# ASTM D7440-08 (Reapproved 2015)e1

Designation D7440 08 Reapproved 20151Standard Practice forCharacterizing Uncertainty in Air Quality Measurements1This standard is issued under the fixed designation D7440; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon indicates an editorial change since the last revision or reapproval.1NOTEEditorial corrections were made throughout in July 2015.1. Scope1.1 This practice is for assisting developers and users of airquality s for sampling concentrations of both airborneand settled materials in characterizing measurements as touncertainty. Where possible, analysis into uncertainty compo-nents as recommended in the ISO Guide to the Expression ofUncertainty in Measurement ISO GUM, 12 is suggested.Aspects of uncertainty estimation particular to air qualitymeasurement are emphasized. For example, air quality assess-ment is often complicated by the difficulty of taking replicatemeasurements owing to the large spatio-temporal variation inconcentration values to be measured; systematic error or bias,both corrected and uncorrected; and the rare non-normaldistribution of errors. This practice operates mainly throughexample. Background and mathematical development are rel-egated to appendices for optional reading.1.2 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards3D1356 Terminology Relating to Sampling and Analysis ofAtmospheresD3670 Guide for Determination of Precision and Bias ofs of Committee D22D6246 Practice for uating the Perance of DiffusiveSamplersD6552 Practice for Controlling and Characterizing Errors inWeighing Collected Aerosols2.2 Other International StandardsISO GUM Guide to the Expression of Uncertainty inMeasurement, ISO Guide 98, 1995 See Ref 1, for anadditional measurement uncertainty resource.4ISO 7708 Air QualityParticle Size Fraction Definitions forHealth-Related Sampling4ISO 15767 Workplace AtmospheresControlling and Char-acterizing Errors in Weighing Collected Aerosol4ISO 16107 Workplace AtmospheresProtocol for uat-ing the Perance of Diffusive Samplers, 20074EN 482 Workplace AtmospheresGeneral Requirementsfor the Perance of Procedures for the Measurement ofChemical Agents43. Terminology3.1 DefinitionsFor definitions of terms used in thispractice, see Terminology D1356.3.2 Other terms defined as follows are taken from ISO GUMunless otherwise noted3.2.1 accuracycloseness of agreement between the resultof a measurement and a true value of the measurand.3.2.2 combined standard uncertainty, ucstandard uncer-tainty of the result of a measurement when that result isobtained from the values of a number of other quantities, equalto the positive square root of a sum of terms, the terms beingthe variances or covariances of these other quantities weightedaccording to how the measurement result varies with changesin these quantities.3.2.2.1 DiscussionAs within ISO GUM, the “other quan-tities” are designated uncertainty components ujfrom source j.The component ujis taken as the standard deviation estimatefrom source j in the case of a source of random variation.3.2.3 coverage factor, knumerical factor used as a multi-plier of the combined standard uncertainty uc in order toobtain an expanded uncertainty U.1This practice is under the jurisdiction ofASTM Committee D22 on Air Qualityand is the direct responsibility of Subcommittee D22.01 on Quality Control.Current edition approved July 1, 2015. Published July 2015. Originally approvedin 2008. Last previous edition approved in 2008 as D7440 08. DOI 10.1520/D7440-08R15E01.2The boldface numbers in parentheses refer to the list of references at the end ofthis standard.3For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume ination, refer to the standards Document Summary page onthe ASTM website.4BIPM version available for download from http//www.bipm.org/en/publications/guides/gum.html. ISO version available from American NationalStandards Institute ANSI, 25 W. 43rd St., 4th Floor, New York, NY 10036,http//www.ansi.org.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.2.3.1 DiscussionThe factor k depends on the specificmeaning attributed to the expanded uncertainty U. However,for simplicity this practice adopts the now nearly traditionalcoverage factor as the value 2, determining the specificmeaning of the expanded uncertainty U in different circum-stances. Other coverage factors if needed are then easilyimplemented simply by multiplication of the traditional ex-panded uncertainty U see 7.1 7.4.3.2.3.2 DiscussionThe use of a single coverage factor,often through approximation, avoids the overly conservativeuse of individual component confidence limits rather than rootvariance estimates as uncertainty components.3.2.4 error of measurementresult of a measurementminus a true value of the measurand.3.2.5 expanded uncertainty, Uquantity defining an inter-val about the result of a measurement that may be expected toencompass a large fraction of the distribution of values thatcould reasonably be attributed to the measurand.3.2.5.1 DiscussionThis definition has the breadth to en-compass a wide variety of conceptions.3.2.5.2 DiscussionThe expanded uncertainty U in somecases is expressed in absolute terms, but sometimes as relativeto the measurement result. What is meant is generally clearfrom the context.3.2.6 influence quantityquantity that is not the measurandbut that affects the result of the measurement.3.2.7 measurandparticular quantity subject to measure-ment.3.2.8 measurand valueadapted from ISO GUM, un-known quantity whose measurement is sought, often called thetrue value. Examples are the concentration mg/m3ofasubstance in the air at a particular time and place, thetime-weighted average of a concentration at a particularposition, or the expected mean concentration estimate asobtained by a reference at a specific time and position.3.2.9 population variance of a random variabletheexpectation of the square of the centered random variable.3.2.10 random errorresult of a measurement minus themean that would result from an infinite number of measure-ments of the same measurand carried out under the samerepeatability conditions of measurement.3.2.10.1 DiscussionRandom error is equal to error minussystematic error.3.2.11 sample variancethe sum of the squared devia-tions of observations from their average divided by one lessthan the number of observations.3.2.11.1 DiscussionThe sample variance is an unbiasedestimator of the population variance.3.2.12 standard deviationpositive square root of the vari-ance.3.2.13 symmetric accuracy range Athe range symmetricabout true measurand values containing 95 of measure-ment estimates. A is a specific quantification of accuracy. 2ISO 161073.2.14 systematic error biasmean that would result froman infinite number of measurements of the same measurandcarried out under repeatability conditions minus a true value ofthe measurand.3.2.15 Type A uation of uncertainty of u-ation of uncertainty by the statistical analysis of series ofobservations.3.2.16 Type B uation of uncertainty of u-ation of uncertainty by means other than the statistical analysisof series of observations.4. Background Ination4.1 Uncertainty in a measurement result can be taken as therange about an estimate, corrected for bias if known, contain-ing the true, or mean reference valuein the language of ISOGUM, the measurand value at given confidence. Uncertaintyaccounts not only for variation in a s results atapplication, but also for incomplete characterization of the when uated. In accordance with ISO GUM,uncertainty may often usefully be analyzed into individualcomponents.4.2 There are several aspects of uncertainty characterizationspecific to air quality measurements. One of these aspectsconcerns known, that is, correctible, systematic error or meanbias of a measurement relative to a true measurand value.Several measurement s exist with such bias left uncor-rected because of policy, tradition, or other reason. Uncertaintydeals only with what is unknown about a measurement, and assuch does not include correctible known bias. The magnitudeof the difference between estimate and measurand value iscovered by accuracy as defined qualitatively in ISO GUM,rather than uncertainty, particularly when the bias is known,but uncorrected. Such s require specification of bothuncertainty and as much as is known of the uncorrected bias, oralternatively the adoption of an accuracy measure.4.3 Often bias is known to exist, but with unknown value. Inthe case where only limits may be placed on the magnitude ofthe bias, ISO GUM generally recommends treating the bias asunily distributed within the known limits. Such a distri-bution refers to independent situations, for example,calibrations, where bias may arise see 7.4 and Appendix X2,rather than variation at the point of application. Eventhough such an equal-likelihood bias distribution may beunrealistic, nevertheless a standard deviation estimate may bemade that reveals the limits on the bias. If the even-distributionapproximation is clearly invalid for a relevant set ofmeasurements, the procedure may be adjusted slightly byadopting an accuracy measure tailored to the assumed limits.4.4 Another issue concerns the distribution of measure-ments. ISO GUM deals only with normally distributed first-order that is, “small” variations relative to measurand values.An example to the contrary is afforded by normally distributeddata confounded by a small number of apparent outliers 3,which may not detract from the perance seeAppendix X4 for details. Another example is the determina-tion of an aerosol concentration at one location perhaps at aworkers lapel as an estimate of the concentration at a separateD7440 08 201512point such as a breathing zone. In this case the variations canbe of the order of the estimate itself and may have the characterof a log-normal distribution.4.5 The spatial inhomogeneity alluded to in 4.4 relates toanother point regarding the focus of this practice. The spatio-temporal variations in air quality characteristics are generallyso large 4 as to preclude uation of a duringapplication through the use of replicate measurements. In thiscase, often an initial single uation is undertakenwith the purpose of determining uncertainty present in subse-quent applications of the . Confidence in such anuation can be specified and relates to the concept ofprediction-intervals 5 see 7.2.4.6 A related subject is measurement system control. Themeasurement system must remain in a state of statisticalcontrol if an introductory uation is to characterize laterpractical applications of the . Measurement systemcontrol is uated using an ongoing quality control program,testing critical perance aspects for detecting problemswhich may develop in the .5. Summary of Practice5.1 The essential idea behind ISO GUM is the analysis tothe fullest extent practical of the elemental sources of what isunknown in the estimate of a measurand value. This contrastswith a global or top-down determination of uncertainty, whichcould for example be done ideally by comparing replicateestimates to known measurand values over all conditionsexpected in application of the . Although a globaluncertainty uation may sometimes seem inexpensive, thereis a difficulty in covering essential contingencies of the application.5.2 Uncertainty component analysis further has severalspecific advantages over global analysis. The results may beapplicable to a variety of situations. For example, an aerosolsampler might be globally uated as to particle-size-dependent error by side-by-side comparison to a referencesampler in several coal mines. The knowledge obtained maynot be as easily applied for sampler use in iron mines, forexample, as more detailed ination on how the samplerpers over given dust size distributions may be needed.Furthermore, specific problem areas of a given may bepinpointed. The detailed itemization of uncertainty sourcesleads to a transparency in covering the essential problems of ameasurement . Examples of potentially significant un-certainty components are listed in Table 1.5.3 Type A and B Uncertainty Components5.3.1 Components that have been statistically uatedduring application may be classified as Type A. SeeSection 7 for specific examples.5.3.2 Some components are often statistically uatedduring an initial uation, rather than at application.Also acknowledged is a common situation that componentsmay not have been characterized in a statistically valid mannerand therefore may require professional judgment for itemizing.Such components are termed Type B uncertainties. Type Buncertainties are often associated with unknown systematicerror or bias; however, random variation may also fall into thiscategory. For example, a common assumption see, forexample, EN 482 regarding personal sampling in the work-place is that the relative standard deviation associated withpersonal sampling pump variations is 3 std from mean 0.001.X4.4.2 The factor q is chosen to approximate A2/A2asproportional to a chi-square variable with effective degrees offreedom effchosen as with 11-10 to reproduce the varianceof A2/A2, given the variances of bias, TRSD, and r. The result,applying lowest order propagation of errors, is that the factor qis given byq25 eff/0.05 eff2X4.4defined in terms of a chi-square 5 -quantile at effectivenumber of degrees of freedom effeff215 21u4TRSD2TRSD212bias2A 41outlier21X4.5outlier215 r1 2 r23n21u22Expu2 30.952X4.6X4.4.3 The numerator in Eq X4.5 is simplified if bias isnegligible or if corrected by means of the uation, leadingtoEbias2TRSD2/ nX4.4.4 At r 4,u increases by 30 over 1.960, andmore importantly, the uncertainty in the estimate r when thenumber n of data points in the uation 50 for examplebrings the confidence limit on r close to the wall at r 5,with the effective number outlierof degrees of freedomdropping to about 4, with marked effect on A95 . However,outlierextremely rapidly increases with decreasing r, equalingabo