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IS 8161-6-1 (1983): Guide for equipment reliabilitytesting, Part 6: Tests for validity of a constant failurerate assumption, Section 1: Chi-square test [LITD 2:Reliability of Electronic and Electrical Components andEquipment]

© BIS 2006

B U R E A U O F I N D I A N S T A N D A R D SMANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG

NEW DELHI 110002

IS : 8161 (Part 6/Sec 1) - 1983(Reaffirmed 2001)

Edition 1.1(1987-11)

Price Group 3

Indian StandardGUIDE FOR

EQUIPMENT RELIABILITY TESTINGPART 6 TESTS FOR VALIDITY OF A CONSTANT

FAILURE RATE ASSUMPTION

Section 1 Chi-square Test

(Incorporating Amendment No. 1)

UDC 621.38.038+621.31:621–[92/026]

IS : 8161 (Part 6/Sec 1) - 1983

© BIS 2006

BUREAU OF INDIAN STANDARDS

This publication is protected under the Indian Copyright Act (XIV of 1957) andreproduction in whole or in part by any means except with written permission of thepublisher shall be deemed to be an infringement of copyright under the said Act.

Indian StandardGUIDE FOR

EQUIPMENT RELIABILITY TESTINGPART 6 TESTS FOR VALIDITY OF A CONSTANT

FAILURE RATE ASSUMPTION

Section 1 Chi-square Test

Reliability of Electronic and Electrical Components andEquipment Sectional Committee, LTDC 3

ChairmanPROF S. SAMPATH

Indian Institute of Technology,Kanpur

Members RepresentingADDITIONAL DIRECTOR, STAN-

DARDS (S & T), RDSORailway Board (Ministry of Railways)

JOINT DIRECTOR, STANDARDS(S & T)-III, RDSO ( Alternate )

DR K. CHANDRA National Physical Laboratory (CSIR), New DelhiSHRI CHARANJIT SINGH Hindustan Aeronautics Ltd, Hyderabad

SQN LDR I. M. GANDOTRA ( Alternate )CONTROLLER, CIL Ministry of Defence (DGI)

LT-COL V. K. KHANNA ( Alternate )DR P. K. DUTTA Peico Electronics and Electricals Ltd, Bombay

SHRI V. NARAYANAN ( Alternate )SHRI B. P. GHOSH National Test House, Calcutta

SHRI B. C. MUKHERJEE ( Alternate )SHRI A. P. GUPTA Instrumentation Ltd, Kota

SHRI I. S. SULAKH ( Alternate )SHRI S. P. KULKARNI The Radio Electronic & Television Manufacturers’

Association, BombaySHRI S. M. KHURSALE ( Alternate )

SHRI H. C. MATHUR Posts and Telegraphs Board, New DelhiSHRI U. R. G. ACHARYA ( Alternate )

SHRI D. C. MEHTA Directorate General of Civil Aviation, New DelhiSHRI R. V. ISRANI ( Alternate )

SHRI S. R. MEHTA Indian Electrical Manufacturers’ Association,Bombay

SHRI T. C. GOSALIA ( Alternate )

( Continued on page 2 )

IS : 8161 (Part 6/Sec 1) - 1983

2

( Continued from page 1 )

Members RepresentingDR K. B. MISRA In personal capacity ( Department of Electrical

Engineering, Indian Institute of Technology,Kharagpur )

SHRI E. G. NAGARAJAN Department of Electronics, New DelhiSHRI K. R. ANANDAKUMARAN NAIR Lucas-TVS Ltd, Madras

SHRI C. RANGANATHAN ( Alternate )SHRI D. V. PETKAR Bhabha Atomic Research Centre, Trombay, Bombay

SHRI A. K. BABAR ( Alternate )SHRI V. B. PRADHAN All India Radio, New Delhi

SHRI N. J. NAIR ( Alternate )SHRI P. S. K. PRASAD Bharat Electronics Ltd, BangaloreSHRI K. RAMGOPAL ISRO Satellite Centre (ISAC), Bangalore

SHRI SIHARAN DE ( Alternate )SHRI K. S. PRAKASA RAO Electronics Corporation of India Ltd, Hyderabad

SHRI S. S. SONWALKAR ( Alternate )SHRI R. SOMASUNDARAM Directorate of Technical Development and Prod-

uction (AIR), Ministry of DefenceSHRI R. N. SHARMA ( Alternate )

COL J. VARGHESE Ministry of Defence (R & D)SHRI P. K. SHUKLA ( Alternate )

SHRI B. VIRESALINGAM Indian Telephone Industries Ltd, BangaloreSHRI V. MUTHAIAH ( Alternate )

SHRI R. C. JAIN,Head (Electronics) ( Secretary )

Director General, ISI ( Ex-officio Member )

Study of Statistical Problems of Reliability of Electronic andElectrical Items Subcommittee, LTDC 3 : 1

ConvenerDR P. K. DUTTA Peico Electronics & Electricals Ltd, Bombay

MembersLT-COL V. K. KHANNA Ministry of Defence (DGI)

MAJ S. P. MURGAI ( Alternate )SHRI V. NARAYANA Indian Statistical Institute, HyderabadSHRI P. S. K. PRASAD Bharat Electronics Ltd, BangaloreSHRI K. RAMGOPAL ISRO Satellite Centre (ISAC), BangaloreDR Y. V. SOMAYAJULU National Physical Laboratory (CSIR), New Delhi

SHRI V. N. SHARMA ( Alternate )COL J. VARGHESE Ministry of Defence (R & D)

SHRI P. K. SHUKLA ( Alternate )

IS : 8161 (Part 6/Sec 1) - 1983

3

Indian StandardGUIDE FOR

EQUIPMENT RELIABILITY TESTINGPART 6 TESTS FOR VALIDITY OF A CONSTANT

FAILURE RATE ASSUMPTION

Section 1 Chi-square Test

0. F O R E W O R D0.1 This Indian Standard (Part 6/Sec 1) was adopted by the IndianStandards Institution on 25 January 1983, after the draft finalized bythe Reliability of Electronic and Electrical Components andEquipment Sectional Committee had been approved by the Electronicsand Telecommunication Division Council.0.2 This standard gives recommended numerical methods for testingthe statistical validity of the constant failure rate assumption is thesixth in the series of Indian Standards for equipment reliabilitytesting. To be able to write a detailed reliability test specification andperform a reliability test, the test engineer will need additionalinformation which are dealt with in detail in other standards in thisseries. A list of standards envisaged in this series some of which areunder consideration is given on page 10.0.3 This standard is largely based on IEC Document 56 (Secretaries)144 Draft IEC Standard 605 Equipment reliability testing: Part 6 Testfor validity of a constant failure rate assumption, issued by theInternational Electrotechnical Commission.0.4 This edition 1.1 incorporates Amendment No. 1 (November 1987).Side bar indicates modification of the text as the result ofincorporation of the amendment.0.5 For the purpose of deciding whether a particular requirement of thisstandard is complied with, the final value, observed or calculated,expressing the result of a test, shall be rounded off in accordance withIS : 2-1960*. The number of significant places retained in the rounded offvalue should be the same as that of the specified value in this standard.

1. SCOPE1.1 This standard (Part 6/Sec 1) gives recommended numericalmethods for testing the statistical validity of the constant failure rate

*Rules for rounding off numerical values ( revised ).

IS : 8161 (Part 6/Sec 1) - 1983

4

assumption underlying the methods used in IS : 8161 (Part IV)* andIS : 8161 (Part VII)-1977*.1.2 Recommendations are given for action to be taken in case theassumption is rejected. If stated in the detailed reliability testspecification, one of these tests shall be applied before a conclusion ismade from reliability testing of failure rate or mean time betweenfailures.

2. GENERAL2.1 The validity test shall be performed at the end of the reliability testusing all relevant failures observed in the reliability test.2.2 It shall be recognized that the validity tests are of a statistical natureand give results associated with certain small risks to reject theassumption when it is true. The tests in this standard (Part 6/Sec 1) aredesigned to a level of significance of 10 percent, that is, 10 percent riskto reject the assumption even if it is true. The risk should be consideredtogether with the producer’s risk of a reliability compliance test if thevalidity test is called for by the equipment contract or specification.2.3 The proposed tests are the most generally accepted statisticalvalidity tests for constant failure rate against any other (but unknown)distribution. The literature contains many tests of a constant failurerate against specific alternative hypotheses, such as increasing failurerate or decreasing failure rate. These tests could be used instead of themethods given in this standard.2.4 Two tests are given one of which is limited to a large number offailures. Each test becomes more sensitive the more failures areobserved. None of the tests is powerful enough to give significantdeparture from constant failure rate when the number of failures issmall.2.5 In case the reliability test is terminated before 3 relevant failureshave been observed, for example, at compliance testing with acceptanceat 0, 1 or 2 failures, and further testing is unfeasible because ofeconomic, time or other reasons, an assumption of a constant failurerate may be permitted without the need of the test of validity.2.6 If the testing is extended for information beyond the decision pointof a reliability compliance test, the result of a validity test using all thedata should only be used for information purposes. The additional datashould not affect the decision made.

*Guide for equipment reliability testing:Part IV Procedure for determining estimates and confidence limits from equip-

ment reliability determination tests ( under preparation ).Part VII Compliance test plans for failure rate and mean time between failures

assuming constant failure rate.

IS : 8161 (Part 6/Sec 1) - 1983

5

3. LIST OF SYMBOLS3.1 The following symbols are used in this standard:

4. ACCUMULATED RELEVANT TEST TIME T AND PARAMETER d4.1 The validity test are based on the accumulated relevant test timesto failures with the addition of any relevant test time accumulatedbetween the latest failure and the point of time at which the validitytest is applied. Each of these accumulated times is the sum of therelevant test times of all the individual test items as recorded byelapsed time meters, work cycle counters or other appropriate means.4.2 The relevant test time for the individual test items is defined in thedetailed reliability test specification in accordance with 10.5 ofIS : 8161 (Part I)-1976*.4.3 If the validity test is to be carried out immediately after a numberof failures, r, have occurred, that is, coinciding with a failure, the valueof T equals Tr and the parameter d, related to the number of failures,equals r–1. If the validity test does not coincide with a failure, theparameter d equals r.4.4 If the validity test is used in conjunction with a truncatedsequential test covered in IS : 8161 (Part VII)-1977*, the sameaccumulated relevant test times may be used in the validity test.

n = total number of test items;r = total number of failures;

Tk = accumulated relevant test time up to the k : th failure;Tr = accumulated relevant test time up to the latest failure;T = total accumulated relevant test time;d = parameter related to number of failures: if the validity

test is done at a point in time coinciding with a failured = r–1; if not d = r;

χ2 = calculated value of the test statistic;χ2

p ( v ) = theoretical value of the χ2 distribution with v degrees offreedom at the fractile of order p;

u = number of intervals in the large sample test;oi = observed number of failures in the i : th interval;ei = expected number of failures in the i : th interval;

(i = 1, ..........,u); andwi = width of the i : th interval measured in accumulated time.

*Guide for equipment reliability testing:Part I Principles and procedures.Part VII Compliance test plans for failure rate and mean time failures assuming

constant failure rate.

IS : 8161 (Part 6/Sec 1) - 1983

6

5. PREFERRED TEST

5.1 The following test is recommended if the number of failures isbetween 3 and 30; if the number of failures exceeds 30 this test or thetest recommended in 6 which is easier to calculate, may be used.5.1.1 The accumulated relevant test times Tk, k = 1, 2,........, r, and Tare calculated. The following test statistic shall be calculated:

5.1.2 In the case of a constant failure rate, χ2 is distributed as χ2 ( 2d )with 2d degrees of freedom.

5.1.3 The calculated value χ2 of the appropriate test statistic iscompared with the theoretical values of ( v ) given in Table 1. Thetwo-sided test to be performed requires the values of p to be 5 percentand 95 percent for the level of significance of 10 percent. The numberof degrees of freedom v = 2d.

Ifχ2 <χ 2

5 (v)

the assumption of a constant failure rate shall be rejected. The failurerate is likely to be increasing.

Ifχ2 >χ 2

9 5(v)

the assumption of a constant failure rate shall also be rejected. Thefailure rate is likely to be decreasing.

6. ALTERNATIVE TEST FOR A LARGE NUMBER OF FAILURES

6.1 For a large number of failures, at least 30, a χ2 goodness-of-fit testmay be used instead of the test in 5. The test is based on theaccumulated relevant test times described in 4.6.2 The period between time zero and the accumulated time T at thevalidity test is divided into u intervals of width wi which need not allbe equal. The expected number of failures in the i : th intervale = wi d/T shall be equal to or greater than 5 with oi being the observernumber of failures in the i : th interval. The following test statisticshall be calculated:

χ2 2 1η TTk-------

k=1

d

∑=

χ2p

χ2 oi ei–( )2

ei----------------------

i=1

U∑=

IS : 8161 (Part 6/Sec 1) - 1983

7

TABLE 1 χχχχ2 VALUES

( Clauses 5.1.3 and 6.3 )

DEGREES OF FREEDOM ( v )

(v) ( v ) (v)

(1) (2) (3) (4)

123

0.0040.1030.35

2.714.616.25

3.845.997.81

456

0.711.151.64

7.789.24

10.6

9.4911.112.6

789

2.172.733.33

12.013.414.7

14.115.516.9

101112

3.944.575.23

16.017.318.5

18.319.721.0

131415

5.896.577.26

19.821.122.3

22.423.725.0

161718

7.968.679.39

23.524.826.0

26.327.628.9

192021

10.110.911.6

27.228.429.6

30.131.432.7

222324

12.313.113.8

30.832.033.2

33.935.236.4

252627

14.615.416.2

34.435.636.7

37.738.940.1

282930

16.917.718.5

37.939.140.3

41.342.643.8

313233

19.320.120.9

41.442.643.7

45.046.247.4

343536

21.722.523.3

44.946.147.2

48.649.851.0

( Continued )

χ25 χ9 0

2 χ9 52

IS : 8161 (Part 6/Sec 1) - 1983

8

6.3 The calculated value χ2 is compared with the theoretical values ofχ 2

p( v ) given in Table 1. The one-sided test to be performed requiresthe value of p to be 90 percent for the level of significance of 10 percent.The number of degrees of freedom v = u–1.

Ifχ2 > (v)

the assumption of a constant failure rate shall be rejected. In this caseit is not possible to assess whether the failure rate is decreasing orincreasing.

TABLE 1 χχχχ2 VALUES — Contd

DEGREES OF FREEDOM ( v )

(v) ( v ) (v)

(1) (2) (3) (4)

373839

24.124.925.7

48.449.550.7

52.253.454.6

404142

26.527.328.1

51.852.954.1

55.856.958.1

434445

29.029.830.6

55.256.457.5

59.360.561.7

464748

31.432.333.1

58.659.860.9

62.864.065.2

495051

33.934.835.6

62.063.264.3

66.367.568.7

525354

36.437.338.1

65.466.567.7

69.871.072.2

555657

39.039.840.6

68.869.971.0

73.374.575.6

585960

41.542.343.2

72.273.374.4

76.877.979.1

NOTE — For degrees of freedom v>60, use where z is thecorresponding percentage of the standard normal distribution.

χ25 χ9 0

2 χ9 52

χ2p Z 2v 1–+( )2[ ] 2⁄=

χ9 02

IS : 8161 (Part 6/Sec 1) - 1983

9

7. ACTIONS TO BE TAKEN IF THE ASSUMPTION IS REJECTED

7.1 If the assumption of a constant failure rate is rejected by either ofthe validity tests, it may be useful to further analyse the data in orderto determine what caused the rejection and to obtain information forthe judgement of appropriate action to be taken.

7.2 An immediate conclusion from the rejection of a constant failurerate assumption is that the prerequisites for the reliability compliancetests covered in IS : 8161 (Part VII)-1977* are not fulfilled and thatdecisions based on those tests shall be questioned. The same appliesfor those estimates of IS : 8161 (Part IV)* of this standard that arebased on constant failure rate and exponential distribution of times toor between failures. Another more appropriate distributionassumption may be found and the data statistically treatedaccordingly.

7.3 If the validity test shows that the failure rate is likely to bedecreasing, indicating the existence of an early failure period, possibleaction would be to improve the quality control procedures of theequipment production or to institute burn-in of all the equipments.

7.4 If the validity test indicates an increasing failure rate, for example,due to wear-out failures, a possible action is to institute preventivemaintenance by scheduled replacement of wearing parts prior tofailure or to make design changes in order to avoid these parts.

7.5 Changes in design, production or preventive maintenance as wellas introducing burn-in are all actions that have effect on the behaviourof the equipment. After any such action, new compliance,determination and/or validity tests should be performed on themodified equipment.

*Guide for equipment reliability testing:Part VII Compliance test plans for failure rate and mean time failures assuming

constant failure rate.Part IV Procedures for determining point estimates and confidence limits equip-

ment reliability determination tests ( under preparation ).

10

INDIAN STANDARDS

ON

EQUIPMENT RELIABILITY TESTING

IS : 8161 Guide for equipment reliability testing:(Part I)-1976 Principles and proceduresPart II Design for test cycles ( under preparation )Part III Preferred test conditions for equipment

reliability testing ( under consideration )Part IV Procedure for determining estimates and

confidence limits from equipment reliabilitydetermination tests ( under preparation )

(Part V)-1981 Compliance test plans for success ratio(Part VI/Sec 1)-

1983Tests for validity of a constant failure rateassumption Section 1 Chi-square Test

(Part VII)-1977 Compliance test plans for failure rate and meantime between failures assuming constant failurerate

Part VIII Tests for validity of a non-constant failure rateassumption ( under consideration )

Part IX Compliance test plans assuming Weiliulldistribution of times to failure ( underconsideration )

Part X Compliance test plans assuming normaldistribution of times to failures ( underconsideration )

(Part XI)-1983 Flow chart describing preparations for andexecution of reliability tests

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Amendments Issued Since Publication

Amend No. Date of IssueAmd. No. 1 November 1987

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