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है”ह”ह

IS 8161-6-2 (1987): Guide for equipment reliabilitytesting, Part 6: Tests for validity of a constant failurerate assumption, Section 2: Kolmogorov-smirnov test [LITD2: Reliability of Electronic and Electrical Components andEquipment]

IS : 8161 ( Part 6/Set 2 ) - 1987

Indian Standard

GUIDE FOR EQUIPMENT RELIABILITY TESTING

PART 6 TEST FOR VALIDITY OF A CONSTANT FAILURE RATE ASSUMPTION

Section 2 Kolmogorov-Smirnov Test

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

Chairman

PROP S. SAMPATH Members

Representing

Indian Institute of Technology, Kanpur

ADDITIONAL DIRECTOR, STANDARDS (S&T),RDSO

Railway Board ( Ministry of Transport )

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

SHRI S. P. BHATIKAR All India Radio, New Delhi SHRI J. S. SOOD ( Alternate )

SHRI V. C. BHANDARI SHRI I. S. SULAKH ( Alternate j

Instrumentation Ltd, Kota

SHRI CHARANJIT SINGH‘ ’ Hindustan Aeronautics Ltd, Hyderabad SHRI R. PARTHASARATHY ( Alternate )

DR P. K. DUTTA Peico Electronics and Electricals Ltd, Bombay SHRI S. P. GHOKKALINGARAM ( Alternate )

SHRI P. K. JAIN Ministry of Defence, Electronics Components Standardization Organization, Bangalore

SHRI G. S. PAI ( Alternate ) SHRI R. K. JAYADEV Overseas Communicatiun Services, Bombay

SHRI DALJIT SINGH ( Alternate ) SHRI M. S. KAMATH Indian Electrical Manufacturers* Association,

Bombay SHRI T. C. GOSALIA ( Alternate )

SHRI C. KRISHNAMURTHY Bharat Elecronics Ltd, Bangalore SHRI N. DHANSEKHARAN ( Alternate )

SHRI S. P_. KULKARNI Radio Electronic & Television Manufacturers’ Association, Bombay

SHRI S. M. KHURSALE ( Alternate ) ( Continued on page 2 )

@ Copyright 1987 BUREAU OF INDIAN STANDARDS

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

IS : 8161 ( Part 6/Set 2 ) - 1987

( Continuedfrom page 1 )

Members Representing

&RI LAKSHMINARAYANA Indian Telephone Industries Ltd, Bangalore SHRI V. MUTHAIAH ( Alternate )

SHRI K. M. MANKAD Eletronics Corporation of India Ltd, Hyderabad SHRI B. BHASKAR RAO ( AIternute )

BRIG R. K. MEHRA Ministry of Defence ( DGI ) LT-COL S. P. MURGAI (Alternate )

SHRI D. C. M~HTA Directe;;; General of Civil Aviation, New

SHRI R. V. ISRANI ( Alternate ) DR K. B. MISRA In personal capacity ( Department of Electrical

Engineering, Indian Institute of Techno- logy, Kharagpur )

SHRI B. C. MUKHERJEE National Test House, Calcutta SHRI R. N. MUKHERJEE ( Alternate )

SHRI K. R. ANANDAKUMARAN NAIR Lucas-TVS Ltd, Madras SHRI C. RANGANATHAN ( Alternate )

SHRI D. V. PETKAR Bhabha Atomic Research Centre, Trombay SHRI A. K. BABAR ( Alternate )

SHRI K. RAMGOPAL ISRO Satellite Centre ( ISAC ), Bangalore SHRI SIHARAN DE ( Alternate )

SHRI P. K. SHUKLA Centre for Aeronautical Systems Studies and Analysis ( CASSA ), Ministry of Defence

SHRI R. SOMASUNDARAM Directorate of Technical Development and Production (AIR) ( Ministry of Defence ), New Delhi

SHRI R. S. YADAV ( Alternate ) SHRI H. C. TEWARI Central Electricity Authority, New Delhi

SHRI D. P. SINHA ( AIternate ) SHRI T. S. VASUDEVAN Department of Telecommunication, New Delhi

SHRI A. V. KOUNDINYA ( Alternate ) DR R. P. WADHWA Department of Electronics, New Delhi

SHRI E. G. NAGARAJAN ( Alternate ) SHRI N. SRINIVASAN, Director General, BIS ( Ex-ojikio Member )

Director ( Electronics ) ( Secretary )

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

Convener

DR P. K. DUTTA

Members

L-r-COL B. S. JAGGI

Peico Electronics & Electricals Ltd, Bombay

EMEDzkrectorate, Ministry of Defence, New

MAJ A. PUNNUCHAMY ( Alternafe ) SHRI P. K. JAIN Ministry of Defence, Electronics Components

Standardization Organization, Bangalore SHRI G. S. PAI ( Alternate )

SHRI C. KRISHNAMURTHY Bharat Electronics Ltd, Bangalore SHRI N. DHP~NASEKHARAN ( Alternate )

( Continued on page 10 )

2

IS : 8161( PM ~/SW 2 ) 1987

Indian Standard GUIDE FOR

EQUIPMENT RELIABILITY TESTING

PART 6 TEST FOR VALID~ITY OF A CONSTANT FAILURE RATE -ASSUMPTION

Section 2 Kolmogorov-Smirnov Test

0. FOREWORD

0.1 This Indian Standard ( Part 6/Section 2 ) was adopted by the Indian Standards Institution on 19 March 1987, after the draft finalized by the Reliability of Electronics and Electrical Components and Equipment Sectional Committee had been approved by the Electronics and Tele- communication Division Council.

0.2 This standard is one of the series of Indian Standards for equipment reliability testing. This standard gives KolmogorovSmirnov Test for validity of a constant failure rate assumption. To bz able to write a detailed reliability test specification and perform a reliability test, the test engineer will need additional information which are dealt with in detail in other standards in this series. A list of standards envisaged in this series are given on page 11.

0.3 For the purpose of deciding whether a particular requirement of this standard is complied with, the final value, observed or calculated, expressing the result of a test, shall be rounded off in accordance with IS : 2-1960*. The number of significant places retained in the rounded off value should be the same as that of the specified value in this standard.

1. SCOPE

1.1 This standard ( Part 6/Section 2 )’ gives Kolmogorov-Smirnbv ( K-S ) test for testing the statistical validity of the constant failure rate

*Rules for rounding 0-T numerical values ( revised ).

3

IS : 8161 ( Part 6/Set 2 ) - 1987

assumption underlaying the methods used in IS : 8161 ( Part 4 )-1985* and IS : 8161 (Part 7 )-1977’1.

2. KOLMOGOROV-SMIRNOV T-EST

2.1 The validity of many statistical techniques used in the calculation, analysis or prediction of reliability depends upon the distribution of failure times. The techniques are sensitive to the departures from the assumed distributions and may lead to seriously wrong results. There- fore, in order to determine whether or not certain techniques are appli- cable in a particular situation, some judgement is to be made as to the underlying probability distribution of failures ( or failure times ). The Kolmogorov-Smirnov or ‘d-test’ is one of the tests designed for the purpose.

2.2 The test is accomplished by finding the theoretical cumulative fre- quency distribution which would be expected under the null hypothesis [F(X)] and comparing it with the observed cumulative frequency distri- bution [S(X)]. Under the null hypothesis, that the sample has been drawn from the specified theoretical distribution, it is expected that for every value of X, &(X) should be fairly close to F(X), that is, the differences between the theoretical and observed distribution should be small and within the limits of random errors. The point at which these two distributions, theoretical and observed, show the maximum deviation is determined.

Let D = Maximum F(X) - Sri(X) 1

2.3 This value of D is calculated and compared with the critical value given in Table 1 for desired level of significance. The null hypothesis is rejected if the ca!culated value of D is greater than the critical value, otherwise not.

3. APPLICATION

3.1 The application of K-S test is described in Table 2 which was designed to test the hypothesis that a complex system having a mixed type of population of components with continuing failed part replacement pro- gramme follows an exponential distribution. Column 1 gives the data

*Guide for equipment reliability testing: Part 4 Procedure for determining point estimates and confidence limits from equipment reliability determination test.

tGuide for equipment reliability testing: Part 7 Compliance test plans for failure rate and mean time between failures assuming constant failure rate.

4

IS : 8161 ( Part 6/Set 2 ) - 1987

on time to failure of tested systems where in 81 failures were recorded and the same are arranged in 8 intervals. The frequency falling in each interval is indicated in co1 2 and ~01-3 gives the cumulative frequency. The probabilities in co1 4 are obtained after dividing the respective entry in co1 3 by 81. The theoretical distribution of co1 5 is obtained as a standard exponential function with an MTBF of 43’8 days. ( Total accumulated time 3 548 days and failures 81. ) The difference of observed and expected probabilities is shown in co1 5. The maximum difference 0’042 is less than the tabulated entry at 5 percent level of significance. This indicates that there is no significant difference between the actual observed data and the assumed exponential distribution. The example has been worked out in detail in Appendix A.

TABLE 1 CRITICAL VALUES OF D IN THE KOLMOGOROV-SMIRNOV TEST

( Clause 2.3 )

SAMPLE SIZE LEVEL OF SIGNIFICANCE (n) T__---- ‘I

5 Percent 1 Percent

1 0.975 0 995

2 0,842 0.929

3 0.708 0.828

4 0.624 0.733

5 0.565 0.669

6 0,521 0.618

7 0.486 0.517 8 0.457 0.543

9 0.432 0.514

10 0.410 0.490

11 0.391 0.468

12 0.375 0.450

13 0.361 0.433

14 0.349 0.418

15 0.338 0.404

16 0.328 0.392

17 0.318 0.381

18 0.309 0.371

19 0.301 0.363

20 0.294 0.356

25 0.27 0.32

30 0.24 0.29

35 0.23 0.27

1.36 1.63

Over 35 & v:

IS : 8161( Part 6/Set 2 ) - 1987

TABLE 2 EXPONENTIAL FIT ON TIME BETWEEN FAILURES DATA

(Chlse 3.1 )

INTERVAL FREQUENCY CUMULATED OBSERVED EXPECTED DIFFERENCE FREQUENCY PROBABILITY PROBABILITY

(1) (2) (3) (4) (5) (6) 0.5-30.5 40 40 0.494 0’502 0’008

30.5-60.5 24 64 0.79 0.749 0’041

60 5-90.5 9 73 0’901 0’873 0.028 90.5-120.5 4 77 0.951 0’936 0’015

120.5-150.5 1 18 0.963 0.968 0’005 150.5-180.5 0 78 0.963 ~0.984 0.021

180’5-210.5 1 79 0’975 0.992 0’017 2105-240.5 2 81 1 *ooo 0.996 0.004

Maximum of difference between observed and expected probability = 0.041

Critical difference at 5 percent level of significant-e 1.36 =-

l/n = 0.15

4. COMPARISON OF X2 AND K-S TESTS

4.1 The K-S test is more effective when sample size is small and can be used even with too small samples. ( around 30 or more > is required.

In case of X2-test, large sample size

4.2 The K-S test is useful only when the assumed distribution is conti- nuous and the distribution parameters are assumed. It is used when the hypothetical distribution is completely specified giving the form and the numerical values of all the parameters of the distribution.

The X2-test is applicable to either continuous or discrete distribution_ It is applicable where only the form of the distribution is hypothesized and same parameters remain to be estimated from the sample. It also requires assumption that observed frequencies are normally distributed about their expected frequencies.

4.3 The K-S test can be used for un-grouped data where as X2-test can be partitioned and added.

4.4 The K-S test can be used to test deviations in a given direction whereas X2-test canbe used only for a two sided test.

4.5 The K-S test shall be used where there are good reasons to believe that this is more sensitive test than the X2-test.

6

IS : 8161 ( Part 6/Set 2) - 1987

APPENDIX A

( Clause 3.1 )

ILLUSTRATIVE EXAMPLE

A-O. Sixteen comp!ex systems having a mixed type of components popu- lation were tested for 221’75 days continuously. During this period, 81 failures were observed. The failed parts were replaced without any loss of time. A ‘day’ was taken as unit for time and failure times were logged. The grouped failure frequency data is as under:

,

Time Interval in Days

b’5- 30’5 30’S 60’5 60’5- 90’5 90’5-120’5

120’5-150’5 150’5-180’5 180’5-210’5 210’5-240’5

No. of Failures Observed

40 24

9 4 1 0 1 2

The hypothesis that this programme is following an exponential failure distribution is to be verified using K-S test. c

A-l. STEP l- CALCULATION FOR MEAN TIME BETWEEN FAILURE ( MTBF )

A-l.1 Accumulated system days = 3 548 days Number of failures observed = 81

3 548 Therefore, MTBF = 81 = 43’8 days.

A-2. STEP 2 - CALCULATION FOR OBSERVED CUMULATIVE FAILURE FREQUENCY PROBABILITY h(X)

A-2.1 The observed cumulative failure frequency is given below in co1 3 by adding total number of failures up to the end of a given time interval as illustrated.

7

IS : 8161 ( Part 6/Set 2 ) - 1987

[ The failure probability given in co1 4 is generated by dividing each entry of co1 3 by the total number of failures, that is, 8 I: ]

Time Interval Number of Observed Observed in Days Observed CumNlative Cumulative

Failures Failures Failure Frequency Probability

S,(X) (1) (2) (3) (4)

0’5 30’5 40 0 + 40 = 40 0’494 30’5 60’5 24 40 i- 24 = 64 0’790 ~60’5- 90’5 9 64 + 9 = 73 0’901 90’5-120’5 4 73 + 4 = 77 0’951

120’5-150’5 I 505-l 80’5 :,

77 + 1 = 78 0’963 78 + 0 = 78 0’963

180’5-210’5 :

78 + 1 - 79 0’975 210’5-240’5 79+ 2=81 1’000

A-3. STEP 3 - CALCULATION FOR THEORETICAL CUMULAI IVE FAILURE FREQUENCY PROBABILITY F(X)

A-3.1 The following reliability equation describes the relat.ion between MTBF acd time duration during which the reliability is calculated. Also, the probability for given time is probability of success.

R(1) = e - t/MTBF

where R(t) is probability of success for time, t.

Therefore, probability of failure during time, t

F(X) = 1 _ R(t) = ml _ e - t/MTBF

Here, only t is variable as it is assumed that MTBF is constant. Thus, the cumulative failure probability up to the end of each time interval is calculated below using the above relation:

Time Interval in Days

End of lTTme

Interval

t/MTBF F(X)=(l-e-tlMTBF)

0’5-30’5 30’5 30’5

__ = 0’696 43’8

0’502

30’0-60’5 60’5 60’5

---= 1’380 43’8

0’749

60’5-90’5 90’5 90’5 z8= 2’066 0’873

90’5- 120’5 120’5 120.5 43.8= 2’751 0’936

.

8

IS : 8161 ( Part 6/Set 2 ) - 1987

Time Interval End of t/MTBF F(x)=(~-~-‘/MTBF) in Days Time

Interval

120 5-150’5 150’5 ‘z= 3’436 0’968

150’5-180’5 180’5 z= 4’121 0’984

180’5-210’5 210’5 2i:;= 4’806 0’992

210’5-240’5 240’5 s= 5’493 0’996

A-4. STEP 4 - CALCULATION FOR CHECKING THE ABSOLUTE DIFFERENCE IN THE OBSERVED AND THEORETICAL CUMULATIVE FAILURE FREQUENCY AND CONCLUSION

A-4.1 The difference in S,(X) and F(X) as calculated in step 2 and step 3 are given below:

WX) F(X) F(X)---&(X) 0’494 0’502 0’008 0’790 0’749 0’04 1 0’901 0’873 0’028 0’951 0’936 0’015 0’963 0’963 0’005 0’963 0’984 0’021 ‘0’975 0’992 0’017 1’000 0’996 0’004

Here, the largest difference is 0’041. As there is no entry for the sample size of 81 in the given K-S table, the critical difference at 5 per-

1’36 cent level of significance is calculated using the relation -

2/n where

n is the sample size.

Therefore, critical difference at = 1’36 _ 1’36 = o.15 5 percent level 2/81- 9

A-5. CONCLUSION

A-5.1 As the observed difference of 0’041 is less than calculated K-S value of 0’15, it is concluded that the data is exponentially distributed with MTBF = 43’8 days.

9

IS : 8161 ( Part B/Set 2 ) - 1987

( Continued frompage 2 )

Members Representing

SHRIV.NARAYANA Indian Statistical Institute, Calcutta BRIG J. S. RAJU Electronic Regional Test Laboratory ( North )*

New Delhi SHRI S. MITRA ( Alternate )

SHRI K. RAMGOPAL ISRO Satellite Centre ( ISAC ), Bangalore SHRI NARAYAN RAO Ministry of Defence ( DGI ), Bangalore

MAI RAJAN MANNIL ( Alternate ) SHRI T. S. VASUDEVAN TelecorD:pcation Research Centre, New

SHRI K. V. KOUNDINYA ( Alternate )

10

INDIAN STANDARDS ON EQUIPMENT RELIABILITY TESTING

IS : 8161 Guide for equipment reliability testing : ( Part 1 )-1976 Principles and procedures ( Part 2 )-1986 Design for test cycles ( Part 3 ) Preferred test conditions for equipment reliability testing :

Section I-1986 Indoor portable equipment ( low degree of simulation ) Section 2-1986 Equipment for stationary use in weather protected locations

( high degree of simulation ) ( Part 4 )-I985 Procedure for determining point estimates and confidence limits

from equipment reliability determination tests ( Part 5 )-I981 Compliance test plans for success ratio ( Part 6 )-1983 Tests for validity of a constant failure rate assumption

Set 2 Kolmogorov-Smirnov test Set 3 Bartlett’s test ( under consideration )

( Part 7 )-1977 Compliance test plans for failures rate and mean time between failures assuming constant failure rate

( Part 8 ) Tests for the validity of a non-constant failure rate assumption ( under considerution )

( Part 9 ) Compliance test plans assuming Weibull distribution of times of failure ( under consideration )

( Part 10 ) Compliance test plans assuming normal distribution of times to failure ( under consideration )

(Part 11 )-1983 Flow chart describing preparations for and execution of reliabi- lity tests

11

INTERNATIONAL SYSTEM OF UNITS ( SI UNITS 1

Base Units

Quantity

Length

Mass

Time

Electric current

Thermodynamic temperature

Luminous intensity

Amount of substance

Supplementary Units

Quantity

Plane angle

.Solid angle

Derived Units

Quantity

Force

Energy

Power

Flux

Flux density

Frequency

Electric conductance

Electromotive force

Pressure, stress

Unit

metre

kilogram

second

ampere

kelvin

candela

mole

Unit

radian

steradian

Unit

newton

joule

watt

weber

tesla

hertz

siemens

volt

Pascal

Symbol

m

kg

S

A

K

cd

mol

Symbol

rad

sr

Symbol

N

J

W

Wb

T

HZ

S

V

Pa

DeJinition

1 N = 1 kg,m/s*

1J = 1 N,m

1 W - 1 J/s

1 Wb = 1 v.s

1T = 1 Wb/m8

1 Hz = 1 c/s( s-1)

1s = 1 A/V

1v = 1 W/A

1 Pa = 1 N/m2