EN ISO 20501:2022

SLOVENSKI STANDARD
01-marec-2023
Nadomešča:
SIST EN 843-5:2007
Fina keramika (sodobna keramika, sodobna tehnična keramika) - Weibullova
statistika za podatke o trdnosti (ISO 20501:2019)
Fine ceramics (advanced ceramics, advanced technical ceramics) - Weibull statistics for
strength data (ISO 20501:2019)
Hochleistungskeramik - Weibullstatistik von Festigkeitswerten (ISO 20501:2019)
Céramiques techniques - Analyse statistique de Weibull des données de résistance à la
rupture (ISO 20501:2019)
Ta slovenski standard je istoveten z: EN ISO 20501:2022
ICS:
81.060.30 Sodobna keramika Advanced ceramics
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EN ISO 20501
EUROPEAN STANDARD
NORME EUROPÉENNE
December 2022
EUROPÄISCHE NORM
ICS 81.060.30 Supersedes EN 843-5:2006
English Version
Fine ceramics (advanced ceramics, advanced technical
ceramics) - Weibull statistics for strength data (ISO
20501:2019)
Céramiques techniques - Analyse statistique de Hochleistungskeramik - Weibullstatistik von
Weibull des données de résistance à la rupture (ISO Festigkeitswerten (ISO 20501:2019)
20501:2019)
This European Standard was approved by CEN on 20 December 2022.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this
European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references
concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN
member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by
translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management
Centre has the same status as the official versions.

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United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2022 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 20501:2022 E
worldwide for CEN national Members.

Contents Page
European foreword . 3

European foreword
The text of ISO 20501:2019 has been prepared by Technical Committee ISO/TC 206 "Fine ceramics” of
the International Organization for Standardization (ISO) and has been taken over as EN ISO 20501:2022
by Technical Committee CEN/TC 184 “Advanced technical ceramics” the secretariat of which is held by
DIN.
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by June 2023, and conflicting national standards shall be
withdrawn at the latest by June 2023.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document supersedes EN 843-5:2006.
Any feedback and questions on this document should be directed to the users’ national standards body.
A complete listing of these bodies can be found on the CEN website.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,
Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of
North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye and the
United Kingdom.
Endorsement notice
The text of ISO 20501:2019 has been approved by CEN as EN ISO 20501:2022 without any modification.

INTERNATIONAL ISO
STANDARD 20501
Second edition
2019-03
Fine ceramics (advanced ceramics,
advanced technical ceramics) —
Weibull statistics for strength data
Céramiques techniques — Analyse statistique de Weibull des données
de résistance à la rupture
Reference number
ISO 20501:2019(E)
©
ISO 2019
ISO 20501:2019(E)
© ISO 2019
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting
on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address
below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Fax: +41 22 749 09 47
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii © ISO 2019 – All rights reserved

ISO 20501:2019(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
3.1 Defect populations . 1
3.2 Mechanical testing . 3
3.3 Statistical terms . 3
3.4 Weibull distributions . 4
4 Symbols . 5
5 Significance and use . 6
6 Method A: maximum likelihood parameter estimators for single flaw populations .7
6.1 General . 7
6.2 Censored data . 8
6.3 Likelihood functions . 8
6.4 Bias correction . 8
6.5 Confidence intervals .10
7 Method B: maximum likelihood parameter estimators for competing flaw populations .13
7.1 General .13
7.2 Censored data .14
7.3 Likelihood functions .14
8 Procedure.15
8.1 Outlying observations .15
8.2 Fractography .15
8.3 Graphical representation .16
9 Test report .18
Annex A (informative) Converting to material-specific strength distribution parameters .19
Annex B (informative) Illustrative examples .21
Annex C (informative) Test specimens with unidentified fracture origin .28
Annex D (informative) Fortran program .31
Bibliography .36
ISO 20501:2019(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso
.org/iso/foreword .html.
This document was prepared by Technical Committee ISO/TC 206, Fine ceramics.
This second edition cancels and replaces the first edition (ISO 20501:2003), which has been technically
revised. It also incorporates the Technical Corrigendum ISO 20501:2003/Cor.1:2009.
The main changes compared to the previous edition are as follows:
— the terms and definitions in Clause 3 have been updated and modified;
— a method to treat a higher number of specimens (N > 120) has been introduced for method A:
maximum likelihood parameter estimators for single flaw populations;
— in Annex D, example codes have been added for calculating the maximum likelihood parameters of
the Weibull distribution with modern analysis software.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/members .html.
iv © ISO 2019 – All rights reserved

ISO 20501:2019(E)
Introduction
Measurements of the strength at failure are taken for one of two reasons: either for a comparison of
the relative quality of two materials regarding fracture strength, or the prediction of the probability
of failure for a structure of interest. This document permits estimates of the distribution parameters
which are needed for either. In addition, this document encourages the integration of mechanical
property data and fractographic analysis.
INTERNATIONAL STANDARD ISO 20501:2019(E)
Fine ceramics (advanced ceramics, advanced technical
ceramics) — Weibull statistics for strength data
1 Scope
This document covers the reporting of uniaxial strength data and the estimation of probability
distribution parameters for advanced ceramics which fail in a brittle fashion. The failure strength of
advanced ceramics is treated as a continuous random variable. Typically, a number of test specimens
with well-defined geometry are brought to failure under well-defined isothermal loading conditions.
The load at which each specimen fails is recorded. The resulting failure stresses are used to obtain
parameter estimates associated with the underlying population distribution.
This document is restricted to the assumption that the distribution underlying the failure strengths is
the two-parameter Weibull distribution with size scaling. Furthermore, this document is restricted to
test specimens (tensile, flexural, pressurized ring, etc.) that are primarily subjected to uniaxial stress
states. Subclauses 6.4 and 6.5 outline methods of correcting for bias errors in the estimated Weibull
parameters, and to calculate confidence bounds on those estimates from data sets where all failures
originate from a single flaw population (i.e. a single failure mode). In samples where failures originate
from multiple independent flaw populations (e.g. competing failure modes), the methods outlined in 6.4
and 6.5 for bias correction and confidence bounds are not applicable.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
NOTE See also Reference [1].
3.1 Defect populations
3.1.1
flaw
inhomogeneity, discontinuity or (defect) feature in a material, which acts as stress concentrator due to
a mechanical load and has therefore a certain risk of mechanical failure
Note 1 to entry: The flaw becomes critical if it acts as fracture origin in a failed specimen.
ISO 20501:2019(E)
3.1.2
censored data
strength measurements (i.e. a sample) containing suspended observations such as that produced by
multiple competing or concurrent flaw populations
Note 1 to entry: Consider a sample where fractography clearly established the existence of three concurrent
flaw distributions (although this discussion is applicable to a sample with any number of concurrent flaw
distributions). The three concurrent flaw distributions are referred to here as distributions A, B, and C. Based
on fractographic analyses, each specimen strength is assigned to a flaw distribution that initiated failure. In
estimating parameters that characterize the strength distribution associated with flaw distribution A, all
specimens (and not just those that failed from type-A flaws) shall be incorporated in the analysis to ensure
efficiency and accuracy of the resulting parameter estimates. The strength of a specimen that failed by a
type-B (or type-C) flaw is treated as a right censored observation relative to the A flaw distribution. Failure
due to a type-B (or type-C) flaw restricts, or censors, the information concerning type-A flaws in a specimen
[2]
by suspending the test before failure occurs by a type-A flaw . The strength from the most severe type-A
flaw in those specimens that failed from type-B (or type-C) flaws is higher than (and thus to the right of) the
observed strength. However, no information is provided regarding the magnitude of that difference. Censored
data analysis techniques incorporated in this document utilize this incomplete information to provide efficient
and relatively unbiased estimates of the distribution parameters.
3.1.3
competing failure modes
distinguishably different types of fracture initiation events that result from concurrent (competing)
flaw distributions
3.1.4
compound flaw distribution
any form of multiple flaw distribution that is neither pure concurrent, nor pure exclusive
Note 1 to entry: A simple example is where every specimen contains the flaw distribution A, while some fraction
of the specimens also contains a second independent flaw distribution B.
3.1.5
concurrent flaw distribution
competing flaw distribution
type of multiple flaw distribution in a homogeneous material where every specimen of that material
contains representative flaws from each independent flaw population
Note 1 to entry: Within a given specimen, all flaw populations are then present concurrently and are competing
to each other to cause failure.
3.1.6
exclusive flaw distribution
mixture flaw distribution
type of multiple flaw distribution created by mixing and randomizing specimens from two or more
versions of a material where each version contains a different single flaw population
Note 1 to entry: Thus, each specimen contains flaws exclusively from a single distribution, but the total data set
reflects more than one type of strength-controlling flaw.
3.1.7
extraneous flaw
strength-controlling flaw observed in some fraction of test specimens that cannot be present in the
component being designed
Note 1 to entry: An example is machining flaws in ground bend specimens that will not be present in as-sintered
components of the same material.
2 © ISO 2019 – All rights reserved

ISO 20501:2019(E)
3.2 Mechanical testing
3.2.1
effective gauge section
that portion of the test specimen geometry included within the limits of integration (volume, area or
edge length) of the Weibull distribution function
Note 1 to entry: In tensile specimens, the integration may be restricted to the uniformly stressed central gauge
section, or it may be extended to include transition and shank regions.
3.2.2
fractography
analysis and characterization of patterns generated on the fracture surface of a test specimen
Note 1 to entry: Fractography can be used to determine the nature and location of the critical fracture origin
causing catastrophic failure in an advanced ceramic test specimen or component.
3.3 Statistical terms
3.3.1
confidence interval
interval within which one would expect to find the true population parameter
Note 1 to entry: Confidence intervals are functionally dependent on the type of estimator utilized and the sample
size. The level of expectation is associated with a given confidence level. When confidence bounds are compared
to the parameter estimate one can quantify the uncertainty associated with a point estimate of a population
parameter.
3.3.2
confidence level
probability that the true population parameter falls within a specified confidence interval
3.3.3
estimator
function for calculating an estimate of a given quantity based on observed data
Note 1 to entry: The resulting value for a given sample may be an estimate of a distribution parameter (a point
estimate) associated with the underlying population, e.g. the arithmetic average of a sample is an estimator of
the distribution mean.
3.3.4
population
collection of data or items under consideration
3.3.5
probability density function
pdf
function f (x) for the continuous random variable X if
fx ≥ 0 (1)
()
and
∞
fx dx = 1 (2)
()
∫
−∞
Note 1 to entry: The probability that the random variable X assumes a value between a and b is given by
b
Pr aX< () ()
∫
a
ISO 20501:2019(E)
3.3.6
cumulative distribution function
function F (x) describing the probability that a continuous random variable X takes a value less than or
equal to a number x
Note 1 to entry: Therefore, the cumulative distribution function (cdf) is related to the probability density
function f (x) by
x
Fx =−Pr ∞< Xx< = fx´´dx (4)
() () ()
∫
−∞
Differentiating Formula (4) with respect to x shows that the pdf is simple the derivative of the cdf:
dF x
()
fx = (5)
()
dx
Note 2 to entry: According to 3.3.5, F (x) is a monotonically increasing function in the range between 0 and 1.
3.3.7
ranking estimator
function that estimates the probability of failure to a particular strength measurement within a
ranked sample
3.3.8
sample
collection of measurements or observations taken from a specified population
3.3.9
statistical bias
type of consistent numerical offset in an estimate relative to the true underlying value, inherent to most
estimates
3.3.10
unbiased estimator
estimator that has been corrected for statistical bias error
3.4 Weibull distributions
3.4.1
Weibull distribution
continuous distribution function which can be used to describe empirical data from measurements
where continuous random variable x has a two-parameter Weibull distribution if the probability
density function is given by
mm−1
 
mx   x 
 
fx = expw− henx ≥0 (6)
()
    
ββ β
 
    
 
or
fx =<00when x (7)
()
and the cumulative distribution function is given by
m
 
 
x
 
Fx =−10exp − whenx ≥ (8)
()
 
β
 
 
 
4 © ISO 2019 – All rights reserved

ISO 20501:2019(E)
or
Fx()=<00whenx (9)
where
m is the Weibull modulus (or the shape parameter) (>0);
β is the Weibull scale parameter (>0).
Note 1 to entry: The random variable representing uniaxial tensile strength of an advanced ceramic will assume
only positive values. If the random variable representing uniaxial tensile strength of an advanced ceramic is
characterized by Formulae (6) to (9), then the probability that this advanced ceramic will fail under an applied
uniaxial tensile stress σ is given by the cumulative distribution function.
m
 
 
σ
 
P =−10exp − whenσ ≥ (10)
 
f
 σ 
 q 
 
P =<00whenσ (11)
f
where
P is the probability of failure;
f
σ is the Weibull characteristic strength.
θ
Note 2 to entry: The Weibull characteristic strength is dependent on the uniaxial test specimen (tensile, flexural,
or pressurized ring) and will change with specimen geometry. In addition, the Weibull characteristic strength
has units of stress, and has to be reported using SI-units of Pa, or adequately in MPa or GPa.
Note 3 to entry: An alternative expression for the probability of failure is given by
m
 
 
1 σ
 
P =−1 exp − dV whenσ >0 (12)
 
f
∫
 V 
V σ
00 
 
P =≤00whenσ (13)
f
The integration in the exponential is performed over all tensile regions of the specimen volume (V) if the
strength-controlling flaws are randomly distributed through the volume of the material, or over all tensile
regions of the specimen area if flaws are restricted to the specimen surface. The integration is sometimes carried
out over an effective gauge section instead of over the total volume or area. In Formula (12), σ is the Weibull
material scale parameter and can be described as the Weibull characteristic strength of a specimen with unit
volume or area loaded in uniform uniaxial tension. For a given specimen geometry, Formulae (10) and (12) can be
1/m
combined, to yield an expression relating σ and σ (this means: σσV = ). Further discussion related to
0 θ
q 0
this issue can be found in Annex A.
4 Symbols
A specimen area
b gauge section dimension, base of bend test specimen
d gauge section dimension, depth of bend test specimen
f (x) probability density function
ISO 20501:2019(E)
F(x) cumulative distribution function
L likelihood function
L length of the inner load span for a bend test specimen
i
L length of the outer load span for a bend test specimen
o
m Weibull modulus
estimate of the Weibull modulus

m
unbiased estimate of the Weibull modulus

m
U
N number of specimens in a sample
P probability of failure
f
q intermediate quantity defined in 6.5.1, used in calculation of confidence bounds
r number of specimens that failed from the flaw population for which the Weibull estimators
are being calculated
t intermediate quantity defined by Formula (22), used in calculation of confidence bounds
UF unbiasing factor
V tensile loaded region of specimen volume
V unit size volume
V effective volume
eff
x realization of a random variable X
X random variable
β Weibull scale parameter
σ uniaxial tensile stress
estimate of mean strength

σ
σ maximum stress in the j th test specimen at failure
j
σ Weibull material scale parameter (strength relative to unit size) defined in Formula (12)
estimate of the WeibuII material scale parameter

σ
σ Weibull characteristic strength (associated with a test specimen) defined in Formula (10)
θ
estimate of the Weibull characteristic strength

σ
q
5 Significance and use
5.1 This document enables the experimentalist to estimate Weibull distribution parameters from
failure data. These parameters permit a description of the statistical nature of fracture of fine ceramic
materials for a variety of purposes, particularly as a measure of reliability as it relates to strength data
6 © ISO 2019 – All rights reserved

ISO 20501:2019(E)
utilized for mechanical design purposes. The observed strength values are dependent on specimen size
 
and geometry. Parameter estimates can be computed for a given specimen geometry (,m σ ) but it is
q
suggested that the parameter estimates be transformed and reported as material-specific parameters
 
(,m σ ). In addition, different flaw distributions (e.g. failures due to inclusions or machining damage)
may be observed, and each will have its own strength distribution parameters. The procedure for
transforming parameter estimates for typical specimen geometries and flaw distributions is outlined in
Annex A.
5.2 This document provides two approaches, method A and method B, which are appropriate for
different purposes.
Method A provides a simple analysis for circumstances in which the nature of strength-defining flaws
is either known or assumed to be from a single population. Fractography to identify and group test
items with given flaw types is thus not required. This method is suitable for use for simple material
screening.
Method B provides an analysis for the general case in which competing flaw populations exist. This
method is appropriate for final component design and analysis. The method requires that fractography
be undertaken to identify the nature of strength-limiting flaws and assign failure data to given flaw
population types.
5.3 In method A, a strength data set can be analysed and values of the Weibull modulus and
 
characteristic strength (,m σ ) are produced, together with confidence bounds on these parameters. If
q
necessary, the estimate of the mean strength can be computed. Finally, a graphical representation of the
failure data along with a test report can be prepared. It should be noted that the confidence bounds are
frequently widely spaced, which indicates that the results of the analysis should not be used to extrapolate
far beyond the existing bounds of probability of failure. A necessary assumption for a valid extrapolation
(with respect to the tested effective volume V and/or small probabilities of failure) is that the flaw
eff
populations in all considered strength test pieces are of the same type.
5.4 In method B, begin by performing a fractographic examination of each failed specimen in order
to characterize fracture origins. Screen the data associated with each flaw distribution for outliers. If all
failures originate from a single flaw distribution compute an unbiased estimate of the Weibull modulus,
and compute confidence bounds for both the estimated Weibull modulus and the estimated Weibull
characteristic strength. If the failures originate from more than one flaw type, separate the data sets
associated with each flaw type, and subject these individually to the censored analysis. Finally, prepare a
graphical representation of the failure data along with a test report. When using the results of the analysis
for design purposes it should be noted that there is an implicit assumption that the flaw populations in
the strength test pieces and the components are of the same types.
6 Method A: maximum likelihood parameter estimators for single flaw
populations
6.1 General
This document outlines the application of parameter estimation methods based on the maximum
likelihood technique (see also References [13], [14], [20] and [21]). This technique has certain
advantages. The parameter estimates obtained using the maximum likelihood technique are unique
(for a two-parameter Weibull distribution), and as the size of the sample increases, the estimates
statistically approach the true values of the population more efficiently than other parameter
estimation techniques.
ISO 20501:2019(E)
6.2 Censored data
The application of the techniques presented in this document can be complicated by the presence of
test specimens that fail from extraneous flaws, fractures that originate outside the effective gauge
section, and unidentified fracture origins. If these complications arise, the strength data from these
specimens should generally not be discarded. Strength data from specimens with fracture origins
[3]
outside the effective gauge section and from specimens with fractures that originate from extraneous
flaws should be censored, and the maximum likelihood methods presented in method B (Clause 7)
are applicable. It is imperative that the number of unidentified fracture origins, and how they were
classified, be stated in the test report. A discussion of the appropriateness of each option can be found
in 7.2.2.
Applying the censored analysis implies that it is assumed that the flaw populations are concurrent.
This is a choice, which should be indicated in the test report.
6.3 Likelihood functions
The likelihood function for the two-parameter Weibull distribution of a sample with a single flaw
[4]
population is defined by Formula (14):

m−1  m
N

 
 σ   σ 
m
i  i 
L=   exp − (14)
   
∏
   
σ σ σ
qq   q 
 
i==1
 
 
NOTE σ is the maximum stress in the i th test specimen at failure and N is the number of test specimens in
i

the sample being analysed. The parameter estimates (the Weibull modulus, m and the characteristic strength,

σ ) are determined by taking the partial derivatives of the logarithm of the likelihood function with respect to
q
 
m and σ ) and equating the resulting expressions to zero.
q
The system of formulae obtained by differentiating the log likelihood function for a sample with a single
[5]
flaw population is given by
N

m
σσln
() ()
∑ i i
N
i=1
− ln σ −=0 (15)
()
∑ i
N

N

m
m
i=1
σ
()
∑ i
i=1
and

N
  m
 

m

 
σσ=   (16)
q ()
∑ i
 
N
 
i=1 
 
 
Formula (15) is solved first for m . Subsequently σ is computed from Formula (16). Obtaining a closed
q

form solution of Formula (15) for m is not possible. This expression shall be solved numerically.
Since the characteristic strength also reflects specimen geometry and stress gradients, this document
 
suggests reporting the estimated Weibull material scale parameter, σ Expressions that relate σ to
0 q
the Weibull material scale parameter σ for typical specimen geometries are given in Annex A.
6.4 Bias correction
6.4.1 The procedures described herein, to correct for statistical bias errors and to compute confidence
bounds, are appropriate only for data sets where all failures originate from a single population (i.e. an
uncensored sample). Procedures for bias correction and confidence bounds in the presence of multiple
active flaw populations are not well developed. It is well-known that the maximum likelihood estimators
8 © ISO 2019 – All rights reserved

ISO 20501:2019(E)
with respect to the two-parametric Weibull distribution are biased, but consistent (i.e. the shift of
expectation values of the estimated Weibull parameters goes to zero with increasing sample size). The

statistical bias associated with the estimator σ is minimal (<0,3 % for 20 test specimens, as opposed to
q

≈7 % bias for m with the same number of specimens). Therefore, this document allows the assumption

that σ is an unbiased estimator of the true population parameter. The parameter estimate of the
q

Weibull modulus, m , generally exhibits statistical bias. The amount of statistical bias depends on the
 
number of specimens in the sample. An unbiased estimate of m shall be obtained by multiplying m by
[6]
unbiasing factors . This procedure is discussed in 6.4.2. Statistical bias associated with the maximum
likelihood estimators presented in this document can be reduced by increasing the sample size.
6.4.2 An unbiased estimator produces nearly zero statistical bias between the value of the true
parameter and the point estimate. The amount of deviation can be quantified either as a percent
difference or with unbiasing factors. In keeping with the accepted practice in the open literature, this
document quantifies statistical bias through the use of unbiasing factors, denoted here as UF. Depending

on the number of specimens in a given sample, the point estimate of the Weibull modulus, m , may exhibit

significant statistical bias. An unbiased estimate of the Weibull modulus (denoted as m is obtained by
U
multiplying the biased estimate with an appropriate unbiasing factor.

mm=×UF (17)
U

Unbiasing factors for m are listed in Table 1. Alternatively, the table values can be approximated by a
simple analytical function f defined by
UF
−1,04033
fN =−11,61394×N (18)
()
UF
This function interpolates the tabulated values in Table 1 with errors smaller than 1 %, but it is also
[17]
applicable to sample sizes N > 120 .
An example in Annex B demonstrates both the use of Table 1 and of Formula (18) in correcting a biased
estimate of the Weibull modulus.
Table 1 — Unbiasing factor for the maximum likelihood estimate of the Weibull modulus
Number of Unbiasing factor, Number of Unbiasing factor,
specimens, N UF specimens, N UF
5 0,700 42 0,968
6 0,752 44 0,970
7 0,792 46 0,971
8 0,820 48 0,972
9 0,842 50 0,973
10 0,859 52 0,974
11 0,872 54 0,975
12 0,883 56 0,976
13 0,893 58 0,977
14 0,901 60 0,978
15 0,908 62 0,979
16 0,914 64 0,980
18 0,923 66 0,980
20 0,931 68 0,981
22 0,938 70 0,981
ISO 20501:2019(E)
Table 1 (continued)
Number of Unbiasing factor, Number of Unbiasing factor,
specimens, N UF specimens, N UF
24 0,943 72 0,982
26 0,947 74 0,982
28 0,951 76 0,983
30 0,955 78 0,983
32 0,958 80 0,984
34 0,960 85 0,985
36 0,962 90 0,986
38 0,964 100 0,987
40 0,966 120 0,990
6.5 Confidence intervals
6.5.1 Confidence bounds quantify the uncertainty associated with a point estimate of a population
parameter. The size of the confidence bounds for maximum likelihood estimates of both Weibull
parameters will diminish with increasing sample size. The values used to construct confidence bounds
are based on percentile distributions obtained by Monte Carlo simulation; e.g. the 90 % confidence

bound on the Weibull modulus is obtained from the 5 and 95 percentile distributions of the ratio of m to
the true population value m. For a point estimate of the Weibull modulus, the normalized values

qm= /m necessary to construct the 90 % confidence bounds are listed in Table 2. Consequently, the
()
upper and lower confidence bounds for m are given by

mm= /q (19)
upper
00, 5
and

mm= /q (20)
lower
09, 5
respectively.
The example in Annex B demonstrates the use of Table 2 in constructing the upper and lower bounds.
Note that the statistically biased estimate of the Weibull modulus shall be used here. Again, this
procedure is not appropriate for censored statistics.
A convenient way to calculate confidence bounds is the usage of an interpolating function of the form
a a a
a a a
12//32 52/
1 2 3
qN()=+1 ++ ++ + (21)
p
2 3
3 5
N N N
N
N N
which can also be used for larger samples with N > 120. The corresponding coefficients for a are given
i
in Table 3. Formula (21) is asymptotically correct for large N and the deviation with respect to the
[17]
tabulated values in Table 2 is smaller than 0,5 % . Additionally, the coefficients for calculation of the
confidence bounds with respect to 95 % confidence level are listed in Table 4.
10 © ISO 2019 – All rights reserved

ISO 20501:2019(E)
6.5.2 Confidence bounds can be constructed for the estimated Weibull characteristic strength. However,
the percentile distributions needed to construct the bounds do not involve the same normalized ratios or
the same tables as those used for the Weibull modulus. Define the function:

 
 σ
q
tm= ln (22)
 
σ
q
 
The 90 % confidence bound on the characteristic strength is obtained from the 5 and 95 percentile
distributions of t, so that the upper and lower confidence bounds are given by:
 
 
σσ= exp −tm/ (23)
qq
 
() () 00, 5
 
upper
 
 
σσ= exp −tm/ (24)
qq  
() () 09, 5
 
lower
For the point estimate of the characteristic strength, these percentile distributions are listed in Table 5.
Again an interpolating function for t is defined by
p
b b b
b b b
12//32 52/
1 2
tN =+ ++ ++ (25)
()
p
2 3
3 5
N N N
N
N N
which allows an analytic and accurate calculation of the confidence bounds. In Table 6 and Table 7 the
corresponding coefficients b are listed with respect to 90 % and 95 % confidence level respectively.
i

An example in Annex B demonstrates the use of Table 5 in constructing upper and lower bounds on σ .
q
Note that the biased estimate of the Weibull modulus (i.e. the result of the maximum likelihood
procedure [see Formula (15)] shall be used here. This procedure is not appropriate for censored

statistics. Formula (22) is not applicable for developing confidence bounds on σ therefore the

confidence bounds on σ . should not be converted through the use of Formulae (10) and (12).
q
Table 2 — Normalized upper and lower bounds on the maximum likelihood estimate of the
Weibull modulus — 90 % confidence interval
Number of q q Number of q q
0,05 0,95 0,05 0,95
specimens, N specimens, N
5 0,683 2,779 42 0,842 1,265
6 0,697 2,436 44 0,845 1,256
7 0,709 2,183 46 0,847 1,249
8 0,720 2,015 48 0,850 1,242
9 0,729 1,896 50 0,852 1,235
10 0,738 1,807 52 0,854 1,229
11 0,745 1,738 54 0,857 1,224
12 0,752 1,682 56 0,859 1,218
13 0,759 1,636 58 0,861 1,213
14 0,764 1,597 60 0,863 1,208
15 0,770 1,564 62 0,864 1,204
16 0,775 1,535 64 0,866 1,200
17 0,779 1,510 66 0,868 1,196
18 0,784 1,487 68 0,869 1,192
19 0,788 1,467 70 0,871 1,188
20 0,791 1,449 72 0,872 1,185
22 0,798 1,418 74 0,874 1,182
ISO 20501:2019(E)
Table 2 (continued)
Number of q q Number of q q
0,05 0,95 0,05 0,95
specimens, N specimens, N
24 0,805 1,392 76 0,875 1,179
26 0,810 1,370 78 0,876 1,176
28 0,815 1,351 80 0,878 1,173
30 0,820 1,334 85 0,881 1,166
32 0,824 1,319 90 0,883 1,160
34 0,828 1,306 95 0,886 1,155
36 0,832 1,294 100 0,888 1,150
38 0,835 1,283 110 0,893 1,141
40 0,839 1,273 120 0,897 1,133
Table 3 — Coefficients according to Formula (16) for the normalized upper and lower bounds on
the maximum likelihood estimate of the Weibull modulus — 90 % confidence interval
a a a a a a
1/2 1 3/2 2 5/2 3
p = 0,05 −1,280 61 2,088 03 −2,365 01 −1,941 65 13,623 8 −14,666 1
p = 0,95 1,283 79 2,136 0 3,451 5 8,520 81 −19,55 11 65,739 1
Table 4 — Coefficients according to Formula (16) for the normalized upper and lower bounds on
the maximum likelihood estimate of the Weibull modulus — 95 % confidence interval
a a a a a a
1/2 1 3/2 2 5/2 3
p = 0,025 −1,523 97 2,531 61 −2,673 06 −4,644 68 22,357 7 −22,903 6
p = 0,975 1,521 37 2,993 89 −0,318 37 44,728 8 −123,859 202,815
Table 5 — Normalized upper and lower bounds on the function t — 90 % confidence interval
Number
...