Bsi bs en 61649 2008

BS EN 61649:2008

BSI British Standards

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Copyright European Committee for Electrotechnical Standardization
ELEC

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BS EN 61 649:2008

BRITISH STANDARD
National foreword
This British Standard is the UK implementation of EN 61 649:2008. It is
identical to IEC 61 649:2008. It supersedes BS IEC 61 649:1 997 which is
withdrawn.
The UK participation in its preparation was entrusted by Technical Committee
DS/1 , Dependability and terotechnology, to Subcommittee DS/1 /1 ,
Dependability.
A list of organizations represented on this committee can be obtained on
request to its secretary.
This publication does not purport to include all the necessary provisions of a
contract. Users are responsible for its correct application.
© BSI 2009
ISBN 978 0 580 54368 5

ICS 03.1 20.01 ; 03.1 20.30

Compliance with a British Standard cannot confer immunity from
legal obligations.
This British Standard was published under the authority of the Standards
Policy and Strategy Committee on 2 8 February 2009.

Amendments issued since publication
Amd. No.

Copyright European Committee for Electrotechnical Standardization
ELEC

Date

Text affected


EUROPEAN STANDARD
NORME EUROPÉENNE
EUROPÄISCHE NORM

EN 61 649

BS EN 61 649:2008

November 2008

ICS 03.1 20.01 ; 03.1 20.30

English version

Weibull analysis
(IEC 61 649:2008)
Analyse de Weibull
(CEI 61 649:2008)

Weibull-Analyse
(IEC 61 649:2008)

This European Standard was approved by CENELEC on 2008-1 0-01 . 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 - 1 050 Brussels
© 2008 CENELEC -

All rights of exploitation in any form and by any means reserved worldwide for CENELEC members.
Ref. No. EN 61 649:2008 E

Copyright European Committee for Electrotechnical Standardization
ELEC


BS EN 61 649:2008
EN 61 649:2008

–2–

Foreword
The text of document 56/1 269/FDIS, future edition 2 of IEC 61 649, prepared by IEC TC 56,
Dependability, was submitted to the IEC-CENELEC parallel vote and was approved by CENELEC as
EN 61 649 on 2008-1 0-01 .
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)

– latest date by which the national standards conflicting
with the EN have to be withdrawn

(dow) 201 1 -1 0-01

2009-07-01

Annex ZA has been added by CENELEC.
__________

Endorsement notice
The text of the International Standard IEC 61 649:2008 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 61 703

NOTE Harmonized as EN 61 703:2002 (not modified).

__________

Copyright European Committee for Electrotechnical Standardization
ELEC


BS EN 61 649:2008
EN 61 649:2008

–3–

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
IEC 60050-1 91

Year
1 990

IEC 60300-3-5

2001

IEC 61 81 0-2

- 1)

ISO 2854

1 976

ISO 3534-1

2006

1)

Un dated reference.

2)

Valid edition at date of issue.

Copyright European Committee for Electrotechnical Standardization
ELEC

Title
International Electrotechnical Vocabulary
(IEV) Chapter 1 91 : Dependability and quality of
service

Dependability management Part 3-5: Application guide - Reliability test
conditions and statistical test principles
Electromechanical elementary relays Part 2: Reliability
Statistical interpretation of data - Techniques
of estimation and tests relating to means and
variances
Statistics - Vocabulary and symbols Part 1 : General statistical terms and terms
used in probability

EN/HD
-

Year
-

-

-

EN 61 81 0-2

2005 2)

-

-

-

-

BS EN 61 649:2008
–2–

61 649

© I EC: 2008

CON TENTS
I N TRODU CTI ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 7
1
2

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 8
N orm ati ve references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3

Term s, defi n iti ons, abbrevi ations an d sym bols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. 1 Term s and d efi n i ti ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. 2 Abbrevi ati ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0
3. 3 Sym bols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0
Appl icati on of the techn i q ues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
The Weibu l l d istribu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
5. 1 The two-param eter Weibu ll distri bu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
5. 2 The three-param eter Weibu ll d istributi on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3
Data consi derations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3

4

5

6

7

8

6. 1 Data types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3
6. 2 Tim e to first fail ure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3
6. 3 Material ch aracteristics and th e Weibu ll d istri bution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3
6. 4 Sam ple size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3
6. 5 Censored an d suspend ed d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4
Graph ical m ethods an d g ood ness-of-fi t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4
7. 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4
7. 2 H ow to m ake the probabil ity pl ot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4
7. 2. 1 Rankin g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5
7. 2. 2 The Weibu l l probabil i ty pl ot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5
7. 2. 3 Dealin g wi th suspensi ons or censored d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5
7. 2. 4 Probabil i ty plotti n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7
7. 2. 5 Ch eckin g the fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7
7. 3 H azard plotti n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8
I n terpreti n g th e Weibu l l probabi li ty pl ot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9
8. 1 The bathtub curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9
8. 1 . 1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9
< 1 – I m pli es earl y fai lu res . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9
8. 1 . 2

β

8. 1 . 3
8. 1 . 4

β
β

8. 2
8. 3
8. 4
8. 5

9

= 1 – I m pl i es constan t i n stantaneous fai l ure rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
> 1 – I m pl i es wear-out. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

U nkn own Weibu ll m od es m ay be "m asked". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Sm al l sam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Outli ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
I n terpretation of non-l in ear plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
8. 5. 1 Distribu ti ons oth er than the Weibu ll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
8. 5. 2 Data inconsistenci es an d m ultim od e fai l ures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Com pu tati on al m ethod s an d g ood n ess-of-fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
9. 1 I n trod ucti on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
9. 2 Assum ptions and con d iti ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9. 3 Lim itations an d accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9. 4 I n put an d ou tpu t d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Copyright European Committee for Electrotechnical Standardization
ELEC

BS EN 61 649:2008
61 649

© I EC: 2008

9. 5
9. 6

–3–

Good n ess-of-fi t test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
MLE – poi nt estim ates of the d istribu ti on param eters β an d η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9. 7 Poi n t estim ate of th e m ean tim e to fai lure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9. 8 Poi n t estim ate of th e fractil e (1 0 %) of th e tim e to fail ure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9. 9 Poi n t estim ate of th e rel i abi l ity at tim e t ( t ≤ T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
9. 1 0 Software program s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1 0 Confid ence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1 0. 1 I n terval estim ation of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1 0. 2 I n terval estim ation of η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1 0. 3 M RR Beta-bin om ial boun ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1 0. 4 Fisher's M atrix boun d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1 0. 5 Lower confi dence l im it for B 1 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1 0. 6 Lower confidence l im it for R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1 1 Com parison of m ed ian rank regression (MRR) an d m axim um likel ih ood (MLE)
estim ation m eth od s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1 1 . 1 Graph ical d ispl ay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1 1 . 2 B l ife estim ates som etim es kn own as B or L percenti l es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1 1 . 3 Sm all sam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1 1 . 4 Shape param eter β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1 1 . 5 Confi d ence i ntervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1 1 . 6 Sin g l e fail ure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1 1 . 7 Mathem atical ri gor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1 1 . 8 Presen tation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1 2 WeiBayes approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 2. 1 Descri ption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 2. 2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 2. 3 WeiBayes wi th ou t fai l ures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 2. 4 WeiBayes wi th fai lures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 2. 5 WeiBayes case stu d y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1 3 Sud d en d eath m eth od . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1 4 Oth er d istributi ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Ann ex A (inform ati ve) Exam pl es and case stud i es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Ann ex B (inform ative) Exam pl e of com putati ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Ann ex C (inform ative) M ed i an rank tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Ann ex D (n orm ati ve) Statistical Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Ann ex E (inform ati ve) Spreadsheet exam pl e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Ann ex F (i nform ative) Exam ple of Weibu ll probabi l ity paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Ann ex G (i nform ati ve) M ixtures of several fai l ure m odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Ann ex H (i nform ative) Three-param eter Weibu l l exam pl e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Ann ex I (i nform ati ve) Constructi n g Weibu l l paper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Ann ex J (i nform ative) Techn ical backgroun d an d references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Bibli ograph y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Fi gu re 1 – Th e PDF sh apes of th e Weibu l l fam il y for ? = 1 , 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2
Fi gu re 2 – Total test tim e (in m in utes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6
Fi gu re 3 – Typi cal bath tu b curve for an item . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9
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Fi gu re 4 – Weibu l l fai l ure m od es m ay be “m asked ” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Fi gu re 5 – Sam ple size: 1 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Fi gu re 6 – Sam ple size: 1 00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Fi gu re 7 – An exam ple showi n g lack of fit with a two-param eter Weibu l l d istri buti on . . . . . . . . . . . . . 23
Fi gu re 8 – Th e sam e d ata plotted wi th a three-param eter Weibu ll distri bu tion sh ows a
good fit with 3 m onths offset (locati on – 2, 99 m on ths) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Fi gu re 9 – Exam pl e of estim ati ng t0 by eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Fi gu re 1 0 – N ew com pressor d esi g n WeiBayes versus old d esi g n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Fi gu re A. 1 – M ain oil pu m p low tim es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Fi gu re A. 2 – Au gm enter pum p bearin g fail ure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Fi gu re A. 3 – Steep

β valu es h id e problem s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Fi gu re B. 1 – Pl ot of com pu tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Fi gu re E. 1 – Weibu ll plot for g raphical anal ysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Fi gu re E. 2 – Weibu l l pl ot of censored d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Fi gu re E. 3 – Cum ul ati ve hazard plot for d ata of Table E. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Fi gu re E. 4 – Cum ul ative hazard pl ots for Table E. 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Fi gu re H . 1 – Steel-fracture tou gh n ess – Curved d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figu re H . 2 – t0 im proves th e fit of Fi g ure H . 1 d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Tabl e 1 – G u id ance for u si ng th is I n tern ati onal Stan d ard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1
Tabl e 2 – Ranked fl are failure rivet d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5
Tabl e 3 – Adj usted ranks for suspen ded or censored data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6
Table 4 – Subgrou p si ze to estim ate tim e to X % fail ures usi ng th e sud d en death
m ethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 36
Tabl e 5 – Ch ain d ata: cycl es to fai lure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Tabl e B. 1 – Tim es to fai l ure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Tabl e B. 2 – Summ ary of resu lts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Tabl e D. 1 – Val u es of th e g amm a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Tabl e D. 2 – Fracti les of the norm al d istri bu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Tabl e E. 1 – Practical anal ysis exam pl e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Tabl e E. 2 – Spreadsheet set-u p for anal ysis of censored d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Tabl e E. 3 – Exam pl e of Weibull an al ysis for suspen d ed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Table E. 4 – Exam pl e of Spreadsh eet appl ication for censored d ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Tabl e
Table
Tabl e
Tabl e

E. 5 – Exam pl e spread sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
E. 6 – A relay d ata provi d ed by I SO/TC94 an d H azard anal ysis for fail ure m od e 1 . . . . . . . 53
I . 1 – Construction of ord i nate ( Y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
I . 2 – Constructi on of abscissa ( t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Tabl e I . 3 – Content of d ata en tered into a spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Copyright European Committee for Electrotechnical Standardization
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–7–

I N TRODUCTI ON
The Weibu l l distribu tion is used to m odel data reg ard less of wh eth er th e fai lure rate is
i ncreasing , d ecreasi ng or constant. The Weibu l l d i stri bution is fl exibl e an d ad aptable to a wi d e
ran g e of d ata. The tim e to fai l ure, cycles to fail u re, m i l eag e to fai l ure, m ech an ical stress or
sim il ar con tin u ous param eters n eed to be record ed for all i tem s. A l ife d istri bu tion can be
m odell ed even if n ot all the item s have fai led.
Gu id ance is gi ven on h ow to perform an an al ysis usin g a spreadsh eet program . G u i d ance is
also g i ven on h ow to an al yse d ifferen t fai l ure m odes separatel y an d i den tify a possible weak
popu lation. U sin g th e three-param eter Weibu l l d istri buti on can g i ve inform ati on on tim e to first
fai l ure or m in im um en d urance i n the sam pl e.

Copyright European Committee for Electrotechnical Standardization
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WEIBULL ANALYSIS
1 Scope
This I n tern ati onal Stan d ard provi d es m ethod s for an al ysin g data from a Weibul l d istri bu tion
usin g conti nu ous param eters such as tim e to fai lure, cycles to fai l ure, m ech an ical stress, etc.
This stand ard is appl icable wh en ever d ata on stren gth param eters, e. g. tim es to fai l ure,
cycl es, stress, etc. are avail able for a rand om sam ple of item s operati ng un d er test con di tions
or i n-service, for th e purpose of estim ati n g m easures of reli abi lity perform ance of the
popu lation from wh ich th ese item s were drawn .
This stan dard is appl icabl e wh en th e d ata bei ng an al ysed are i n depend entl y, id entical l y
d istribu ted . Th is sh ou l d eith er be tested or assum ed to be true (see I EC 60300-3-5).
I n th is stan d ard , num eri cal m ethods and graphi cal m ethods are d escribed to pl ot d ata, to
m ake a goodn ess-of-fit test, to estim ate the param eters of th e two- or three-param eter
Weibull d istri buti on an d to plot confi dence l im its. Guid ance is g i ven on h ow to in terpret th e
plot i n term s of risk as a function of tim e, fai l ure m odes and possi bl e weak popu l ation an d
tim e to first fai lure or m i n im um en durance.

2 Normative references
The fol l owi ng referenced d ocum ents are i n d ispen sabl e for th e appl icati on of th is d ocum ent.
For d ated references, on l y the ed i ti on ci ted appli es. For u n dated references, th e l atest ed i tion
of th e referenced d ocum en t (i nclu d ing an y am end m ents) appl ies.
I EC 60050-1 91 : 1 990,

In tern a tion a l Electrotech n ica l Voca b ula ry – Pa rt 1 91 : De pe n da b ility a n d

qua lity of service

I EC 60300-3-5: 2001 ,

De pe n da b ility ma n a ge me n t – Pa rt 3-5: A p p lica tion guide – Relia b ility

test conditions a nd sta tistica l test princip les

I EC 61 81 0-2,

Electrom ech a n ica l ele m en ta ry re la ys – Pa rt 2: Re lia b ility

I SO 2854: 1 976,

Sta tistica l

in te rpreta tion

of da ta

–

Te ch n ique s

of estima tio ns

and

tests

re la tin g to m ea ns a n d va ria nces

I SO 3534-1 : 2006,

Sta tistics – Voca b ula ry a n d sym b ols – Pa rt 1 : Ge n era l sta tistica l terms a n d

te rms in pro b a b ility

3 Terms, definitions, abbreviations and symbols
For the purposes of th i s d ocum ent, th e d efin iti ons, abbrevi ati ons an d sym bols g i ven in
I EC 60050-1 91 and I SO 3534-1 appl y, together wi th th e fol l owi ng .

3.1 Terms and definitions
3.1 .1
censoring

term i natin g a test after eith er a g i ven durati on or a g i ven num ber of fail ures
N OTE A test term in ated wh en there are stil l u nfai l ed item s m ay be cal l ed a “cen sored test", an d test tim e d ata
from such tests m ay be referred to as “censored d ata”.
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3. 1 . 2
s u s p en d ed i tem

item u pon wh i ch testi n g has been curtai led with out rel evan t fai l ure
N OTE 1

Th e i tem m ay not h ave fai l ed , or i t m ay h ave fai l ed i n a m od e oth er th an th at un d er i n vesti g ati on .

N OTE 2 An “earl y suspen si on” is on e th at was su spen d ed before th e first fail u re. A “l ate suspension ” is
su spend ed after th e l ast fai l u re.
3. 1 . 3
l i fe te s t

test cond ucted to estim ate or verify th e d urabil ity of a prod uct
N OTE The en d of the usefu l l i fe wil l often be d efi n ed as th e ti m e wh en a certai n percentage of th e item s have
fai l ed for n on -repai rabl e item s and as the ti m e wh en th e fai l u re i ntensi ty h as i ncreased to a speci fi ed l evel for
repai rabl e i tem s.
3. 1 . 4
n on -rep ai rab l e i te m

item th at can not, u n der g i ven con di tions, after a fail ure, be return ed to a state in wh i ch i t can
perform as requ ired
N OTE

Th e g i ven con d iti ons m ay be tech ni cal , econ om ic, ecol og ical an d /or oth ers.

3. 1 . 5
ope rati n g ti m e

tim e i nterval for wh ich th e i tem is in an operatin g state
N OTE ”Operatin g ti m e” is g en eric, an d shou l d be expressed i n u nits appropriate to th e item concern ed , e. g.
cal en d ar ti m e, operati n g cycl es, d i stance run , etc. an d th e u n i ts sh oul d al ways be cl earl y stated .
3. 1 . 6
re l evan t fai l u re

fai l ure that shou ld be i n cl ud ed in i nterpreti ng test or operation al resu lts or i n calcu l ati ng the

valu e of a reliabi l ity perform ance m easure
N OTE Th e criteri a for i ncl u sion shou l d be stated .
3. 1 . 7
re l i ab i l i ty te s t

experim en t carried ou t i n order to m easure, qu an tify or classify a reli abi lity m easure or
property of an i tem
N OTE 1 Rel i abi l i ty testi ng i s d i fferent from envi ron m en tal testi n g wh ere th e ai m i s to prove th at the i tem s u nd er
test can survi ve extrem e cond i ti on s of storag e, transportati on an d u se.
N OTE 2 Rel i abi l i ty tests m ay i ncl ud e en vi ron m ental testi ng .
3. 1 . 8
re pa i rab l e i te m

item that can, un d er g i ven con d itions, after a fai lure, be return ed to a state i n wh ich i t can
perform as requ ired
N OTE

Th e g i ven con d iti ons m ay be tech ni cal , econ om ic, ecol og ical an d /or oth ers.

3. 1 . 9
ti m e to fai l u re

operati n g tim e accum ul ated from the first use, or from restorati on , u ntil fai lure
N OTE I n appl icati on s wh ere the ti m e in storage or on stan d by is si gn ifican tl y g reater th an “operati ng ti m e”, th e
ti m e to fai l ure m ay be based on th e ti m e i n the speci fi ed servi ce.

Copyright European Committee for Electrotechnical Standardization
ELEC

BS EN 61 649:2008
– 10 –

61 649

© I EC: 2008

3.1 .1 0
time between failures

tim e d uration between consecu ti ve fail ures
N OTE 1

The ti m e between fai l u res incl ud es th e u p tim e an d th e d own ti m e.

N OTE 2 I n appl i cati ons where the ti m e i n storag e or on stan d by i s si gn i fi cantl y g reater th an operati n g ti m e, the
ti m e to fai l ure m ay be based on th e ti m e i n the speci fi ed servi ce.

3.1 .1 1
B life
L percentiles

ag e at wh ich a g i ven percen tag e of item s h ave fai l ed
N OTE "B 1 0 " l i fe i s the ag e at wh i ch 1 0 % of i tem s (e. g . beari ngs) h ave fai l ed . Som eti m es i t i s d en oted by th e
L (l i fe) val ue. B l i ves m ay be read d i rectl y from th e Wei bu l l pl ot or d eterm i ned m ore accu ratel y from th e Wei bul l
eq u ati on . The age at wh i ch 50 % of the i tem s fai l , th e B 50 l i fe, i s the m ed i an ti m e to fai l u re.

3.2 Abbreviations
ASI C

appl ication specific in tegrated circuit

BG A

ball gri d array

CDF

cum ul ati ve d istri bu tion fu ncti on

PDF

probabi l i ty d ensity fu nction

M LE

m axim um l ikel ihood esti m ation

M RR

m ed ian rank regressi on

M TTF

m ean tim e to fai l ure

3.3
t

η

β

t0

Symbols
tim e – vari abl e
Weibul l characteristic l ife or scal e param eter
Weibul l shape param eter
startin g poin t or ori g in of th e d istri buti on , fail ure free tim e

2

coeffici en t of d eterm in ati on

f( t)

probabi l i ty density fu ncti on

F( t)

cum ul ati ve d istri bu tion fu ncti on

h( t)

hazard fu ncti on

? ( t)

instan tan eous fai lure rate

H ( t)

cum ul ati ve h azard functi on

F1

num ber of failures with fai lure m od e 1

F2

num ber of failures with fai lure m od e 2

F3

num ber of failures with fai lure m od e 3

r

Copyright European Committee for Electrotechnical Standardization
ELEC


BS EN 61 649:2008
61 649

© I EC: 2008

– 11 –

4 Application of the techniques

Tabl e 1 sh ows th e circu m stances in wh ich particu lar aspects of th is stand ard are appl icabl e. I t
shows th e th ree m ai n m eth ods for estim ati n g param eters from the Weibu ll d istribu tion , n am el y
graph ical , com putati on al an d WeiBayes, an d in d icates th e type of d ata req u irem en ts for each
of th ese three m eth ods.

Table 1 – Guidance for using IEC 61 649
Method/
Kinds of data

Graphical
methods

Computational
methods

WeiBayes

I n terval cen sored

?

NC

?

Mu l ti pl e censored

?

NC

?

Si n gl y cen sored

?

?

?

NC

NC

?

Sm al l sam pl e ( ? 20)

?

NC

?

Large sam pl e

?

?

NC

Curved d ata

?

NC

NC

Com pl ete d ata

?

?

?

Zero fai l u res

N OTE

?

N C m ean s not covered i n th i s stand ard .

5 The Weibull distribution
5.1 The two-parameter Weibull distribution
The two-param eter Weibu ll d istribu ti on is by far the m ost wi d el y used d istribu tion for l ife d ata

an al ysis. The Weibu ll probabi l ity d ensi ty function (PDF) is sh own i n Eq u ati on (1 ):
?t

β − 1 − ?η
f( ) = β ⋅
⋅
ηβ
?

t

t

e

?
?
?

β
(1 )

wh ere
t

η
β

is the tim e, expressed as a vari able;
is the ch aracteristic life or scal e param eter;

is the sh ape param eter.

The Weibu l l cum u lative distribu ti on function (CDF) h as an explicit eq u ati on as sh own in
Equ ati on (2):
F(t )

=

1 -e

η

-(t/ )

β

(2)

The two param eters are η , th e ch aracteristic life, an d β , th e sh ape param eter. Th e shape
param eter i n dicates th e rate of chan g e of th e i nstantaneous fai l ure rate with tim e. Exam pl es
i ncl u d e: i nfant m ortal i ty, ran dom or wear-ou t. I t d eterm in es wh ich m ember of the Weibul l
fam il y of distri bu tions is m ost appropriate. Differen t m em bers have wid el y d ifferen t sh aped
PDFs (see Fig ure 1 ). Th e Weibu l l d istribu ti on fits a broad ran g e of l ife d ata com pared with
oth er d istributi ons. The variabl e t is g en eric an d can have vari ous m easures such as tim e,
d istance, num ber of cycl es or m echan ical stress appl ications.

Copyright European Committee for Electrotechnical Standardization
ELEC

BS EN 61 649:2008
– 12 –

61 649

© I EC: 2008

3,5

3,0

β = 0,5

2,5

2,0
f(t)

β = 3,44
1 ,5

β=1
1 ,0

0,5

0
0

0,5

1 ,0

1 ,5

2,0

2,5

3,0

3,5

Datum in time
IEC

1 321 /08

Figure 1 – The PDF shapes of the Weibull family for ? = 1 ,0
From Fi gu re 1 , th e PDF shape for β = 3, 44 (i nd icated) l ooks l ike the n orm al d istri buti on : it is
a fair approxim ati on, except for th e tai ls of the d istributi on .
The i nstan tan eous fai l ure rate ? ( t) (or h( t), th e h azard fu ncti on) of th e two-param eter Weibul l
d istri bu tion is shown in Equ ati on (3):
β −1
t
? ( t) = h(t )
(3)
β

=β⋅

Three ran g es of val u es of th e sh ape param eter,

•
•
•

η

β , are sali ent:

for β = 1 , 0 th e Weibu l l d i stri buti on is i d en tical to the expon enti al d istribu ti on an d the
i nstan tan eous fail ure rate, ? ( t), th en becom es a constan t eq u al to the reci procal of the
scale param eter, η ;

β
β

> 1 , 0 is th e case of increasin g i nstantaneous fail u re rate; and
< 1 , 0, is th e case of decreasin g i nstan tan eous fai l ure rate.

Ch aracteristic life, η , is th e tim e at which 63, 2 % of the item s are expected to fai l. This is tru e
for al l Weibu l l d istribu tions, regard l ess of the sh ape param eter, β . I f th ere i s repl acem ent of
item s, then 63, 2 % of th e tim es to failure are expected to be lower or equ al to the
characteristic life, η . Further d iscussi on of th e issues concern in g repair and non-repairabl e
item s can be fou n d in I EC 60300-3-5. The 63, 2 % com es from setti ng t = η in Eq u ati on (2)
wh ich resu lts i n Eq uati on (4):

η =

F( )

Copyright European Committee for Electrotechnical Standardization
ELEC

1 -e

ηη

-( / )

β

=

1 -e

-(1 )

β

=

1 - (1 /e)

=

0,63 2

(4)

BS EN 61 649:2008
61 649

© I EC: 2008

– 13 –

5.2 The three-parameter Weibull distribution
Equ ati on (5) sh ows th e CDF of th e three-param eter Weibu l l d istributi on:

F(t )

The param eter

t0

=

-(
1 -e

t

− t )β

η

0

(5)

is call ed th e failure-free tim e, l ocation param eter or m i n i m um life.

The effect of l ocati on param eter is typical l y n ot un d erstood wel l u nti l a poor fi t is observed
with a 2-param eter Weibu ll plot. When a lack of fit is observed, en g in eers attem pt to use oth er
d istribu tions that m ay provid e them with a better fit. H owever, th e l ack of fit can be reconci led
wh en th e d ata is plotted with a 3-param eter Weibu ll distri bu tion (see 8. 5). U sing th e l ocation
param eter, it becom es evi dent th at th e product fail ures are offset by a fixed peri od of tim e,
called th e threshol d. Th e effect of l ocation param eter is norm al l y observed wh en a product
sees “sh elf-l ife” after wh i ch th e first fai l ure occurs. A good in d icator of the effect of a l ocati on
param eter is th e con vex shape of a plot.

6 Data considerations
6.1 Data types
Life data are related to item s that "ag e" to fai l ure. Weibu l l fai lu re d ata are usu al l y l ife d ata but
m ay also d escri be m ateri al data wh ere th e “ag in g” m ay be stress, force or tem perature. "Ag e"
m ay be operati ng tim e, starts an d stops, l an d ings, takeoffs, low-cycl e fatig ue cycl es, m il eag e,
shelf or storage tim e, cycles or tim e at h ig h stress or h i gh tem peratu re, or m an y oth er
con tin u ous param eters. I n th is stand ard th e “ag e” param eter will be called tim e. When
req u ired , “tim e” can be substituted by an y of th e “ag e” param eters l isted above.

6.2 Time to first failure
The Weibu l l “tim e” vari able is usu all y consid ered to be a m easure of life consum pti on. Th e
fol l owi ng in terpretations can be used:
–

tim e to first fai lure of a repairable item ;

–
–

tim e to fai lu re of an n on-repairabl e i tem ;
tim e from new to each fail ure of a repairable system if a n on-repairabl e item in th e
system fails m ore th an once d uri ng th e period of observati on . I t has to be assum ed
th at th e repair (ch ang e of th e i tem ) d oes not introduce a n ew fai lure, so th at th e
system after the repair can , wi th an approxim ati on, be reg arded as h avi n g th e sam e
reli abili ty as imm ediatel y before the fail ure (comm onl y referred as th e “bad as ol d”
assum pti on);
tim e to first fai l ure of a n on-repai rable i tem , fol lowi ng sch ed u l ed m ain ten an ce, wi th th e
assum pti on th at th e fai l ure is rel ated to th e previ ous m ai n ten ance.

–

6.3 Material characteristics and the Weibull distribution

Material characteristics such as creep, stress rupture or breakag e an d fati g ue are often
plotted on Weibu ll probabil i ty paper. H ere the hori zon tal scale m ay be stress, cycles, l oad ,
num ber of load repetiti on s or tem perature.

6.4 Sample size
U ncertai nty wi th regard to the Weibu l l param eter estim ation is related to the sam pl e si ze an d
th e n um ber of rel evant failures. Weibu ll param eters can be estim ated using as few as two
fai l ures; h owever, th e u ncertai n ty of such an estim ate woul d be excessi ve and could not
confirm th e applicabi lity of the Weibu l l m od el. Whatever the sam ple si ze, confi d ence l im its
shou l d be calcu l ated an d pl otted i n ord er to assess the uncertai nty of th e estim ati ons.
Copyright European Committee for Electrotechnical Standardization
ELEC

BS EN 61 649:2008
– 14 –

61 649

© I EC: 2008

As with al l statistical an al ysis, th e m ore data th at is avai lable, the better th e estim ati on bu t if
th e d ata set is l im ited, th en refer to th e ad vice g i ven in 1 1 . 3.
6. 5

C en s o red an d s u s p en d ed d ata

When an al ysi n g l ife d ata i t is n ecessary to i nclud e d ata on those item s in the sam ple th at
have n ot fai led , or have n ot fai led by a fail ure m ode anal ysis. Th is data is referred to as
censored or suspen d ed data (see I EC 60300-3-5). When the tim es to fai l ure of al l item s are
observed, the d ata are said to be com pl ete.
An item on test th at has not fai led by the fai l ure m od e in q uestion is a suspensi on or censored
item . I t m ay h ave fail ed by a d ifferen t fai lure m od e or n ot fail ed at al l . An "earl y suspensi on" is
on e th at was suspen d ed before th e first fai lu re ti m e. A "l ate suspensi on" is suspen d ed after
th e l ast fai lure. Suspensi ons between failures are call ed rand om or progressi ve suspensions.
I f item s rem ai n u nfai l ed , th en th e correspon d in g d ata are said to be censored. I f a test is
term i nated at a specifi ed tim e, T, before all item s have fail ed , then the data are said to be
tim e censored. I f a test i s term in ated after a specified num ber of fail ures h ave occurred, then
th e d ata are sai d to be failure censored .
Furth er d iscussion of cen sorin g is covered i n I EC 60300-3-5.
7
7. 1

G raph i cal m eth od s an d g ood n e ss-of-fi t
O vervi ew

Graph ical anal ysis consi sts of plotti ng th e d ata on Weibu l l probabil i ty paper, fitti n g a lin e
throug h th e d ata, i nterpreti ng th e plot and estim ati n g the param eters usin g special probabi l ity
paper d eri ved by transform ing th e Weibu l l eq uati on into a l in ear form . Th is is illustrated in
Ann ex I .
Data is pl otted after first organ i zin g it from earli est to l atest, a process call ed ranki n g. Th e
tim e to fai lu re d ata are pl otted as th e X coordin ate on the Weibul l probabi l ity paper.
The Y coord in ate is the m edi an rank as specifi ed in 7. 2. 1 . For sam pl e sizes above 30 th e
m edian rank is, in practi ce, th e sam e as th e per cen t of fail ures. I f th e pl otted d ata fol low a
l in ear tren d, a regressi on l i n e m ay be d rawn .
The param eters m ay th en be read off the plot. Th e ch aracteristic l ife,
of th e item s faili n g, cal led th e “B63, 2 Life”. The shape param eter,
sl ope on Weibu l l paper.

η , is the tim e to 63, 2 %
β , is estim ated as th e

Med i an rank regressi on (MRR) is a m ethod for estim ating th e param eters of th e d istri buti on
usin g lin ear regressi on tech n i qu es wi th the vari ables bei n g th e m ed ian rank and lifetim e or
stress , etc.
Anoth er graph ical m eth od that is used for estim atin g param eters of a Wei bu l l d istributi on is
call ed h azard pl ottin g. Th is is d escri bed i n 7. 3.
7. 2

H ow to m ake th e p rob a bi l i ty p l ot

I n ord er to m ake a probabil i ty plot, a sequ ence of steps n eeds to be carri ed ou t. These steps
are descri bed in d etai l bel ow.

Copyright European Committee for Electrotechnical Standardization
ELEC


BS EN 61 649:2008
61 649

© I EC: 2008

7.2.1

– 15 –

Ranking

To m ake the Weibu ll plot, rank th e d ata from the lowest to the hi g hest tim es to failu re. This
ranki n g wi l l set up th e pl otti n g posi tions for the tim e, t, axis an d th e ordin ate, F( t), i n
percentage val ues. Th ese wi ll provi d e i nform ati on for th e constructi on of the Weibu l l li ne
shown i n Equ ati on (6).
Med ian ranks are g iven i n An n ex C. En ter as an exam pl e the tabl es for 50 % m edi an rank, for
a sam ple si ze of five, an d fi nd th e m ed ian ranks shown i n Table 2 for fi ve failu re tim es sh own
in th e m id d le colum n. The m ed ian rank pl ottin g positi ons in An n ex C are used with all types of
probabi l i ty paper, i. e. Wei bu l l, log-n orm al, norm al, an d extrem e val u e.
N OTE 1

I f two d ata poi nts have th e sam e ti m e, they are pl otted at d i fferen t m ed i an ran k val ues.

Table 2 – Ranked flare failure rivet data
Order number

Failure time

I

t

Median rank

m i n (X)

% (Y)

30

1 2, 94

2

49

31 , 38

3

82

50, 00

4

90

68, 62

5

96

87, 06

1

The m edi an estim ate i s preferred to th e m ean or average valu e for n on-sym m etrical
d istri bu tions. M ost l ife d ata d istri buti ons are skewed and , th erefore, th e m edi an plays an
im portant rol e.
I f a table of m ed ian ranks an d a m eans to calcu l ate m edi an ranks usi ng the Beta d istri buti on
is n ot avai labl e, then Ben ard ’s approxim ati on, Eq u ati on (6), m ay be used:
Fi

wh ere

N

(6)

is the sam pl e si ze and i is th e ranked position of th e data item of i nterest.

N OTE 2 Th i s equ ati on is m ostl y used for
negl ected : Fi = (i /N) × 1 00 % .

7.2.2

= (i − 0, 3) %
( N + 0, 4)

N

≤ 30;

for

N

>30 the correcti on of the cu m ul ati ve freq uen cy can be

The Weibull probability plot

After transform ing th e d ata, th e pl ot can be constructed usin g three d ifferent m eth ods:
–

Weibul l probabi l i ty paper – Ann ex F sh ows Weibul l probabi l ity paper;

–
–

a com puter spreadsh eet program – An nex E g i ves a spreadsh eet exam ple;
comm ercial off-the-sh elf software.

7.2.3

Dealing with suspensions or censored data

N on-fai l ed item s or item s that fai l by a d ifferen t fai l ure m od e are "censored " or "suspen d ed "
item s, respecti vel y. Th ese d ata can n ot be i gn ored . The tim es on suspen d ed i tem s have to be
i ncl u ded i n the an al ysis.
The form ula below g i ves th e ad justed ranks wi thout th e n eed for calcu latin g rank increm ents.
I t is used for every fai l ure an d req ui res an add itional colum n for reverse ranks. The proced ure
is to rank th e d ata wi th th e suspensi ons and to use Equ ation (7) to d eterm i ne th e ranks,
adj usted for th e presence of th e suspensions.
Copyright European Committee for Electrotechnical Standardization
ELEC


BS EN 61 649:2008
– 16 –
Adj usted rank

=

61 649

(Reverse rank) x (Previous adj usted rank)
(Reverse rank)

+

+ (N +

© I EC: 2008

1)

(7)

1

The rank ord er n um bers are adj usted for th e effect of the three suspend ed item s in Table 3.

Table 3 – Adjusted ranks for suspended or censored data
Rank

Time

Status

Reverse
rank

Median
rank

Adjusted rank

1

10

Suspensi on

8

Suspen d ed . . .

2

30

Fail u re

7

[7

3

45

Suspensi on

6

Su spen d ed …

4

49

Fail u re

5

[5

5

82

Fai l u re

4

6

90

Fail u re

3

7

96

Fai l u re

2

× 1 , 1 25 +(8+1 )] / (5+1 ) = 2, 438
[4 × 2, 438 +(8+1 )] / (4+1 ) = 3, 750

[3 × 3, 750 +(8+1 )] / (3+1 ) = 5, 063
[2 × 5, 063 +(8+1 )] / (2+1 ) = 6, 375

8

1 00

Su spension

1

Su spen d ed . . .

×0

%

9, 8

+(8+1 )] / (7+1 ) = 1 , 1 25

25, 5
41 , 1
56, 7
72, 3

I n th is exam ple, th e adj u sted ranks use Ben ard 's approxim ati on to calcu late th e m ed i an ranks
as i t is easier th an in terpolatin g i n the tabl e. Th e resu l ts i n Table 3 are plotted in Fi gure 2.
N OTE I f two i tem s fai l at th e sam e age, they are assi g n ed seq u en ti al rank ord er n u m bers. I n case of l ater
su spen si on, th e proced u re i s to be repeated for the rest of th e fai l u res.

90
70
50

Weibull CDF

30

Without suspensions

With suspensions

10
5

2
1
1

10

1 00

1 000

Total test time
IEC

Figure 2 – Total test time (in minutes)

Copyright European Committee for Electrotechnical Standardization
ELEC

1 322/08


BS EN 61 649:2008
61 649

© I EC: 2008

– 17 –

Ben ard's approxim ation for th e m ed ian rank is suffici entl y accurate wh en u si ng suspensi on for
plotti ng Weibu ll d istri butions an d estim atin g the param eters. H ere " i " is th e adj usted rank an d
" N" is the sum of fai l ures and suspensions. The m edi an ranks are con verted to percentages
for pl ottin g on Weibu l l paper. For exam pl e, for th e first fai lure i n Tabl e 3 wi th an adj usted rank
of 1 , 1 25:
Median Rank (%)

= (1,1 25 − 0, 3) × 1 00 = 9, 82%
(8 + 0, 4)

(8)

Fi gu re 2 shows th e correct Weibu l l plot as i t i nclu des th e suspensions.
The fol l owi ng are th e steps to plot data sets wi th suspensions:
a) rank th e tim es, both fai l u res and suspensi ons, from earli est to latest;
b) calcu l ate the adj usted ranks for th e fai l ures (suspensi ons are not plotted);

c) use Ben ard's approxim ati on to calcu late th e m ed i an ranks;
d) plot th e fail ure tim es ( x ) versus th e m ed i an ranks ( y ) on Weibu l l paper;

η by read in g the B63, 2 l ife from th e plot;
estim ate β with a ru ler or use special beta scal es usu al l y g i ven on th e Weibu ll

e) estim ate
f)

probabil ity

paper;

g) i nterpret th e pl ot.
7. 2. 4

Probabil ity plotti ng

Plotti n g the d ata on Weibu l l paper by h an d or on a com pu ter m ay be sufficien t for ch ecki ng
th e goodn ess-of-fit. Su bj ecti vel y fittin g a straig ht li ne by eye can g i ve an i nd icati on of
goodn ess-of-fit.
7. 2. 5

Checki ng the fi t

I f th e d ata cluster arou n d a straig ht l i ne on a probabil i ty pl ot, it is evid ence that th e d ata is
represented by the su bj ect d istributi on. H owever, sm all sam pl es m ake i t di fficul t to g au g e th e
goodn ess-of-fit. There are statistical m easures of good ness-of-fi t such as Ch i-sq u ared ,
Kolm ogorov-Sm irn off an d N ancy Man n's tests. Th is stand ard uses the correlation coeffici ent
squ ared call ed th e coeffi ci en t of determ i nation .

This can be calcu l ated u si ng Eq u ati on (9):
?
?

?

?

r

2

=

?

?= ?=
N

?=
N

i 1

xi yi

−

N

xi

i 1

i 1

N

?

?

?
N

? i =1
?

xi

2

− N( x )

2

??
?
??
?

?=
N

i 1

yi

2

?
yi ?
?
?
?
?

2

− N( y)

2

?
?
?

(9)

wh ere x an d y are th e m ed i an rank an d th e fail ure tim e, respecti vel y, x an d y are averag es

of x an d y an d N is th e sam pl e si ze.
i

i

i

i

r2

is th e proporti on of vari ati on in the d ata that can be explai n ed by the Weibul l h ypoth esis.
The cl oser th is is to 1 , the better th e data are fi tted to a Weibu ll distribu ti on ; the cl oser to 0
i nd icates a poor fi t. The correl ation coefficien t, " r , " is i n ten ded to m easure th e streng th of a
linear rel ati onsh i p between two vari ables. " r" is a num ber between -1 an d +1 , d epen di n g on
the slope. Al tern ativel y, if the Weibu l l plot is constructed usin g a spreadsh eet, th en wh en
usin g li near regressi on tech n i qu es the correlati on coeffici ent is often g i ven as part of th e
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ELEC


BS EN 61 649:2008
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ou tpu t wh en d ata are fitted with a straig ht l i ne (usu all y a selective opti on). Sim il arl y,
comm ercial software packages provi de th e coeffici en t of determ i nation .

This sh ou l d be used wi th care and on l y if visu al i n spection concurs wi th observation .

7.3 H azard plotting
Weibull probabi lity plotti ng tech n i qu es first esti m ate th e cum u lati ve proportion failed , F (t) ,
usin g m ed i an ranks, an d th en pl ot tim es to fai l ure ag ai nst respecti ve estim ated cum u lative
probabi l i ti es on the Weibu ll probabi l ity paper.
The hazard plotti ng tech n iq u e beg ins with the estim ati on of th e i nstantan eous fai l ure rate or
th e h azard functi on ,

λ( ) =
t

( )=

h t

f (t )

[1

− F (t )]

,

(1 0)

usin g an estim ate of th e cum ul ati ve h azard functi on .
The cum ul ative h azard fu ncti on ,

( )=? ( )

t

H t

0

h t dt

= − ln ??1 − F ( t ) ??

,

(1 1 )

is estim ated by th e cum u l ati ve sum of the estim ated hazard fu ncti ons.
For the Weibu ll distri bu ti on ,

( ) = ?η
?

H t

?

t ?

?
?

β

,

(1 2)

taking natural l ogarithm s of both si des, yiel ds th e fol l owi ng l in ear relati onsh ip for ln H
ag ai nst l n t:
ln( H

( ) =β
t

)

ln(t )

− β ln(η )

()
t

(1 3)

The Weibu ll h azard paper is therefore l n-ln paper. The n om in al slope of the fitted l i n e to th e
data is , and t
when H t 1 .

β

=η

( )=

N OTE Ei th er natu ral l ogs or base 1 0 l og s can be used for Wei bu l l h azard paper.

Thou g h the Weibu l l h azard paper is avai l abl e for som e cou n tri es, th is tech n iq u e can be used
wi th th e usual Weibul l probabil i ty paper appl yi n g the transform ati on :

( )=

F t

1

− e−

H (t )

This can be very easi l y i m plem en ted usi ng a spreadsh eet program .
The hazard pl ottin g proced ure is as fol lows:
a) sort th e tim es, both fai lures and suspensi ons sim ultaneousl y, from earl i est to l atest;
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ELEC

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b) for each fai lure, calcu late th e i nstan tan eous fai lure rate, g iven by 1 /(n um ber of item s
rem ai n in g after the previous fai lure or censori ng);
c) for each failure, calcu l ate the cum ul ati ve sum of instantaneous fail ure rates, as an
estim ate of th e hazard fu nction ;
d) plot th e estim ated cum u l ati ve h azard s ag ainst th e tim e of fai lure on eith er l og-l og paper or
l n-l n paper;
e) fit a straig ht l in e to th e pl ots;
f) estim ate th e param eters.
Ann ex E g i ves worked exam ples. I EC 61 81 0-2 also provid es exam pl es of usi ng h azard
plotti ng to estim ate th e param eters of a Weibu l l d i stri buti on .

8 Interpreting the Weibull probability plot
8.1 The bathtub curve
8.1 .1 General
The often-used bath tub curve (see Fi gu re 3) sh ows th e relationsh ip between th e Weibu l l
shape param eter, β , an d th e hazard fu ncti on throug h ou t th e l ife of an item . N ot all item s,
however, d ispl ay al l el em en ts of th e bathtu b curve d urin g their l ifetim es.

Hazard function

Useful life
(Random period)

β>1

h(t)

β<1

β=1
Time
Infant
mortality
(early life)

Wear-out

IEC

1 323/08

Figure 3 – Typical bathtub curve for an item
8.1 .2

β <1 – Implies early failures

Both el ectron ic and m echan ical system s m ay i n itiall y have hi g h fail ure rates. M anufacturers
con d uct prod ucti on process con trol , prod ucti on acceptance tests, "burn-i n, " or rel iabi li ty stress
screen ing (RSS), to preven t earl y failu res before d el i very to th e custom er. Therefore, sh ape
param eters of l ess th an on e i n d icate th e foll owin g :

−
−
−
−
−

l ack of ad eq u ate process control;
i nad eq uate burn -i n or stress screen i n g;
prod ucti on probl em s, m is-assem bl y, poor q ual i ty con trol ;
overh aul problem s;
m ixture of popu l ati ons;

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ELEC


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−

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run -i n or wear-in .

Man y el ectron ic com pon en ts d urin g th eir usefu l l ife sh ow a d ecreasi n g i n stantaneous fai l ure
rate, th us featurin g sh ape param eters l ess th an 1 . Preven tive m ai ntenance on such a
com pon ent is n ot appropri ate, as ol d parts are better th an n ew.

8.1 .3

β

= 1 – Implies constant instantaneous failure rate

This is often called th e ran d om fail ure period as th e fai lures occur ran dom l y i n tim e. These
fai lure m od es are consi d ered in d epend ent of tim e. I n th is case, preven ti ve m ainten ance wou ld
not im prove the system .
Therefore, an y of th e fol lowi ng m ig ht be suspected:

−
−
−
−

ran d om m ain ten ance errors, h um an errors;
ran d om overl oad ;
fai l ures d u e to nature, foreig n obj ect d am age, l i gh tn in g strikes;
m ixtures of data from three or m ore fail ure m od es (assum in g they have d ifferent valu es
of β ) where n o fai lure m echan ism d om inates th e failure beh avi our.

H ere ag ai n, preven ti ve m ain ten ance is n ot appropri ate. Th e Weibu l l d istributi on with β = 1 is
i dentical to th e expon enti al d istribu tion . Of those that survi ve to tim e, t, a constan t percen tag e
fai ls i n the n ext peri od of tim e. Th is is kn own as a constan t i nstantan eous fail ure rate.

8.1 .4

β

> 1 – Implies wear-out

Som e typi cal exam pl es of th ese cases are as foll ows:

−
−
−
−
−
−
−
−

wear;
corrosion;
crack propagation ;
fatig u e;
m oistu re absorpti on ;
d iffusi on ;
evaporati on (wei gh t l oss);
d am age accum ul ati on.

Desi gn m easu res h ave to ensure th at those phenom en a d o not si gn ificantl y contri bu te to th e
probabi l i ty of product fai l ure duri n g th e expected operati onal l ife.
Three-param eter Weibu l l d istribu tion estim ates m i nim um tim e to first fai lu re, wh ich is h ig h l y
ad vantageous i n cases where the shape param eter is greater th an 1 (see 5. 2 or 8. 5).

8.2 Unknown Weibull modes may be "masked"
The “m asked" ph en om enon m ig ht be encou ntered wh en th ere are two or m ore com petin g
fai l ure m odes wi th h i g h valu es of sh ape param eters an d hig h l y d ifferent scal e param eters.
This m eans that for a sm all sam ple size, m ost or all of th e sam ples wou ld fai l i n th e fai lure
m od e wi th lower scale param eter. The oth er fai lu re m od es m ay not be i dentifi ed u n ti l th e first
fai l ure m od e is el im in ated. An exam pl e of two com peti n g fai l ure m od es is shown i n Fi g ure 4.

Copyright European Committee for Electrotechnical Standardization
ELEC


BS EN 61 649:2008
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© I EC: 2008

– 21 –

99
90
70
Weibull CDF

50
30

Failure
mode A

Failure
mode B

10
5
2
1

0,1

1

10
Time to failure

1 00

1 000
IEC

1 324/08

Figure 4 – Weibull failure modes may be “masked”
8.3 Small samples
Weibul l an al ysis is possi ble from sm all sam ples. H owever, confi d ence l imits are affected b y
sam pl e size. Sm al l sam pl es wi l l increase th e u ncertai nty i n estim atin g the life param eters.
I m provem en t i n uncertai nty with i ncreasing sam pl e si ze is i l l ustrated i n Fi gures 5 and 6. (Th e
“90 % B life bou nds” (con fid ence l im its) con tai n th e u nknown “B life” with a freq uency of 90 % .
The i nterval is m uch sm al ler in Figu re 6. )

99
90
70
Weibull CDF

50
30

90 % B
life bounds

10
5
2
1

10

Time to failure

1 0 000
IEC

Figure 5 – Sample size: 1 0

Copyright European Committee for Electrotechnical Standardization
ELEC

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BS EN 61 649:2008
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99
90
70
Weibull CDF

50
30
90 % B
life bounds

10
5
2
1

1

1 0 000

Time to failure

IEC

F i g u re 6 – S a m p l e s i z e :

1 326/08

1 00

The degree of uncertainty i s l argest i n th e tai l of the Weibul l d istribution , as i nd icated by sm al l

valu es of F (t) wh en th e fracti on fai li ng is sm al l . Sam pl es of greater th an 20 fail ures an d
suspensions are req uired to d ifferenti ate the Weibu ll from oth er d istri buti on s.
Decisi ons based upon resu lts from a sm al l sam pl e sh ou ld consi d er th e u n certain ty an d, wh ere
possible, sh ou l d be refi n ed by coll ecti on and an al ysi s of ad d iti onal d ata.
8. 4

O u t l i e rs

Som etim es, the first or l ast point in a d ata set is a wil d point an d not a m ember of th e d ata set
for som e reason ; such poin ts are term ed outl iers. These poi nts m ay be i m portan t to th e l ife
data anal ysis, an d th erefore requ ire in vesti g ati on of th e en g in eerin g aspects of data
record i ng , test records, i nstrum entation cal ibrations, etc. in ord er to i d en tify the cause of
extrem e scatter of th e poi nt. Som etim es, th e outl i ers m ay i nd icate a weak popul ation or
process fl aws, and are th erefore hig h l y si gn ificant from the rel i abil i ty assurance point of vi ew.
8. 5

I n t e rp re t a t i o n o f n o n - l i n e a r p l o t s

I f th e d ata on a Weibu l l pl ot appears curved , as il l ustrated i n Fi gure 7, th is i nd icates that th e t0
param eter m ay be non-zero. Before m od ifyin g the plot, it is n ecessary to check if th e plot
con tai ns m ore than on e fail ure m ode. I f so, refer to An n ex G an d i n vesti g ate furth er wh eth er
th ose m od es are com peting with each oth er or are sim ple m ixtures.

Copyright European Committee for Electrotechnical Standardization
ELEC


BS EN 61 649:2008
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© I EC: 2008

– 23 –

99
90

Weibull CDF

70
50
30
20
10
5
3
2
1

Table of statistic
Shape = 1 ,870
Scale = 57,656 1

0,01
0,1

1 ,0
1 0,0
Time in service (months)

1 00,0
IEC

1 327/08

Figure 7 – An example showing lack of fit with a two-parameter Weibull distribution
M in im um life d oes n ot m ean at “zero tim e”, bu t rath er that th ere exists a m inim um life or a
m inim um en du rance. “Zero age” is wh ere non e of th e wear-out fail ure m echan ism s of an i tem
has started to operate whereas “zero tim e” is wh ere an i tem h as seen n o operati n g tim e. For
exam ple, it m ay be ph ysical l y im possible for th e fail ure m ode to prod uce fail ures
i nstan tan eousl y, or earl y i n life. Fig ure 8 sh ows the sam e d ata as used i n Fi gure 7 bu t wi th
th e ori g i n sh ifted 2, 99 mon ths. Th is plot shows a l in ear fit to th e data and is i n terpreted as a
fai lure-free peri od (3 m on ths), wi th in wh ich th e probabi l ity of fai l ure is zero.

Copyright European Committee for Electrotechnical Standardization
ELEC