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RELIABILITY & LIFE TESTING HANDBOOK
VOLUME 1
TABLE OF CONTENTS
CHAPTER 1- OBJECTIVES, TYPES, SCHEDULING AND
MANAGEMENT OF RELIABILITY AND
LIFE TESTING ..................................................................................................... 1
1.1- OBJECTIVES OF RELIABILITY AND LIFE TESTING .................................................. 1
1.2- WHY RELIABILITY AND LIFE TESTING? .................................................................... 4
1.3- TEST TYPES, THEIR APPLICATIONS AND BENEFITS ............................................... 4
1.4- TEST PLANNING, SCHEDULING AND DOCUMENTATION ...................................... 7
CHAPTER 2- RELIABILITY AND LIFE DATATHEIR ACQUISITION AND PROCESSING ................................................ 21
2.1- OBJECTIVES ..................................................................................................................... 21
2.2- RELIABILITY AND LIFE DATA NEEDED .................................................................... 21
2.3- RELIABILITY AND LIFE DATA ACQUISITION FORMS ........................................... 22
2.4- FINAL FORMAT OF DATA ............................................................................................. 30
2.5- FINAL REDUCED DATA - RESULTS ............................................................................ 30
2.6- RELIABILITY DATA SOURCES ..................................................................................... 35
2.7- RELIABILITY DATA ANALYSIS, FEEDBACK,
AND CORRECTIVE ACTION SYSTEM ......................................................................... 36
CHAPTER 3- FIVE VERY IMPORTANT ANALYTICAL
FUNCTIONS IN RELIABILITY AND LIFE
TESTING ........................................................................................................... 49
3.1- THE FIVE FUNCTIONS ................................ .………………………………………….. 49
3.2-THE FAILURE PROBABILITY DENSITY FUNCTION ...................................................50
3.3-THE RELIABILITY FUNCTION ........................................................................................58
3.4- THE CONDITIONAL RELIABILITY FUNCTION ...........................................................62
3.5- THE FAILURE RATE FUNCTION ....................................................................................64
3.6-THE MEAN LIFE FUNCTION ............................................................................................69
3.7-ADDITIONAL COVERAGE ................................................................................................73
CHAPTER 4- FAILURE FREQUENCY, FAILURE RATE AND
RELIABILITY DETERMINATION FROM FIELD
DATA ................................................................................................................... 81
4.1- SIMPLE DATA ANALYSIS ........................................................................................... 81
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4.2- EARLY FAILURE CHARACTERISTICS ..................................................................... 88
4.3- USEFUL LIFE CHARACTERISTICS ............................................................................ 88
4.4- WEAROUT LIFE CHARACTERISTICS ....................................................................... 93
4.5- MORE SOPHISTICATED DATA ANALYSES ............................................................. 93
CHAPTER 5- CHI-SQUARE, STUDENT'S t AND
F DISTRIBUTIONS ......................................................................................... 105
5.1- THE CHI-SQUARE DISTRIBUTION ............................................................................. 105
5.2- THE STUDENT'S t DISTRIBUTION .............................................................................. 116
5.3- THE F DISTRIBUTION ................................................................................................... 123
CHAPTER 6- THE EXPONENTIAL DISTRIBUTION ....................................................... 133
6.1-EXPONENTIAL DISTRIBUTION CHARACTERISTICS .............................................. 133
6.2- EXPONENTIAL RELIABILITY CHARACTERISTICS ................................................ 143
6.3- EXPONENTIAL FAILURE RATE AND MEAN-TIMEBETWEEN-FAILURES CHARACTERISTICS ............................................................. 148
6.4- DETERMINATION OF THE EXPONENTIAL
FAILURE RATE AND MTBF FROM
INDIVIDUAL TIME-TO-FAILURE DATA ................................................................... 150
6.5- DETERMINATION OF THE EXPONENTIAL
FAILURE RATE AND MTBF BY
PROBABILITY PLOTTING ............................................................................................ 151
6.6 -A BETTER ESTIMATE OF
RELIABILITY ................................................................................................................. 156
CHAPTER 7- MTBF CONFIDENCE LIMITS, TEST TIME
AND OC CURVES FOR THE EXPONENTIAL
CASE .................................................................................................................. 173
7.1-CONFIDENCE LEVEL AND CONFIDENCE LIMITS .................................................. 173
7.2-DETERMINATION OF THE CONFIDENCE LIMITS ON m …………………………..174
7.3-ONE-SIDED CONFIDENCE LIMITS ON m IN A
FAILURE TERMINATED TEST .................................................................................... 184
7.4-TWO-SIDED AND ONE-SIDED CONFIDENCE LIMITS
ON m IN A TIME TERMINATED TEST ........................................................................ 186
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7.5- DETERMINATION OF THE CONFIDENCE LIMITS ON
THE RELIABILITY FOR THE EXPONENTIAL CASE .................................................. 192
7.6- MTBF AND RELIABILITY DETERMINATION WHEN NO
FAILURES ARE OBSERVED IN A TEST ....................................................................... 194
7.7- COMPARISON OF THE LOWER, ONE-SIDED
CONFIDENCE LIMIT ON m WHEN ONE FAILURE
OCCURS IN A FAILURE TERMINATED TEST WITH THE
CASE WHEN NO FAILURES OCCUR IN A TIME
TERMINATED TEST ........................................................................................................ 196
7.8- EFFECT OF CONFIDENCE LEVEL, SAMPLE SIZE, AND
OCCURRENCE OF FAILURES ON THE CONFIDENCE
LIMITS ON m .................................................................................................................... 197
7.9- THE SKEWNESS OF THE DISTRIBUTION OF TIIE
MEAN TIME BETWEEN FAILURES .............................................................................. 198
7.10-TEST DURATION AND SAMPLE SIZE FOR VARIOUS
TEST TYPES AND CONFIDENCE LEVELS .................................................................. 198
7.11-EXPECTED WAITING TIMES FOR RELIABILITY TESTS
IN THE EXPONENTIAL CASE ........................................................................................ 207
7.12-EXPECTED NUMBER OF FAILURES FOR RELIABILITY
TESTS IN THE EXPONENTIAL CASE ........................................................................... 213
7.13-OPERATING CHARACTERISTIC CURVES
FOR RELIABILITY TESTS IN THE
EXPONENTIAL CASE ...................................................................................................... 217
7.14-BINOMIAL RELIABILITY DEMONSTRATION TEST
WHEN THE TEST TIME IS DIFFERENT THAN
THE MISSION TIME, FOR THE EXPONENTIAL CASE ............................................. 223
CHAPTER 8- TESTS OF COMPARISON OF THE MEAN LIFE
FOR THE EXPONENTIAL CASE ............................................................... 255
8.1- INTRODUCTION ............................................................................................................ 255
8.2- TEST PROCEDURE ........................................................................................................ 256
8.3- CHOICE OF THE SIGNIFICANCE LEVEL ................................................................... 263
8.4- QUANTITATIVE BASES FOR SELECTING A MANUFACTURER…………………264
CHAPTER 9- THE NORMAL DISTRIBUTION .................................................................. 269
9.1- NORMAL DISTRIBUTION CHARACTERISTICS ...................................................... 269
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9.2- COMPUTATIONAL METHODS FOR THE
DETERMINATION OF THE PARAMETERS OF THE
NORMAL DISTRIBUTION .......................................................................................... 273
9.3- DETERMINATION OF THE PARAMETERS
OF THE NORMAL DISTRIBUTION BY
PROBABILITY PLOTTING .......................................................................................... 282
9.4- NORMAL RELIABILITY CHARACTERISTICS ......................................................... 288
9.5- NORMAL FAILURE RATE CHARACTERISTICS ...................................................... 294
CHAPTER 10- CONFIDENCE INTERVAL ON THE MEAN LIFE
AND ON THE RELIABILITY OF NORMALLY
DISTRIBUTED DATA .................................................................................. 303
10.1-TWO-SIDED CONFIDENCE LIMITS ON THE MEAN
LIFE WHEN  T IS KNOWN, OR N  25 ..................................................................... 303
10.2-ONE-SIDED CONFIDENCE LIMITS ON THE MEAN
LIFE WHEN  T IS KNOWN, OR N  25 ..................................................................... 308

10.3- CONFIDENCE LIMITS ON T WHEN  T IS
UNKNOWN, OR N < 25 ............................................................................................... 311
10.4- RANGE OF LIFE ABOUT THE MEAN LIFE WITH A GIVEN
PROBABILITY WHEN  T IS KNOWN, OR N  25 ................................................... 318
10.5- RANGE OF LIFE ABOUT THE MEAN LIFE FOR A GIVEN
PROBABILITY WHEN  T IS UNKNOWN, OR N < 25 ............................................ 320
10.6- TOLERANCE LIMITS ON LIFE .................................................................................. 322
10.7- LOWER, ONE-SIDED CONFIDENCE LIMIT ON THE
RELIABILITY FOR THE NORMAL pdf CASE ......................................................... 326
CHAPTER 11- THE LOGNORMAL DISTRIBUTION ....................................................... 333
11.1-LOGNORMAL DISTRIBUTION CHARACTERISTICS .............................................. 333
11.2- PROBABILITY PLOTTING OF THE LOGNORMAL
DISTRIBUTION ........................................................................................................... 347
11.3- LOGNORMAL RELIABILITY CHARACTERISTICS ................................................ 353
11.4- LOGNORMAL FAILURE RATE CHARACTERISTICS ............................................. 357
11.5- APPLICATIONS OF THE LOGNORMAL
DISTRIBUTION ............................................................................................................. 361
11.6- PHENOMENOLOGICAL CONSIDERATIONS FOR
USING THE LOGNORMAL DISTRIBUTION ......................................................... 361
CHAPTER 12- THE WEIBULL DISTRIBUTION ............................................................... 373
12.1- WEIBULL DISTRIBUTION CHARACTERISTICS .................................................... 373
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12.2- WEIBULL RELIABILITY CHARACTERISTICS ....................................................... 381
12.3-WEIBULL FAILURE RATE CHARACTERISTICS .................................................... 383
12.4- ESTIMATION OF THE PARAMETERS OF THE
WEIBULL DISTRIBUTION BY PROBABILITY
PLOTTING………………………… ............................................................................ 384
12.5- CONSTRUCTION OF THE WEIBULL
PROBABILITY PAPER ................................................................................................ 413
12.6- PROBABILITY OF PASSING A RELIABILITY TEST .............................................. 415
12.7- APPLICATIONS OF THE WEIBULL DISTRIBUTION ............................................. 416
12.8- PHENOMENOLOGICAL CONSIDERATIONS FOR
USING THE WEIBULL DISTRIBUTION ................................................................... 416
12.9- ANALYSIS OF PROBABILITY PLOTS AND
CHOOSING THE RIGIIT PROBABILITY PLOTTING
PAPER ........................................................................................................................... 416
12.10-THE LEAST SQUARES, MATCHING
MOMENTS AND MAXIMUM LIKELIHOOD ESTIMATORS
METHODS OF DETERMINING THE PARAMETERS OF
THE TWO-PARAMETER WEIBULL DISTRIBUTION ........................................... 416
12.11-THE LINEAR ESTIMATION OF THE PARAMETERS
OF THE WEIBULL DISTRIBUTION - RELATIONSHIP
BETWEEN THE WEIBULL AND THE EXTREME
VALUE DISTRIBUTION ............................................................................................ 443
12.12-THE GENERAL LINEAR MODEL OF THE
ORDER STATISTICS ................................................................................................ 444
12.13-THE BEST LINEAR UNBIASED ESTIMATE
(BLUE) OF  AND  ............................................................................................... 445
12.14-ESTIMATING THE WEIBULL PARAMETERS'
BLUE’S ....................................................................................................................... 447
12.15-THE BEST LINEAR INVARIANT ESTIMATE
(BLIE) OF  AND  .................................................................................................. 448
CHAPTER 13- CONFIDENCE LIMITS ON THE RELIABILITY
WITH WEIBULL DISTRIBUTED TIMES TO
FAILURE, AS WELL AS ON THE MEAN
LIFE, MISSION DURATION,  AND  ................................................... 465
13.1- PROCEDURE ............................................................................................................... 465
13.2- MISSION DURATION FOR GIVEN RELIABILITY
AND CONFIDENCE LEVEL ...................................................................................... 469
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13.3- MISSION RANGE FOR A GIVEN CONFIDENCE
LEVEL AND RELIABILITY ..................................................................................... 469
13.4- THE TRUE WEIBULL SLOPE,  ............................................................................... 470
13.5-THE WEIBULL POPULATION MEAN AND ITS
CONFIDENCE LIMITS ................................................................................................ 473
13.6- CONFIDENCE INTERVALS ON THE PARAMETERS
OF THE WEIBULL DISTRIBUTION .......................................................................... 476
CHAPTER 14- RANKS NOT AVAILABLE IN TABLES ................................................... 517
14.1- COMPUTATION OF ANY RANK ................................................................................ 517
14.2- ALTERNATE METHODS FOR THE COMPUTATION OF
THE MEDIAN RANK .................................................................................................. 519
CHAPTER 15- TESTS OF COMPARISON FOR THE WEIBULL
DISTRIBUTION ............................................................................................. 525
15.1- APPLICABILITY ........................................................................................................... 525
I5.2- METHODOLOGY AT THE MEAN LIFE ..................................................................... 525
15.3- TEST OF COMPARISON AT THE MEAN LIFE
USING A NOMOGRAPH ............................................................................................. 531
15.4- TEST OF COMPARISON AT THE MEAN LIFE WHEN
THE  ’S ARE UNEQUAL ......................................................................................... 535
15.5- TEST OF COMPARISON AT THE 10% FAILED LEVEL .......................................... 540
15.6- TEST OF COMPARISON AT THE 10% FAILED LEVEL
USING A NOMOGRAPH .............................................................................................. 542
15.7- CONSTRUCTION OF THE NOMOGRAPH GIVEN IN
FIG. 15.6a ...................................................................................................................... 545
15.8- EQUATION FOR CONSTRUCTING THE NOMOGRAPH
GIVEN IN FIG. 15.13 ................................................................................................... 555
CHAPTER 16- THE GAMMA DISTRIBUTION ................................................................. 565
16.1- GAMMA DISTRIBUTION CHARACTERISTICS ....................................................... 565
16.2- ESTIMATION OF THE PARAMETERS OF THE GAMMA
DISTRIBUTION BY PROBABILITY PLOTTING ..................................................... 571
16.3- GAMMA RELIABILITY CHARACTERISTICS ......................................................... 574
16.4- GAMMA DISTRIBUTION FAILURE RATE
CHARACTERISTICS ................................................................................................... 586
16.5- APPLICATIONS OF THE GAMMA DISTRIBUTION ............................................... 588
16.6- PHENOMENOLOGICAL CONSIDERATIONS FOR
USING THE GAMMA DISTRIBUTION ..................................................................... 588
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CHAPTER 17- THE BETA DISTRIBUTION ....................................................................... 593
17.1-BETA DISTRIBUTION'S CHARACTERISTICS .......................................................... 593
17.2- BETA DISTRIBUTION'S RELIABILITY
CHARACTERISTICS ................................................................................................... 598
17.3- BETA DISTRIBUTION'S FAILURE RATE
CHARACTERISTICS ................................................................................................... 600
17.4- ESTIMATION OF THE PARAMETERS OF THE BETA
DISTRIBUTION ........................................................................................................... 601
17.5- APPLICATIONS OF THE BETA DISTRIBUTION ..................................................... 607
CHAPTER 18-METHODS OF PARAMETER ESTIMATION .......................................... 613
18.1- INTRODUCTION .......................................................................................................... 613
18.2- IDENTIFICATION OF DATA OUTLIERS .................................................................. 616
18.3- LEAST SQUARES METHOD OF PARAMETER
ESTIMATION ................................................................................................................ 633
18.4- MATCHING MOMENTS METHOD OF PARAMETER
ESTIMATION ................................................................................................................ 647
18.5- MAXIMUM LIKELIHOOD METHOD OF PARAMETER
ESTIMATION ............................................................................................................... 653
18.6- MODIFIED MOMENTS METHOD OF PARAMETER
ESTIMATION ............................................................................................................... 658
CHAPTER 19- CHI-SQUARED GOODNESS-OF-FIT TEST ............................................ 679
19.1- WHEN AND HOW TO APPLY .................................................................................... 679
19.2- COMMENTS ON CHI-SQUARED GOODNESS-OF-FIT
TEST .............................................................................................................................. 689
CHAPTER 20- KOLMOGOROV-SMIRNOV GOODNESSOF-FIT TEST .................................................................................................. 697
20.1- INTRODUCTION .......................................................................................................... 697
20.2- STEP-BY-STEP METHOD FOR THE STANDARD K-S
GOODNESS-OF-FIT TEST WITH THE DISTRIBUTION
PARAMETERS KNOWN ............................................................................................. 698
20.3- THE REFINED K-S GOODNESS-OF-FIT TEST FOR
CONTINUOUS DISTRIBUTIONS WITH
PARAMETERS KNOWN ............................................................................................. 709
20.4- MODIFIED K-S GOODNESS-OF-FIT TEST
FOR CONTINUOUS DISTRIBUTIONS WITH
PARAMETERS UNKNOWN ......................................................................................... 725
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20.5- COMMENTS ON THE KOLMOGOROV - SMIRNOV
GOODNESS-OF-FIT TEST ............................................................................................ 729
CHAPTER 21- ANDERSON - DARLING AND CRAMER VON MISES GOODNESS-OF-FIT TESTS ................................................. 739
21.1-INTRODUCTION ............................................................................................................ 739
21.2-THE STANDARD ANDERSON - DARLING AND
CRAMER - VON MISES GOODNESS-OF-FIT TESTS
FOR CONTINUOUS DISTRIBUTIONS WITH
PARAMETERS KNOWN ............................................................................................... 740
21.3-STEP-BY-STEP METHOD FOR THE STANDARD
ANDERSON - DARLING AND CRAMER - VON MISES
GOODNESS-OF-FIT TESTS WITH THE
PARAMETERS KNOWN ............................................................................................... 742
21.4-MODIFIED AD AND CvM GOODNESSOF-FIT TEST FOR CONTINUOUS DISTRIBUTIONS
WITH PARAMETERS UNKNOWN .............................................................................. 772
21.5-COMMENTS ON THE ANDERSON DARLING AND CRAMER - VON MISES
GOODNESS-OF-FIT TESTS .......................................................................................... 784
21.6-RECOMMENDATIONS FOR THE SELECTION
OF GOODNESS-OF-FIT TESTS .................................................................................... 784
APPENDICES ............................................................................................................................ 789
APPENDIX A- RANK TABLES ............................................................................................... 790
APPENDIX B- STANDARDIZED NORMAL DISTRIBUTION'S
AREA TABLES ............................................................................................... 804
APPENDIX C- STANDARDIZED NORMAL DISTRIBUTION'S
ORDINATE VALUES, OR PROBABILITY
DENSITIES ...................................................................................................... 812
APPENDIX D-CHI-SQUARE DISTRIBUTION'S PERCENTILE
VALUES ........................................................................................................... 813
APPENDIX E- STUDENT'S t DISTRIBUTION'S PERCENTILE
VALUES ........................................................................................................... 817
APPENDIX F- F DISTRIBUTION'S PERCENTAGE POINTS ............................................... 818
APPENDIX G- KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT
TEST CRITICAL VALUE TABLES ................................................................ 837
APPENDIX H-TWO-SIDED TOLERANCE FACTORS
FOR NORMAL DISTRIBUTIONS ................................................................. 847
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APPENDIX I- ONE-SIDED TOLERANCE FACTORS
FOR NORMAL DISTRIBUTIONS ................................................................. 849
APPENDIX J- ONE-SIDED TOLERANCE FACTORS
FOR NORMAL DISTRIBUTIONS WITH
50% PROBABILITY OF INCLUSION ........................................................... 851
APPENDIX K- COEFFICIENTS OF THE BLUES FOR THE
EXTREME VALUE DISTRIBUTION ............................................................ 852
APPENDIX L- WEIGHTS FOR OBTAINING THE BEST
LINEAR INVARIANT ESTIMATES OF
THE PARAMETERS OF THE
WEIBULL DISTRIBUTION ............................................................................ 856
APPENDIX M- BINOMIAL OR ATTRIBUTE RELIABILITY
TESTING CURVES FOR CONFIDENCE LEVELS
OF 50%, 80%, 90%, 95% and 99% .................................................................. 896
INDEX ....................................................................................................................................... 901
ABOUT THE AUTHOR ............................................................................................................ 914
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