ASTM D8293-22

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: D8293 − 22
Standard Guide for
Evaluating and Expressing the Uncertainty of
Radiochemical Measurements
This standard is issued under the fixed designation D8293; 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
1.1 This guide provides concepts, terminology, symbols, 2.1 ASTM Standards:
and recommendations for the evaluation and expression of the D1129 Terminology Relating to Water
uncertainty of radiochemical measurements of water and other D7902 Terminology for Radiochemical Analyses
environmental media by testing laboratories. It applies to E177 Practice for Use of the Terms Precision and Bias in
measurements of radionuclide activities, including gross ASTM Test Methods
activities, regardless of whether they involve chemical prepa- E288 Specification for Laboratory Glass Volumetric Flasks
ration of the samples. E438 Specification for Glasses in Laboratory Apparatus
E456 Terminology Relating to Quality and Statistics
1.2 This guide does not provide a complete tutorial on
E542 Practice for Gravimetric Calibration of Laboratory
measurement uncertainty. Interested readers should refer to the
Volumetric Instruments
documents listed in Section 2 and References for more
E617 Specification for Laboratory Weights and Precision
information. See, for example, GUM, QUAM, Taylor and
Mass Standards
Kuyatt (1) , and Chapter 19 of MARLAP (2).
E898 Practice for Calibration of Non-Automatic Weighing
1.3 The system of units for this guide is not specified.
Instruments
Dimensional quantities in the guide are presented only as
E969 Specification for Glass Volumetric (Transfer) Pipets
illustrations of calculation methods. The examples are not
E1272 Specification for Laboratory Glass Graduated Cylin-
binding on products or test methods treated.
ders
1.4 This standard does not purport to address all of the
E2655 Guide for Reporting Uncertainty of Test Results and
safety concerns, if any, associated with its use. It is the
Use of the Term Measurement Uncertainty in ASTM Test
responsibility of the user of this standard to establish appro- Methods
priate safety, health, and environmental practices and deter-
2.2 ANSI Standards:
mine the applicability of regulatory limitations prior to use.
ANSI N42.23 Measurement and Associated Instrumentation
1.5 This international standard was developed in accor-
Quality Assurance for Radioassay Laboratories
dance with internationally recognized principles on standard-
2.3 BIPM Documents:
ization established in the Decision on Principles for the
GUM: JCGM 100:2008 Evaluation of measurement data—
Development of International Standards, Guides and Recom- 5
Guide to the expression of uncertainty in measurement
mendations issued by the World Trade Organization Technical
JCGM 101:2008 Evaluation of measurement data—
Barriers to Trade (TBT) Committee.
1 3
This guide is under the jurisdiction of ASTM Committee D19 on Water and is For referenced ASTM standards, visit the ASTM website, www.astm.org, or
the direct responsibility of Subcommittee D19.04 on Methods of Radiochemical contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Analysis. Standards volume information, refer to the standard’s Document Summary page on
Current edition approved July 15, 2022. Published February 2023. Originally the ASTM website.
approved in 2019. Last previous edition approved in 2019 as D8293 – 19. DOI: Available from American National Standards Institute (ANSI), 25 W. 43rd St.,
10.1520/D8293-22. 4th Floor, New York, NY 10036, http://www.ansi.org.
2 5
The boldface numbers in parentheses refer to the list of references at the end of Available from www.bipm.org/utils/common/documents/jcgm/JCGM_100_
this standard. 2008_E.pdf, accessed January 2021.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D8293 − 22
Supplement 1 to the “Guide to the expression of uncer- 3.2.2 subsampling factor, F , n—ratio of the massic or
S
tainty in measurement”—Propagation of distributions us- volumic activity of a subsample to that of the sample from
ing a Monte Carlo method which it is taken.
JCGM 102:2011 Evaluation of measurement data—
4. Summary of Practice
Supplement 2 to the “Guide to the expression of uncer-
tainty in measurement”—Extension to any number of 4.1 General rules and recommendations for evaluating and
quantities
expressing measurement uncertainty are given, followed by
JCGM 200:2008 International vocabulary of metrology— more detailed discussions of uncertainty evaluation,
Basic and general concepts and associated terms (VIM)
propagation, and reporting. Topics include Type A and Type B
2.4 OIML Documents: evaluations of uncertainty, correlations, coverage factors,
OIML D 28: 2004 (E) Conventional value of the result of
rounding rules, and shorthand formats for expressing uncer-
weighing in air tainty. Guidelines for determining the practical significance of
2.5 Eurachem Guides:
uncertainty components are presented. Next, some of the most
QUAM Quantifying Uncertainty in Analytical commonly encountered components of uncertainty in radio-
Measurement, Eurachem/CITAC Guide CG 4, Third edi- chemical measurements are discussed, with suggested methods
tion
of evaluation and examples. Topics include counting
uncertainty, background, chemical yield, counting efficiency
3. Terminology
(calibration), aliquot sizes, decay and ingrowth factors, and
3.1 Definitions: subsampling. A few other miscellaneous topics, such as the
3.1.1 conventional mass, n—property of a body equal to the calculation of weighted averages with uncertainties and non-
mass of a standard of density 8000 kg/m that exactly balances Poisson counting, are also included. Special topics such as
that body when weighed in air of density 1.2 kg/m at 20 °C, uncertainty budgets and evaluation of uncertainties for mass
as defined in Specification E617 and International Document measurements are presented in the appendices, followed by
OIML D 28. several applications and worked-out examples.
3.1.2 index of dispersion, J, n—ratio of the variance of a
5. Significance and Use
random variable or probability distribution to its mean; also
5.1 This guide is intended to help testing laboratories and
called simply the variance-to-mean ratio.
the developers of methods and software for those laboratories
3.1.3 normalized absolute difference, NAD, n—absolute
to apply the concepts of measurement uncertainty to radio-
value of the normalized difference.
chemical analyses.
3.1.4 normalized difference, n—quotient of the difference
5.2 The result of a laboratory measurement never exactly
between two measured values and the combined standard
equals the true value of the measurand. The difference between
uncertainty of that difference.
the two is called the error of the measurement. An estimate of
3.1.4.1 Discussion—The normalized difference is similar to
the possible magnitude of this error is called the uncertainty of
a zeta score (ζ score) as that term is commonly used in
the measurement. While the error is primarily a theoretical
proficiency testing. Other terms may be used for the same
concept, since its value is never known, the uncertainty has
concept.
practical uses. Together, the measured value and its uncertainty
3.1.5 relative sensitivity factor, n—ratio of the relative
allow one to place bounds on the likely true value of the
change in an output quantity to a small relative change in a
measurand.
specified input quantity.
5.3 Reliable measurement-based decision making requires
3.1.6 For definitions of many other terms used in this guide,
not only measured values but also an indication of their
refer to Terminology D1129, Terminology D7902, Practice
uncertainty. Traditionally, significant figures have been used
E177, Terminology E456, Guide E2655, Test Method E898,
with varying degrees of success to indicate implicitly the order
JCGM 200, and the GUM.
of magnitude of measurement uncertainties; however, report-
3.2 Definitions of Terms Specific to This Standard:
ing an explicit uncertainty estimate with each result is more
3.2.1 minimum detectable value, n—smallest true value of a
reliable and informative, and is considered an industry-
nonnegative statistical parameter that ensures a specified high
standard best practice.
probability of a positive result in a specified hypothesis test for
that parameter.
6. Procedure
6.1 General Rules and Recommendations:
Available from www.bipm.org/utils/common/documents/jcgm/JCGM_101_
6.1.1 Whenever a laboratory reports the result of a radio-
2008_E.pdf, accessed January 2021.
analytical measurement, the report should include an explicit
Available from www.bipm.org/utils/common/documents/jcgm/JCGM_102_
estimate of the measurement uncertainty. The measured value
2011_E.pdf, accessed January 2021.
and its uncertainty together constitute the overall result of the
Available from www.bipm.org/utils/common/documents/jcgm/JCGM_200_
2012.pdf, accessed January 2021.
measurement. General guidance for evaluating and expressing
Available from www.oiml.org/en/files/pdf_d/d028-e04.pdf, accessed January
measurement uncertainty is provided in the GUM and in Guide
2021.
E2655. Supplemental guidance is given in JCGM 101 and
Available from eurachem.org/index.php/publications/guides/quam, accessed
January 2021. JCGM 102. More specific guidance for radiochemical
D8293 − 22
measurements, including a list of recommended practices for 6.2.1 Let the measurement model be given abstractly by the
radiation laboratories, can be found in MARLAP (2). Guidance equation:
for chemical measurement laboratories, much of which is also
Y 5 f~X ,X ,…,X ! (1)
1 2 N
applicable to radiochemistry, is provided in QUAM. Ref (3)
where Y denotes the output quantity, which is also the
also provides a set of detailed examples related to radiochemi-
measurand, X , X , …, X denote the input quantities, and f is
cal analyses.
1 2 N
the measurement function. In practice the measurement model
6.1.2 Laboratories performing radiochemical analyses
may be implemented as one or more equations—for example,
should follow the guidance of the GUM and its supplements,
in a spreadsheet or specialized software application. What
which provide standards for terminology, notation, and meth-
matters is that there are unambiguous rules for calculating the
odology. The use of standard terminology and notation pro-
output quantity from the input quantities. For a less abstract
motes clear communication between laboratories and their
example of a measurement model, see Eq 38 in 6.11.
clients. Furthermore, the use of common methodologies pro-
motes comparability of results and effective decision-making
NOTE 1—The distinction between input quantities and output quantities
based on those results.
depends on context. An input quantity in one measurement may be an
output quantity from another measurement.
6.1.3 Generally, the reported uncertainty represents an esti-
mate of the “total uncertainty” of the measurement, which
6.2.2 When the laboratory makes a measurement, it finds
accounts for all significant sources of inaccuracy in the result.
particular values x , x , …, x for the input quantities. These
1 2 N
However, for legal or contractual reasons, a laboratory may
values may be called input estimates. The lab applies the
sometimes be required to report only a partial uncertainty
measurement function to the input estimates to calculate the
estimate. For example, U.S. laboratories analyzing drinking
output estimate.
water for compliance with the U.S. EPA’s National Primary
y 5 f~x ,x ,…,x ! (2)
1 2 N
Drinking Water Regulations may be required to report the
counting uncertainty for each result (see 6.12).
This output estimate y is the measured value.
6.1.4 The estimate of total uncertainty accounts for both
NOTE 2—In Eq 1, upper-case symbols (Y or X ) denote random variables
i
random and systematic effects in the measurement process, but
or abstract quantities, while in Eq 2, lower-case symbols (y or x ) denote
i
not spurious errors such as those due to instrument malfunc-
particular values of those random variables or quantities. This distinction
tions and human blunders, which represent a loss of statistical
is maintained when describing techniques for uncertainty evaluation and
propagation; however, in most applications of these equations, the
control of the process. Statistical control of the measurement
distinction is dropped, and the same symbols are used for both the random
process is a prerequisite for meaningful uncertainty evalua-
variables and their values.
tions.
6.2.3 When the laboratory determines each input estimate
6.1.5 The uncertainty of a measured value should always be
x , it determines the associated standard uncertainty u(x ), as
positive, never zero or negative.
i i
described in 6.3. If necessary, the lab also estimates the
6.1.6 Typically, the result of a radiochemical analysis is not
covariance of any pair of correlated input estimates, x and x .
measured directly but is instead calculated from other mea- i j
Given this information, the laboratory then mathematically
sured quantities, called input quantities, using a mathematical
combines the uncertainties and covariances using standard
model of the measurement process. In this context, the final
techniques for uncertainty propagation (6.4 – 6.6) to obtain the
calculated result is called the output quantity. The uncertainty
combined standard uncertainty u (y).
of each input is first estimated in the form of a standard c
6.2.4 The laboratory optionally multiplies u (y) by a cover-
deviation, called the standard uncertainty. The laboratory then
c
age factor k, described in 6.7, to obtain an expanded uncer-
obtains the standard uncertainty of the final result by combin-
tainty U. It then rounds and reports the result y with either the
ing the standard uncertainties of the inputs according to general
combined standard uncertainty u (y) or the expanded uncer-
mathematical rules applied to the measurement model. Math-
c
tainty U, as described in 6.8.
ematically combining uncertainties in this manner is called
propagation of uncertainty. A standard uncertainty obtained by
6.3 Evaluating Measurement Uncertainties:
uncertainty propagation is also called a combined standard
6.3.1 The GUM classifies methods for direct evaluation of
uncertainty.
uncertainty as either Type A or Type B. A Type A evaluation is
6.1.7 The laboratory may report the uncertainty of the result
an evaluation of standard uncertainty by the statistical analysis
either as the combined standard uncertainty or a specified
of one or more series of observations. By definition, any
multiple thereof, called an expanded uncertainty. The analysis
evaluation of standard uncertainty that is not Type A is Type B.
report should always specify which type of uncertainty is being
6.3.2 An uncertainty evaluated by a Type A method may be
reported, and if it is an expanded uncertainty, the report should
called a Type A uncertainty, and an uncertainty evaluated by a
specify the multiplicative factor, called the coverage factor and
Type B method may be called a Type B uncertainty. However,
denoted by k. For an expanded uncertainty, the report should
the rules of uncertainty propagation make no distinction
also state the approximate coverage probability, defined as the
between the two types: all uncertainties are propagated in the
probability p that the interval about the measured value
same manner.
described by its expanded uncertainty will contain the true
6.3.3 Any Type A evaluation of uncertainty has a well-
value of the measurand.
defined number of statistical degrees of freedom, as indicated
6.2 Overview of Procedure: in the examples that follow.
D8293 − 22
6.3.4 One of the simplest examples of a Type A evaluation uncertainty evaluation. On the other hand, the use of s(q )
k
of uncertainty is the estimation of the standard uncertainty of a presumes that all the observations q as well as future obser-
k
measured value q by the experimental standard deviation of vations from the same measurement process have the same
repeated observations made in the same manner. If the ob- variance.
served values are q , q , …, q , the arithmetic mean (average)
1 2 n
6.3.8 A Type A evaluation of the covariance of two mea-
and the experimental standard deviation are given by:
sured quantities involves a statistical analysis of a series of
n n
paired observations of those quantities. To evaluate the experi-
1 1
‾q 5 q and s q 5 ~q 2 q‾! (3)
~ ! Œ
( k k ( k
mental covariance of a pair of observed values, use the
n n 2 1
k51 k51
equation:
The standard uncertainty of any single observation, u(q ),
k
n
equals s(q ). The number of degrees of freedom for this
k s~q ,r ! 5 ~q 2 q‾!~r 2 r‾! (7)
k k ( k k
n 2 1
k51
evaluation is n − 1.
Example—To evaluate the repeatability of an electronic
where q and r denote two simultaneously observed values
k k
balance (see 6.17.4), an analyst makes a series of 20 measure-
and q¯ and r¯ denote the average values. To evaluate the
ments of a 1-gram weight, obtaining the values w , w , …, w
1 2 20 experimental covariance of the means, divide the preceding
listed below (all values in grams).
estimate by n, as shown below:
1.0002 0.9997 0.9999 1.0001 0.9999
n
1.0000 1.0000 0.9996 0.9997 0.9999
s q‾,r‾ 5 q 2 q‾ r 2 r‾ (8)
~ ! ~ !~ !
( k k
1.0000 0.9999 0.9998 0.9997 1.0000
n n 2 1
~ ! k51
1.0000 1.0002 1.0000 1.0000 0.9999
6.3.9 Line- or curve-fitting by ordinary least squares (OLS)
The analyst then calculates the average and standard devia-
can also be used for a Type A evaluation of standard uncer-
tion of the values as follows:
tainty. Since OLS is most properly applied to homoscedastic
w‾ data, in which the variance for each data point is the same,
5 w
( k
k51
good examples of its use in radiochemistry seem to be rare.
19.9985 g (4)
However, X6.5 describes an example in which a logarithmic
transformation is applied to data assumed to have approxi-
5 0.999925 g
mately the same relative variance, resulting in transformed data
with nearly constant variance. OLS is then used to fit a straight
s~w ! 2
k
5Π~w 2 0.999925 g!
line to the transformed data and to estimate its Type A standard
( k
20 2 1
k51
uncertainty.
27 2
(5)
4.975 × 10 g
5Π6.3.10 A Type B evaluation of uncertainty typically involves
an assumed probability distribution for the quantity being
5 0.00016 g
estimated. The distribution is determined by the estimated
value q and in most cases by one or more other parameters,
The standard deviation s(w ) is an estimate of the balance’s
k
often including a tolerance that describes an interval about q
repeatability. (See also X2.1.)
within which the true value is believed to lie. The standard
6.3.5 Another simple example of a Type A evaluation is the
uncertainty u(q) equals the standard deviation of the assumed
estimation of the uncertainty of an average measured value, q¯ ,
distribution.
by the experimental standard deviation of the mean, s(q¯ ), also
6.3.11 Type B evaluations are commonly used for the
known as the “standard error” of the mean. Given repeated
observations q , q , …, q , the experimental standard deviation uncertainties of inputs for which repeated observations are
1 2 n
impractical. Examples of such inputs include the values for
of the mean is given by:
certified reference materials, radioactive half-lives, and capaci-
n
s~q ! 1
ties of volumetric glassware. Since it is common for a testing
k
s~q‾! 5 5Œ ~q 2 q‾! (6)
( k
n~n 2 1!
k51 lab to count a prepared sample test source only once, Type B
=n
(Poisson) evaluations of counting uncertainty are also com-
When the average value q¯ is used to estimate a quantity, u~q¯ !
mon. In fact, Type B evaluations of uncertainty are probably
equals s(q¯ ). The number of degrees of freedom is n − 1.
far more common than Type A evaluations at most testing labs.
6.3.6 A typical use for s(q¯ ) is to evaluate the uncertainty of
6.3.12 Appendix X1 describes Type B evaluations based on
a particular measured quantity from repeated observations of
normal, rectangular, triangular, and U distributions. Evalua-
that quantity. A typical use for s(q ) is to estimate the
k tions based on Poisson distributions are described in 6.12.
uncertainties of unrepeated future observations of particular
6.3.13 The statistical analysis that implements a Type A
quantities of the same type measured by the same process, as
evaluation of a standard uncertainty u(x ) always has an
i
in the example of balance repeatability above.
associated number of degrees of freedom, ν . The larger the
i
6.3.7 One may use s(q¯ ) to estimate the standard uncertainty number ν , the smaller the relative standard uncertainty of u(x )
i i
of an average q¯ even when the individual observations q have as an estimator for the true standard deviation of the distribu-
k
different variances, although in such cases the Type A degrees tion of X . Based on the approximate mathematical relationship
i
of freedom, n − 1, may overestimate the quality of the between Type A degrees of freedom and the relative standard
D8293 − 22
uncertainty of u(x ), the number of degrees of freedom for a gation are applied first. For example, see 6.11, where four of
i
Type B evaluation is defined to be: the input estimates are calculated from other measured values.
6.4.5 The combined standard uncertainty of the output
1 ∆u x
~ !
i
ν 5 (9)
S D estimate y, denoted by u (y), is found using the law of
i
c
2 u~x !
i
propagation of uncertainty (also called simply the uncertainty
where u(x ) denotes the standard uncertainty of u(x ), and
i i propagation formula).
u(x ) / u(x ) is its relative standard uncertainty—the “uncer-
i i
N
] f
tainty of the uncertainty.” While this equation is only approxi-
u y 5 u x 1· · · (10)
~ ! ΠS D ~ !
c ( i
] x
mately true for Type A degrees of freedom, it serves as a i51
i
definition for Type B degrees of freedom.
The partial derivatives ∂f / ∂x , called sensitivity coeffıcients,
i
6.3.14 Table 1 gives examples of Type B degrees of
are the first partial derivatives of f, evaluated at or near the
freedom calculated using Eq 9. Note that the calculated number
observed values of the x . Each sensitivity coefficient ∂f / ∂x
i i
ν may not be an integer. It may also be infinite if u(x ) is
i i
represents the ratio of the change in the value of the output y
considered to be zero, a case which is most likely when the
to a tiny change in the value of a single input x . In
i
Type B uncertainty is based on a rectangular distribution with
measurement reports, a sensitivity coefficient ∂f / ∂x is com-
i
a well-known tolerance, as described in Appendix X1—for
monly denoted by c .
i
example, the uncertainty due to rounding a number. For the
NOTE 3—The partial derivative ∂f / ∂x may also be denoted by ∂y / ∂x .
degrees of freedom associated with Poisson uncertainty i i
evaluations, see 6.12.11. For other Type B evaluations, the 6.4.6 Ideally, the partial derivatives ∂f / ∂x would be evalu-
i
value of u(x ) is often based on available knowledge and
ated at the true values of the input quantities X , if they were
i i
professional judgment. known, but since they are unknown, the evaluation uses the
observed values x . In some cases, especially when measure-
i
6.4 Propagation of Uncertainty:
ments are performed for purposes of quality control, there may
6.4.1 After the uncertainties and covariances of the input
be better prior estimates of the true values, which can be used
estimates are determined, they are combined mathematically
to evaluate the sensitivity coefficients.
using uncertainty propagation to obtain the combined standard
6.4.7 Fig. 1 illustrates a partial derivative of a function of
uncertainty of the output estimate.
two variables. The figure depicts the function as a curved
6.4.2 A component of the combined standard uncertainty is
surface, y5f~x ,x !. A plane perpendicular to the x -axis slices
1 2 1
a portion of the total uncertainty attributed to a particular
the surface at a chosen value of x , intersecting the surface in
cause, such as counting statistics, standards, tracers, volumetric
a curve. The value of ∂f / ∂x at any point on the curve equals
glassware, and subsampling, to name only a few. The com-
the slope of the tangent line at that point.
bined standard uncertainty can be interpreted as a combination
of its components.
6.4.8 The ellipsis (. . .) in Eq 10 indicates the possibility of
6.4.3 Components of the combined standard uncertainty do
additional terms under the radical. Additional terms appear
not add linearly in the manner of a simple sum; instead, they
when input estimates are correlated with each other (see 6.5).
add “in quadrature.” To add components in quadrature, one
6.4.9 The component of the combined standard uncertainty
squares each component, adds the squares, and takes the square u (y) generated by the uncertainty of an input estimate x can be
c i
root of the resulting sum. This operation is described below in
found as follows:
more detail.
] f
6.4.4 The laboratory evaluates the standard uncertainty of u ~y! 5 u~x ! 5 c u~x ! (11)
U U
i i ? i? i
] x
i
each input estimate x . An uncertainty u(x ) may be evaluated
i i
directly by a Type A or Type B method, as described in 6.3 and
Appendix X1. However, it often happens that an input estimate
x is obtained as the output estimate from another
i
measurement, in which case its uncertainty is typically ob-
tained by uncertainty propagation. In fact, when the measure-
ment function f(X , X , …, X ) is complicated enough, it is
1 2 N
common to break the full expression down into
subexpressions, each of which in effect represents a simpler
measurement function to which the rules of uncertainty propa-
TABLE 1 Type B Degrees of Freedom
∆u(x ) / u(x ) ν
i i i
50 % 2
33.3 % 4.5
25 % 8
20 % 12.5
10 % 50
5 % 200
0 % `
FIG. 1 Partial Derivative ]f ⁄ ]x of a Function y = f (x , x )
2 1 2
D8293 − 22
Here u (y) denotes the component of u (y) generated by u(x ).
i c i
The law of propagation of uncertainty can then be written
explicitly in terms of uncertainty components, as shown below:
N
u y 5Œ u y 1· · · (12)
~ ! ~ !
c ( i
i51
6.4.10 After the combined standard uncertainty u (y) is
c
calculated, it may be multiplied by a coverage factor, k, to
obtain an expanded uncertainty, U5k·u y . The expanded
~ !
c
uncertainty describes an interval y 6 U that is believed to have
a high probability of containing the true value of the mea-
FIG. 3 Combining Three Uncertainty Components
surand. The value of k should be greater than 1 and typically
ranges from 2 to 3. If the distribution of the result is normal
(Gaussian) and if the standard uncertainty is a good estimate of u~x ,x ! 5 r~x ,x !u~x !u~x ! (14)
i j i j i j
the true standard deviation, a coverage factor of 2 provides
where r(x , x ) denotes the estimated correlation coeffıcient of
i j
approximately 95 % coverage probability, and a coverage
x and x :
i j
factor of 3 provides more than 99 % coverage probability. See
u~x ,x !
6.7 for more information about determining a coverage factor
i j
r~x ,x ! 5 (15)
i j
u x u x
~ ! ~ !
to provide a specified coverage probability p. i j
NOTE 4—A testing lab should choose the coverage factor k = 2 or the
The correlation coefficient is always a dimensionless real
coverage probability p = 95 % by default unless there are compelling
number between −1 and +1.
reasons to do otherwise.
NOTE 5—The input estimates x and x are correlated if r x ,x fi0 and
6.4.11 Fig. 2 applies a geometric analogy to illustrate ~ !
i j i j
uncorrelated if r~x ,x !50. The correlation is strong if |r~x ,x !| is near 1
uncertainty propagation for a hypothetical measurement equa- i j i j
and weak if it is near 0.
tion y5f~x ,x ! with two input quantities. The width and height
1 2
6.5.3 Typically, few inputs in a radiochemical measurement
of the rectangle represent the two uncertainty components
are significantly correlated and usually only when they are
while the diagonal represents the combined standard uncer-
calculated from the same data. Common examples include
tainty. In this analogy, Eq 12 for two uncorrelated input
calibration parameters, especially when the calibration in-
estimates is equivalent to the Pythagorean Theorem: u ~y!
c
2 2
volves a multi-parameter curve or when calibrations for two
5u y 1u y .
~ ! ~ !
1 2
radionuclides are based on one standard solution (for example,
6.4.12 Fig. 3 applies the same analogy in three dimensions
90 90 90
Sr and Y calibrations using the same Sr standard, as
for a measurement equation y5f~x ,x ,x ! with three inputs.
1 2 3
described in X5.3). Another common example is a correlation
Here the three uncertainty components are identified with the
between the counting efficiency and the chemical yield, as
length, width, and height of a box, and the combined standard
described in 6.16.
uncertainty is identified with the box’s diagonal. In this case,
6.5.4 When two inputs x and x are correlated only because
2 2 2 2 i j
u y 5u y 1u y 1u y .
~ ! ~ ! ~ ! ~ !
c 1 2 3
they are calculated from other variables w , w , …, w , the
1 2 m
6.5 Correlations:
covariance u(x , x ) may be calculated as described below in
i j
6.5.1 When input estimates may be correlated with each
6.5.8. Alternatively, one may recast the measurement equation
other, the complete law of propagation of uncertainty is written
y5f~x ,x ,…,x !, replacing x and x by the expressions used to
1 2 N i j
as follows:
calculate them, thereby eliminating the correlated inputs x and
i
x from the equation and including new inputs w , w , …, w , as
N N21 N j 1 2 m
] f ] f ] f
shown below:
u ~y! 5Πu ~x !12 u~x ,x ! (13)
S D
c ( i ( ( i j
] x ] x ] x
i51 i51 j5i11
i i j
y 5 f~x ,…,x ,x ,…,x ,x ,…,x ,w ,…,w ! (16)
1 i21 i11 j21 j11 N 1 m
In this equation, u(x , x ) denotes the estimated covariance of
i j
input estimates x and x . Note that u~x ,x !5u~x ,x ! and 6.5.5 To evaluate the covariance of two input quantities
i j j i
i j
experimentally, see 6.3.8.
u~x ,x !5u ~x !.
i i i
6.5.6 Generally, one estimates the covariance of two inputs
6.5.2 The covariance u(x , x ) can also be written as:
i j
only when there is a reason to suspect it may be nonzero. One
may suspect a correlation whenever two inputs are measured
using the same devices, or using different devices that are
affected by the same influence quantities, such as ambient
temperature, pressure, and humidity. Even when a correlation
exists, if the uncertainty components generated by the two
inputs are both small, the effect of the correlation is necessarily
small, too, and might be ignored.
6.5.7 Fig. 2 graphically illustrated the propagation of uncer-
tainty for two uncorrelated input estimates by analogy with the
FIG. 2 Combining Two Uncertainty Components Pythagorean Theorem. Fig. 4 extends the geometric analogy to
D8293 − 22
6.5.10 The covariance of any pair of parameters may be
divided by the product of the two parameters’ uncertainties to
obtain the estimated correlation coefficient, which is generally
more convenient for record-keeping.
6.5.11 See X5.3 for more examples of the use of Eq 18.
6.6 Alternatives for Uncertainty Propagation:
6.6.1 It is often possible to implement Eq 10 without
explicit calculation of derivatives using only the rules of
uncertainty propagation for sums, differences, products, and
quotients. Uncertainty propagation for sums and differences
follows the pattern:
2 2 2
u ~x 6x 6· · ·6x ! 5 =u x 1u x 1· · ·1u x (20)
~ ! ~ ! ~ !
c 1 2 n 1 2 n
6.6.2 More generally, if the variables x are multiplied by
i
any constants a , the pattern becomes:
i
FIG. 4 Combining Uncertainties of Two Correlated Inputs
u ~a x 1a x 1· · ·1a x ! 5
c 1 1 2 2 n n
(21)
2 2 2 2 2 2
=a u x 1a u x 1· · ·1a u x
~ ! ~ ! ~ !
1 1 2 2 n n
NOTE 6—Eq 21 may be derived from the fact that:
cases where the two input estimates are correlated. A correla-
tion transforms the rectangle of Fig. 2 into a parallelogram with ]
~a x 1a x 1· · ·1a x ! 5 a (22)
1 1 2 2 n n i
acute and obtuse angles, as shown. For either of the parallelo- ] x
i
grams in Fig. 4, the cosine of the indicated angle has the same
6.6.3 For products and quotients, the pattern for uncertainty
absolute value as the correlation coefficient r(x , x ), with a
1 2
propagation is:
sign that depends on whether the correlation increases the total
uncertainty (a, positive cosine) or decreases it (b, negative
x x …x x x …x
1 2 n 1 2 n
u 5
S D U U
c
cosine).
w w …w w w …w
1 2 m 1 2 m
(23)
6.5.8 The covariance of two output estimates y and z
2 2 2 2
u ~x ! u ~x ! u ~w ! u ~w !
1 n 1 m
calculated from input estimates x , x , …, x can be calculated × 1· · ·1 1 1· · ·1
Œ
1 2 N 2 2 2 2
x x w w
1 n 1 m
using a formula similar to the law of propagation of uncer-
tainty. Let y5f~x ,x ,…,x ! and z5g~x ,x ,…,x !. Then:
1 2 N 1 2 N provided that none of the factors x or w is zero. One may
i j
N N also rewrite Eq 23 in terms of relative variances:
] f ] g
u~y,z! 5 u~x ,x ! (17)
( ( i j
] x ] x x x …x
i51j51
i j 1 2 n
2 2
5 u x 1· · ·1u x
u ~ ! ~ !
S D rel 1 rel n
rel
w w …w
(24)
1 2 m
If x , x , …, and x are uncorrelated, the preceding equation
1 2 N
2 2
1u w 1· · ·1u w
~ ! ~ !
rel 1 rel m
reduces to:
N
where u (x ) denotes the relative standard uncertainty,
] f ] g
rel i
u~y,z! 5 u x (18)
~ !
i
(
u x ⁄ |x |.
] x ] x ~ !
i51
i i i i
NOTE 7—Eq 23 may be derived from the fact that if y5
Example—Suppose two sources are counted separately using
~x x …x ! ⁄ ~w w …w !, where none of the factors is zero, then:
1 2 n 1 2 m
a liquid scintillation counter and the results are background-
] y x …x x …x y
corrected using the same background estimate. If the two gross 1 i21 i11 n
5 5 (25)
] x w w …w x
i 1 2 m i
count rates are R and R and the background count rate is
S1 S2
and
R , the net count rates are given by R = R − R and R =
B N1 S1 B N2
] y x x …x y
R − R . Since the variable R appears in both expressions, its
S2 B B 1 2 n
5 2 5 2 (26)
uncertainty generates a covariance for R and R . ] w w …w …w w
N1 N2 j 1 j m j
] R ] R
N1 N2
2 6.6.4 If some of the x may be zero, Eq 23 must be
u~R ,R !
5 u R i
N1 N2 ~ !
B
] R ] R
B B
rearranged to avoid division by those factors. For example, if
(19)
5 ~21!~21!u R
~ !
B
x could be zero, one might write either:
5 u R
~ !
B
x x …x x x …x
1 2 n 2 3 n
u 5 u x
S D FS D ~ !
6.5.9 The covariance of two calibration parameters calcu- c 1
w w …w w w …w
1 2 m 1 2 m
2 2 2
lated by least squares should be determined as part of the
x x …x u x u x
~ ! ~ !
1 2 n 2 n
1 1 · · · 1 (27)
S D S
least-squares analysis. The method of weighted least squares 2 2
w w …w x x
1 2 m 2 n
provides not only a solution vector but also the estimated 2 2 1/2
u w u w
~ ! ~ !
1 m
1 1 · · · 1
covariance matrix for the solution. The diagonal entries of this DG
2 2
w w
1 m
matrix are the variances of the estimated parameters, and the
off-diagonal entries are the covariances. or
D8293 − 22
x x …x x x …x where:
1 2 n 2 3 n
u 5U U
S D
c
w w …w w w …w
1 2 m 1 2 m
D 5 f~x ,…, x 1u~x !, …,x ! 2 f~x ,…,x ! (35)
i 1 i i N 1 N
2 2 2 2
u x u x u w u w
~ ! ~ ! ~ ! ~ !
2 n 1 m
2 2
6.6.10 Another method for uncertainty propagation is
׌u ~x !1x 1· · ·1 1 1· · ·1
S D
1 1 2 2 2 2
x x w w
2 n 1 m
Monte Carlo simulation, described in JCGM 101. The Monte
(28)
Carlo method evaluates both the output estimate y and its
standard uncertainty u(y) by analyzing the distribution of
The last two equations are equivalent to Eq 23 when all the
results produced in many trials of a computer simulation of the
factors are nonzero, but both are still valid if x happens to be
measurement. In each trial, the algorithm generates pseudo-
zero. For a less abstract example, see Eq 39 in 6.11.2, where
random values for the input quantities X by sampling from
the uncertainty equation is written to avoid division by the
i
their estimated or assumed distributions, and uses the measure-
variable R .
N
ment function f(x , x , …, x ) to calculate a value for the output
6.6.5 More generally, if p , p , …, p are any exponents, not
1 2 N
1 2 n
quantity Y. The output estimate y and its standard uncertainty
necessarily 61, and if none of the factors x is zero:
i
u(y) are then given by the average and experimental standard
p p p p p p
1 2 n 1 2 n
u ~x x …x ! 5 x x …x
c 1 2 n ? 1 2 n ?
deviation of all the trial results. For details of the method and
2 2 2
u x u x u x (29)
~ ! ~ ! ~ !
1 2 n a discussion of its advantages and disadvantages in relation to
2 2 2
× p 1p 1· · ·1p
Œ
1 2 2 2 n 2
x x x
1 2 n
the standard GUM approach, see JCGM 101.
In terms of relative variances:
6.7 Calculating Coverage Factors:
2 p p p 2 2 2 2 2 2
1 2 n 6.7.1 As discussed earlier, the combined standard uncer-
u ~x x …x ! 5 p u ~x !1p u ~x !1· · ·1p u ~x ! (30)
rel 1 2 n 1 rel 1 2 rel 2 n rel n
tainty u (y) may be multiplied by a coverage factor k to obtain
c
6.6.6 These simplified rules do not apply if the same input
an expanded uncertainty U = k · u (y) such that the coverage
c
quantity appears more than once in the expression whose
interval from y − U to y + U is believed to contain the true
uncertainty is being evaluated. For example, the variance of x
value of the measurand with high probability. The coverage
2 2 2
+ x is not u (x) + u (x), or 2u (x), as one might infer incorrectly
factor most commonly used in reports of radiochemical mea-
2 2
from Eq 20; it actually equals u (2x), or 4u (x), as implied by
c
surement results is k = 2, which is usually assumed to produce
Eq 21. When a variable appears more than once, it may be
a coverage interval with approximately 95 % coverage prob-
possible to recast the expression algebraically so that each
ability. For some purposes, especially internal laboratory
variable appears only once, and the simplified rules can still be
quality control, the coverage factor k = 3 is often used. The
used. However, in some cases algebra fails and calculus is
coverage probability at k = 3 is assumed to be more than 99 %,
needed.
so that the interval from y − U to y + U should “almost always”
6.6.7 For measurement functions with more than a few
contain the true value of the measurand.
input quantities, an explicit expression for the combined
6.7.2 If the distribution of the measurement result y is
standard uncertainty can become quite complex, even using the
normal (Gaussian) and if its combined standard uncertainty
rules described above. A number of software applications have
u (y) is a sufficiently accurate estimate of the standard devia-
c
been developed and are available to assist in reliably
tion of that distribution, then the coverage factor k that
p
developing, testing, and documenting uncertainty models.
provides a specified coverage probability p is approximated by
6.6.8 Other options for calculating the combined standard
the (1 + p) / 2-quantile of the standard normal distribution,
uncertainty without calculus often involve approximations for
z . Table 2 lists several values of k based on this
(1 + p) / 2 p
the sensitivity coefficients. For i = 1, 2, …, N, the sensitivity
approximation.
coefficient ∂f / ∂x may be approximated by:
i
6.7.3 Laboratories often use the approximation described
] f 1
above by default regardless of whether the uncertainty is
‾
' @ f x ,…, x 1u x , …,x M_
~ ~ ! !
1 i i N
] x 2u~x !
well-known. In most cases, the combined standard uncertainty
i i
(31)
‾ u (y) is not known well enough to justify retaining more than
c
M_ 2 f~x ,…, x 2 u~x !, …,x !#
1 i i N
two significant figures in the value of k .
p
Recognizing this fact, the GUM allows the approximation:
6.7.4 As discussed in 6.3.13, one may calculate the number
of degrees of freedom ν for the Type A or Type B standard
N
i
u ~y!'ΠZ (32) uncertainty u(x ) of each input estimate. Given the degrees of
c ( i i
i51
freedom ν for all the input estimates, it is possible to calculate
i
where:
the effective degrees of freedom, ν , for the combined standard
eff
uncertainty and to use ν to estimate a somewhat better
eff
‾
Z 5 @ f x ,…, x 1u x , …,x M_
~ ~ ! !
i 1 i i N
coverage factor k for a specified coverage probability p.
2 p
(33)
‾
M_ 2 f~x ,…, x 2 u~x !, …,x !#
1 i i N
TABLE 2 Coverage Factors: Well-Characterized Uncertainty
6.6.9 The Kragten spreadsheet method for propagating
p (1+p) / 2 k = z
p (1+p) / 2
uncertainty (4) is based on a similar one-sided approximation
95.0 % 0.975 2.0
of each sensitivity coefficient:
99.0 % 0.995 2.6
99.5 % 0.9975 2.8
N
2 99.7 % 0.9985 3.0
u ~y!'ΠD (34)
c ( i
i51
D8293 − 22
6.7.5 Assuming that none of the input estimates are corre- figure unless the resulting figure is a 1, in which case the
lated with each other and that the combined standard uncer- uncertainty is rounded to two figures. A simpler rule, recom-
tainty u (y) is not dominated either by a Type A uncertainty mended by ANSI N42.23 and MARLAP (2), is to round the
c
component with only a few degrees of freedom or by a Type B uncertainty to two figures in all cases. This simpler rule is also
component based on a distribution that is very different from a recommended by the NIST Weights and Measures Division in
normal distribution, one may use the Welch-Satterthwaite Good Laboratory Practice (5) and is used at the “NIST
formula, shown below, to calculate the effective degrees of Reference on Constants, Units, and Uncertainty” (6).
freedom for u (y).
6.8.3.2 Never round the reported result of a measurement to
c
a power of 10 that is larger than the combined standard
u ~y!
c
ν 5 (36)
N
eff 4 uncertainty. Such rounding would introduce significant addi-
u y
~ !
i
( tional uncertainty, which might dominate all other components.
ν
i51
i
6.8.3.3 Do not round intermediate results of calculations
Note that if the number of Type B degrees of freedom v for
i
unnecessarily. Round only final results and their uncertainties
th
an input estimate x is infinite, the i term of the sum in the
i
as described above.
denominator of Eq 36 is zero and should be omitted.
6.8.4 Shorthand Formats:
6.7.6 Given ν , estimate the coverage factor k by the
eff p
6.8.4.1 When the laboratory reports a measurement result, it
(1 + p) / 2-quantile of the t-distribution with ν degrees of
eff
may present the measured value and the uncertainty as distinct,
freedom.
clearly identified items—for example, in different columns of a
k 5 t ν (37)
~ !
p ~11p! ⁄ 2 eff
table. Alternatively, it may present the value and its uncertainty
in a single expression using one of the shorthand formats
Round k to either 2 or 3 figures. Table 3 provides examples
p
described below.
of k calculated in this manner.
p
6.8.4.2 The most common shorthand format for reporting a
NOTE 8—The calculated value of ν is generally not an integer. Some
eff
result with its combined standard uncertainty places the digits
software packages may be able to calculate the quantile t (ν ) for
(1 + p) / 2 eff
of the rounded uncertainty in parentheses immediately after the
non-integral degrees of freedom. When such software is unavailable,
final digits of the rounded measured value. For example, a
either interpolate between values of t for integral degrees of freedom or
truncate ν to a whole integer.
eff value of 0.124 Bq/g with combined standard uncertainty 0.037
Bq/g is represented by 0.124(37) Bq/g. When scientific nota-
6.8 Reporting the Uncertainty:
−1
tion is used, the expression becomes 1.24(37) × 10 Bq/g. For
6.8.1 The report of the measurement should indicate
examples of the use of this format, see the “NIST Reference on
whether the uncertainty is the combined standard uncertainty
Constants, Units, and Uncertainty” (6).
or an expanded uncertainty. For an expanded uncertainty, the
report should state the coverage factor k and the approximate 6.8.4.3 It is also possible to put the entire expression for the
coverage probability p. The report should also include a combined standard uncertainty insi
...