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SEMI E10-0304 E © SEMI 1986, 2004 13 Table A1-1 1 -Sided Lower Confidence Bound Fac tors for the MTBF p (Time or Cycle Censored Data or F ixed Length Test) Use for time or cycle censored data to multiply the MTBF p or MC…

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SEMI E10-0304
E
© SEMI 1986, 2004 12
APPENDIX 1
CONFIDENCE BOUND FACTORS
NOTICE: This appendix was approved as an official part of SEMI E10 by full letter ballot procedure. It offers
detailed information related to Section 7.
A1-1 Introduction
A1-1.1 E-MTBF
p
may be substituted for MTBF
p
in all
calculations in this section.
A1-1.2 Tables A1-1 and A1-2 contain factors that
multiply an MTBF
p
point estimate to obtain upper and
lower confidence limits. Table A1-1 applies in the
common case where the equipment is observed for a
fixed period of time and the number of failures that will
occur is unknown in advance (time censored data). The
alternative is failure censored data, where the number
of failures is specified in advance and the equipment is
observed until that many failures occur. Table A1-3
contains lower limit factors for failure censored data.
Since failure censored data rarely occurs in tool or
equipment reliability measurement, Table A1-3 is only
included for completeness. The upper limit factors
given in Table A1-2 apply to both kinds of censored
data.
A1-1.3 Table A1-4 can be used to plan equipment
assessment or qualification tests in order to be able to
demonstrate a desired MTBF
p
at a given confidence
level. In order to use Table A1-4, you must first choose
a maximum number of failures, r, you might observe
during the test period and still be able to meet the
required MTBF
p
objective.
A1-1.4 For reference, here are the formulas for the
lower and upper confidence limit factors for time
censored data found in Tables A1-1 and A1-2:
MTBF
L
OWER
=
2
r
X
2r
+
2;1
−
α
2
×
MTBF
p
where r
=
# of
f
ailures
MTBF
UPPER
=
2r
X
2r;
α
2
×
MTBF
p
A1-1.5 In both cases, the confidence level is 100 × (1 –
α) that the true MTBF
p
is above MTBF
LOWER
and below
MTBF
UPPER
and chi square distribution tables are used.
A1-1.6 For 0 fails, use:
MTBF
LOWER
=
productive time
-log e
α
A1-1.7 Factors to use when there are 0 failures based
on this formula are given in the first row of Table A1-1.
A1-1.8 For failure censored data, MTBF
UPPER
is the
same, but the lower limit factor in Table A1-3 is:
M
TB
F
L
OWER
=
2
r
X
2
r
;
1-
α
2
×
M
TB
F
p
SEMI E10-0304
E
© SEMI 1986, 2004 13
Table A1-1 1-Sided Lower Confidence Bound Factors for the MTBF
p
(Time or Cycle Censored Data or Fixed
Length Test)
Use for time or cycle censored data to multiply the MTBF
p
or MCBF estimate to obtain a lower bound at the given confidence level.
For 0 failures, multiply the operating hours or cycles by the factor corresponding to the desired confidence level.
CONFIDENCE LEVEL
# FAILS
r
60% 70% 80% 85% 90% 95% 97.5%
0 1.091 0.831 0.621 0.527 0.434 0.334 0.271
1 0.494 0.410 0.334 0.297 0.257 0.211 0.179
2 0.644 0.553 0.467 0.423 0.376 0.318 0.277
3 0.718 0.630 0.544 0.499 0.449 0.387 0.342
4 0.763 0.679 0.595 0.550 0.500 0.437 0.391
5 0.795 0.714 0.632 0.589 0.539 0.476 0.429
6 0.817 0.740 0.661 0.618 0.570 0.507 0.459
7 0.834 0.760 0.684 0.642 0.595 0.532 0.485
8 0.848 0.777 0.703 0.662 0.616 0.554 0.508
9 0.859 0.790 0.719 0.679 0.634 0.573 0.527
10 0.868 0.802 0.733 0.694 0.649 0.590 0.544
12 0.883 0.821 0.755 0.718 0.675 0.617 0.572
15 0.899 0.841 0.780 0.745 0.704 0.649 0.606
20 0.916 0.864 0.809 0.777 0.739 0.688 0.647
30 0.935 0.892 0.844 0.816 0.783 0.737 0.700
50 0.953 0.918 0.879 0.856 0.829 0.790 0.759
100 0.969 0.943 0.915 0.897 0.877 0.847 0.822
500 0.987 0.976 0.962 0.954 0.944 0.929 0.916
Table A1-2 1-Sided Upper Confidence Bound Factors for the MTBF
p
Use to multiply the MTBF
p
estimate to obtain an upper bound at the given confidence level (time censored or failure censored data).
CONFIDENCE LEVEL
# FAILS
r
60% 70% 80% 85% 90% 95% 97.5%
1 1.958 2.804 4.481 6.153 9.491 19.496 39.498
2 1.453 1.823 2.426 2.927 3.761 5.628 8.257
3 1.313 1.568 1.954 2.255 2.722 3.669 4.849
4 1.246 1.447 1.742 1.962 2.293 2.928 3.670
5 1.205 1.376 1.618 1.795 2.055 2.538 3.080
6 1.179 1.328 1.537 1.687 1.904 2.296 2.725
7 1.159 1.294 1.479 1.610 1.797 2.131 2.487
8 1.144 1.267 1.435 1.552 1.718 2.010 2.316
9 1.133 1.247 1.400 1.507 1.657 1.917 2.187
10 1.123 1.230 1.372 1.470 1.607 1.843 2.085
12 1.108 1.203 1.329 1.414 1.533 1.733 1.935
15 1.093 1.176 1.284 1.357 1.456 1.622 1.787
20 1.077 1.147 1.237 1.296 1.377 1.509 1.637
30 1.060 1.115 1.185 1.231 1.291 1.389 1.482
50 1.044 1.085 1.137 1.170 1.214 1.283 1.347
100 1.029 1.058 1.093 1.115 1.144 1.189 1.229
500 1.012 1.025 1.039 1.049 1.060 1.078 1.094
SEMI E10-0304
E
© SEMI 1986, 2004 14
Table A1-3 1-Sided Lower Confidence Bound Factors for the MTBF
p
(Failure Censored Data)
Use for failure censored data to multiply the MTBF
p
estimate to obtain a lower bound at the given confidence level. Failure censored data means
the test or observation period lasts as long as needed to obtain a preset number of failures.
CONFIDENCE LEVEL
# FAILS
r
60% 70% 80% 85% 90% 95% 97.5%
1 1.091 0.831 0.621 0.527 0.434 0.334 0.271
2 0.989 0.820 0.668 0.593 0.514 0.422 0.359
3 0.966 0.830 0.701 0.635 0.564 0.477 0.415
4 0.958 0.840 0.725 0.665 0.599 0.516 0.456
5 0.955 0.849 0.744 0.688 0.626 0.546 0.488
6 0.954 0.856 0.759 0.706 0.647 0.571 0.514
7 0.953 0.863 0.771 0.721 0.665 0.591 0.536
8 0.954 0.869 0.782 0.734 0.680 0.608 0.555
9 0.954 0.874 0.791 0.745 0.693 0.623 0.571
10 0.955 0.878 0.799 0.755 0.704 0.637 0.585
12 0.956 0.886 0.812 0.771 0.723 0.659 0.610
15 0.958 0.895 0.828 0.790 0.745 0.685 0.639
20 0.961 0.906 0.846 0.812 0.772 0.717 0.674
30 0.966 0.920 0.870 0.841 0.806 0.759 0.720
50 0.971 0.935 0.896 0.872 0.844 0.804 0.772
100 0.978 0.952 0.923 0.906 0.885 0.855 0.830
500 0.989 0.978 0.964 0.956 0.945 0.930 0.918
Table A1-4 Test Length Guide
Use to determine the test time needed to demonstrate a desired MTBF
p
at a given confidence level if r failures occur. Multiply the desired MTBF
p
by the k factor corresponding to r and the confidence level.
k FACTOR FOR GIVEN CONFIDENCE LEVELS
# FAILS
r
50% 60% 75% 80% 90% 95%
0 0.693 0.916 1.39 1.61 2.30 3.00
1 1.68 2.02 2.69 2.99 3.89 4.74
2 2.67 3.11 3.92 4.28 5.32 6.30
3 3.67 4.18 5.11 5.52 6.68 7.75
4 4.67 5.24 6.27 6.72 7.99 9.15
5 5.67 6.29 7.42 7.90 9.28 10.51
6 6.67 7.35 8.56 9.07 10.53 11.84
7 7.67 8.38 9.68 10.23 11.77 13.15
8 8.67 9.43 10.80 11.38 13.00 14.43
9 9.67 10.48 11.91 12.52 14.21 15.70
10 10.67 11.52 13.02 13.65 15.40 16.96
15 15.67 16.69 18.48 19.23 21.29 23.10
20 20.68 21.84 23.88 24.73 29.06 30.89