14: Analysis of Test Data
“Testing leads to failure, and failure
leads to understanding."
Definitions
• Failure rate = the number of failures per million
hours of operation λ =1/MTBF
• Failure: the state of inability of an item to
perform its required function
• Reliability = the probability that an item will
perform a required function, under specified
conditions, for a specified period of time, at a
desired confidence level. (MTBF or Failure rate, λ)
• Minimum life = time to first failure
Definitions, Continued
• Confidence level = probability that a given
statement is correct
• Confidence limits = the extremes of a
confidence interval within which the unknown
has a designated probability of being included
• The minimum life of a device = the time of
occurrence of the first failure.
• Pareto analysis = plot of individual failures
versus the frequency of the failures.
Example 1: MTBF
• An EEG machine has a MTBF of 4380 hours.
What is the failure rate?
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λ = 1/MTBF
λ = 1/4380 failures per hour
λ = 0.000228 failures per hour
λ = 228 failures per million hours
Example 2: MTBF
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10 power supplies are put on test, to be terminated after each
has completed 1000 hours of operation. Two power supplies fail,
one at 420 hours and the other at 665 hours. What is the failure
rate of the power supplies?
• Eight units completed 1000 hours.
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Total test time = 8(1000) + 420 + 665 = 9085 hours
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λ = number of failures/total test time
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λ = 2/9085 = 0.000220 failures per hour
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λ = 220 failures per million hours
General MTBF Tests in practice:
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Time terminated, failed parts replaced
Time terminated, no replacement (last…)
Failure terminated, failed parts replaced
Failure terminated, no replacement (easy)
No failures observed during the test
…SOME EXAMPLES FOLLOW
Time terminated, failed parts replaced:
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MTBF = N(td)/r
where
N = number of units tested
td = test duration
r = number of failures
…continued
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The performance of ten pressure monitors is monitored while
operating for a period of 1200 hours. The test results are listed below.
Every failed unit is replaced immediately. What is the MTBF?
Unit Number
Time of Failure (hours)
#failures
1
650
1
2
420
1
3
130 and 725
2
4
585
1
5
630 and 950
2
6
390
1
7
No Failure
0
8
880
1
9
No Failure
0
10
220 and 675
2
11
finished …
• N = 10 sets of tests
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r = 11 failues
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td = 1200 hours (test duration)
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MTBF = N(td)/r
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= 10(1200)/11
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= 1091 hours
Failure Terminated, Failed Parts
Replaced
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Six TENS units were placed on test until all units failed, the last
occurring at 850 hours. The test results are listed below. Every
failed unit, except the last one, is replaced immediately. What is
the MTBF?
Unit Number Time of Failure (hours)
Failures
1
130
1
2
850
1
3
120 and 655
2
4
440
1
5
725
1
6
580
1
MTBF = N(td)/r
= 6(850)/7 = 729 hours
No failures observed during the test
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For the case where no failures are observed, an MTBF
value cannot be calculated. A lower one-sided confidence
limit must be calculated and the MTBF stated to be greater
than that value.
ml = 2(Ta)/Χ2 α;2
where
ml = lower one-sided confidence limit
Ta = total test time
Χ2 α;2= the chi-square value from the table in Appendix
1, where α is the risk level and 2 is the # of
degrees of freedom
continued …
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10 ventilators are tested for 1000 hours without failure. What is the
MTBF at a 90% confidence level?
N = 10 test units
td = 1000 = test duration
r = 0 = failures
1 - % = 0.90 = confidence level
% = 0.10 = risk level
Ta = N(td) = 10(1000) = 10000 = total test time
ml = 2(Ta)/Χ2 α;2
= 2(10000)/Χ210;2
= 20000/4.605
= 4343 hours = lower 1 sided confidence limit…
We can then state that the MTBF > 4343 hours, with 90% confidence.
Reliability
• Reliability = exp(- λt) = exp (-t/MTBF)
• Example: test time =3200 hours, λ = 220 failures/
million hours
• Reliability = exp(-3200*220/1000000)
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= exp(-.704) = .495
• or ~ half will fail in 3200 hours….
Confidence Limits
• Time Terminated Confidence Limits
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mL = 2(Ta)/Χ2α/2;2r+2
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where
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mL = lower confidence limit
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Ta = total test time
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Χ2α/2;2r+2 = Chi square value from Appendix 1
for α risk level α and 2r+2 degrees of freedom
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mU = 2(Ta)/Χ21-α/2;2r
Confidence Limits, continued
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mL = 2(Ta)/Χ2α/2;2r
and
mU = 2(Ta)/Χ21-α/2;2r
• … left to the reader -
Pareto Analysis
Field Data Plots:
Final comments
The data examples given are for estimate
purposes only, most of the reliability estimates
will likely be slightly on the low side, which
can be a good thing…