EN 61649:2008

SLOVENSKI STANDARD
01-januar-2009
Weibullova analiza (IEC 61649:2008)
Weibull analysis (IEC 61649:2008)
Weibull-Analyse (IEC 61649:2008)
Analyse de Weibull (CEI 61649:2008)
Ta slovenski standard je istoveten z: EN 61649:2008
ICS:
03.120.30 8SRUDEDVWDWLVWLþQLKPHWRG Application of statistical
methods
29.020 Elektrotehnika na splošno Electrical engineering in
general
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EUROPEAN STANDARD
EN 61649
NORME EUROPÉENNE
November 2008
EUROPÄISCHE NORM
ICS 03.120.01; 03.120.30
English version
Weibull analysis
(IEC 61649:2008)
Analyse de Weibull Weibull-Analyse
(CEI 61649:2008) (IEC 61649:2008)

This European Standard was approved by CENELEC on 2008-10-01. CENELEC 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 Central Secretariat or to any CENELEC 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 CENELEC member into its own language and notified
to the Central Secretariat has the same status as the official versions.

CENELEC members are the national electrotechnical committees of Austria, Belgium, Bulgaria, Cyprus, the
Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia,
Lithuania, Luxembourg, Malta, the Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain,
Sweden, Switzerland and the United Kingdom.

CENELEC
European Committee for Electrotechnical Standardization
Comité Européen de Normalisation Electrotechnique
Europäisches Komitee für Elektrotechnische Normung

Central Secretariat: rue de Stassart 35, B - 1050 Brussels

© 2008 CENELEC - All rights of exploitation in any form and by any means reserved worldwide for CENELEC members.
Ref. No. EN 61649:2008 E
Foreword
The text of document 56/1269/FDIS, future edition 2 of IEC 61649, prepared by IEC TC 56,
Dependability, was submitted to the IEC-CENELEC parallel vote and was approved by CENELEC as
EN 61649 on 2008-10-01.
The following dates were fixed:
– latest date by which the EN has to be implemented
at national level by publication of an identical
national standard or by endorsement (dop) 2009-07-01
– latest date by which the national standards conflicting
with the EN have to be withdrawn (dow) 2011-10-01
Annex ZA has been added by CENELEC.
__________
Endorsement notice
The text of the International Standard IEC 61649:2008 was approved by CENELEC as a European
Standard without any modification.
In the official version, for Bibliography, the following notes have to be added for the standards indicated:
IEC 60300-1 NOTE  Harmonized as EN 60300-1:2003 (not modified).
IEC 60300-2 NOTE  Harmonized as EN 60300-2:2004 (not modified).
IEC 60300-3-1 NOTE  Harmonized as EN 60300-3-1:2004 (not modified).
IEC 60300-3-2 NOTE  Harmonized as EN 60300-3-2:2005 (not modified).
IEC 60300-3-4 NOTE  Harmonized as EN 60300-3-4:2008 (not modified).
IEC 61703 NOTE  Harmonized as EN 61703:2002 (not modified).
__________
– 3 – EN 61649:2008
Annex ZA
(normative)
Normative references to international publications
with their corresponding European publications

The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.

NOTE  When an international publication has been modified by common modifications, indicated by (mod), the relevant EN/HD
applies.
Publication Year Title EN/HD Year

IEC 60050-191 1990 International Electrotechnical Vocabulary - -
(IEV) -
Chapter 191: Dependability and quality of
service
IEC 60300-3-5 2001 Dependability management - - -
Part 3-5: Application guide - Reliability test
conditions and statistical test principles

1) 2)
IEC 61810-2 - Electromechanical elementary relays - EN 61810-2 2005
Part 2: Reliability
ISO 2854 1976 Statistical interpretation of data - Techniques - -
of estimation and tests relating to means and
variances
ISO 3534-1 2006 Statistics - Vocabulary and symbols - - -
Part 1: General statistical terms and terms
used in probability
1)
Undated reference.
2)
Valid edition at date of issue.

IEC 61649
Edition 2.0 2008-08
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Weibull analysis
Analyse de Weibull
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
PRICE CODE
INTERNATIONALE
XB
CODE PRIX
ICS 03.120.01; 03.120.30 ISBN 2-8318-9954-0
– 2 – 61649 © IEC:2008
CONTENTS
FOREWORD.5
INTRODUCTION.7
1 Scope.8
2 Normative references .8
3 Terms, definitions, abbreviations and symbols.8
3.1 Terms and definitions .8
3.2 Abbreviations .10
3.3 Symbols .10
4 Application of the techniques.11
5 The Weibull distribution .11
5.1 The two-parameter Weibull distribution.11
5.2 The three-parameter Weibull distribution .13
6 Data considerations.13
6.1 Data types.13
6.2 Time to first failure .13
6.3 Material characteristics and the Weibull distribution .13
6.4 Sample size .13
6.5 Censored and suspended data .14
7 Graphical methods and goodness-of-fit .14
7.1 Overview .14
7.2 How to make the probability plot.14
7.2.1 Ranking.15
7.2.2 The Weibull probability plot .15
7.2.3 Dealing with suspensions or censored data .15
7.2.4 Probability plotting.17
7.2.5 Checking the fit .17
7.3 Hazard plotting.18
8 Interpreting the Weibull probability plot.19
8.1 The bathtub curve .19
8.1.1 General .19
8.1.2 β < 1 – Implies early failures.19
8.1.3 β = 1 – Implies constant instantaneous failure rate.20
8.1.4 > 1 – Implies wear-out.20
β
8.2 Unknown Weibull modes may be "masked".20
8.3 Small samples.21
8.4 Outliers .22
8.5 Interpretation of non-linear plots.22
8.5.1 Distributions other than the Weibull .25
8.5.2 Data inconsistencies and multimode failures .25
9 Computational methods and goodness-of-fit .25
9.1 Introduction .25
9.2 Assumptions and conditions .26
9.3 Limitations and accuracy .26
9.4 Input and output data .26

61649 © IEC:2008 – 3 –
9.5 Goodness-of-fit test.27
9.6 MLE – point estimates of the distribution parameters β and η .27
9.7 Point estimate of the mean time to failure.28
9.8 Point estimate of the fractile (10 %) of the time to failure.28
9.9 Point estimate of the reliability at time t (t ≤ T).28
9.10 Software programs .28
10 Confidence intervals.28
10.1 Interval estimation of β .28
10.2 Interval estimation of η .29
10.3 MRR Beta-binomial bounds .30
10.4 Fisher's Matrix bounds .30
10.5 Lower confidence limit for B .31
10.6 Lower confidence limit for R .31
11 Comparison of median rank regression (MRR) and maximum likelihood (MLE)
estimation methods .31
11.1 Graphical display.31
11.2 B life estimates sometimes known as B or L percentiles .31
11.3 Small samples.32
11.4 Shape parameter β.32
11.5 Confidence intervals.32
11.6 Single failure .32
11.7 Mathematical rigor.32
11.8 Presentation of results .32
12 WeiBayes approach.33
12.1 Description.33
12.2 Method.33
12.3 WeiBayes without failures .33
12.4 WeiBayes with failures .33
12.5 WeiBayes case study .34
13 Sudden death method .35
14 Other distributions .37
Annex A (informative) Examples and case studies .38
Annex B (informative) Example of computations .40
Annex C (informative) Median rank tables.42
Annex D (normative) Statistical Tables .47
Annex E (informative) Spreadsheet example.48
Annex F (informative) Example of Weibull probability paper.55
Annex G (informative) Mixtures of several failure modes.56
Annex H (informative) Three-parameter Weibull example.59
Annex I (informative) Constructing Weibull paper.61
Annex J (informative) Technical background and references.64
Bibliography.67
Figure 1 – The PDF shapes of the Weibull family for Ș = 1,0 .12
Figure 2 – Total test time (in minutes).16
Figure 3 – Typical bathtub curve for an item .19

– 4 – 61649 © IEC:2008
Figure 4 – Weibull failure modes may be “masked” .21
Figure 5 – Sample size: 10 .21
Figure 6 – Sample size: 100 .22
Figure 7 – An example showing lack of fit with a two-parameter Weibull distribution .23
Figure 8 – The same data plotted with a three-parameter Weibull distribution shows a
good fit with 3 months offset (location – 2,99 months).24
Figure 9 – Example of estimating t by eye .25
Figure 10 – New compressor design WeiBayes versus old design .35
Figure A.1 – Main oil pump low times.38
Figure A.2 – Augmenter pump bearing failure .39
β values hide problems .39
Figure A.3 – Steep
Figure B.1 – Plot of computations .41
Figure E.1 – Weibull plot for graphical analysis.49
Figure E.2 – Weibull plot of censored data.51
Figure E.3 – Cumulative hazard plot for data of Table E.4 .52
Figure E.4 – Cumulative hazard plots for Table E.6 .54
Figure H.1 – Steel-fracture toughness – Curved data.59
Figure H.2 – t improves the fit of Figure H.1 data .60
Table 1 – Guidance for using this International Standard .11
Table 2 – Ranked flare failure rivet data .15
Table 3 – Adjusted ranks for suspended or censored data .16
Table 4 – Subgroup size to estimate time to X % failures using the sudden death
method .36
Table 5 – Chain data: cycles to failure .36
Table B.1 – Times to failure .40
Table B.2 – Summary of results .41
Table D.1 – Values of the gamma function.47
Table D.2 – Fractiles of the normal distribution .47
Table E.1 – Practical analysis example.48
Table E.2 – Spreadsheet set-up for analysis of censored data .50
Table E.3 – Example of Weibull analysis for suspended data.50
Table E.4 – Example of Spreadsheet application for censored data .51
Table E.5 – Example spreadsheet.52
Table E.6 – A relay data provided by ISO/TC94 and Hazard analysis for failure mode 1 .53
Table I.1 – Construction of ordinate (Y).62
Table I.2 – Construction of abscissa (t).62
Table I.3 – Content of data entered into a spreadsheet.62

61649 © IEC:2008 – 5 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
WEIBULL ANALYSIS
FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
international co-operation on all questions concerning standardization in the electrical and electronic fields. To
this end and in addition to other activities, IEC publishes International Standards, Technical Specifications,
Technical Reports, and Guides (hereafter referred to as “IEC Publication(s)”). Their preparation is entrusted to
technical committees; any IEC National Committee interested in the subject dealt with may participate in this
preparatory work. International, governmental and non-governmental organizations liaising with the IEC also
participate in this preparation. IEC collaborates closely with the International Organization for Standardization
(ISO) in accordance with conditions determined by agreement between the two organizations.
2) The formal decisions or agreements of IEC on technical matters express, as nearly as possible, an international
consensus of opinion on the relevant subjects since each technical committee has representation from all
interested IEC National Committees.
3) IEC Publications have the form of recommendations for international use and are accepted by IEC National
Committees in that sense. While all reasonable efforts are made to ensure that the technical content of IEC
Publications is accurate, IEC cannot be held responsible for the way in which they are used or for any
misinterpretation by any end user.
4) In order to promote international uniformity, IEC National Committees undertake to apply IEC Publications
transparently to the maximum extent possible in their national and regional publications. Any divergence
between any IEC Publication and the corresponding national or regional publication shall be clearly indicated in
the latter.
5) IEC provides no marking procedure to indicate its approval and cannot be rendered responsible for any
equipment declared to be in conformity with an IEC Publication.
6) All users should ensure that they have the latest edition of this publication.
7) No liability shall attach to IEC or its directors, employees, servants or agents including individual experts and
members of its technical committees and IEC National Committees for any personal injury, property damage or
other damage of any nature whatsoever, whether direct or indirect, or for costs (including legal fees) and
expenses arising out of the publication, use of, or reliance upon, this IEC Publication or any other IEC
Publications.
8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
indispensable for the correct application of this publication.
9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of
patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
International Standard IEC 61649 has been prepared by IEC technical committee 56:
Dependability.
This second edition cancels and replaces the first edition, published in 1997, and constitutes
a technical revision.
The main changes with respect to the previous edition are as follows:
– the title has been shortened and simplified to read “Weibull analysis”;
– provision of methods for both analytical and graphical solutions have been added.
The text of this standard is based on the following documents:
FDIS Report on voting
56/1269/FDIS 56/1281/RVD
Full information on the voting for the approval of this standard can be found in the report on
voting indicated in the above table.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.

– 6 – 61649 © IEC:2008
The committee has decided that the contents of this publication will remain unchanged until
the maintenance result date indicated on the IEC web site under "http://webstore.iec.ch" in
the data related to the specific publication. At this date, the publication will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
61649 © IEC:2008 – 7 –
INTRODUCTION
The Weibull distribution is used to model data regardless of whether the failure rate is
increasing, decreasing or constant. The Weibull distribution is flexible and adaptable to a wide
range of data. The time to failure, cycles to failure, mileage to failure, mechanical stress or
similar continuous parameters need to be recorded for all items. A life distribution can be
modelled even if not all the items have failed.
Guidance is given on how to perform an analysis using a spreadsheet program. Guidance is
also given on how to analyse different failure modes separately and identify a possible weak
population. Using the three-parameter Weibull distribution can give information on time to first
failure or minimum endurance in the sample.

– 8 – 61649 © IEC:2008
WEIBULL ANALYSIS
1 Scope
This International Standard provides methods for analysing data from a Weibull distribution
using continuous parameters such as time to failure, cycles to failure, mechanical stress, etc.
This standard is applicable whenever data on strength parameters, e.g. times to failure,
cycles, stress, etc. are available for a random sample of items operating under test conditions
or in-service, for the purpose of estimating measures of reliability performance of the
population from which these items were drawn.
This standard is applicable when the data being analysed are independently, identically
distributed. This should either be tested or assumed to be true (see IEC 60300-3-5).
In this standard, numerical methods and graphical methods are described to plot data, to
make a goodness-of-fit test, to estimate the parameters of the two- or three-parameter
Weibull distribution and to plot confidence limits. Guidance is given on how to interpret the
plot in terms of risk as a function of time, failure modes and possible weak population and
time to first failure or minimum endurance.
2 Normative references
The following referenced documents are indispensable for the application of this document.
For dated references, only the edition cited applies. For undated references, the latest edition
of the referenced document (including any amendments) applies.
IEC 60050-191:1990, International Electrotechnical Vocabulary – Part 191: Dependability and
quality of service
IEC 60300-3-5:2001, Dependability management – Part 3-5: Application guide – Reliability
test conditions and statistical test principles
IEC 61810-2, Electromechanical elementary relays – Part 2: Reliability
ISO 2854:1976, Statistical interpretation of data – Techniques of estimations and tests
relating to means and variances
ISO 3534-1:2006, Statistics – Vocabulary and symbols – Part 1: General statistical terms and
terms in probability
3 Terms, definitions, abbreviations and symbols
For the purposes of this document, the definitions, abbreviations and symbols given in
IEC 60050-191 and ISO 3534-1 apply, together with the following.
3.1 Terms and definitions
3.1.1
censoring
terminating a test after either a given duration or a given number of failures
NOTE A test terminated when there are still unfailed items may be called a “censored test", and test time data
from such tests may be referred to as “censored data”.

61649 © IEC:2008 – 9 –
3.1.2
suspended item
item upon which testing has been curtailed without relevant failure
NOTE 1 The item may not have failed, or it may have failed in a mode other than that under investigation.
NOTE 2 An “early suspension” is one that was suspended before the first failure. A “late suspension” is
suspended after the last failure.
3.1.3
life test
test conducted to estimate or verify the durability of a product
NOTE The end of the useful life will often be defined as the time when a certain percentage of the items have
failed for non-repairable items and as the time when the failure intensity has increased to a specified level for
repairable items.
3.1.4
non-repairable item
item that cannot, under given conditions, after a failure, be returned to a state in which it can
perform as required
NOTE The given conditions may be technical, economic, ecological and/or others.
3.1.5
operating time
time interval for which the item is in an operating state
NOTE ”Operating time” is generic, and should be expressed in units appropriate to the item concerned, e.g.
calendar time, operating cycles, distance run, etc. and the units should always be clearly stated.
3.1.6
relevant failure
failure that should be included in interpreting test or operational results or in calculating the
value of a reliability performance measure
NOTE The criteria for inclusion should be stated.
3.1.7
reliability test
experiment carried out in order to measure, quantify or classify a reliability measure or
property of an item
NOTE 1 Reliability testing is different from environmental testing where the aim is to prove that the items under
test can survive extreme conditions of storage, transportation and use.
NOTE 2 Reliability tests may include environmental testing.
3.1.8
repairable item
item that can, under given conditions, after a failure, be returned to a state in which it can
perform as required
NOTE The given conditions may be technical, economic, ecological and/or others.
3.1.9
time to failure
operating time accumulated from the first use, or from restoration, until failure
NOTE In applications where the time in storage or on standby is significantly greater than “operating time”, the
time to failure may be based on the time in the specified service.

– 10 – 61649 © IEC:2008
3.1.10
time between failures
time duration between consecutive failures
NOTE 1 The time between failures includes the up time and the down time.
NOTE 2 In applications where the time in storage or on standby is significantly greater than operating time, the
time to failure may be based on the time in the specified service.
3.1.11
B life
L percentiles
age at which a given percentage of items have failed
NOTE "B " life is the age at which 10 % of items (e.g. bearings) have failed. Sometimes it is denoted by the
L (life) value. B lives may be read directly from the Weibull plot or determined more accurately from the Weibull
equation. The age at which 50 % of the items fail, the B life, is the median time to failure.
3.2 Abbreviations
ASIC application specific integrated circuit
BGA ball grid array
CDF cumulative distribution function
PDF probability density function
MLE maximum likelihood estimation
MRR median rank regression
MTTF mean time to failure
3.3 Symbols
t time – variable
η
Weibull characteristic life or scale parameter
β Weibull shape parameter
t starting point or origin of the distribution, failure free time
r coefficient of determination
f(t) probability density function
F(t) cumulative distribution function
h(t) hazard function
Ȝ(t)
instantaneous failure rate
H(t) cumulative hazard function
F number of failures with failure mode 1
F number of failures with failure mode 2
F number of failures with failure mode 3
61649 © IEC:2008 – 11 –
4 Application of the techniques
Table 1 shows the circumstances in which particular aspects of this standard are applicable. It
shows the three main methods for estimating parameters from the Weibull distribution, namely
graphical, computational and WeiBayes, and indicates the type of data requirements for each
of these three methods.
Table 1 – Guidance for using IEC 61649
Method/ Graphical Computational
WeiBayes
Kinds of data methods methods
Interval censored ¥ NC ¥
Multiple censored ¥ NC ¥
Singly censored ¥¥ ¥
Zero failures NC NC ¥
Small sample (”20) ¥ NC ¥
Large sample ¥¥ NC
Curved data ¥ NC NC
Complete data ¥¥ ¥
NOTE NC means not covered in this standard.
5 The Weibull distribution
5.1 The two-parameter Weibull distribution
The two-parameter Weibull distribution is by far the most widely used distribution for life data
analysis. The Weibull probability density function (PDF) is shown in Equation (1):
β
§·t
−
β −1
¨¸
t
η
©¹
f(te)=⋅β ⋅ (1)
β
η
where
t is the time, expressed as a variable;
η is the characteristic life or scale parameter;
β is the shape parameter.
The Weibull cumulative distribution function (CDF) has an explicit equation as shown in
Equation (2):
β
-(t/η )
F(t) = 1 - e (2)
The two parameters are η , the characteristic life, and β , the shape parameter. The shape
parameter indicates the rate of change of the instantaneous failure rate with time. Examples
include: infant mortality, random or wear-out. It determines which member of the Weibull
family of distributions is most appropriate. Different members have widely different shaped
PDFs (see Figure 1). The Weibull distribution fits a broad range of life data compared with
other distributions. The variable t is generic and can have various measures such as time,
distance, number of cycles or mechanical stress applications.

– 12 – 61649 © IEC:2008
3,5
3,0
β = 0,5
2,5
2,0
β = 3,44
1,5
β = 1
1,0
0,5
0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
Datum in time
IEC  1321/08
Figure 1 – The PDF shapes of the Weibull family for Ș = 1,0
From Figure 1, the PDF shape for β = 3,44 (indicated) looks like the normal distribution: it is
a fair approximation, except for the tails of the distribution.
The instantaneous failure rate Ȝ(t) (or h(t), the hazard function) of the two-parameter Weibull
distribution is shown in Equation (3):
β −1
t
Ȝ(t) = h(t)=⋅β (3)
β
η
Three ranges of values of the shape parameter, β , are salient:
• for β = 1,0 the Weibull distribution is identical to the exponential distribution and the
instantaneous failure rate, Ȝ(t), then becomes a constant equal to the reciprocal of the
scale parameter, η ;
• β > 1,0 is the case of increasing instantaneous failure rate; and
• β < 1,0, is the case of decreasing instantaneous failure rate.
Characteristic life, η , is the time at which 63,2 % of the items are expected to fail. This is true
for all Weibull distributions, regardless of the shape parameter, β . If there is replacement of
items, then 63,2 % of the times to failure are expected to be lower or equal to the
η. Further discussion of the issues concerning repair and non-repairable
characteristic life,
items can be found in IEC 60300-3-5. The 63,2 % comes from setting t = η in Equation (2)
which results in Equation (4):
ββ
-(ηη/ ) -(1)
F(η) == 1 - e 1 - e = 1 - (1/e) = 0,632 (4)
f(t)
61649 © IEC:2008 – 13 –
5.2 The three-parameter Weibull distribution
Equation (5) shows the CDF of the three-parameter Weibull distribution:
tt−
β
-( )
η
F(t) = 1 - e
(5)
The parameter t is called the failure-free time, location parameter or minimum life.
The effect of location parameter is typically not understood well until a poor fit is observed
with a 2-parameter Weibull plot. When a lack of fit is observed, engineers attempt to use other
distributions that may provide them with a better fit. However, the lack of fit can be reconciled
when the data is plotted with a 3-parameter Weibull distribution (see 8.5). Using the location
parameter, it becomes evident that the product failures are offset by a fixed period of time,
called the threshold. The effect of location parameter is normally observed when a product
sees “shelf-life” after which the first failure occurs. A good indicator of the effect of a location
parameter is the convex shape of a plot.
6 Data considerations
6.1 Data types
Life data are related to items that "age" to failure. Weibull failure data are usually life data but
may also describe material data where the “aging” may be stress, force or temperature. "Age"
may be operating time, starts and stops, landings, takeoffs, low-cycle fatigue cycles, mileage,
shelf or storage time, cycles or time at high stress or high temperature, or many other
continuous parameters. In this standard the “age” parameter will be called time. When
required, “time” can be substituted by any of the “age” parameters listed above.
6.2 Time to first failure
The Weibull “time” variable is usually considered to be a measure of life consumption. The
following interpretations can be used:
– time to first failure of a repairable item;
– time to failure of an non-repairable item;
– time from new to each failure of a repairable system if a non-repairable item in the
system fails more than once during the period of observation. It has to be assumed
that the repair (change of the item) does not introduce a new failure, so that the
system after the repair can, with an approximation, be regarded as having the same
reliability as immediately before the failure (commonly referred as the “bad as old”
assumption);
– time to first failure of a non-repairable item, following scheduled maintenance, with the
assumption that the failure is related to the previous maintenance.
6.3 Material characteristics and the Weibull distribution
Material characteristics such as creep, stress rupture or breakage and fatigue are often
plotted on Weibull probability paper. Here the horizontal scale may be stress, cycles, load,
number of load repetitions or temperature.
6.4 Sample size
Uncertainty with regard to the Weibull parameter estimation is related to the sample size and
the number of relevant failures. Weibull parameters can be estimated using as few as two
failures; however, the uncertainty of such an estimate would be excessive and could not
confirm the applicability of the Weibull model. Whatever the sample size, confidence limits
should be calculated and plotted in order to assess the uncertainty of the estimations.

– 14 – 61649 © IEC:2008
As with all statistical analysis, the more data that is available, the better the estimation but if
the data set is limited, then refer to the advice given in 11.3.
6.5 Censored and suspended data
When analysing life data it is necessary to include data on those items in the sample that
have not failed, or have not failed by a failure mode analysis. This data is referred to as
censored or suspended data (see IEC 60300-3-5). When the times to failure of all items are
observed, the data are said to be complete.
An item on test that has not failed by the failure mode in question is a suspension or censored
item. It may have failed by a different failure mode or not failed at all. An "early suspension" is
one that was suspended before the first failure time. A "late suspension" is suspended after
the last failure. Suspensions between failures are called random or progressive suspensions.
If items remain unfailed, then the corresponding data are said to be censored. If a test is
terminated at a specified time, T, before all items have failed, then the data are said to be
time censored. If a test is terminated after a specified number of failures have occurred, then
the data are said to be failure censored.
Further discussion of censoring is covered in IEC 60300-3-5.
7 Graphical methods and goodness-of-fit
7.1 Overview
Graphical analysis consists of plotting the data on Weibull probability paper, fitting a line
through the data, interpreting the plot and estimating the parameters using special probability
paper derived by transforming the Weibull equation into a linear form. This is illustrated in
Annex I.
Data is plotted after first organizing it from earliest to latest, a process called ranking. The
time to failure data are plotted as the X coordinate on the Weibull probability paper.
The Y coordinate is the median rank as specified in 7.2.1. For sample sizes above 30 the
median rank is, in practice, the same as the per cent of failures. If the plotted data follow a
linear trend, a regression line may be drawn.
The parameters may then be read off the plot. The characteristic life, η , is the time to 63,2 %
of the items failing, called the “B63,2 Life”. The shape parameter, β , is estimated as the
slope on Weibull paper.
Median rank regression (MRR) is a method for estimating the parameters of the distribution
using linear regression techniques with the variables being the median rank and lifetime or
stress ,etc.
Another graphical method that is used for estimating parameters of a Weibull distribution is
called hazard plotting. This is described in 7.3.
7.2 How to make the probability plot
In order to make a probability plot, a sequence of steps needs to be carried out. These steps
are described in detail below.

61649 © IEC:2008 – 15 –
7.2.1 Ranking
To make the Weibull plot, rank the data from the lowest to the highest times to failure. This
ranking will set up the plotting positions for the time, t, axis and the ordinate, F(t), in
percentage values. These will provide information for the construction of the Weibull line
shown in Equation (6).
Median ranks are given in Annex C. Enter as an example the tables for 50 % median rank, for
a sample size of five, and find the median ranks shown in Table 2 for five failure times shown
in the middle column. The median rank plotting positions in Annex C are used with all types of
probability paper, i.e. Weibull, log-normal, normal, and extreme value.
NOTE 1 If two data points have the same time, they are plotted at different median rank values.
Table 2 – Ranked flare failure rivet data
Order number Failure time t Median rank
I
min (X) % (Y)
1 30 12,94
2 49 31,38
3 82 50,00
4 90 68,62
5 96 87,06
The median estimate is preferred to the mean or average value for non-symmetrical
distributions. Most life data distributions are skewed and, therefore, the median plays an
important role.
If a table of median ranks and a means to calculate median ranks using the Beta distribution
is not available, then Benard’s approximation, Equation (6), may be used:
(0i −,3)
F = % (6)
i
(0N +,4)
where N is the sample size and i is the ranked position of the data item of interest.
NOTE 2 This equation is mostly used for N ≤30; for N >30 the correction of the cumulative frequency can be
neglected: F = (i/N) × 100 %.
i
7.2.2 The Weibull probability plot
After transforming the data, the plot can be constructed using three different methods:
– Weibull probability paper – Annex F shows Weibull probability paper;
– a computer spreadsheet program – Annex E gives a spreadsheet example;
– commercial off-the-shelf software.
7.2.3 Dealing with suspensions or censored data
Non-failed items or items that fail by a different failure mode are "censored" or "suspended"
items, respectively. These data cannot be ignored. The times on suspended items have to be
included in the analysis.
The formula below gives the adjusted ranks without the need for calculating rank increments.
It is used for every failure and requires an additional column for reverse r
...