ASTM E2536-15

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.

General Information

• Status: Historical
• Publication Date: 31-Dec-2014
• 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

• Replaces: ASTM E2536-14 - Standard Guide for Assessment of Measurement Uncertainty in Fire Tests
• Effective Date: 01-Jan-2015
• Referred By: ASTM E535-19 - Standard Practice for Preparation of Fire-Test-Response Standards
• Effective Date: 01-Jan-2015
• Referred By: ASTM E176-21ae1 - Standard Terminology of Fire Standards
• Effective Date: 01-Jan-2015
• Referred By: ASTM E2989-19a - Standard Guide for Assessment of Continued Applicability of Reaction to Fire Test Reports Used in Building Regulation
• Effective Date: 01-Jan-2015

Standards Content (Sample)

NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: E2536 − 15 AnAmerican 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
1.2 The guidelines presented in this standard can also be
uncertainty of measurements of fire test methods developed
and maintained byASTM International, based on the approach 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 Jan. 1, 2015. Published January 2015. Originally
1.3 Application of this guide is limited to tests that provide
approved in 2006. Last previous edition approved in 2014 as E2536-14. DOI:
10.1520/E2536-15. 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 − 15
example, methods for measuring the heat release rate of 3.2.1 accuracy of measurement, n—closeness of the agree-
burning specimens based on oxygen consumption calorimetry, mentbetweentheresultofameasurementandthetruevalueof
such as Test Method E1354. the measurand.
1.4 This guide does not apply to tests that provide results in 3.2.2 combined standard uncertainty, n—standard uncer-
the form of indices or binary results (for example, pass/fail). tainty of the result of a measurement when that result is
For example, the uncertainty of the Flame Spread Index obtainedfromthevaluesofanumberofotherquantities,equal
obtained according to Test Method E84 cannot be determined. to the positive square root of a sum of terms, the terms being
the variances or covariances of these other quantities weighted
1.5 In some cases additional guidance is required to supple-
according to how the measurement result varies with changes
ment this standard. For example, the expression of uncertainty
in these quantities.
of heat release rate measurements at low levels requires
additionalguidanceanduncertaintiesassociatedwithsampling 3.2.3 coverage factor, n—numerical factor used as a multi-
plierofthecombinedstandarduncertaintyinordertoobtainan
are not explicitly addressed.
expanded uncertainty.
1.6 Thisfirestandardcannotbeusedtoprovidequantitative
measures. 3.2.4 error (of measurement), n—result of a measurement
minus the true value of the measurand; error consists of two
1.7 The values stated in SI units are to be regarded as
components: random error and systematic error.
standard. No other units of measurement are included in this
standard. 3.2.5 expanded uncertainty, n—quantitydefininganinterval
about the result of a measurement that may be expected to
2. Referenced Documents
encompass a large fraction of the distribution of values that
could reasonably be attributed to the measurand.
2.1 ASTM Standards:
E84Test Method for Surface Burning Characteristics of
3.2.6 measurand, n—quantity subject to measurement.
Building Materials
3.2.7 precision, n—variability of test result measurements
E119Test Methods for Fire Tests of Building Construction
around reported test result value.
and Materials
E176Terminology of Fire Standards 3.2.8 random error, n—result of a measurement minus the
mean that would result from an infinite number of measure-
E230Specification and Temperature-Electromotive Force
(EMF) Tables for Standardized Thermocouples ments of the same measurand carried out under repeatability
conditions.
E691Practice for Conducting an Interlaboratory Study to
Determine the Precision of a Test Method
3.2.9 repeatability (of results of measurements),
E1354Test Method for Heat and Visible Smoke Release
n—closeness of the agreement between the results of succes-
Rates for Materials and Products Using an Oxygen Con-
siveindependentmeasurementsofthesamemeasurandcarried
sumption Calorimeter
out under repeatability conditions.
2.2 ISO Standards:
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
2.3 CEN Standard: 3.2.11 reproducibility (of results of measurements), n—
closeness of the agreement between the results of measure-
EN 13823Reaction to fire tests for building products –
Building products excluding floorings exposed to the ments of the same measurand carried out under reproducibility
conditions.
thermal attack by a single burning item
3.2.12 reproducibility conditions, n—on identical test mate-
3. Terminology
rial using the same measurement procedure, but different
3.1 Definitions: For definitions of terms used in this guide
observer(s) and measuring instrument(s) in different laborato-
and associated with fire issues, refer to the terminology
ries performed during a short period of time.
contained in Terminology E176. For definitions of terms used
3.2.13 standard deviation, n—a quantity characterizing the
in this guide and associated with precision issues, refer to the
dispersion of the results of a series of measurements of the
terminology contained in Practice E691.
same measurand; the standard deviation is proportional to the
3.2 Definitions of Terms Specific to This Standard:
square root of the sum of the squared deviations of the
measured values from the mean of all measurements.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
3.2.14 standard uncertainty, n—uncertainty of the result of
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
a measurement expressed as a standard deviation.
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website.
3.2.15 systematic error (or bias), n—meanthatwouldresult
Available from International Organization for Standardization, P.O. Box 56,
from an infinite number of measurements of the same mea-
CH-1211, Geneva 20, Switzerland.
surandcarriedoutunderrepeatabilityconditionsminusthetrue
Available from European Committee for Standardization (CEN), Avenue
Marnix 17, B-1000, Brussels, Belgium, http://www.cen.eu. value of the measurand.
E2536 − 15
3.2.16 type A evaluation (of uncertainty), n—method of y 5 Y1ε 5 Y1ε 1ε 1…1ε (4)
1 2 N
evaluation of uncertainty by the statistical analysis of series of
where:
observations.
ε = contribution to the total measurement error from the
3.2.17 type B evaluation (of uncertainty), n—method of
error associated with x.
i
evaluation of uncertainty by means other than the statistical
6.2 A possible approach to determine the uncertainty of y
analysis of series of observations.
involvesalargenumber(n)ofrepeatmeasurements.Themean
3.2.18 uncertainty of measurement, n—parameter, associ-
value of the resulting distribution y¯ is the best estimate of the
~ !
ated with the result of a measurement, that characterizes the
measurand.Theexperimentalstandarddeviationofthemeanis
dispersion of the values that could reasonably be attributed to
the best estimate of the standard uncertainty of y, denoted by
the measurand.
u(y):
4. Summary of Guide
n
~y 2 y¯!
4.1 Thisguideprovidesconceptsandcalculationmethodsto 2 ( k
s ~y!
k51
u y '=s y¯ 5 5 (5)
~ ! ~ ! Œ
assess the uncertainty of measurements obtained from fire !
n n n 2 1
~ !
tests.
where:
4.2 Appendix X1 of this guide contains an example to
u = standard uncertainty,
illustrate application of this guide by assessing the uncertainty
s = experimental standard deviation,
of heat release rate measured in the Cone Calorimeter (Test
n = number of observations;
Method E1354).
th
y =k measured value, and
k
y¯ = mean of n measurements.
5. Significance and Use
The number of observations n shall be large enough to
5.1 Users of fire test data often need a quantitative indica-
ensure that y¯ provides a reliable estimate of the expectation µ
y
tion of the quality of the data presented in a test report. This
of the random variable y, and that s ~y¯! provides a reliable
quantitative indication is referred to as the “measurement
estimate of the variance σ y¯ 5σ y /n. If the probability distri-
~ ! ~ !
uncertainty”. There are two primary reasons for estimating the
bution of y is normal, then standard deviation of s y¯ relative
~ !
uncertainty of fire test results.
1/2
to σ ~y¯! is approximately [2(n-1)]− . Thus, for n=10 the
5.1.1 ISO/IEC17025 requires that competent testing and
relative uncertainty of s y¯ is 24 %t, while for n=50itis10
~ !
calibration laboratories include uncertainty estimates for the
%.Additional values are given in Table E.1 in annex E of the
results that are presented in a report.
GUM.
5.1.2 Fire safety engineers need to know the quality of the
input data used in an analysis to determine the uncertainty of
6.3 Unfortunately it is often not feasible or even possible to
the outcome of the analysis.
performasufficientlylargenumberofrepeatmeasurements.In
those cases, the uncertainty of the measurement can be
6. Evaluating Standard Uncertainty
determined by combining the standard uncertainties of the
input estimates. The standard uncertainty of an input estimate
6.1 A quantitative result of a fire test Y is generally not
x is obtained from the distribution of possible values of the
obtained from a direct measurement, but is determined as a i
inputquantity X.Therearetwotypesofevaluationsdepending
function f from N input quantities X , … , X :
i
1 N
on how the distribution of possible values is obtained.
Y 5 f X ,X ,…,X (2)
~ !
1 2 N
6.3.1 Type A evaluation of standard uncertainty—A type A
where:
evaluation of standard uncertainty of x is based on the
i
Y = measurand; frequency distribution, which is estimated from a series of n
f = functional relationship between the measurand and the
repeated observations x (k = 1 … n). The resulting equation
i,k
input quantities; and
is similar to Eq 5:
X = input quantities (i=1 … N).
i
n
6.1.1 The input quantities are categorized as: x 2 x¯
~ !
2 ( i,k i
s x
~ ! k51
i
6.1.1.1 quantities whose values and uncertainties are di-
u~x !'=s ~x¯ ! 5Œ 5 (6)
!
i i
n n~n 2 1!
rectly determined from single observation, repeated observa-
tion or judgment based on experience, or where:
th
6.1.1.2 quantities whose values and uncertainties are
x =k measured value; and
i,k
brought into the measurement from external sources such as
x¯ = mean of n measurements.
i
reference data obtained from handbooks.
6.3.2 Type B evaluation of standard uncertainty:
6.1.2 An estimate of the output, y, is obtained from Eq 2
6.3.2.1 A type B evaluation of standard uncertainty of x is
usinginputestimates x , x , …, x forthevaluesofthe Ninput i
1 2 N
not based on repeated measurements but on an a priori
quantities:
frequency distribution. In this case the uncertainty is deter-
y 5 f x ,x ,…, x (3)
~ !
1 2 N
minedfrompreviousmeasurementsdata,experienceorgeneral
Substituting Eq 2 and 3 into Eq 1 leads to: knowledge, manufacturer’s specifications, data provided in
E2536 − 15
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
]f
2 2 2
specification, handbook or other source is stated to be a
u ~y! 5Π# u ~x ![Π@c u~x !# (9)
F ?
c ( xi i ( i i
]X
i5l i5l
i
particular multiple of a standard deviation, the standard uncer-
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
c
level of confidence. Assuming a normal distribution this
c = sensitivity coefficients.
i,
corresponds to a multiplier of two, that is, the stand
...

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 − 14 E2536 − 15 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:
ε = measurement error;
y = measured value of the measurand; and
Y = true value of the measurand.
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.
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 Feb. 1, 2014Jan. 1, 2015. Published March 2014January 2015. Originally approved in 2006. Last previous edition approved in 20092014 as
E2536-09.E2536-14. DOI: 10.1520/E2536-14.10.1520/E2536-15.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2536 − 15
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.
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 and 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
2.2 ISO Standards:
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
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.
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.
Available from European Committee for Standardization (CEN), Avenue Marnix 17, B-1000, Brussels, Belgium, http://www.cen.eu.
E2536 − 15
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.
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
1 2 N
quantities:
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
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):
E2536 − 15
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
i i
are 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 quote
...