ASTM D7440-23

Standard Practice for Characterizing Uncertainty in Air Quality Measurements

SIGNIFICANCE AND USE
6.1 A primary use intended for this practice is for qualifying ASTM International Standards as Standard Test Methods. In the past, a “Precision and Bias” report has been required. However, recently a statement of uncertainty has become an acceptable alternative to Guide D3670. Inclusion of such a statement with a method description simplifies comparison of ASTM Test Methods to analogous ISO and Committee for European Normalization (CEN) standards, now required to have uncertainty statements.  
6.2 Standardizing the characterization of sampling/analytical method performance is expected to be useful in other applications as well. For example, performance details are a necessity for justifying compliance decisions based on experimental air quality assessments (7). Documented uncertainty can form a basis for specific criteria defining acceptable sampling/analytical method performance.  
6.3 Furthermore, high quality atmospheric measurements are vital for making decisions as to how hazardous substances are to be controlled. Valid data are required for drawing reasonable epidemiological conclusions, for making sound decisions as to acceptable limits, as well as for determining the efficacy of a hazard control system.  
6.4 Finally, because of developing world-wide acceptance of ISO GUM for detailing measurements when statistics are simple, the practice should be useful in comparing ASTM International Test Methods to other published methods. The codification of statistical procedures may in fact minimize the difficulty in interpreting a plethora of individual, albeit possibly valid, approaches.
SCOPE
1.1 This practice is for assisting developers and users of air quality methods for sampling concentrations of both airborne and settled materials in characterizing measurements as to uncertainty. Where possible, analysis into uncertainty components as recommended in the International Organization for Standardization (ISO) Guide to the Expression of Uncertainty in Measurement (ISO GUM, (1)2) is suggested. Aspects of uncertainty estimation particular to air quality measurement are emphasized. For example, air quality assessment is often complicated by: the difficulty of taking replicate measurements owing to the large spatio-temporal variation in concentration values to be measured; systematic error or bias, both corrected and uncorrected; and the (rare) non-normal distribution of errors. This practice operates mainly through example. Background and mathematical development are relegated to appendices for optional reading.  
1.2 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.  
1.3 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.4 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

General Information

• Status: Published
• Publication Date: 31-Aug-2023
• ICS: 13.040.20 - Ambient atmospheres
• Technical Committee: D22 - Air Quality
• Drafting Committee: D22.01 - Quality Control

Relations

• Replaces: ASTM D7440-08(2015)e1 - Standard Practice for Characterizing Uncertainty in Air Quality Measurements
• Effective Date: 01-Sep-2023
• Referred By: ASTM E3203-21 - Standard Test Method for Determination of Lead in Dried Paint, Soil, and Wipe Samples by Inductively Coupled Plasma-Optical Emission Spectroscopy (ICP-OES)
• Effective Date: 01-Sep-2023
• Referred By: ASTM D7659-21 - Standard Guide for Strategies for Surface Sampling of Metals and Metalloids for Worker Protection
• Effective Date: 01-Sep-2023
• Referred By: ASTM D7439-21 - Standard Test Method for Determination of Elements in Airborne Particulate Matter by Inductively Coupled Plasma–Mass Spectrometry
• Effective Date: 01-Sep-2023
• Referred By: ASTM D4532-22 - Standard Test Method for Respirable Dust in Workplace Atmospheres Using Cyclone Samplers
• Effective Date: 01-Sep-2023
• Referred By: ASTM D7911-19 - Standard Guide for Using Reference Material to Characterize Measurement Bias Associated with Volatile Organic Compound Emission Chamber Test
• Effective Date: 01-Sep-2023
• Referred By: ASTM D7035-21 - Standard Test Method for Determination of Metals and Metalloids in Airborne Particulate Matter by Inductively Coupled Plasma Atomic Emission Spectrometry (ICP-AES)
• Effective Date: 01-Sep-2023

Overview

ASTM D7440-23: Standard Practice for Characterizing Uncertainty in Air Quality Measurements sets forth comprehensive guidance for developers and users of air quality sampling and analysis methods. Developed by ASTM International, this standard addresses the characterization and documentation of uncertainty in measurements of both airborne and settled materials. By advocating for the inclusion of uncertainty statements-aligned with the ISO Guide to the Expression of Uncertainty in Measurement (ISO GUM)-ASTM D7440-23 facilitates comparison with analogous international standards and enhances the reliability and transparency of air quality data.

Key Topics

• Measurement Uncertainty: Guidance on identifying, quantifying, and reporting uncertainty components related to air quality sampling and analysis.
• Types of Uncertainty: Distinction between Type A (statistical analysis-based) and Type B (other means, professional judgment) uncertainty evaluations.
• Influence Quantities: Consideration of external factors such as temperature, humidity, atmospheric pressure, and aerosol size distribution that impact measurement results.
• Bias and Accuracy: Addressing systematic error (bias), its impact on uncertainty, and how known biases are reported in tandem with uncertainty statements.
• Reporting Requirements: Clear instructions on documenting uncertainty, listing sources, detailing the evaluation approach, and providing expanded uncertainty with relevant coverage factors.
• Alignment with ISO GUM: Encourages usage of ISO GUM protocols, supporting consistent global expressions of uncertainty.
• Quality Assurance: Emphasizes the necessity for well-documented performance and ongoing measurement system control.

Applications

The practical applications of ASTM D7440-23 span a broad spectrum within the air quality domain:

• Method Development and Validation: Essential for laboratories, regulatory agencies, and test method developers when designing or evaluating air quality measurement methods.
• Regulatory Compliance: Clarifies measurement performance for compliance with air quality regulations by requiring documented uncertainty, supporting defensible compliance and enforcement decisions.
• International Comparability: Eases comparison between ASTM, ISO, and CEN test methods, an essential aspect with increasing international harmonization of measurement standards.
• Performance Specifications: Provides a rational basis for defining performance criteria and acceptable thresholds in air quality sampling and analysis.
• Epidemiological Studies: Supports the generation of high-quality, traceable data necessary for robust epidemiological studies and health impact assessments.
• Industrial Hygiene and Environmental Monitoring: Informs risk assessments and efficiency of hazard control systems by ensuring that measurement uncertainty is well-understood and properly reported.
• Quality Control Programs: Aids implementation of routine quality control, ensuring results remain valid and reliable over time.

Related Standards

The following standards are closely related and may be referenced for complementary guidance:

• ASTM D1356: Terminology Relating to Sampling and Analysis of Atmospheres
• ASTM D3670: Guide for Determination of Precision and Bias of Methods of Committee D22
• ASTM D6246: Practice for Evaluating the Performance of Diffusive Samplers
• ASTM D6552: Practice for Controlling and Characterizing Errors in Weighing Collected Aerosols
• ISO GUM: Guide to the Expression of Uncertainty in Measurement
• ISO 7708: Air Quality - Particle Size Fraction Definitions for Health-Related Sampling
• ISO 15767: Workplace Atmospheres - Controlling and Characterizing Errors in Weighing Collected Aerosols
• ISO 16107: Workplace Atmospheres - Protocol for Evaluating the Performance of Diffusive Samplers
• EN 482: General Requirements for Performance of Procedures for Measurement of Chemical Agents

Summary

ASTM D7440-23 is a vital practice standard for anyone involved in air quality measurements. By standardizing how uncertainty is characterized, reported, and compared across national and international boundaries, it enhances the reliability, comparability, and scientific integrity of air quality data-ensuring that measurement results support sound decisions in environmental protection, public health, and regulatory compliance. Incorporating this standard enables laboratories and professionals to meet current best-practice expectations for measurement uncertainty in air quality assessments.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: D7440 − 23
Standard Practice for
Characterizing Uncertainty in Air Quality Measurements
This standard is issued under the fixed designation D7440; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2. Referenced Documents
2.1 ASTM Standards:
1.1 This practice is for assisting developers and users of air
D1356 Terminology Relating to Sampling and Analysis of
quality methods for sampling concentrations of both airborne
Atmospheres
and settled materials in characterizing measurements as to
D3670 Guide for Determination of Precision and Bias of
uncertainty. Where possible, analysis into uncertainty compo-
Methods of Committee D22
nents as recommended in the International Organization for
D6246 Practice for Evaluating the Performance of Diffusive
Standardization (ISO) Guide to the Expression of Uncertainty
2 Samplers
in Measurement (ISO GUM, (1) ) is suggested. Aspects of
D6552 Practice for Controlling and Characterizing Errors in
uncertainty estimation particular to air quality measurement are
Weighing Collected Aerosols
emphasized. For example, air quality assessment is often
2.2 ISO Standards:
complicated by: the difficulty of taking replicate measurements
ISO GUM Guide to the Expression of Uncertainty in
owing to the large spatio-temporal variation in concentration
Measurement, ISO Guide 98, 1995
values to be measured; systematic error or bias, both corrected
ISO 7708 Air Quality — Particle Size Fraction Definitions
and uncorrected; and the (rare) non-normal distribution of
for Health-Related Sampling
errors. This practice operates mainly through example. Back-
ISO 15767 Workplace Atmospheres — Controlling and
ground and mathematical development are relegated to appen-
Characterizing Errors in Weighing Collected Aerosol
dices for optional reading.
ISO 16107 Workplace Atmospheres — Protocol for Evalu-
1.2 The values stated in SI units are to be regarded as ating the Performance of Diffusive Samplers
ISO 16702 Workplace air quality — Determination of total
standard. No other units of measurement are included in this
organic isocyanate groups in air using 1-(2-
standard.
methoxyphenyl)piperazine and liquid chromatography
1.3 This standard does not purport to address all of the
safety concerns, if any, associated with its use. It is the 3. Terminology
responsibility of the user of this standard to establish appro-
3.1 Definitions—For definitions of terms used in this
priate safety, health, and environmental practices and deter-
practice, see Terminology D1356.
mine the applicability of regulatory limitations prior to use.
3.2 Other Terms Defined as Follows are Taken from ISO
1.4 This international standard was developed in accor-
GUM, Unless Otherwise Noted:
dance with internationally recognized principles on standard-
3.2.1 accuracy—closeness of agreement between the result
ization established in the Decision on Principles for the
of a measurement and a true value of the measurand.
Development of International Standards, Guides and Recom-
3.2.2 combined standard uncertainty, u —standard uncer-
c
mendations issued by the World Trade Organization Technical
tainty of the result of a measurement when that result is
Barriers to Trade (TBT) Committee.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
This practice is under the jurisdiction of ASTM Committee D22 on Air Quality Standards volume information, refer to the standard’s Document Summary page on
and is the direct responsibility of Subcommittee D22.01 on Quality Control. the ASTM website.
Current edition approved Sept. 1, 2023. Published October 2023. Originally BIPM version available for download from http://www.bipm.org/en/
ɛ1
approved in 2008. Last previous edition approved in 2015 as D7440 – 08 (2015) . publications/guides/gum.html. ISO version available from American National
DOI: 10.1520/D7440-23. Standards Institute (ANSI), 25 W. 43rd St., 4th Floor, New York, NY 10036,
The boldface numbers in parentheses refer to the list of references at the end of http://www.ansi.org.
this standard.
See Ref (1), for an additional measurement uncertainty resource.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D7440 − 23
obtained from the values of a number of other quantities, equal 3.2.11 (sample) variance—the sum of the squared devia-
to the positive square root of a sum of terms, the terms being tions of observations from their average divided by one less
the variances or covariances of these other quantities weighted than the number of observations.
according to how the measurement result varies with changes 3.2.11.1 Discussion—The sample variance is an unbiased
in these quantities.
estimator of the population variance.
3.2.2.1 Discussion—As within ISO GUM, the “other quan-
3.2.12 standard deviation—positive square root of the vari-
tities” are designated uncertainty components u from source j.
j
ance.
The component u is taken as the standard deviation estimate
j
3.2.13 symmetric accuracy range A—the range symmetric
from source j in the case of a source of random variation.
about (true) measurand values containing 95 % of measure-
3.2.3 coverage factor, k—numerical factor used as a multi-
ment estimates.
plier of the combined standard uncertainty (u ) in order to
c
3.2.13.1 Discussion—A is a specific quantification of accu-
obtain an expanded uncertainty (U).
racy (2). ISO 16107
3.2.3.1 Discussion—The factor k depends on the specific
3.2.14 systematic error (bias)—mean that would result from
meaning attributed to the expanded uncertainty U. However,
an infinite number of measurements of the same measurand
for simplicity this practice adopts the now nearly traditional
carried out under repeatability conditions minus a true value of
coverage factor as the value 2, determining the specific
the measurand.
meaning of the expanded uncertainty U in different circum-
3.2.15 Type A evaluation (of uncertainty)—method of evalu-
stances. Other coverage factors if needed are then easily
ation of uncertainty by the statistical analysis of series of
implemented simply by multiplication of the traditional ex-
observations.
panded uncertainty U (see 7.1 – 7.4).
3.2.3.2 Discussion—The use of a single coverage factor,
3.2.16 Type B evaluation (of uncertainty)—method of evalu-
often through approximation, avoids the overly conservative
ation of uncertainty by means other than the statistical analysis
use of individual component confidence limits rather than root
of series of observations.
variance estimates as uncertainty components.
4. Background Information
3.2.4 error (of measurement)—result of a measurement
minus a true value of the measurand.
4.1 Uncertainty in a measurement result can be taken as the
3.2.5 expanded uncertainty, U—quantity defining an inter- range about an estimate, corrected for bias if known, contain-
val about the result of a measurement that may be expected to ing the true, or mean reference value—in the language of ISO
encompass a large fraction of the distribution of values that GUM, the measurand value at given confidence. Uncertainty
could reasonably be attributed to the measurand. accounts not only for variation in a method’s results at
3.2.5.1 Discussion—This definition has the breadth to en- application, but also for incomplete characterization of the
method when evaluated. In accordance with ISO GUM,
compass a wide variety of conceptions.
uncertainty may often usefully be analyzed into individual
3.2.5.2 Discussion—The expanded uncertainty U in some
components.
cases is expressed in absolute terms, but sometimes as relative
to the measurement result. What is meant is generally clear
4.2 There are several aspects of uncertainty characterization
from the context.
specific to air quality measurements. One of these aspects
concerns known, that is, correctible, systematic error or mean
3.2.6 influence quantity—quantity that is not the measurand
but that affects the result of the measurement. bias of a measurement relative to a true measurand value.
Several measurement methods exist with such bias left uncor-
3.2.7 measurand—particular quantity subject to measure-
rected because of policy, tradition, or other reason. Uncertainty
ment.
deals only with what is unknown about a measurement, and as
3.2.8 measurand value—(adapted from ISO GUM), un-
such does not include correctible (known) bias. The magnitude
known quantity whose measurement is sought, often called the
of the difference between estimate and measurand value is
true value.
covered by accuracy as defined qualitatively in ISO GUM,
3.2.8.1 Discussion—Examples are the concentration
rather than uncertainty, particularly when the bias is known,
(mg/m ) of a substance in the air at a particular time and place,
but uncorrected. Such methods require specification of both
the time-weighted average of a concentration at a particular
uncertainty and as much as is known of the uncorrected bias, or
position, or the expected mean concentration estimate as
alternatively the adoption of an accuracy measure.
obtained by a reference method at a specific time and position.
4.3 Often bias is known to exist, but with unknown value. In
3.2.9 (population) variance (of a random variable)—the
the case where only limits may be placed on the magnitude of
expectation of the square of the centered random variable.
the bias, ISO GUM generally recommends treating the bias as
3.2.10 random error—result of a measurement minus the
uniformly distributed within the known limits. Such a distri-
mean that would result from an infinite number of measure- bution refers to independent situations, for example,
ments of the same measurand carried out under the same
calibrations, where bias may arise (see 7.4 and Appendix X2),
(repeatability) conditions of measurement.
rather than variation at the point of method application. Even
3.2.10.1 Discussion—Random error is equal to error minus though such an equal-likelihood bias distribution may be
systematic error. unrealistic, nevertheless a standard deviation estimate may be
D7440 − 23
made that reveals the limits on the bias. If the even-distribution measurement method. Examples of potentially significant un-
approximation is clearly invalid for a relevant set of certainty components are listed in Table 1.
measurements, the procedure may be adjusted slightly by
5.3 Type A and B Uncertainty Components:
adopting an accuracy measure tailored to the assumed limits.
5.3.1 Components that have been statistically evaluated
4.4 Another issue concerns the distribution of measure- during method application may be classified as Type A. (See
ments. ISO GUM deals only with normally distributed first- Section 7 for specific examples.)
order (that is, “small”) variations relative to measurand values. 5.3.2 Some components are often statistically evaluated
An example to the contrary is afforded by normally distributed during an initial method evaluation, rather than at application.
data confounded by a small number of apparent outliers (3), Also acknowledged is a common situation that components
which may not detract from the method performance (see may not have been characterized in a statistically valid manner
Appendix X4 for details). Another example is the determina- and therefore may require professional judgment for itemizing.
tion of an aerosol concentration at one location (perhaps at a Such components are termed Type B uncertainties. Type B
worker’s lapel) as an estimate of the concentration at a separate uncertainties are often associated with unknown systematic
point (such as a breathing zone). In this case the variations can error or bias; however, random variation may also fall into this
be of the order of the estimate itself and may have the character category. For example, a common assumption regarding per-
sonal sampling in the workplace is that the relative standard
of a log-normal distribution.
deviation associated with personal sampling pump variations is
4.5 The spatial inhomogeneity alluded to in 4.4 relates to
<5 % at essentially 100 % confidence.
another point regarding the focus of this practice. The spatio-
temporal variations in air quality characteristics are generally 5.4 Intrinsic versus Environmentally Associated Compo-
nents: Influence Quantities:
so large (4) as to preclude evaluation of a method during
application through the use of replicate measurements. In this 5.4.1 Some uncertainties may be intrinsic to a method. For
example, estimates from aerosol samplers may depend criti-
case, often an initial single method evaluation is undertaken
with the purpose of determining uncertainty present in subse- cally on sampler dimensions, which if variable leads to
intersampler estimate variation.
quent applications of the method. Confidence in such an
evaluation can be specified and relates to the concept of 5.4.2 On the other hand, a sampler’s performance may
prediction-intervals (5) (see 7.2). depend on the environment. For example, suppose a sampler is
sensitive to temperature changes that are impractical to mea-
4.6 A related subject is measurement system control. The
sure in the field; that is, sampler estimates are not temperature-
measurement system must remain in a state of statistical
corrected. Then measurement of this sensitivity during method
control if an introductory evaluation is to characterize later
evaluation together with knowledge of the temperature varia-
practical applications of the method. Measurement system
tion expected for a given field application can be used to
control is evaluated using an ongoing quality control program,
determine the uncertainty associated with this effect.
testing critical performance aspects for detecting problems
which may develop in the method.
5. Summary of Practice TABLE 1 Common Potential Uncertainty Components
Sampling
5.1 The essential idea behind ISO GUM is the analysis to
Personal sampling pump flow rate: setting the pump and subsequent drift
the fullest extent practical of the elemental sources of what is
Sampling rate of diffusive sampler
unknown in the estimate of a measurand value. This contrasts Sampler dimension (aerosol and diffusive sampling)
Collection efficiency of a sampler or sampling medium
with a global or top-down determination of uncertainty, which
(also, see (6))
could for example be done ideally by comparing replicate
Analytical
estimates to known measurand values over all conditions Aerosol weighing
Recovery (for example, chromatographic or spectroscopic methods)
expected in application of the method. Although a global
Poisson counting (for example, in XRD methods)
uncertainty evaluation may sometimes seem inexpensive, there
Instrument or sensor variation
Operator effects giving inter-lab differences (if data from several labs are to
is a difficulty in covering essential contingencies of the method
be used)
application.
Sample
Sample stability
5.2 Uncertainty component analysis further has several
Sample preparation (for example, handling silica quasi-suspensions)
specific advantages over global analysis. The results may be
Sample loss during transport or storage
applicable to a variety of situations. For example, an aerosol Evaluation
Calibration material uncertainty
sampler might be (globally) evaluated as to particle-size-
Evaluation chamber concentration uncertainty
dependent error by side-by-side comparison to a reference
Other bias-correction uncertainty
sampler in several coal mines. The knowledge obtained may
Environmental Influence Parameters
Temperature (inadequacy of correction, if correction is made as with diffusive
not be as easily applied for sampler use in iron mines, for
samplers)
example, as more detailed information on how the sampler
Atmospheric pressure
performs over given dust size distributions may be needed. Humidity
Aerosol size distribution (if not measured by a given aerosol sampling method)
Furthermore, specific problem areas of a given method may be
Ambient wind velocity
pinpointed. The detailed itemization of uncertainty sources
Sampled concentration magnitude itself (for example, sorbent loading)
leads to a transparency in covering the essential problems of a
D7440 − 23
5.4.3 A quantity such as the temperature is known as an 6.2 Standardizing the characterization of sampling/
influence quantity. A common example where influence vari- analytical method performance is expected to be useful in other
ables are important involves diffusive monitors, where wind applications as well. For example, performance details are a
velocity, temperature, pressure, and fluctuating workplace necessity for justifying compliance decisions based on experi-
concentrations can affect diffusive monitor uptake rates (Prac- mental air quality assessments (7). Documented uncertainty
tice D6246, ISO 16107). can form a basis for specific criteria defining acceptable
5.4.4 Situations exist for which the distribution of an sampling/analytical method performance.
influence quantity is unknown. For example, the deviation
6.3 Furthermore, high quality atmospheric measurements
between aerosol concentration estimates and samples taken
are vital for making decisions as to how hazardous substances
according to accepted convention (for example, ISO 7708)
are to be controlled. Valid data are required for drawing
generally depend on the aerosol size distribution sampled.
reasonable epidemiological conclusions, for making sound
Only limits on the distribution of size distributions (the
decisions as to acceptable limits, as well as for determining the
influence quantity) may be known. In this case, the ISO GUM
efficacy of a hazard control system.
approach is generally to assume a uniform distribution (see
6.4 Finally, because of developing world-wide acceptance
7.4).
5.4.5 On the other hand, the size distribution may be known of ISO GUM for detailing measurements when statistics are
simple, the practice should be useful in comparing ASTM
to be constant over a set of measurements. In this case, the
constant-distribution assumption leads to an abstract perfor- International Test Methods to other published methods. The
codification of statistical procedures may in fact minimize the
mance characterization. Alternatively, a quantity known as the
symmetric accuracy range A (Appendix X1 and Section X4.2) difficulty in interpreting a plethora of individual, albeit possi-
bly valid, approaches.
in the case of unknown, but large limited |bias|, may be used to
establish intervals bracketing the (true) values of measurand
and thus represents the expanded uncertainty. 7. Specific Examples
NOTE 1—Some of the above concepts can be illuminated through
5.5 Combined and Expanded Uncertainty—The essential
example. Application to more complicated situations is then possible.
ISO GUM approach then is to obtain estimates u of the
j
7.1 Standard Deviation σ Known Exactly:
standard deviation (often designated as s as computed on most
handheld calculators) associated with the jth uncertainty 7.1.1 Suppose the method yields unbiased estimates m in
source. The estimates u may be designated as uncertainty measuring unknown M so that:
j
components. Then if the sources are independent, that is, if the
m 5 M1M·ε (4)
variations are uncorrelated, a combined standard uncertainty u
c
where ε is normally distributed about 0 with known standard
estimating the net standard deviation may be computed as:
deviation σ, sometimes designated the true relative standard
u 5 u (1)
c Œ( j deviation TRSD. For example, suppose the method has been
j
evaluated with essentially an infinite number of measurements
5.5.1 Finally, an expanded uncertainty U is calculated at
of a calibration standard, giving a tight estimate of σ. Then
coverage factor k as:
estimates m are distributed normally about M so that:
U 5 k·u (2)
c M 2 1.960 × M·σ,m,M11.960 × M·σ at probability 5 95 % (5)
5.5.2 The purpose of the expanded uncertainty U is to
7.1.2 Thus, to first order in σ, the true value M is bracketed
bracket the unknown measurand value. For example, for an
by:
unknown mass M, given an estimate m, a coverage factor could
m 2 1.960 × m·σ,M,m11.960 × m·σ at probability 5 95 % (6)
be selected so that:
m 2 U,M,m1U for 95 % of estimates m of measurand value M 7.1.3 Therefore, the (relative) expanded uncertainty U
would be consistent with Eq 3, if the coverage factor k is
(3)
chosen as:
5.5.3 However, this practice suggests use of the nearly
k 5 1.960 (7)
traditional value k = 2, permitting the meaning in terms of
confidence levels to float.
as a factor of combined standard uncertainty u :
c
u 5 σ (8)
6. Significance and Use
c
6.1 A primary use intended for this practice is for qualifying
in other words:
ASTM International Standards as Standard Test Methods. In
U 5 1.960 ×σ (9)
the past, a “Precision and Bias” report has been required.
However, recently a statement of uncertainty has become an
7.1.4 Eq 7 is consistent with the traditional selection k = 2.
acceptable alternative to Guide D3670. Inclusion of such a
NOTE 2—Although the measurement variation depicted in Eq 4 is very
statement with a method description simplifies comparison of
common in air quality measurements, at decreasing values of M, generally
ASTM Test Methods to analogous ISO and Committee for
a constant variation (that is, independent of M) becomes significant,
European Normalization (CEN) standards, now required to
leading to non-zero limits of quantitation and detection. (See, for example,
have uncertainty statements. ISO 15767 and Practice D6552.)
D7440 − 23
2 1/2
7.2 Standard Deviation σ Estimated Initially by n Replicates k 5 1.960 υ/χ (13)
~ !
υ 0.05
(Type B Uncertainty):
7.2.3 In Fig. 1 the coverage factor k of Eq 13 is plotted
7.2.1 Almost as simple as 7.1 is the situation in which a
versus degrees of freedom υ and is seen to approach 1.960 as
(relative) standard deviation estimate s is obtained through an
υ → ∞ corresponding to 7.1. However, Fig. 1 indicates that
initial n measurements of a calibration standard prior to the
over a wide range of degrees of freedom adopted in practical
method’s multiple subsequent uses without re-evaluation.
method evaluations, k is of the order of 3 in order to achieve
Variations of this situation are common in air quality measure-
95 % evaluation confidence.
ment. For example, diffusive samplers may be evaluated
initially by a vendor followed by many applications without
NOTE 3—Specification of an evaluation confidence level together with
coverage probability (both taken here to equal 95 %) relates to the
re-evaluation (see ISO 16107 or Practice D6246). Suppose Eq
statistical theory of tolerance or prediction intervals (5).
4-6 still hold, except that that now σ is unknown but is
estimated by s with υ = n – 1 degrees of freedom. What is
7.3 Continual Method Evaluation (Type A Uncertainty):
known is that σ is limited by:
7.3.1 Preferred, though often not practical in air quality
2 1/2
measurements, is an n-measurement calibration giving an
σ, υ/χ s (10)
~ !
υ 0.05
estimate s for σ with υ = n − 1 degrees of freedom every time
at 95 % confidence in the evaluation/calibration experiment,
a practical method is applied. Then it is possible to show that
where χ is the chi-square 5 % quantile at υ degrees of
υ 0.05
the true value M is bracketed by:
freedom (obtainable from statistics tables or programs).
m 2 t × m·s,M,m1t × m·s at probability 5 95 %
υ 0.975 υ 0.975
Therefore, at 95 % confidence in the evaluation, the unknown
M is bracketed by: (14)
2 1/2 2 1/2
m 2 1.960 υ/χ × m·s,M,m11.960 υ/χ × m·s
~ ! ~ !
υ 0.05 υ 0.05
where t is the student-t 97.5 % quantile at υ (also found
υ 0.975
(11)
in statistical sources).
7.3.2 Therefore the coverage factor k is now given by:
for greater than 95 % of measurements.
7.2.2 In this case, the combined (relative) uncertainty u is:
k 5 t (15)
c
υ 0.975
u 5 s (12)
c
7.3.3 In Fig. 1 this coverage factor is plotted versus the
but if the meaning of Eq 3 is sought, the coverage k factor in number υ of degrees of freedom in the evaluation. As can be
Eq 11 is now:
seen from the figure, with continual method evaluation, the
FIG. 1 Comparison of Coverage Factors k for Single Initial Method Evaluation versus Continual Evaluation with υ Degrees of Freedom
D7440 − 23
coverage factor is close to 2 over a range of values for υ. In
u 5Œ ∆ (20)
fact, this is the reason behind the now nearly traditional use of ∆ max
the value 2 for the coverage factor.
and again the random uncertainty component u is:
random
7.3.4 The use of the traditional coverage factor = 2 simply
gives intervals bracketing the unknown measurand with inter-
u 5 s (21)
random
pretation specific to the measurement circumstances. Of
with combined uncertainty u given by:
c
course, as alluded to in Section 3, if U is actually reported with
the traditional coverage factor 2, then, if needed, an expanded 2 2
u 5 =u 1u (22)
c ∆ random
uncertainty with 95 % evaluation confidence is easily obtained
and:
by multiplication (by about 3/2).
U 5 k u (23)
7.4 Uncertainty Characterization of Unknown Bias or Sys- c
tematic Error (Type B Uncertainty):
7.4.3 Finite-Calibration Uncertainty—Similarly to 7.4.2,
7.4.1 Unknown systematic error or bias in a measurement
correcting bias by a single n-mean estimate m of a reference
ref
may originate in several ways. For example, if a method is not
mass M (again with estimated corrected standard deviation
ref
re-calibrated at each application, then error from the finiteness
s) and then calibrating subsequent application measurements
of an initial calibration may be present as a non-random
by a calibration factor M /m leads to an uncertainty com-
ref ref
variable in subsequent applications. Even if re-calibrated, bias
ponent u given by:
n
may result from repeated use of a reference material or method,
1/2
u 5 s/n (24)
n
itself with unknown bias. In either case, the uncertainty
component corresponding to uncertain method bias may be
7.4.3.1 In this case the two components u and u are
n random
taken as the uncertainty in the bias itself.
not independent if u is given by Eq 21.
random
7.4.2 Reference Uncertainty—As an example, suppose a
NOTE 4—If an n-measurement calibration is effected at each application
method is repeatedly calibrated by a reference method that is
measurement, then the value in Eq 24 still appears as part of the
itself biased, though is negligibly variable (as example). Then
calibration uncertainty, but now refers to a random rather than systematic
the estimated mass in measuring unknown M may be repre-
variation.
sented as:
7.4.4 Large Bias Magnitude of Unknown Sign—There are
m 5 M 11∆ 1M·ε (16)
~ !
ref examples in air quality measurement where the range of
unknown bias may be large relative to the variable components
where the standard deviation estimate s for the normally
of uncertainty. For example, aerosol samplers used for mea-
distributed random variable ε may be obtained from the
suring dust concentrations according to one of the international
calibration experiment, and where ∆ is the unknown bias of
ref
sampling conventions (ISO 7708), for example, respirable,
the reference method. (See Appendix X2 for details on a
thoracic or inhalable, generally differ in particle-size accep-
similar situation, including finite-calibration bias.)
tance from convention. Therefore, in sampling a particular site
7.4.2.1 Suppose that all that is known about the reference
with aerosol of unknown particle size distribution range, an
bias ∆ is that it is bounded by a constant positive quantity
ref
unknown and sometimes large bias relative to convention is
∆ , often a matter of judgment, so that:
max
possible.
∆ ,∆ (17)
? ref? max 7.4.4.1 With large bias magnitude, the ISO GUM approach
of 7.4.2 of combining uncertainty components squared may be
7.4.2.2 ISO GUM generally suggests handling this situation
replaced by a linear combination of bias magnitude uncertainty
by approximating (evaluation to evaluation) ∆ as uniformly
ref
and variability uncertainty. On the other hand, the usual ISO
distributed between 6∆ . Then it is simple to compute an
max
GUM approach (with coverage factor k = 2) gives similar
inter-evaluation variance as:
uncertainty values. For an example, see Section X2.3.
Var ∆ 5 ∆ (18)
@ #
ref max 7.5 Analysis of a Round Robin Evaluation:
7.5.1 Analysis of a specific round robin evaluation of a
7.4.2.3 ∆ characterizes a shortcoming in the method
max
measurement method as applied by several independent labs is
evaluation, as does an imperfect initial d
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
´1
Designation: D7440 − 08 (Reapproved 2015) D7440 − 23
Standard Practice for
Characterizing Uncertainty in Air Quality Measurements
This standard is issued under the fixed designation D7440; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Editorial corrections were made throughout in July 2015.
1. Scope
1.1 This practice is for assisting developers and users of air quality methods for sampling concentrations of both airborne and
settled materials in characterizing measurements as to uncertainty. Where possible, analysis into uncertainty components as
recommended in the ISO International Organization for Standardization (ISO) Guide to the Expression of Uncertainty in
Measurement (ISO GUM, (1) ) is suggested. Aspects of uncertainty estimation particular to air quality measurement are
emphasized. For example, air quality assessment is often complicated by: the difficulty of taking replicate measurements owing
to the large spatio-temporal variation in concentration values to be measured; systematic error or bias, both corrected and
uncorrected; and the (rare) non-normal distribution of errors. This practice operates mainly through example. Background and
mathematical development are relegated to appendices for optional reading.
1.2 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.3 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety and healthsafety, health, and environmental practices and determine
the applicability of regulatory limitations prior to use.
1.4 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
D1356 Terminology Relating to Sampling and Analysis of Atmospheres
D3670 Guide for Determination of Precision and Bias of Methods of Committee D22
D6246 Practice for Evaluating the Performance of Diffusive Samplers
D6552 Practice for Controlling and Characterizing Errors in Weighing Collected Aerosols
2.2 Other International ISO Standards:
ISO GUM Guide to the Expression of Uncertainty in Measurement, ISO Guide 98, 1995 (See Ref (1), for an additional
measurement uncertainty resource.)
This practice is under the jurisdiction of ASTM Committee D22 on Air Quality and is the direct responsibility of Subcommittee D22.01 on Quality Control.
Current edition approved July 1, 2015Sept. 1, 2023. Published July 2015October 2023. Originally approved in 2008. Last previous edition approved in 20082015 as D7440
ɛ1
– 08. 08 (2015) . DOI: 10.1520/D7440-08R15E01.10.1520/D7440-23.
The boldface numbers in parentheses refer to the list of references at the end of this standard.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
BIPM version available for download from http://www.bipm.org/en/publications/guides/gum.html. ISO version available from American National Standards Institute
(ANSI), 25 W. 43rd St., 4th Floor, New York, NY 10036, http://www.ansi.org.
See Ref (1), for an additional measurement uncertainty resource.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D7440 − 23
ISO 7708 Air Quality—ParticleQuality — Particle Size Fraction Definitions for Health-Related Sampling
ISO 15767 Workplace Atmospheres—ControllingAtmospheres — Controlling and Characterizing Errors in Weighing Collected
Aerosol
ISO 16107 Workplace Atmospheres—ProtocolAtmospheres — Protocol for Evaluating the Performance of Diffusive Samplers,
2007Samplers
EN 482ISO 16702 Workplace Atmospheres—General Requirements for the Performance of Procedures for the Measurement of
Chemical Agentsair quality — Determination of total organic isocyanate groups in air using 1-(2-methoxyphenyl)piperazine
and liquid chromatography
3. Terminology
3.1 Definitions—For definitions of terms used in this practice, see Terminology D1356.
3.2 Other terms defined as follows are taken from ISO GUM unless otherwise noted:Terms Defined as Follows are Taken from
ISO GUM, Unless Otherwise Noted:
3.2.1 accuracy—closeness of agreement between the result of a measurement and a true value of the measurand.
3.2.2 combined standard uncertainty, u —standard uncertainty of the result of a measurement when that result is obtained from
c
the values of a number of other quantities, equal to the positive square root of a sum of terms, the terms being the variances or
covariances of these other quantities weighted according to how the measurement result varies with changes in these quantities.
3.2.2.1 Discussion—
As within ISO GUM, the “other quantities” are designated uncertainty components u from source j. The component u is taken
j j
as the standard deviation estimate from source j in the case of a source of random variation.
3.2.3 coverage factor, k—numerical factor used as a multiplier of the combined standard uncertainty (u ) in order to obtain an
c
expanded uncertainty (U).
3.2.3.1 Discussion—
The factor k depends on the specific meaning attributed to the expanded uncertainty U. However, for simplicity this practice adopts
the now nearly traditional coverage factor as the value 2, determining the specific meaning of the expanded uncertainty U in
different circumstances. Other coverage factors if needed are then easily implemented simply by multiplication of the traditional
expanded 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 conservative use of individual component
confidence limits rather than root variance estimates as uncertainty components.
3.2.4 error (of measurement)—result of a measurement minus a true value of the measurand.
3.2.5 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.
3.2.5.1 Discussion—
This definition has the breadth to encompass a wide variety of conceptions.
3.2.5.2 Discussion—
The expanded uncertainty U in some cases is expressed in absolute terms, but sometimes as relative to the measurement result.
What is meant is generally clear from the context.
3.2.6 influence quantity—quantity that is not the measurand but that affects the result of the measurement.
3.2.7 measurand—particular quantity subject to measurement.
3.2.8 measurand value—(adapted from ISO GUM), unknown quantity whose measurement is sought, often called the true value.
Examples are the concentration (mg/m ) of a substance in the air at a particular time and place, the time-weighted average of a
concentration at a particular position, or the expected mean concentration estimate as obtained by a reference method at a specific
time and position.
3.2.8.1 Discussion—
Examples are the concentration
D7440 − 23
(mg/m ) of a substance in the air at a particular time and place, the time-weighted average of a concentration at a particular
position, or the expected mean concentration estimate as obtained by a reference method at a specific time and position.
3.2.9 (population) variance (of a random variable)—the expectation of the square of the centered random variable.
3.2.10 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 the same (repeatability) conditions of measurement.
3.2.10.1 Discussion—
Random error is equal to error minus systematic error.
3.2.11 (sample) variance—the sum of the squared deviations of observations from their average divided by one less than the
number of observations.
3.2.11.1 Discussion—
The sample variance is an unbiased estimator of the population variance.
3.2.12 standard deviation—positive square root of the variance.
3.2.13 symmetric accuracy range A—the range symmetric about (true) measurand values containing 95 % of measurement
estimates. A is a specific quantification of accuracy.(2)
3.2.13.1 Discussion—
A is a specific quantification of accuracy(2). ISO 16107
3.2.14 systematic error (bias)—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.
3.2.15 Type A evaluation (of uncertainty)—method of evaluation of uncertainty by the statistical analysis of series of observations.
3.2.16 Type B evaluation (of uncertainty)—method of evaluation of uncertainty by means other than the statistical analysis of
series of observations.
4. Background Information
4.1 Uncertainty in a measurement result can be taken as the range about an estimate, corrected for bias if known, containing the
true, or mean reference value—in the language of ISO GUM, the measurand value at given confidence. Uncertainty accounts not
only for variation in a method’s results at application, but also for incomplete characterization of the method when evaluated. In
accordance with ISO GUM, uncertainty may often usefully be analyzed into individual components.
4.2 There are several aspects of uncertainty characterization specific to air quality measurements. One of these aspects concerns
known, that is, correctible, systematic error or mean bias of a measurement relative to a true measurand value. Several
measurement methods exist with such bias left uncorrected because of policy, tradition, or other reason. Uncertainty deals only
with what is unknown about a measurement, and as such does not include correctible (known) bias. The magnitude of the
difference between estimate and measurand value is covered by accuracy as defined qualitatively in ISO GUM, rather than
uncertainty, particularly when the bias is known, but uncorrected. Such methods require specification of both uncertainty and as
much as is known of the uncorrected bias, or alternatively the adoption of an accuracy measure.
4.3 Often bias is known to exist, but with unknown value. In the case where only limits may be placed on the magnitude of the
bias, ISO GUM generally recommends treating the bias as uniformly distributed within the known limits. Such a distribution 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. Even though such an equal-likelihood bias distribution may be unrealistic, nevertheless a standard
deviation estimate may be made that reveals the limits on the bias. If the even-distribution approximation is clearly invalid for a
relevant set of measurements, the procedure may be adjusted slightly by adopting an accuracy measure tailored to the assumed
limits.
4.4 Another issue concerns the distribution of measurements. 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 distributed data confounded
D7440 − 23
by a small number of apparent outliers (3), which may not detract from the method performance (see Appendix X4 for details).
Another example is the determination of an aerosol concentration at one location (perhaps at a worker’s lapel) as an estimate of
the concentration at a separate point (such as a breathing zone). In this case the variations can be of the order of the estimate itself
and may have the character of a log-normal distribution.
4.5 The spatial inhomogeneity alluded to in 4.4 relates to another point regarding the focus of this practice. The spatio-temporal
variations in air quality characteristics are generally so large (4) as to preclude evaluation of a method during application through
the use of replicate measurements. In this case, often an initial single method evaluation is undertaken with the purpose of
determining uncertainty present in subsequent applications of the method. Confidence in such an evaluation can be specified and
relates to the concept of prediction-intervals (5) (see 7.2).
4.6 A related subject is measurement system control. The measurement system must remain in a state of statistical control if an
introductory evaluation is to characterize later practical applications of the method. Measurement system control is evaluated using
an ongoing quality control program, testing critical performance aspects for detecting problems which may develop in the method.
5. Summary of Practice
5.1 The essential idea behind ISO GUM is the analysis to the fullest extent practical of the elemental sources of what is unknown
in the estimate of a measurand value. This contrasts with a global or top-down determination of uncertainty, which could for
example be done ideally by comparing replicate estimates to known measurand values over all conditions expected in application
of the method. Although a global uncertainty evaluation may sometimes seem inexpensive, there is a difficulty in covering essential
contingencies of the method application.
5.2 Uncertainty component analysis further has several specific advantages over global analysis. The results may be applicable to
a variety of situations. For example, an aerosol sampler might be (globally) evaluated as to particle-size-dependent error by
side-by-side comparison to a reference sampler in several coal mines. The knowledge obtained may not be as easily applied for
sampler use in iron mines, for example, as more detailed information on how the sampler performs over given dust size
distributions may be needed. Furthermore, specific problem areas of a given method may be pinpointed. The detailed itemization
of uncertainty sources leads to a transparency in covering the essential problems of a measurement method. Examples of
potentially significant uncertainty components are listed in Table 1.
5.3 Type A and B Uncertainty Components:
5.3.1 Components that have been statistically evaluated during method application may be classified as Type A. (See Section 7
for specific examples.)
5.3.2 Some components are often statistically evaluated during an initial method evaluation, rather than at application. Also
acknowledged is a common situation that components may not have been characterized in a statistically valid manner and therefore
may require professional judgment for itemizing. Such components are termed Type B uncertainties. Type B uncertainties are often
associated with unknown systematic error or bias; however, random variation may also fall into this category. For example, a
common assumption (see, for example, EN 482) regarding personal sampling in the workplace is that the relative standard
deviation associated with personal sampling pump variations is <5 % at essentially 100 % confidence.
5.4 Intrinsic versus Environmentally Associated Components: Influence Quantities:
5.4.1 Some uncertainties may be intrinsic to a method. For example, estimates from aerosol samplers may depend critically on
sampler dimensions, which if variable leads to intersampler estimate variation.
5.4.2 On the other hand, a sampler’s performance may depend on the environment. For example, suppose a sampler is sensitive
to temperature changes that are impractical to measure in the field; that is, sampler estimates are not temperature-corrected. Then
measurement of this sensitivity during method evaluation together with knowledge of the temperature variation expected for a
given field application can be used to determine the uncertainty associated with this effect.
5.4.3 A quantity such as the temperature is known as an influence quantity. A common example where influence variables are
important involves diffusive monitors, where wind velocity, temperature, pressure, and fluctuating workplace concentrations can
affect diffusive monitor uptake rates (Practice D6246, ISO 16107).
D7440 − 23
TABLE 1 Common Potential Uncertainty Components
Sampling
personal sampling pump flow rate: setting the pump and subsequent drift
Personal sampling pump flow rate: setting the pump and subsequent drift
sampling rate of diffusive sampler
Sampling rate of diffusive sampler
sampler dimension (aerosol and diffusive sampling)
Sampler dimension (aerosol and diffusive sampling)
collection efficiency of a sampler or sampling medium
Collection efficiency of a sampler or sampling medium
(also, see (6))
Analytical
aerosol weighing
Aerosol weighing
recovery (for example, chromatographic or spectroscopic methods)
Recovery (for example, chromatographic or spectroscopic methods)
Poisson counting (for example, in XRD methods)
instrument or sensor variation
Instrument or sensor variation
operator effects giving inter-lab differences (if data from several labs are to
be used)
Operator effects giving inter-lab differences (if data from several labs are to
be used)
Sample
sample stability
Sample stability
sample preparation (for example, handling silica quasi-suspensions)
Sample preparation (for example, handling silica quasi-suspensions)
sample loss during transport or storage
Sample loss during transport or storage
Evaluation
calibration material uncertainty
Calibration material uncertainty
evaluation chamber concentration uncertainty
Evaluation chamber concentration uncertainty
other bias-correction uncertainty
Other bias-correction uncertainty
Environmental Influence Parameters
temperature (inadequacy of correction, if correction is made as with diffusive
samplers)
Temperature (inadequacy of correction, if correction is made as with diffusive
samplers)
atmospheric pressure
Atmospheric pressure
humidity
Humidity
aerosol size distribution (if not measured by a given aerosol sampling method)
Aerosol size distribution (if not measured by a given aerosol sampling method)
ambient wind velocity
Ambient wind velocity
sampled concentration magnitude itself (for example, sorbent loading)
Sampled concentration magnitude itself (for example, sorbent loading)
5.4.4 Situations exist for which the distribution of an influence quantity is unknown. For example, the deviation between aerosol
concentration estimates and samples taken according to accepted convention (for example, ISO 7708) generally depend on the
aerosol size distribution sampled. Only limits on the distribution of size distributions (the influence quantity) may be known. In
this case, the ISO GUM approach is generally to assume a uniform distribution (see 7.4).
5.4.5 On the other hand, the size distribution may be known to be constant over a set of measurements. In this case, the
constant-distribution assumption leads to an abstract performance characterization. Alternatively, a quantity known as the
symmetric accuracy range A (Appendix X1 and Section X4.2) in the case of unknown, but large limited |bias|, may be used to
establish intervals bracketing the (true) values of measurand and thus represents the expanded uncertainty.
5.5 Combined and Expanded Uncertainty—The essential ISO GUM approach then is to obtain estimates u of the standard
j
deviation (often designated as s as computed on most handheld calculators) associated with the jth uncertainty source. The
estimates u may be designated as uncertainty components. Then if the sources are independent, that is, if the variations are
j
uncorrelated, a combined standard uncertainty u estimating the net standard deviation may be computed as:
c
u 5 u (1)
c Œ( j
j
D7440 − 23
5.5.1 Finally, an expanded uncertainty U is calculated at coverage factor k as:
U 5 k·u (2)
c
5.5.2 The purpose of the expanded uncertainty U is to bracket the unknown measurand value (for example, value. For example,
for an unknown mass M,) given an estimate m.m, For example, a coverage factor could be selected so that:
m 2 U,M,m1U for 95 % of estimates m of measurand value M (3)
5.5.3 However, this practice suggests use of the nearly traditional value k = 2, permitting the meaning in terms of confidence levels
to float.
6. Significance and Use
6.1 A primary use intended for this practice is for qualifying ASTM International Standards as Standard Test Methods. In the past,
a “Precision and Bias” report has been required. However, recently a statement of uncertainty has become an acceptable alternative
to D3670 – 91:Guide D3670Guide for Determination of Precision and Bias of Methods of Committee D22. . Inclusion of such a
statement with a method description simplifies comparison of ASTM Test Methods to analogous ISO and CEN Committee for
European Normalization (CEN) standards, now required to have uncertainty statements.
6.2 Standardizing the characterization of sampling/analytical method performance is expected to be useful in other applications
as well. For example, performance details are a necessity for justifying compliance decisions based on experimental air quality
assessments (7). Documented uncertainty can form a basis for specific criteria defining acceptable sampling/analytical method
performance.
6.3 Furthermore, high quality atmospheric measurements are vital for making decisions as to how hazardous substances are to be
controlled. Valid data are required for drawing reasonable epidemiological conclusions, for making sound decisions as to
acceptable limits, as well as for determining the efficacy of a hazard control system.
6.4 Finally, because of developing world-wide acceptance of ISO GUM for detailing measurements when statistics are simple, the
practice should be useful in comparing ASTM International Test Methods to others’other published methods. The codification of
statistical procedures may in fact minimize the difficulty in interpreting a plethora of individual, albeit possibly valid, approaches.
7. Specific Examples
NOTE 1—Some of the above concepts can be illuminated through example. Application to more complicated situations is then possible.
7.1 Standard Deviation σ Known Exactly:
7.1.1 Suppose the method yields unbiased estimates m in measuring unknown M so that:
m 5 M1M·ε (4)
where ε is normally distributed about 0 with known standard deviation σ, sometimes designated the true relative standard
deviation TRSD. For example, suppose the method has been evaluated with essentially an infinite number of measurements of a
calibration standard, giving a tight estimate of σ. Then estimates m are distributed normally about M so that:
M 2 1.960 ×M·σ,m,M11.960 ×M·σ at probability 5 95 % (5)
7.1.2 Thus, to first order in σ, the true value M is bracketed by:
m 2 1.960 ×m·σ,M,m11.960 ×m·σ at probability 5 95 % (6)
7.1.3 Therefore, the (relative) expanded uncertainty U would be consistent with Eq 3, if the coverage factor k is chosen as:
k 51.960 (7)
as a factor of combined standard uncertainty u :
c
u 5σ (8)
c
D7440 − 23
in other words:
U 5 1.960 ×σ (9)
7.1.4 Eq 7 is consistent with the traditional selection k = 2.
NOTE 2—Although the measurement variation depicted in Eq 4 is very common in air quality measurements, at decreasing values of M, generally a
constant variation (that is, independent of M) becomes significant, leading to non-zero limits of quantitation and detection. (See, for example, ISO 15767
and Practice D6552.)
7.2 Standard Deviation σ Estimated Initially by n Replicates (Type B Uncertainty):
7.2.1 Almost as simple as 7.1 is the situation in which a (relative) standard deviation estimate s is obtained through an initial n
measurements of a calibration standard prior to the method’s multiple subsequent uses without re-evaluation. Variations of this
situation are common in air quality measurement. For example, diffusive samplers may be evaluated initially by a vendor followed
by many applications without re-evaluation (see ISO 16107 or Practice D6246). Suppose Eq 4-6 still hold, except that that now
σ is unknown but is estimated by s with υ = n – 1 degrees of freedom. What is known is that σ is limited by:
2 1/2
σ, υ/χ s (10)
~ !
υ 0.05
at 95 % confidence in the evaluation/calibration experiment, where χ is the chi-square 5 % quantile at υ degrees of freedom
υ 0.05
(obtainable from statistics tables or programs). Therefore, at 95 % confidence in the evaluation, the unknown M is bracketed by:
2 1/2 2 1/2
m 2 1.960~υ/χ ! ×m·s,M,m11.960~υ/χ ! ×m·s (11)
υ 0.05 υ 0.05
for greater than 95 % of measurements.
7.2.2 In this case, the combined (relative) uncertainty u is:
c
u 5 s (12)
c
but if the meaning of Eq 3 is sought, the coverage k factor in Eq 11 is now:
2 1/2
k 5 1.960 υ/χ (13)
~ !
υ 0.05
7.2.3 In Fig. 1 the coverage factor k of Eq 13 is plotted versus degrees of freedom υ and is seen to approach 1.960 as υ → ∞
corresponding to 7.1. However, Fig. 1 indicates that over a wide range of degrees of freedom adopted in practical method
evaluations, k is of the order of 3 in order to achieve 95 % evaluation confidence.
NOTE 3—Specification of an evaluation confidence level together with coverage probability (both taken here to equal 95 %) relates to the statistical theory
of tolerance or prediction intervals (5).
7.3 Continual Method Evaluation (Type A Uncertainty):
7.3.1 Preferred, though often not practical in air quality measurements, is an n-measurement calibration giving an estimate s for
σ with υ = n − 1 degrees of freedom every time a practical method is applied. Then it is possible to show that the true value M
is bracketed by:
m 2 t ×m·s,M,m1t ×m·s at probability 5 95 % (14)
υ 0.975 υ 0.975
where t is the student-t 97.5 % quantile at υ (also found in statistical sources).
υ 0.975
7.3.2 Therefore the coverage factor k is now given by:
k 5 t (15)
υ 0.975
7.3.3 In Fig. 1 this coverage factor is plotted versus the number υ of degrees of freedom in the evaluation. As can be seen from
the figure, with continual method evaluation, the coverage factor is close to 2 over a range of values for υ. In fact, this is the reason
behind the now nearly traditional use of the value 2 for the coverage factor.
7.3.4 The use of the traditional coverage factor = 2 simply gives intervals bracketing the unknown measurand with interpretation
D7440 − 23
FIG. 1 Comparison of Coverage Factors k for Single Initial Method Evaluation versus Continual Evaluation with υ Degrees of Freedom
specific to the measurement circumstances. Of course, as alluded to in Section 3, if U is actually reported with the traditional
coverage factor 2, then, if needed, an expanded uncertainty with 95 % evaluation confidence is easily obtained by multiplication
(by about 3/2).
7.4 Uncertainty Characterization of Unknown Bias or Systematic Error (Type B Uncertainty):
7.4.1 Unknown systematic error or bias in a measurement may originate in several ways. For example, if a method is not
re-calibrated at each application, then error from the finiteness of an initial calibration may be present as a non-random variable
in subsequent applications. Even if re-calibrated, bias may result from repeated use of a reference material or method, itself with
unknown bias. In either case, the uncertainty component corresponding to uncertain method bias may be taken as the uncertainty
in the bias itself.
7.4.2 Reference Uncertainty—As an example, suppose a method is repeatedly calibrated by a reference method that is itself biased,
though is negligibly variable (as example). Then the estimated mass in measuring unknown M may be represented as:
m 5 M~11∆ !1M·ε (16)
ref
where the standard deviation estimate s for the normally distributed random variable ε may be obtained from the calibration
experiment, and where ∆ is the unknown bias of the reference method. (See Appendix X2 for details on a similar situation,
ref
including finite-calibration bias.)
7.4.2.1 Suppose that all that is known about the reference bias ∆ is that it is bounded by a constant positive quantity ∆ , often
ref max
a matter of judgment, so that:
∆ ,∆ (17)
? ref? max
7.4.2.2 ISO GUM generally suggests handling this situation by approximating (evaluation to evaluation) ∆ as uniformly
ref
distributed between 6∆ . Then it is simple to compute an inter-evaluation variance as:
max
D7440 − 23
Var@∆ # 5 ∆ (18)
ref max
7.4.2.3 ∆ characterizes a shortcoming in the method evaluation, as does an imperfect initial determination of σ, the standard
max
deviation of ε (see 7.2). Thus, assuring confidence (for example, 95 %) in the calibration with the same (prediction or tolerance)
sense as in 7.2, a coverage factor k can be selected so that an expanded uncertainty given by:
2 2
U 5 kŒ ∆ 1s (19)
max
brackets the unknown M for a high fraction (for example, 95 %) of measurements (see Appendix X2 for details).
7.4.2.4 In other words, the uncertainty component u for the bias is:
∆
u 5 ∆ (20)
Œ
∆ max
and again the random uncertainty component u is:
random
u 5 s (21)
random
with combined uncertainty u given by:
c
2 2
u 5=u 1u (22)
c ∆ random
and:
U 5 k u (23)
c
7.4.3 Finite-Calibration Uncertainty—Similarly to 7.4.2, correcting bias by a single n-mean estimate m of a reference mass M
ref ref
(again with estimated corrected standard deviation s) and then calibrating subsequent application measurements by a calibration
factor M /m leads to an uncertainty component u given by:
ref ref n
1/2
u 5 s/n (24)
n
7.4.3.1 In this case the two components u and u are not independent if u is given by Eq 21.
n random random
NOTE 4—If an n-measurement calibration is effected at each application measurement, then the value in Eq 24 still appears as part of the calibration
uncertainty, but now refers to a random rather than systematic variation.
7.4.4 Large Bias Magnitude of Unknown Sign—There are examples in air quality m
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