.4 aregiven in Table 3.

It can be seen from these examples that carefully appliedgravimetric procedures can produce total uncertainties, includ-ing analytical bias, as small as those for the isotope dilutionmass spectrometric method. The potential for biased results ingravimetric methods is also seen clearly in these examples. Notethat separation step errors in the case of the Na certification andof precipitation errors in the case of Ba account for aconsiderable portion of the total uncertainty reported. Laborato-ries applying either of these methods under less strignentcontrol will produce results with greater associated un-certainties.

The Routine Laboratory Approach

The average laboratory cannot devote the time and effort tomethod development and to evaluating sources of uncertaintythat a national metrology laboratory such as NIST does. Yet theaverage laboratory needs to be able to state uncertainties for itsmeasurements in a way that allows users of the data to judgetheir validity in addition to their repeatability. This is the caseregardless of the use to which the data are put; it is particularlythe case when the data will contribute to a reference samplecharacterization. The magnitude of the uncertainty is dictatedby requirements for the end use of the data, by ‘fitness forpurpose’ considerations. For certification purposes, the un-certainty should be 3–10 times smaller than that of routineanalyses.12

It has been stated previously1,2,5 that many aspects of alaboratory’s quality control/quality assurance program willprovide data that can be used in the estimation of uncertainty.Information from calibration certificates for balances andlaboratory glassware provides Type B estimations of un-certainties due to weighing and volume errors. Information onreference sample certificates of analysis provides ‘recovery ofstandard’ uncertainty estimates and/or calibration error esti-mates, depending on how the laboratory used the referencesample. Similarly, information from control charts, collabor-ative studies and proficiency tests can be used to estimatelaboratory uncertainty. Evaluations could be either Type A orType B, depending on whether sufficient data existed toproduce a statistical standard deviation or whether judgmentinstead was used. These approaches to uncertainty evaluationwill often provide a single combined uncertainty of severalrelated error sources that the metrology laboratory mightevaluate individually.

For example, the laboratory measuring Ba or S gravimet-rically as the BaSO4 precipitate might analyze a number ofreference samples, and determine their precipitation losses andcontaminants (analytical bias) collectively with a Type Aevaluation from the average ‘recovery of standard’ for theseveral RMs, rather than following the NIST process. This is avalid approach, and will result in an uncertainty statement thatsuits the accuracy requirements of the laboratory measure-ments.

The approach will be illustrated with an evaluation ofuncertainty for major oxide data that the author contributed forsix of the Japanese Geological Survey sedimentary RMs, threesediments, two slates and one chert, having analyzed thesamples in replicate (three splits, digested in duplicate with eachdigestion solution analyzed on three different days) by ICP-AES.13 The decomposition was by lithium metaborate–tetra-borate fusion, and a two-point calibration using similarlyprepared USGS rock RMs was employed. Two additionalUSGS rock RMs were included in each experimental run forcontrol purposes.

The control data14 given in Table 4 provide reproducibilitydata based on replicate measurements (n = 26 for AGV-1, n =

22 for BHVO-1, n = 24 for BIR-1, n = 24 for G-2) that weremade at widely spaced times over several years between 1984and 1990. The changed conditions resulting from the timeintervals included such things as replacement torches andmirrors in the spectrometer, different decomposition solutionsfor both the control sample RMs and the calibration RMs andvariations in instrument set-up, including manual wavelengthpeaking and torch height adjustments. However, the sameanalytical procedure and instrument were used in all measure-ments and the same analyst performed the measurements, overthe entire time period involved. The detailed within-runrepeatability data are no longer available, but within-run RSDswere typically only a fraction (less than one fifth) of thereproducibility data shown in Table 4.

Table 4 also provides bias estimations or ‘recovery ofstandard’ data, both as the difference between the mean of thereplicate control sample measurements and the reference valueand as the maximum deviation encountered during the timeperiod involved. There is no case where the bias estimated fromthe mean deviation is statistically significant6 in comparisonwith either the certified value uncertainties (see below) or themeasurement reproducibility. Additionally, it is acceptablysmall from a fitness for purpose perspective to remainuncorrected. Acceptable errors for the method are a function ofconcentration13 and for SiO2 are 1% relative to the referencevalue.

It is presumed that sub-sampling error is not a factor, basedon prior homogeneity studies conducted in establishing refer-ence values for the USGS rock reference materials. Dataaffected by spurious error are absent from the control chart datasummarized in Table 4. Weighing and volume errors areestimated from balance and glassware calibrations, in the samemanner as reported by NIST3,4 and summarized in Tables 2 and3.

In using the control chart data to establish uncertainties inresults for the Japanese Geological Survey sedimentary rocks,the principal factors to consider are how well known the USGSrock references values are and how the difference between thematrix of the sedimentary versus the igneous rocks might affectthe ‘recovery of standard’ value for the analytical samples incomparison with the controls.

The uncertainties in reference values for the USGS rockstandards are comparable to those for NIST SRM certifiedvalues established using two or more reference independentmethods of analysis, but slightly larger than those based ondefinitive method certification.15–17 For example, uncertaintiesfor reference values of SiO2 as stated on the certificates ofanalysis18,19 are 0.20% relative for the NIST obsidian and basaltSRMs 278 and 688. For the USGS rock RMs, the uncertaintiesare estimated as twice the stanard error of compilation means,and range from 0.1% relative for G-1 and W-1 to 0.30% forDTS-1 and PCC-1.15 The uncertainty in the reference values

Table 4 ICP-AES control data14

Deviation (relative) from referencevalue (%)15

Oxide RM

Reproducibilityover multi-yearperiod (as RSD)

of control datamean of extreme value

SiO2 AGV-1 1.58 20.12 +1.7BHVO-1 1.68 +0.60 21.64

Al2O3 BIR-1 1.9 +0.20 +2.6G-2 2.0 +0.5 +3.2

Fe2O3 AGV-1 1.8 20.20 +3.3BIR-1 1.7 +1.9 22.4

Na2O G-2 2.9 21.7 22.7BHVO-1 4.0 +0.4 +2.2

1286 Analyst, November 1997, Vol. 122

shown in Table 5 is the pooled uncertainty for the two RMs usedas controls for SiO2 analysis, a Type A evaluation.

While long experience with the method suggests that there isno correlation between magnitude of bias and matrix type,13,14

this has not specifically been evaluated. To be certain that theuncertainty is not underestimated, a rectangular distribution isassumed, and the uncertainty is stated without specifying theconfidence level. In this case, standard uncertainty for the biasestimate is the worst case control value deviation from thereference value divided by AB3.2,3

It is recognized that the calibration range for the measure-ments does not encompass extreme values, e.g., the 98.8% SiO2of the Japanese sedimentary chert sample JCh-1. A bias largerthan observed for the two control samples is likely to result fromextrapolation of the calibration curve beyond its linear range.13

This is another reason for assuming a rectangular distributionfor ‘recovery of standard’ values, rather than a normaldistribution, for which uncertainty would be based on theaverage deviation of all control results from the referencevalue.

Applying the above to the control data, total uncertaintieswere calculated as shown in Table 5. The expanded uncertaintyshown can be expected to estimate reliably not only the totaluncertainties in results for the two control samples measuredusing the ICP-AES method, but more generally for most rocksamples measured using the same analytical method. Theyincluded not only the random component of uncertainty that iscustomarily reported in all of the analytical literature, but alsothe analytical bias component that is often neglected.

A laboratory’s participation in collaborative measurementprograms can similarly provide a means of uncertaintyestimation. In this case, rather than using control samples run

simultaneously with the Japanese sedimentary RMs to providethe estimates of ‘recovery of standard’, the reference values ofthe newly characterized reference samples20 are used for thatpurpose (see Table 6).

These data reflect a larger relative deviation of meanmeasured results from reference values than the USGS controlsample data had. This may result because uncertainties in thereference values themselves are larger than for the USGS rocks,being based on a 8–10-fold smaller database. Alternatively, itcould result from extrapolation of the calibration curve for someindividual analyses, as noted above. Were we to examine datafrom the proficiency test program, even larger deviations ofindividual laboratory data from the central value would benoted, as would greater uncertainty in the central value.However, all of these approaches to estimating measurementsuncertainties follow the principles of the process outlined in theISO Guide,1 and are recommended to all geoanalyticallaboratories, particularly those participating in collaborativeprograms to establish values for reference materials.

Conclusion

Most laboratories appropriately evaluate and report the repeata-bility of their laboratory measurements. However, repeatabilityis only one component of the total measurement uncertainty, ordoubt about the validity of measurement. Approaches toevaluating measurement uncertainty more completely havebeen presented.

These approaches to evaluating measurement uncertainty,although less rigorous than the approach taken by metrologylaboratories, will help geoanalytical laboratories to identify anyof their standard operating procedures (SOPs) that do notproduce ‘fit for purpose’ data. This will lead to improved SOPs,and therefore to overall improvement of the reliability oflaboratory data. Improved data in turn will permit laboratoriesparticipating in proficiency tests to see continuous improve-ments in their respective z-scores. Alternatively, it will permitlaboratories producing ‘fit-for purpose’ data to refine theiruncertainty estimates to include any acceptably small bias thatcontributes to uncertainty. Were this to be done, comparabilityor harmony of data from collaborative programs would improveand the definition of reference values and their uncertainties forgeochemical RMs would improve accordingly.

References

1 ISO, Guide to the Expression of Uncertainty in Measurement, ISO,Geneva, 1993.

2 Eurachem Working Group, Quantifying Uncertainty in AnalyticalMeasurements, Laboratory of the Government Chemist, Teddington,UK, 1995.

3 Moody, J. R., and Vetter, T. W., J. Res. Natl. Inst. Stand. Technol.,1996, 101, 155.

4 Vetter, T. W., Pratt, K. W., Turk, G. C., Beck, C. M., and Butler, T.A., Analyst, 1995, 120, 2025.

5 Taylor, B. N., and Kuyatt, C. E., Guidelines for Evaluating andExpressing the Uncertainty of NIST Measurement Results, NISTTechnical Note 1297, US Government Printing Office, Washington,DC, 1993.

6 Becker, D., Christiansen, R., Currie, L., Diamondstone, B., Eber-hardt, K., Gills, T., Hertz, H., Klouda, G., Moody, J., Parris, R.,Schaffer, R., Steel, E., Taylor, J., Watters, R., and Zeisler, R., NISTSpec. Publ., 1992, No. 829.

7 Horowitz, A. J., Analyst, 1997, 122, 1193.8 Ramsey, M. J., Analyst, 1997, 122, 1255.9 Kane, J. S., Analyst, 1997, 122, 1289.

10 Fairbairn, H. W., Schlecht, W. G., Stevens, R. E., Dennen, W. H.,Ahrens, L. H., and Cheyes, F., US Geol. Surv. Bull., 1951, No.980.

11 Youden, W. J., Anal. Chem., 1960, 32, 23A.

Table 5 Estimation of uncertainty for SiO2 in control samples AGV-1 andBHVO-114

Type Source of uncertainty Relative standard uncertainty (%)

A Repeatability* 0.34Uncertainty in reference

value150.17

B ‘Recovery of standard’ 0.17/AB3 = 0.98B Weighing errors 0.1B Volume errors 0.02

Combined Type A ABBBBBBBB(0.34)2 + (0.017)2 = 0.0038Combined Type B ABBBBBBBBBB(098)2 + (0.1)2 + (0.02)2 = 0.985Combined uncertainty ABBBBBBBB(0.38)2 + (0.985)2 = 1.06Coverage factor 2Expanded uncertainty (rela-

tive)2.12

* Maximum repeatability value = reproducibility value (Table 4) 30.2.

Table 6 ‘Recovery of standard’ uncertainty estimate for SiO2 in rocks basedon participation in collaborative certification of Japanese sedimentary rockstandards

Standard

Contributed dataStandard (massfraction as %)

Reference value(mass fraction as %)

(mean ± s)

Relative deviationof contributed data

from referencevalue (%)

JSd-1 66.17 66.55 ± 0.354 20.57JSd-2 60.82 60.78± 0.411 +0.066JSd-3 76.03 76.00± 0.606 +0.039JSI-1 59.20 59.47± 0.263 20.46JSI-2 59.8 59.45± 0.223 +0.58

JCh-1 98.15 97.81± 0.484 +0.35

Analyst, November 1997, Vol. 122 1287

12 Uriano, G. A., and Gravatt, C. A., CRC Crit. Rev. Anal. Chem., 1977,6, 361.

13 Kane, J. S., and Dorrzapf, A. F., Jr., in GeoExpo/86: Exploration ofthe North American Cordillera, ed. Elliott, I. L., and Smee, B. M.,Association of Exploration Geochemists, Vancouver, BC, Canada,1987, pp. 184–188.

14 Kane, J. S., unpublished control sample data.15 Gladney, E. S., Burns, C. E., and Roelandts, I., Geostand. Newsl.,

1983, 7, 1988, 12, 253.16 Kane, J. S., J. Geochem. Explor., 1992, 44, 37.17 Kane, J. S., Spectrochim. Acta, Part B, 1991, 46, 1623.

18 Uriano, G. A., NBS Certificate for Analysis for SRM 278, NationalBureau of Standards, Washington, DC, 1981.

19 Uriano, G. A., NBS Certificate for Analysis for SRM 278, NationalBureau of Standards, Washington, DC, 1981.

20 Imai, N., Sakuramachi, H., Terashima, S., Itoh, S., and Ando, A.,Geostand. Newsl., 1996, 20, 165.

Paper 7/04789DReceived July 7, 1997

Accepted September 22, 1997

1288 Analyst, November 1997, Vol. 122