# ASTM D7440-08 (Reapproved 2015)e1

Designation: D7440 − 08 (Reapproved 2015)´1Standard 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.ε1NOTE—Editorial corrections were made throughout in July 2015.1. Scope1.1 This practice is for assisting developers and users of airquality methods 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, (1)2) 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 Standards:3D1356 Terminology Relating to Sampling and Analysis ofAtmospheresD3670 Guide for Determination of Precision and Bias ofMethods of Committee D22D6246 Practice for Evaluating the Performance of DiffusiveSamplersD6552 Practice for Controlling and Characterizing Errors inWeighing Collected Aerosols2.2 Other International Standards:ISO GUM Guide to the Expression of Uncertainty inMeasurement, ISO Guide 98, 1995 (See Ref (1), for anadditional measurement uncertainty resource.)4ISO 7708 Air Quality—Particle Size Fraction Definitions forHealth-Related Sampling4ISO 15767 Workplace Atmospheres—Controlling and Char-acterizing Errors in Weighing Collected Aerosol4ISO 16107 Workplace Atmospheres—Protocol for Evaluat-ing the Performance of Diffusive Samplers, 20074EN 482 Workplace Atmospheres—General Requirementsfor the Performance of Procedures for the Measurement ofChemical Agents43. Terminology3.1 Definitions—For definitions of terms used in thispractice, see Terminology D1356.3.2 Other terms defined as follows are taken from ISO GUMunless otherwise noted:3.2.1 accuracy—closeness of agreement between the resultof a measurement and a true value of the measurand.3.2.2 combined standard uncertainty, uc—standard 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 Discussion—As 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, k—numerical 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 service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s 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 Discussion—The 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 Discussion—The 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 measurement)—result of a measurementminus a true value of the measurand.3.2.5 expanded uncertainty, U—quantity 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 Discussion—This definition has the breadth to en-compass a wide variety of conceptions.3.2.5.2 Discussion—The 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 quantity—quantity that is not the measurandbut that affects the result of the measurement.3.2.7 measurand—particular quantity subject to measure-ment.3.2.8 measurand value—(adapted from ISO GUM), un-known quantity whose measurement is sought, often called thetrue value. Examples are the concentration (mg/m3)ofasubstance 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 method at a specific time and position.3.2.9 (population) variance (of a random variable)—theexpectation of the square of the centered random variable.3.2.10 random error—result of a measurement minus themean that would result from an infinite number of measure-ments of the same measurand carried out under the same(repeatability) conditions of measurement.3.2.10.1 Discussion—Random error is equal to error minussystematic error.3.2.11 (sample) variance—the sum of the squared devia-tions of observations from their average divided by one lessthan the number of observations.3.2.11.1 Discussion—The sample variance is an unbiasedestimator of the population variance.3.2.12 standard deviation—positive square root of the vari-ance.3.2.13 symmetric accuracy range A—the range symmetricabout (true) measurand values containing 95 % of measure-ment estimates. A is a specific quantification of accuracy. (2)ISO 161073.2.14 systematic error (bias)—mean 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 evaluation (of uncertainty)—method of evalu-ation of uncertainty by the statistical analysis of series ofobservations.3.2.16 Type B evaluation (of uncertainty)—method of evalu-ation of uncertainty by means other than the statistical analysisof series of observations.4. Background Information4.1 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 value—in the language of ISOGUM, the measurand value at given confidence. Uncertaintyaccounts not only for variation in a method’s results atapplication, but also for incomplete characterization of themethod when evaluated. 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 methods 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 methods 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 asuniformly 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 method 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 method performance (seeAppendix X4 for details). Another example is the determina-tion of an aerosol concentration at one location (perhaps at aworker’s lapel) as an estimate of the concentration at a separateD7440 − 08 (2015)´12point (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 evaluation of a method duringapplication through the use of replicate measurements. In thiscase, often an initial single method evaluation is undertakenwith the purpose of determining uncertainty present in subse-quent applications of the method. Confidence in such anevaluation 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 evaluation is to characterize laterpractical applications of the method. Measurement systemcontrol is evaluated using an ongoing quality control program,testing critical performance aspects for detecting problemswhich may develop in the method.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 method. Although a globaluncertainty evaluation may sometimes seem inexpensive, thereis a difficulty in covering essential contingencies of the methodapplication.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) evaluated 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 information on how the samplerperforms over given dust size distributions may be needed.Furthermore, specific problem areas of a given method may bepinpointed. The detailed itemization of uncertainty sourcesleads to a transparency in covering the essential problems of ameasurement method. Examples of potentially significant un-certainty components are listed in Table 1.5.3 Type A and B Uncertainty Components:5.3.1 Components that have been statistically evaluatedduring method application may be classified as Type A. (SeeSection 7 for specific examples.)5.3.2 Some components are often statistically evaluatedduring an initial method evaluation, 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 Aˆ2/A2asproportional to a chi-square variable with effective degrees offreedom υeffchosen as with (11-10) to reproduce the varianceof Aˆ2/A2, given the variances of biaˆs, TRSˆD, and rˆ. The result,applying lowest order propagation of errors, is that the factor qis given by:q25 υeff/χ0.05 υeff2(X4.4)defined in terms of a chi-square 5 %-quantile at effectivenumber of degrees of freedom υeff:υeff215 υ21uˆ4TRSˆD2TRSˆD212·biaˆs2Aˆ 41υoutlier21(X4.5)υoutlier215 πrˆ~1 2 rˆ!23n21uˆ22Exp@u2# 30.952(X4.6)X4.4.3 The numerator in Eq X4.5 is simplified if bias isnegligible (or if corrected by means of the evaluation, leadingto:E@biaˆs2# ~TRSD2/ n!X4.4.4 At r =4%,u increases by 30 % over 1.960, andmore importantly, the uncertainty in the estimate rˆ when thenumber n of data points in the evaluation = 50 (for example)brings the confidence limit on r close to the wall at r =5%,with the effective number υoutlierof degrees of freedomdropping to about 4, with marked effect on A95 %. However,υoutlierextremely rapidly increases with decreasing rˆ, equalingabo