ASTM E2536-21

Standard Guide for Assessment of Measurement Uncertainty in Fire Tests

SIGNIFICANCE AND USE
5.1 Users of fire test data often need a quantitative indication of the quality of the data presented in a test report. This quantitative indication is referred to as the “measurement uncertainty”. There are two primary reasons for estimating the uncertainty of fire test results.  
5.1.1 ISO/IEC 17025 requires that competent testing and calibration laboratories include uncertainty estimates for the results that are presented in a report.  
5.1.2 Fire safety engineers need to know the quality of the input data used in an analysis to determine the uncertainty of the outcome of the analysis.
SCOPE
1.1 This guide covers the evaluation and expression of uncertainty of measurements of fire test methods developed and maintained by ASTM International, based on the approach presented in the GUM. The use in this process of precision data obtained from a round robin is also discussed.  
1.2 The guidelines presented in this standard can also be applied to evaluate and express the uncertainty associated with fire test results. However, it may not be possible to quantify the uncertainty of fire test results if some sources of uncertainty cannot be accounted for. This problem is discussed in more detail in Appendix X2.  
1.3 Application of this guide is limited to tests that provide quantitative results in engineering units. This includes, for example, methods for measuring the heat release rate of burning specimens based on oxygen consumption calorimetry, such as Test Method E1354.  
1.4 This guide does not apply to tests that provide results in the form of indices or binary results (for example, pass/fail). For example, the uncertainty of the Flame Spread Index obtained according to Test Method E84 cannot be determined.  
1.5 In some cases additional guidance is required to supplement this standard. For example, the expression of uncertainty of heat release rate measurements at low levels requires additional guidance and uncertainties associated with sampling are not explicitly addressed.  
1.6 This fire standard cannot be used to provide quantitative measures.  
1.7 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.  
1.8 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: 30-Sep-2021
• ICS: 13.220.40 - Ignitability and burning behaviour of materials and products
• Technical Committee: E05 - Fire Standards
• Drafting Committee: E05.31 - Terminology and Services / Functions

Relations

• Refers: ASTM E176-24 - Standard Terminology of Fire Standards
• Effective Date: 01-Jan-2024
• Refers: ASTM E2989-19a(2024) - Standard Guide for Assessment of Continued Applicability of Reaction to Fire Test Reports Used in Building Regulation
• Effective Date: 01-Jan-2024
• Refers: ASTM E84-23d - Standard Test Method for Surface Burning Characteristics of Building Materials
• Effective Date: 01-Dec-2023
• Refers: ASTM E84-23c - Standard Test Method for Surface Burning Characteristics of Building Materials
• Effective Date: 01-Sep-2023
• Refers: ASTM E2989-19a - Standard Guide for Assessment of Continued Applicability of Reaction to Fire Test Reports Used in Building Regulation
• Effective Date: 15-Dec-2019
• Refers: ASTM E119-19 - Standard Test Methods for Fire Tests of Building Construction and Materials
• Effective Date: 01-Oct-2019
• Refers: ASTM E84-19b - Standard Test Method for Surface Burning Characteristics of Building Materials
• Effective Date: 01-Jul-2019
• Refers: ASTM E2989-19 - Standard Guide for Assessment of Continued Applicability of Reaction to Fire Test Reports Used in Building Regulation
• Effective Date: 01-May-2019
• Refers: ASTM E84-19a - Standard Test Method for Surface Burning Characteristics of Building Materials
• Effective Date: 15-Apr-2019
• Refers: ASTM E84-19 - Standard Test Method for Surface Burning Characteristics of Building Materials
• Effective Date: 01-Mar-2019
• Refers: ASTM E176-18a - Standard Terminology of Fire Standards
• Effective Date: 15-Dec-2018
• Refers: ASTM E119-18c - Standard Test Methods for Fire Tests of Building Construction and Materials
• Effective Date: 01-Nov-2018
• Refers: ASTM E119-18ce1 - Standard Test Methods for Fire Tests of Building Construction and Materials
• Effective Date: 01-Nov-2018
• Refers: ASTM E84-18b - Standard Test Method for Surface Burning Characteristics of Building Materials
• Effective Date: 01-Oct-2018
• Refers: ASTM E119-18b - Standard Test Methods for Fire Tests of Building Construction and Materials
• Effective Date: 01-Sep-2018

Overview

ASTM E2536-21 - Standard Guide for Assessment of Measurement Uncertainty in Fire Tests provides a comprehensive methodology for evaluating and expressing measurement uncertainty in fire test methods developed by ASTM International. This guide aligns with international best practices, notably the Guide to the Expression of Uncertainty in Measurement (GUM), and fulfills requirements set by standards like ISO/IEC 17025 for competent testing laboratories. Estimating measurement uncertainty is a critical aspect of fire safety engineering, as it allows users to quantitatively assess the quality of fire test data and supports the reliability of regulatory and safety analyses.

Key Topics

• Measurement Uncertainty Definition: Uncertainty represents the parameter characterizing the dispersion of values that could be attributed to the measured quantity (measurand). It gives a quantitative indication of a measurement's quality.
• Random and Systematic Errors: The guide distinguishes between random errors (variations from repeated measurements) and systematic errors (consistent bias), both of which affect uncertainty estimations.
• Type A and Type B Evaluations:
  • Type A: Statistical evaluation from repeated observations, calculating standard deviation and mean.
  • Type B: Based on information other than repeated experimentation, such as manufacturer data or calibration certificates.
• Law of Propagation of Uncertainty: Details combining uncertainties from multiple input quantities, including use of sensitivity coefficients and consideration of correlated variables.
• Expanded Uncertainty: Involves applying a coverage factor (typically 2 for 95% confidence) to the combined standard uncertainty to define an interval where the true value likely falls.
• Reporting Requirements: Results should be reported with their measurement uncertainty, including clear definitions, confidence levels, and explanations of methods used.

Applications

• Laboratory and Industrial Fire Testing: Testing and calibration laboratories must quantify measurement uncertainty to comply with ISO/IEC 17025, especially when reporting quantitative fire test results, such as heat release rate measurements.
• Fire Safety Engineering and Analysis: Fire safety professionals rely on data with reported uncertainty to conduct safety analyses, risk assessments, and regulatory compliance evaluations. Knowing the degree of uncertainty influences engineering decisions and modeling outcomes.
• Product Development and Regulatory Approval: Manufacturers and specifiers use uncertainty estimates to interpret fire performance data confidently, supporting product certification and regulatory submissions.
• International Standardization: By using a process aligned with GUM and international trade principles (WTO TBT), ASTM E2536-21 fosters global acceptance of fire test results and mutual recognition among testing bodies.

Limitations:

• The guide applies exclusively to tests yielding quantitative results in engineering units (e.g., heat release rate via oxygen consumption calorimetry). It does not cover index or binary outcome tests such as pass/fail results.
• Some sources of uncertainty, particularly those related to sampling and very low-level measurements, may require additional guidance.

Related Standards

ASTM E2536-21 should be used in conjunction with several related standards to ensure comprehensive fire testing and data interpretation:

• ASTM E1354: Standard Test Method for Heat and Visible Smoke Release Rates for Materials and Products Using an Oxygen Consumption Calorimeter
• ASTM E84: Standard Test Method for Surface Burning Characteristics of Building Materials
• ISO/IEC 17025: General requirements for the competence of testing and calibration laboratories
• ISO 29473: Fire tests - Uncertainty of measurements in fire tests
• Guide to the Expression of Uncertainty in Measurement (GUM)
• ASTM E176: Terminology of Fire Standards
• ASTM E691: Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• ASTM E2989: Guide for Assessment of Continued Applicability of Reaction to Fire Test Reports Used in Building Regulation
• EN 13823: Reaction to fire tests for building products (CEN)

These references collectively support the robust assessment, calculation, and communication of measurement uncertainty in fire testing, underpinning credible data for safety-critical applications.

Keywords: measurement uncertainty, fire test, ASTM E2536, fire safety engineering, ISO/IEC 17025, uncertainty analysis, heat release rate, fire testing standards, laboratory accreditation, data quality in fire tests.

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: E2536 − 21 An American National Standard
Standard Guide for
Assessment of Measurement Uncertainty in Fire Tests
This standard is issued under the fixed designation E2536; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
INTRODUCTION
The objective of a measurement is to determine the value of the measurand, that is, the physical
quantity that needs to be measured. Every measurement is subject to error, no matter how carefully
it is conducted. The (absolute) error of a measurement is defined in Eq 1.
All terms in Eq 1 have the units of the physical quantity that is measured. This equation cannot be
used to determine the error of a measurement because the true value is unknown, otherwise a
measurement would not be needed. In fact, the true value of a measurand is unknowable because it
cannot be measured without error. However, it is possible to estimate, with some confidence, the
expected limits of error. This estimate is referred to as the uncertainty of the measurement and
provides a quantitative indication of its quality.
Errors of measurement have two components, a random component and a systematic component.
The former is due to a number of sources that affect a measurement in a random and uncontrolled
manner. Random errors cannot be eliminated, but their effect on uncertainty is reduced by increasing
the number of repeat measurements and by applying a statistical analysis to the results. Systematic
errors remain unchanged when a measurement is repeated under the same conditions. Their effect on
uncertainty cannot be completely eliminated either, but is reduced by applying corrections to account
for the error contribution due to recognized systematic effects. The residual systematic error is
unknown and shall be treated as a random error for the purpose of this standard.
General principles for evaluating and reporting measurement uncertainties are described in the
Guide on Uncertainty of Measurements (GUM). Application of the GUM to fire test data presents
some unique challenges. This standard shows how these challenges can be overcome.An example to
illustrate application of the guidelines provided in this standard can be found in Appendix X1.
ε[y 2 Y (1)
where:
ε = measurement error;
y = measured value of the measurand; and
Y = true value of the measurand.
1. Scope presentedintheGUM.Theuseinthisprocessofprecisiondata
obtained from a round robin is also discussed.
1.1 This guide covers the evaluation and expression of
uncertainty of measurements of fire test methods developed 1.2 The guidelines presented in this standard can also be
andmaintainedbyASTMInternational,basedontheapproach applied to evaluate and express the uncertainty associated with
firetestresults.However,itmaynotbepossibletoquantifythe
uncertainty of fire test results if some sources of uncertainty
ThisguideisunderthejurisdictionofASTMCommitteeE05onFireStandards
cannot be accounted for. This problem is discussed in more
and is the direct responsibility of Subcommittee E05.31 on Terminology and
Services / Functions. detail in Appendix X2.
Current edition approved Oct. 1, 2021. Published November 2021. Originally
1.3 Application of this guide is limited to tests that provide
approved in 2006. Last previous edition approved in 2015 as E2536-15a. DOI:
10.1520/E2536-21. quantitative results in engineering units. This includes, for
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2536 − 21
example, methods for measuring the heat release rate of 2.3 CEN Standard:
burning specimens based on oxygen consumption calorimetry, EN 13823Reaction to fire tests for building products –
Building products excluding floorings exposed to the
such as Test Method E1354.
thermal attack by a single burning item
1.4 This guide does not apply to tests that provide results in
the form of indices or binary results (for example, pass/fail).
3. Terminology
For example, the uncertainty of the Flame Spread Index
3.1 Definitions: For definitions of terms used in this guide
obtained according to Test Method E84 cannot be determined.
and associated with fire issues, refer to the terminology
1.5 In some cases additional guidance is required to supple-
contained in Terminology E176. For definitions of terms used
ment this standard. For example, the expression of uncertainty
in this guide and associated with precision issues, refer to the
of heat release rate measurements at low levels requires
terminology contained in Practice E691.
additionalguidanceanduncertaintiesassociatedwithsampling
3.2 Definitions of Terms Specific to This Standard:
are not explicitly addressed.
3.2.1 accuracy of measurement, n—closeness of the agree-
1.6 Thisfirestandardcannotbeusedtoprovidequantitative
mentbetweentheresultofameasurementandthetruevalueof
measures.
the measurand.
1.7 The values stated in SI units are to be regarded as 3.2.2 combined standard uncertainty, n—standard uncer-
tainty of the result of a measurement when that result is
standard. No other units of measurement are included in this
obtainedfromthevaluesofanumberofotherquantities,equal
standard.
to the positive square root of a sum of terms, the terms being
1.8 This international standard was developed in accor-
the variances or covariances of these other quantities weighted
dance with internationally recognized principles on standard-
according to how the measurement result varies with changes
ization established in the Decision on Principles for the
in these quantities.
Development of International Standards, Guides and Recom-
3.2.3 coverage factor, n—numerical factor used as a multi-
mendations issued by the World Trade Organization Technical
plierofthecombinedstandarduncertaintyinordertoobtainan
Barriers to Trade (TBT) Committee.
expanded uncertainty.
2. Referenced Documents
3.2.4 error (of measurement), n—result of a measurement
2 minus the true value of the measurand; error consists of two
2.1 ASTM Standards:
components: random error and systematic error.
E84Test Method for Surface Burning Characteristics of
3.2.5 expanded uncertainty, n—quantitydefininganinterval
Building Materials
about the result of a measurement that may be expected to
E119Test Methods for Fire Tests of Building Construction
encompass a large fraction of the distribution of values that
and Materials
could reasonably be attributed to the measurand.
E176Terminology of Fire Standards
E230Specification for Temperature-Electromotive Force
3.2.6 measurand, n—quantity subject to measurement.
(emf) Tables for Standardized Thermocouples
3.2.7 precision, n—variability of test result measurements
E691Practice for Conducting an Interlaboratory Study to
around reported test result value.
Determine the Precision of a Test Method
3.2.8 random error, n—result of a measurement minus the
E1354Test Method for Heat and Visible Smoke Release
mean that would result from an infinite number of measure-
Rates for Materials and Products Using an Oxygen Con-
ments of the same measurand carried out under repeatability
sumption Calorimeter
conditions.
E2989Guide for Assessment of Continued Applicability of
3.2.9 repeatability (of results of measurements),
ReactiontoFireTestReportsUsedinBuildingRegulation
n—closeness of the agreement between the results of succes-
2.2 ISO Standards:
siveindependentmeasurementsofthesamemeasurandcarried
ISO 29473Fire tests — Uncertainty of measurements in fire
out under repeatability conditions.
tests
3.2.10 repeatability conditions, n—onidenticaltestmaterial
ISO/IEC17025General requirements for the competence of
using the same measurement procedure, observer(s), and
testing and calibration laboratories
measuring instrument(s) and performed in the same laboratory
GUMGuide to the expression of uncertainty in measure-
during a short period of time.
ment
3.2.11 reproducibility (of results of measurements), n—
closeness of the agreement between the results of measure-
ments of the same measurand carried out under reproducibility
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
conditions.
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.
Available from International Organization for Standardization (ISO), ISO
Central Secretariat, Chemin de Blandonnet 8, CP 401, 1214 Vernier, Geneva, Available from European Committee for Standardization (CEN), Avenue
Switzerland, https://www.iso.org. Marnix 17, B-1000, Brussels, Belgium, http://www.cen.eu.
E2536 − 21
3.2.12 reproducibility conditions, n—on identical test mate-
f = functional relationship between the measurand and the
rial using the same measurement procedure, but different
input quantities; and
observer(s) and measuring instrument(s) in different laborato-
X = input quantities (i=1 … N).
i
ries performed during a short period of time.
6.1.1 The input quantities are categorized as:
3.2.13 standard deviation, n—a quantity characterizing the
6.1.1.1 quantities whose values and uncertainties are di-
dispersion of the results of a series of measurements of the
rectly determined from single observation, repeated observa-
same measurand; the standard deviation is proportional to the
tion or judgment based on experience, or
square root of the sum of the squared deviations of the
6.1.1.2 quantities whose values and uncertainties are
measured values from the mean of all measurements.
brought into the measurement from external sources such as
3.2.14 standard uncertainty, n—uncertainty of the result of
reference data obtained from handbooks.
a measurement expressed as a standard deviation.
6.1.2 An estimate of the output, y, is obtained from Eq 2
3.2.15 systematic error (or bias), n—meanthatwouldresult
usinginputestimates x , x , …, x forthevaluesofthe Ninput
1 2 N
from an infinite number of measurements of the same mea-
quantities:
surandcarriedoutunderrepeatabilityconditionsminusthetrue
y 5 f x ,x ,…, x (3)
~ !
1 2 N
value of the measurand.
3.2.16 type A evaluation (of uncertainty), n—method of Substituting Eq 2 and 3 into Eq 1 leads to:
evaluation of uncertainty by the statistical analysis of series of
y 5 Y1ε 5 Y1ε 1ε 1…1ε (4)
1 2 N
observations.
where:
3.2.17 type B evaluation (of uncertainty), n—method of
ε = contribution to the total measurement error from the
evaluation of uncertainty by means other than the statistical
error associated with x.
i
analysis of series of observations.
3.2.18 uncertainty of measurement, n—parameter, associ- 6.2 A possible approach to determine the uncertainty of y
involvesalargenumber(n)ofrepeatmeasurements.Themean
ated with the result of a measurement, that characterizes the
value of the resulting distribution y¯ is the best estimate of the
dispersion of the values that could reasonably be attributed to ~ !
measurand.Theexperimentalstandarddeviationofthemeanis
the measurand.
the best estimate of the standard uncertainty of y, denoted by
4. Summary of Guide
u(y):
4.1 Thisguideprovidesconceptsandcalculationmethodsto
n
assess the uncertainty of measurements obtained from fire 2
~y 2 y¯!
2 ( k
s ~y!
k51
tests.
u y '=s y¯ 5 5 (5)
~ ! ~ ! Œ
!
n n n 21
~ !
4.2 Appendix X1 of this guide contains an example to
where:
illustrate application of this guide by assessing the uncertainty
of heat release rate measured in the Cone Calorimeter (Test u = standard uncertainty,
s = experimental standard deviation,
Method E1354).
n = number of observations;
th
y =k measured value, and
5. Significance and Use
k
y¯ = mean of n measurements.
5.1 Users of fire test data often need a quantitative indica-
The number of observations n shall be large enough to
tion of the quality of the data presented in a test report. This
quantitative indication is referred to as the “measurement ensure that y¯ provides a reliable estimate of the expectation µ
y
of the random variable y, and that s y¯ provides a reliable
uncertainty”. There are two primary reasons for estimating the ~ !
uncertainty of fire test results. estimate of the variance σ ~y¯!5σ~y!/n. If the probability distri-
bution of y is normal, then standard deviation of s y¯ relative
5.1.1 ISO/IEC17025 requires that competent testing and ~ !
1/2
calibration laboratories include uncertainty estimates for the to σ ~y¯! is approximately [2(n-1)]− . Thus, for n=10 the
relative uncertainty of s y¯ is 24 %t, while for n=50itis10
results that are presented in a report. ~ !
5.1.2 Fire safety engineers need to know the quality of the %.Additional values are given in Table E.1 in annex E of the
GUM.
input data used in an analysis to determine the uncertainty of
the outcome of the analysis.
6.3 Unfortunately it is often not feasible or even possible to
performasufficientlylargenumberofrepeatmeasurements.In
6. Evaluating Standard Uncertainty
those cases, the uncertainty of the measurement can be
6.1 A quantitative result of a fire test Y is generally not
determined by combining the standard uncertainties of the
obtained from a direct measurement, but is determined as a
input estimates. The standard uncertainty of an input estimate
function f from N input quantities X , … , X :
1 N
x is obtained from the distribution of possible values of the
i
inputquantity X.Therearetwotypesofevaluationsdepending
Y 5 f ~X ,X ,…,X ! (2)
i
1 2 N
on how the distribution of possible values is obtained.
where:
6.3.1 Type A evaluation of standard uncertainty—A type A
Y = measurand;
evaluation of standard uncertainty of x is based on the
i
E2536 − 21
frequency distribution, which is estimated from a series of n 6.3.3 Accounting for multiple sources of error—The uncer-
repeated observations x (k = 1 … n). The resulting equation taintyofaninputquantityissometimesduetomultiplesources
i,k
is similar to Eq 5: error.Inthiscase,thestandarduncertaintyassociatedwitheach
source of error has to be estimated separately and the standard
n
2 uncertainty of the input quantity is then determined according
x 2 x¯
~ !
2 ( i,k i
s x
~ ! k51
i
2 to the following equation:
u~x !'=s ~x¯ ! 5Œ 5 (6)
!
i i
n n~n 21!
m
where: u x 5 u x (8)
~ ! Π@ ~ !#
i ( j i
j51
th
x =k measured value; and
i,k
where:
x¯ = mean of n measurements.
i
m = number of sources of error affecting the uncertainty of
6.3.2 Type B evaluation of standard uncertainty:
x; and
i
6.3.2.1 A type B evaluation of standard uncertainty of x is
i
th
u, = standard uncertainty due to j source of error.
j
not based on repeated measurements but on an a priori
frequency distribution. In this case the uncertainty is deter-
7. Determining Combined Standard Uncertainty
minedfrompreviousmeasurementsdata,experienceorgeneral
7.1 The standard uncertainty of y is obtained by appropri-
knowledge, manufacturer’s specifications, data provided in
ately combining the standard uncertainties of the input esti-
calibration certificates, uncertainties assigned to reference data
mates x , x ,…, x . If all input quantities are independent, the
1 2 N
taken from handbooks, etc.
combined standard uncertainty of y is given by:
6.3.2.2 If the quoted uncertainty from a manufacturer
N N
specification, handbook or other source is stated to be a
]f
2 2 2
u y 5 # u x [ c u x (9)
~ ! ΠF ~ ! Π@ ~ !#
particular multiple of a standard deviation, the standard uncer- ?
c ( xi i ( i i
]X
i5l i i5l
tainty u (x) is simply the quoted value divided by the multi-
c i
where:
plier. For example, the quoted uncertainty is often at the 95%
u = combined standard uncertainty, and
level of confidence. Assuming a normal distribution this
c
c = sensitivity coefficients.
corresponds to a multiplier of two, that is, the standard
i,
uncertainty is half the quoted value.
Eq 9 is referred to as the law of propagation of uncertainty
6.3.2.3 Often the uncertainty is expressed in the form of
and based on a first-order Taylor series approximation of Y = f
upper and lower limits. Usually there is no specific knowledge
(X , X , …, X ). When the nonlinearity of f is significant,
1 2 N
about the possible values of X within the interval and one can
higher-order terms must be included (see clause 5.1.2 in the
i
onlyassumethatitisequallyprobablefor X tolieanywherein
GUM for details).
i
it. Fig. 1 shows the most common example where the corre-
7.2 When the input quantities are correlated, Eq 9 must be
sponding rectangular distribution is symmetric with respect to
revised to include the covariance terms. The combined stan-
its best estimate x. The standard uncertainty in this case is
i
dard uncertainty of y is then calculated from:
given by:
u y 5 (10)
~ !
c
∆X
i
u~x ! 5 (7)
i
N N21 N
=3
c u x 12 c c u x u x r x ,x
Π@ ~ !# ~ ! ~ ! ~ !
i i i j i j i j
( ( (
i5l i5l j5i1l
where:
∆X = half-width of the interval.
where:
i
If some information is known about the distribution of the r(x,x) = estimated correlation coeffıcient between X and
i j i
possiblevaluesofX withintheinterval,thatknowledgeisused X.
i j
to better estimate the standard deviation.
Since the true values of the input quantities are not known,
the correlation coefficient is estimated on the basis of the
measured values of the input quantities.
8. Determining Expanded Uncertainty
8.1 It is often necessary to give a measure of uncertainty
that defines an interval about the measurement result that may
beexpectedtoencompassalargefractionofthedistributionof
values that could reasonably be attributed to the measurand.
Thismeasureistermedexpandeduncertaintyandisdenotedby
U. The expanded uncertainty is obtained by multiplying the
combined standard uncertainty by a coverage factor k:
U~y! 5 ku ~y! (11)
c
where:
U = expanded uncertainty, and
FIG. 1 Rectangular Distribution
E2536 − 21
by the units of y and U. Alternatively, the relative expanded
k = coverage factor.
uncertainty U/|y| in percent can be specified instead of the
8.1.1 The value of the coverage factor k is chosen on the
absolute expanded uncertainty. In either case the report shall
basis of the level of confidence required of the interval y – U
describe how the measurand Y is defined, specify the approxi-
to y + U. In general, k will be in the range 2 to 3. Because of
mate confidence level and explain how the corresponding
the Central Limit Theorem, k can usually be determined from:
coverage factor was determined. The former can be done by
k 5 t~ν ! (12)
eff
reference to the appropriate fire test standard.
where:
9.2 The report shall also include a discussion of sources of
t = t-distributionstatisticforthespecifiedconfidencelevel
uncertainty that are not addressed by the analysis.
and degrees of freedom, and
ν = effective degrees of freedom.
eff
10. Summary of Procedure For Evaluating and
Table1givesvaluesofthet-distributionstatisticfordifferent
Expressing Uncertainty
levels of confidence and degrees of freedom.Amore complete
10.1 The procedure for evaluating and expressing uncer-
table can be found in Annex G of the GUM.
tainty of fire test results involves the following steps:
8.1.2 The effective degrees of freedom can be computed
from the Welch-Satterthwaite formula:
10.1.1 Express mathematically the relationship between the
measurand Yandtheinputquantities X uponwhich Ydepends:
u y i
@ ~ !#
c
ν 5 (13)
eff N 4
Y = f(X , X , … , X ).
u x 1 2 N
@ ~ !#
i
(
ν 10.1.2 Determine x, the estimated value for each input
i5l
i i
quantity X.
i
where:
10.1.3 Identify all sources of error for each input quantity
ν = degrees of freedom assigned to the standard uncertainty
i
and evaluate the standard uncertainty u(x) for each input
i
of input estimate x.
i
estimate x.
i
8.1.3 The degrees of freedom ν is equal to n −1 if x is
i i
10.1.4 Evaluate the correlation coefficient for estimates of
estimatedasthearithmeticmeanof nindependentobservations
input quantities that are dependent.
(type A standard uncertainty evaluation). If u(x) is obtained
i
10.1.5 Calculate the result of the measurement, that is, the
from a type B evaluation and it can be treated as exactly
estimate y of the measurand Y from the functional relationship
known, which is often the case in practice, ν →∞.If u(x)is
i i
f using the estimates x of the input quantities X obtained in
not exactly known, ν can be estimated from:
i i
i
10.1.2.
2 22
1 u x 1 ∆u x
@ ~ !# ~ !
c i i
ν ' ' (14)
S D
i 2 10.1.6 Determine the combined standard uncertainty u (y)
2 σ u x 2 u x c
@ ~ ~ !!# ~ !
i i
ofthemeasurementresultyfromthestandarduncertaintiesand
The quantity in large brackets in Eq 14 is the relative uncer-
tainty of u(x), which is a subjective quantity whose value is
correlation coefficients associated with the input estimates as
i
obtained by scientific judgement based on the pool of avail-
described in Section 7.
able information.
10.1.7 Select a coverage factor k on the basis of the desired
8.2 The probability distribution of u (y) is often approxi-
c
levelofconfidenceasdescribedinSection8andmultiply u (y)
c
mately normal and the effective degrees of freedom of u (y)is
c
by this value to obtain the expanded uncertainty U.
of significant size. When this is the case, one can assume that
10.1.8 Report the result of the measurement y together with
taking k = 2 produces an interval having a level of confidence
its expanded uncertainty U as discussed in Section 9.
of approximately 95.5 %, and that taking k = 3 produces an
interval having a level of confidence of approximately 99.7 %.
11. Keywords
9. Reporting Uncertainty
11.1 fire test; fire test laboratory; measurand; measurement
9.1 The result of a measurement and the corresponding
uncertainty; quality
uncertaintyshallbereportedintheformof Y= y 6 Ufollowed
TABLE 1 Selected Values of the t-distribution Statistic
Degrees of Confidence Level Degrees of Confidence Level Degrees of Confidence Level
Freedom Freedom Freedom
95% 99% 95% 99% 95% 99%
1 12.71 63.66 6 2.45 3.71 20 2.09 2.85
2 4.30 9.92 7 2.36 3.50 30 2.04 2.75
3 3.18 5.84 8 2.31 3.36 40 2.02 2.70
4 2.78 4.60 9 2.26 3.25 50 2.01 2.68
5 2.57 4.03 10 2.23 3.17 ` 1.96 2.58
E2536 − 21
APPENDIXES
(Nonmandatory Information)
X1. ILLUSTRATIVE EXAMPLE
∆ h
X1.1 Introduction:
c
˙
Y[Q, X [ , X 5 C, X [∆P, X 5 T , X 5 X , X 5 β
1 2 3 4 e 5 6
O
r 2
o
X1.1.1 Heat release rate measured in the Cone Calorimeter
(X1.3)
according to Test Method E1354 is used here to illustrate the
˙
Note that in a test Q is calculated as a function of time
application of the guidelines provided in this guide.
based on the input quantities measured at discrete time inter-
vals ∆t.
X1.2 Expresstherelationshipbetweenthemeasurand Yand
X1.3 Determine x, the estimated value of X for each input
the input quantities X.
i i
i
quantity.
X1.2.1 The heat release rate is calculated according to Eq 4
X1.3.1 For the purpose of this example a 19 mm thick slab
in Test Method E1354:
ofwesternredcedarwastestedataheatfluxof50kW/m .The
o
X 2 X
O
∆h ∆P 2
O test was conducted in the horizontal orientation with the
c 2
˙
Q 5 1.10C (X1.1)
F G Œ
F G
r T 1.105 21.5X
retainer frame. The spark igniter was used and the test was
o e
O
terminated after 15 min.
where:
X1.3.2 The corresponding measured values of ∆P(X ), T
3 e
˙
Q = heat release rate (kW),
(X ) and X (X ) are shown as a function of time in Figs.
4 O 5
∆h = net heat of combustion (kJ/kg), 2
c
X1.1-X1.3, respectively. Note that the latter is shifted over the
r = stoichiometric oxygen to fuel ratio (kg/kg),
o
1/2 1/2 1/2
delay time of the oxygen analyzer to synchronize X with the
C = orifice coefficient (m ·kg ·K ), 5
other two measured input quantities.
∆P = pressure drop across the orifice plate (Pa),
T = exhaust stack temperature at the orifice plate flow
e
X1.3.3 The first input quantity is estimated as X = ∆h /r ≈
1 c o
meter (K),
13 100 kJ/kg = x , which is based on the average for a large
o 1
X = ambient oxygen mole fraction in dry air (0,2095),
O
number of organic materials (1). The orifice constant was
and
obtainedfromamethanegasburnercalibrationasdescribedin
X = measured oxygen mole fraction in the exhaust duct.
O
section 13.2 of Test Method E1354 and is equal to X = C ≈
1/2 1/2 1/2
Theratioof ∆h to r isreferredtoas“Thornton’sconstant”. 0.04430 m g K = x . Finally, the mid value of 1.5 is used
c o 2
Theaveragevalueofthisconstantis13,100kJ/kgO ,whichis to estimate the expansion factor β.
accurate to within 65% for a large number of organic
X1.4 Identify all sources of error and evaluate the standard
materials (1).
uncertainty for each X.
i
X1.2.2 EqX1.1isbasedontheassumptionthatthestandard
X1.4.1 Standarduncertaintyof ∆h /r -Theaveragevalueof
c o
volume of the gaseous products of combustion is 50 % larger
13 100 kJ/kg is reported in the literature to be accurate to
than the volume of oxygen consumed in combustion. This is
within 65% for a large number of organic materials (1). The
correct for complete combustion of methane. However, for
probability distribution is assumed to be rectangular, which,
purecarbonthereisnoincreaseinvolumebecauseonemoleof
according to Eq 7 leads to:
CO isgeneratedpermoleofO consumed.Forpurehydrogen
2 2
∆x 0.05 313,100 kJ
the volume doubles as two moles
...

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.
Designation: E2536 − 15a E2536 − 21 An American National Standard
Standard Guide for
Assessment of Measurement Uncertainty in Fire Tests
This standard is issued under the fixed designation E2536; 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.
INTRODUCTION
The objective of a measurement is to determine the value of the measurand, that is, the physical
quantity that needs to be measured. Every measurement is subject to error, no matter how carefully
it is conducted. The (absolute) error of a measurement is defined in Eq 1.
All terms in Eq 1 have the units of the physical quantity that is measured. This equation cannot be
used to determine the error of a measurement because the true value is unknown, otherwise a
measurement would not be needed. In fact, the true value of a measurand is unknowable because it
cannot be measured without error. However, it is possible to estimate, with some confidence, the
expected limits of error. This estimate is referred to as the uncertainty of the measurement and
provides a quantitative indication of its quality.
Errors of measurement have two components, a random component and a systematic component.
The former is due to a number of sources that affect a measurement in a random and uncontrolled
manner. Random errors cannot be eliminated, but their effect on uncertainty is reduced by increasing
the number of repeat measurements and by applying a statistical analysis to the results. Systematic
errors remain unchanged when a measurement is repeated under the same conditions. Their effect on
uncertainty cannot be completely eliminated either, but is reduced by applying corrections to account
for the error contribution due to recognized systematic effects. The residual systematic error is
unknown and shall be treated as a random error for the purpose of this standard.
General principles for evaluating and reporting measurement uncertainties are described in the
Guide on Uncertainty of Measurements (GUM). Application of the GUM to fire test data presents
some unique challenges. This standard shows how these challenges can be overcome. An example to
illustrate application of the guidelines provided in this standard can be found in Appendix X1.
ε[y 2 Y (1)
where:
where:
ε = measurement error;
y = measured value of the measurand; and
Y = true value of the measurand.
This guide is under the jurisdiction of ASTM Committee E05 on Fire Standards and is the direct responsibility of Subcommittee E05.31 on Terminology and Services
/ Functions.
Current edition approved Oct. 1, 2015Oct. 1, 2021. Published November 2015November 2021. Originally approved in 2006. Last previous edition approved in 2015 as
E2536-15.E2536-15a. DOI: 10.1520/E2536-15A.10.1520/E2536-21.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2536 − 21
1. Scope
1.1 This guide covers the evaluation and expression of uncertainty of measurements of fire test methods developed and maintained
by ASTM International, based on the approach presented in the GUM. The use in this process of precision data obtained from a
round robin is also discussed.
1.2 The guidelines presented in this standard can also be applied to evaluate and express the uncertainty associated with fire test
results. However, it may not be possible to quantify the uncertainty of fire test results if some sources of uncertainty cannot be
accounted for. This problem is discussed in more detail in Appendix X2.
1.3 Application of this guide is limited to tests that provide quantitative results in engineering units. This includes, for example,
methods for measuring the heat release rate of burning specimens based on oxygen consumption calorimetry, such as Test Method
E1354.
1.4 This guide does not apply to tests that provide results in the form of indices or binary results (for example, pass/fail). For
example, the uncertainty of the Flame Spread Index obtained according to Test Method E84 cannot be determined.
1.5 In some cases additional guidance is required to supplement this standard. For example, the expression of uncertainty of heat
release rate measurements at low levels requires additional guidance and uncertainties associated with sampling are not explicitly
addressed.
1.6 This fire standard cannot be used to provide quantitative measures.
1.7 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.8 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:
E84 Test Method for Surface Burning Characteristics of Building Materials
E119 Test Methods for Fire Tests of Building Construction and Materials
E176 Terminology of Fire Standards
E230 Specification for Temperature-Electromotive Force (emf) Tables for Standardized Thermocouples
E691 Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
E1354 Test Method for Heat and Visible Smoke Release Rates for Materials and Products Using an Oxygen Consumption
Calorimeter
E2989 Guide for Assessment of Continued Applicability of Reaction to Fire Test Reports Used in Building Regulation
2.2 ISO Standards:
ISO 29473 Fire tests — Uncertainty of measurements in fire tests
ISO/IEC 17025 General requirements for the competence of testing and calibration laboratories
GUM Guide to the expression of uncertainty in measurement
2.3 CEN Standard:
EN 13823 Reaction to fire tests for building products – Building products excluding floorings exposed to the thermal attack by
a single burning item
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.
Available from International Organization for Standardization, P.O. Box 56, CH-1211, Geneva 20, Switzerland.Standardization (ISO), ISO Central Secretariat, Chemin
de Blandonnet 8, CP 401, 1214 Vernier, Geneva, Switzerland, https://www.iso.org.
Available from European Committee for Standardization (CEN), Avenue Marnix 17, B-1000, Brussels, Belgium, http://www.cen.eu.
E2536 − 21
3. Terminology
3.1 Definitions: For definitions of terms used in this guide and associated with fire issues, refer to the terminology contained in
Terminology E176. For definitions of terms used in this guide and associated with precision issues, refer to the terminology
contained in Practice E691.
3.2 Definitions of Terms Specific to This Standard:
3.2.1 accuracy of measurement, n—closeness of the agreement between the result of a measurement and the true value of the
measurand.
3.2.2 combined standard uncertainty, n—standard uncertainty of the result of a measurement when that result is obtained from 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.
3.2.3 coverage factor, n—numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an
expanded uncertainty.
3.2.4 error (of measurement), n—result of a measurement minus the true value of the measurand; error consists of two
components: random error and systematic error.
3.2.5 expanded uncertainty, n—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.
3.2.6 measurand, n—quantity subject to measurement.
3.2.7 precision, n—variability of test result measurements around reported test result value.
3.2.8 random error, n—result of a measurement minus the mean that would result from an infinite number of measurements of
the same measurand carried out under repeatability conditions.
3.2.9 repeatability (of results of measurements), n—closeness of the agreement between the results of successive independent
measurements of the same measurand carried out under repeatability conditions.
3.2.10 repeatability conditions, n—on identical test material using the same measurement procedure, observer(s), and measuring
instrument(s) and performed in the same laboratory during a short period of time.
3.2.11 reproducibility (of results of measurements), n— closeness of the agreement between the results of measurements of the
same measurand carried out under reproducibility conditions.
3.2.12 reproducibility conditions, n—on identical test material using the same measurement procedure, but different observer(s)
and measuring instrument(s) in different laboratories performed during a short period of time.
3.2.13 standard deviation, n—a quantity characterizing the dispersion of the results of a series of measurements of the same
measurand; the standard deviation is proportional to the square root of the sum of the squared deviations of the measured values
from the mean of all measurements.
3.2.14 standard uncertainty, n—uncertainty of the result of a measurement expressed as a standard deviation.
3.2.15 systematic error (or bias), n—mean that would result from an infinite number of measurements of the same measurand
carried out under repeatability conditions minus the true value of the measurand.
3.2.16 type A evaluation (of uncertainty), n—method of evaluation of uncertainty by the statistical analysis of series of
observations.
E2536 − 21
3.2.17 type B evaluation (of uncertainty), n—method of evaluation of uncertainty by means other than the statistical analysis of
series of observations.
3.2.18 uncertainty of measurement, n—parameter, associated with the result of a measurement, that characterizes the dispersion
of the values that could reasonably be attributed to the measurand.
4. Summary of Guide
4.1 This guide provides concepts and calculation methods to assess the uncertainty of measurements obtained from fire tests.
4.2 Appendix X1 of this guide contains an example to illustrate application of this guide by assessing the uncertainty of heat
release rate measured in the Cone Calorimeter (Test Method E1354).
5. Significance and Use
5.1 Users of fire test data often need a quantitative indication of the quality of the data presented in a test report. This quantitative
indication is referred to as the “measurement uncertainty”. There are two primary reasons for estimating the uncertainty of fire test
results.
5.1.1 ISO/IEC 17025 requires that competent testing and calibration laboratories include uncertainty estimates for the results that
are presented in a report.
5.1.2 Fire safety engineers need to know the quality of the input data used in an analysis to determine the uncertainty of the
outcome of the analysis.
6. Evaluating Standard Uncertainty
6.1 A quantitative result of a fire test Y is generally not obtained from a direct measurement, but is determined as a function f from
N input quantities X , … , X :
1 N
Y 5 f X ,X ,…,X (2)
~ !
1 2 N
where:
Y = measurand;
f = functional relationship between the measurand and the input quantities; and
X = input quantities (i = 1 … N).
i
6.1.1 The input quantities are categorized as:
6.1.1.1 quantities whose values and uncertainties are directly determined from single observation, repeated observation or
judgment based on experience, or
6.1.1.2 quantities whose values and uncertainties are brought into the measurement from external sources such as reference data
obtained from handbooks.
6.1.2 An estimate of the output, y, is obtained from Eq 2 using input estimates x , x , …, x for the values of the N input quantities:
1 2 N
y 5 f ~x ,x ,…, x ! (3)
1 2 N
Substituting Eq 2 and 3 into Eq 1 leads to:
y 5 Y1ε 5 Y1ε 1ε 1…1ε (4)
1 2 N
where:
ε = contribution to the total measurement error from the error associated with x .
1 i
6.2 A possible approach to determine the uncertainty of y involves a large number (n) of repeat measurements. The mean value
E2536 − 21
of the resulting distribution y¯ is the best estimate of the measurand. The experimental standard deviation of the mean is the best
~ !
estimate of the standard uncertainty of y, denoted by u(y):
n
~y 2 y¯!
2 ( k
s y
~ ! k51
u~y!'=s ~y¯! 5Œ 5 (5)
!
n n~n 2 1!
where:
u = standard uncertainty,
s = experimental standard deviation,
n = number of observations;
th
y = k measured value, and
k
y¯ = mean of n measurements.
The number of observations n shall be large enough to ensure that y¯ provides a reliable estimate of the expectation μ of the
y
2 2
random variable y, and that s y¯ provides a reliable estimate of the variance σ y¯ 5σ y /n. If the probability distribution of y is
~ ! ~ ! ~ !
1/2
normal, then standard deviation of s ~y¯! relative to σ~y¯! is approximately [2(n-1)]− . Thus, for n = 10 the relative uncertainty of
s y¯ is 24 %t, while for n = 50 it is 10 %. Additional values are given in Table E.1 in annex E of the GUM.
~ !
6.3 Unfortunately it is often not feasible or even possible to perform a sufficiently large number of repeat measurements. In those
cases, the uncertainty of the measurement can be determined by combining the standard uncertainties of the input estimates. The
standard uncertainty of an input estimate x is obtained from the distribution of possible values of the input quantity X . There are
i i
two types of evaluations depending on how the distribution of possible values is obtained.
6.3.1 Type A evaluation of standard uncertainty—A type A evaluation of standard uncertainty of x is based on the frequency
i
distribution, which is estimated from a series of n repeated observations x (k = 1 … n). The resulting equation is similar to Eq
i,k
5:
n
~x 2 x¯ !
2 ( i,k i
s x
~ ! k51
i
u~x !'=s ~x¯ ! 5Œ 5 (6)
!
i i
n n~n 2 1!
where:
th
x = k measured value; and
i,k
x¯ = mean of n measurements.
i
6.3.2 Type B evaluation of standard uncertainty:
6.3.2.1 A type B evaluation of standard uncertainty of x is not based on repeated measurements but on an a priori frequency
i
distribution. In this case the uncertainty is determined from previous measurements data, experience or general knowledge,
manufacturer’s specifications, data provided in calibration certificates, uncertainties assigned to reference data taken from
handbooks, etc.
6.3.2.2 If the quoted uncertainty from a manufacturer specification, handbook or other source is stated to be a particular multiple
of a standard deviation, the standard uncertainty u (x ) is simply the quoted value divided by the multiplier. For example, the quoted
c i
uncertainty is often at the 95 % level of confidence. Assuming a normal distribution this corresponds to a multiplier of two, that
is, the standard uncertainty is half the quoted value.
6.3.2.3 Often the uncertainty is expressed in the form of upper and lower limits. Usually there is no specific knowledge about the
possible values of X within the interval and one can only assume that it is equally probable for X to lie anywhere in it. Fig. 1 shows
i i
the most common example where the corresponding rectangular distribution is symmetric with respect to its best estimate x . The
i
standard uncertainty in this case is given by:
ΔX
i
u x 5 (7)
~ !
i
=3
where:
E2536 − 21
FIG. 1 Rectangular Distribution
ΔX = half-width of the interval.
i
If some information is known about the distribution of the possible values of X within the interval, that knowledge is used to
i
better estimate the standard deviation.
6.3.3 Accounting for multiple sources of error—The uncertainty of an input quantity is sometimes due to multiple sources error.
In this case, the standard uncertainty associated with each source of error has to be estimated separately and the standard
uncertainty of the input quantity is then determined according to the following equation:
m
u~x ! 5Π@u ~x !# (8)
i ( j i
j51
where:
m = number of sources of error affecting the uncertainty of x ; and
i
th
u , = standard uncertainty due to j source of error.
j
7. Determining Combined Standard Uncertainty
7.1 The standard uncertainty of y is obtained by appropriately combining the standard uncertainties of the input estimates x , x ,…,
1 2
x . If all input quantities are independent, the combined standard uncertainty of y is given by:
N
N N
]f
2 2 2
u ~y! 5Π# u ~x ![Π@c u~x !# (9)
F ?
c ( xi i ( i i
]X
i5l i5l
i
where:
u = combined standard uncertainty, and
c
c = sensitivity coefficients.
i,
Eq 9 is referred to as the law of propagation of uncertainty and based on a first-order Taylor series approximation of Y = f (X ,
X , …, X ). When the nonlinearity of f is significant, higher-order terms must be included (see clause 5.1.2 in the GUM for details).
2 N
7.2 When the input quantities are correlated, Eq 9 must be revised to include the covariance terms. The combined standard
uncertainty of y is then calculated from:
u ~y!5 (10)
c
N N21 N
Π@c u~x !# 12 c c u~x !u~x !r~x ,x !
( i i ( ( i j i j i j
i5l i5l j5i1l
where:
r(x , x ) = estimated correlation coeffıcient between X and X .
i j i j
Since the true values of the input quantities are not known, the correlation coefficient is estimated on the basis of the measured
values of the input quantities.
E2536 − 21
8. Determining Expanded Uncertainty
8.1 It is often necessary to give a measure of uncertainty that defines an interval about the measurement result that may be
expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand. This
measure is termed expanded uncertainty and is denoted by U. The expanded uncertainty is obtained by multiplying the combined
standard uncertainty by a coverage factor k:
U y 5 ku y (11)
~ ! ~ !
c
where:
U = expanded uncertainty, and
k = coverage factor.
8.1.1 The value of the coverage factor k is chosen on the basis of the level of confidence required of the interval y – U to y + U.
In general, k will be in the range 2 to 3. Because of the Central Limit Theorem, k can usually be determined from:
k 5 t~ν ! (12)
eff
where:
t = t-distribution statistic for the specified confidence level and degrees of freedom, and
ν = effective degrees of freedom.
eff
Table 1 gives values of the t-distribution statistic for different levels of confidence and degrees of freedom. A more complete
table can be found in Annex G of the GUM.
8.1.2 The effective degrees of freedom can be computed from the Welch-Satterthwaite formula:
@u ~y!#
c
ν 5 (13)
eff N
@u~x !#
i
(
ν
i5l
i
where:
ν = degrees of freedom assigned to the standard uncertainty of input estimate x .
i i
8.1.3 The degrees of freedom ν is equal to n −1 if x is estimated as the arithmetic mean of n independent observations (type A
i i
standard uncertainty evaluation). If u(x ) is obtained from a type B evaluation and it can be treated as exactly known, which is often
i
the case in practice, ν → ∞. If u(x ) is not exactly known, ν can be estimated from:
i i i
2 22
1 @u x # 1 Δu x
~ ! ~ !
c i i
ν ' ' (14)
2 S D
i
2@σ~u~x !!# 2 u~x !
i i
The quantity in large brackets in Eq 14 is the relative uncertainty of u(x ), which is a subjective quantity whose value is ob-
i
tained by scientific judgement based on the pool of available information.
8.2 The probability distribution of u (y) is often approximately normal and the effective degrees of freedom of u (y) is of
c c
significant size. When this is the case, one can assume that taking k = 2 produces an interval having a level of confidence of
approximately 95.5 %, and that taking k = 3 produces an interval having a level of confidence of approximately 99.7 %.
9. Reporting Uncertainty
9.1 The result of a measurement and the corresponding uncertainty shall be reported in the form of Y = y 6 U followed by the
TABLE 1 Selected Values of the t-distribution Statistic
Degrees of Confidence Level Degrees of Confidence Level Degrees of Confidence Level
Freedom Freedom Freedom
95% 99% 95% 99% 95% 99%
1 12.71 63.66 6 2.45 3.71 20 2.09 2.85
2 4.30 9.92 7 2.36 3.50 30 2.04 2.75
3 3.18 5.84 8 2.31 3.36 40 2.02 2.70
4 2.78 4.60 9 2.26 3.25 50 2.01 2.68
5 2.57 4.03 10 2.23 3.17 ` 1.96 2.58
E2536 − 21
units of y and U. Alternatively, the relative expanded uncertainty U/|y| in percent can be specified instead of the absolute expanded
uncertainty. In either case the report shall describe how the measurand Y is defined, specify the approximate confidence level and
explain how the corresponding coverage factor was determined. The former can be done by reference to the appropriate fire test
standard.
9.2 The report shall also include a discussion of sources of uncertainty that are not addressed by the analysis.
10. Summary of Procedure For Evaluating and Expressing Uncertainty
10.1 The procedure for evaluating and expressing uncertainty of fire test results involves the following steps:
10.1.1 Express mathematically the relationship between the measurand Y and the input quantities X upon which Y depends: Y =
i
f(X , X , … , X ).
1 2 N
10.1.2 Determine x , the estimated value for each input quantity X .
i i
10.1.3 Identify all sources of error for each input quantity and evaluate the standard uncertainty u(x ) for each input estimate x .
i i
10.1.4 Evaluate the correlation coefficient for estimates of input quantities that are dependent.
10.1.5 Calculate the result of the measurement, that is, the estimate y of the measurand Y from the functional relationship f using
the estimates x of the input quantities X obtained in 10.1.2.
i i
10.1.6 Determine the combined standard uncertainty u (y) of the measurement result y from the standard uncertainties and
c
correlation coefficients associated with the input estimates as described in Section 7.
10.1.7 Select a coverage factor k on the basis of the desired level of confidence as described in Section 8 and multiply u (y) by
c
this value to obtain the expanded uncertainty U.
10.1.8 Report the result of the measurement y together with its expanded uncertainty U as discussed in Section 9.
11. Keywords
11.1 fire test; fire test laboratory; measurand; measurement uncertainty; quality
APPENDIXES
(Nonmandatory Information)
X1. ILLUSTRATIVE EXAMPLE
X1.1 Introduction:
X1.1.1 Heat release rate measured in the Cone Calorimeter according to Test Method E1354 is used here to illustrate the
application of the guidelines provided in this guide.
X1.2 Express the relationship between the measurand Y and the input quantities X .
i
X1.2.1 The heat release rate is calculated according to Eq 4 in Test Method E1354:
o
X 2 X
O
Δh ΔP 2
O
c 2
˙
Q 5 1.10C (X1.1)
Œ
F G F G
r T 1.105 2 1.5X
o e
O
E2536 − 21
where:
Q˙ = heat release rate (kW),
Δh = net heat of combustion (kJ/kg),
c
r = stoichiometric oxygen to fuel ratio (kg/kg),
o
1/2 1/2 1/2
C = orifice coefficient (m ·kg ·K ),
ΔP = pressure drop across the orifice plate (Pa),
T = exhaust stack temperature at the orifice plate flow meter (K),
e
o
X = ambient oxygen mole fraction in dry air (0,2095), and
O
X = measured oxygen mole fraction in the exhaust duct.
O
The ratio of Δh to r is referred to as “Thornton’s constant”. The average value of this constant is 13,100 kJ/kg O , which is
c o 2
accurate to within 65 % for a large number of organic materials (1).
X1.2.2 Eq X1.1 is based on the assumption that the standard volume of the gaseous products of combustion is 50 % larger than
the volume of oxygen consumed in combustion. This is correct for complete combustion of methane. However, for pure carbon
there is no increase in volume because one mole of CO is generated per mole of O consumed. For pure hydrogen the volume
2 2
doubles as two moles of water vapor are generated per mole O consumed. A more accurate form of Eq X1.1 that takes the volume
increase into account is as follows: (2)
o
X 2 X
O
Δh ΔP
2 O
c 2
˙
Q 5 1.10CΠ(X1.2)
F G o
F G
r T 11~β2 1! X 2 β X
o e O
2 O
where:
β = moles of gaseous combustion products generated per mole of O consumed.
This is the equation that is used to estimate the uncertainty of heat release rate measurements in the Cone Calorimeter. Hence,
the output and input quantities are as follows:
Δ h
c
˙
Y[Q, X [ , X 5 C, X [ΔP, X 5 T , X 5 X , X 5 β (X1.3)
1 2 3 4 e 5 6
O
r 2
o
Note that in a test Q˙ is calculated as a function of time based on the input quantities measured at discrete time intervals Δt.
X1.3 Determine x , the estimated value of X for each input quantity.
i i
X1.3.1 For the purpose of this example a 19 mm thick slab of western red cedar was tested at a heat flux of 50 kW/m . The test
was conducted in the horizontal orientation with the retainer frame. The spark igniter was used and the test was terminated after
15 min.
X1.3.2 The corresponding measured values of ΔP (X ), T (X ) and X (X ) are shown as a function of time in Figs. X1.1-X1.3,
3 e 4 O 5
respectively. Note that the latter is shifted over the delay time of the oxygen analyzer to synchronize X with the other two
measured input quantities.
X1.3.3 The first input quantity is estimated as X = Δh /r ≈ 13 100 kJ/kg = x , which is based on the average for a large number
1 c o 1
of organic materials (1). The orifice constant was obtained from a methane gas burner calibration as described in section 13.2 of
1/2 1/2 1/2
Test Method E1354 and is equal to X = C ≈ 0.04430 m g K = x . Finally, the mid value of 1.5 is used to estimate the
2 2
expansion factor β.
The boldface numbers in parentheses refer to a list of references at the end of this standard.
E2536 − 21
FIG. X1.1 Differential Pressure Measurements
FIG. X1.2 Stack Temperature Measurements
X1.4 Identify all sources of error and evaluate the standard uncertainty for each X .
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X1.4.1 Standard uncertainty of Δh /r - The average value of 13 100 kJ/kg is reported in the literature to be accurate to within
c o
65 % for a large number of organic materials (1). The probability distribution is assumed to be rectangular, which, according to
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