Practical Use of Failure Rates and Mean Time to Failure Data (2)

1 By John Forsberg, Spectrum Quality, Copyright 2013
Spectrum Quality
Practical Use of Failure Rates and
Mean Time to Failure Data
Based on Field or Accelerated Life Test Data and the use of the Chi
Square Distribution for the Calculation of Confidence Limits
[Basic Reliability Applications]

Contents
Section 1.1 Field Failure Rates............................................................................3
Section 1.2 Estimating Field Failure Rate and MTBF based on Field %/Month .7
Section 1.3 Accelerated Life Tests ......................................................................7
What Are Accelerated Life Tests: .............................................................................................................................. 7
Why Accelerated Life Testing:................................................................................................................................... 7
Several areas where Accelerated Life Testing can be effective: ................................................................................ 8
Field Equivalent Hours............................................................................................................................................... 8
Accelerated Life Test Example: ................................................................................................................................. 8
Section 2.1 Chi Square and the MTBF Confidence Limit Formulas ...................9
Section 2.2 Comparing Different Products, Vintages or Competitive Products.10
Data 1, Comparison of Product Vintages: .......................................................................11
Graph 1, Comparison of Product Vintages:.....................................................................11
Section 2.3 One Sided Confidence Limits for MTBF........................................12
Graph 2.............................................................................................................................13
Section 2.4 Failure Rate Confidence Limits, Two Sided...................................14
Tables 1 through 4, Chi Square Values and Chi Square Factors for MTBF
Limits, for both two sided and one sided:..........................................................15

Practical Use of Failure Rates and Mean Time to Failure Data
Section 1.1 Field Failure Rates
Many companies collect actual Field Failure Data on their products in use by Customers typically within a
warranty period, in many cases a 1 year (12 month) period. This method can be inaccurate. Companies
with effective field failure tracking systems are able to record the exact number of failures for a specific
model and manufacturing date code. This is greatly helped through the use of a physical or electronic serial
number with the month and date of manufacture on the product. If products remain in storage or
transportation before use, this delay also needs to be accounted for. If records are maintained, a Failure rate
can be calculated by counting the failures against items shipped for a specific month/year date code.
An effective method is to count all failed units manufactured for the previous 12 months that occurred in
the most recent month period, against the same 12 month rolling count of units shipped. Failure rates are
then calculated.
To illustrate with a simplified example, say a company’s model started shipping in January of the previous
year and 1,000 units shipped each month. Units were date coded using a serial number system. Estimates
of the failure rate is 0.7%/month based on similar products already in use. So around 7 units per month will
be returned under warranty for each 1,000 units shipped. For the year ending in December, a cumulative
12,000 units were shipped with manufacturing date codes of January through December. In the following
January, 84 units were confirmed failures in the month of December, with date codes from the previous 12
months.
A snapshot of the initial month for units returned and shipped and with one month of time accumulated
might be:
FR (%/M) = (7 units returned) / (1,000 units shipped) X 100 = 0.70 %/month
An estimate of mean failure rate for the previous year with 12 months of accumulated history and with all
12 date codes from our example:
FR (%/M) = (84 units returned in recent month) / (12,000 cumulative units shipped) X 100 =
0.70 %/month
Within the 84 failures in the most recent month, a larger number of failures would be expected from date
codes shipped initially and fewer from the most recent months. The number of failures would be 7 in the
first 1,000 shipped plus 14 in the second month with 7 from the first 1,000 shipped and 7 from the second
1,000 or 2,000 in total and so on, so that the cumulative number of failures for all 12 months on average
would be:
Cum. Failures first 12 months = 7 + 14 + 21 + 28 + 35 + 42 + 49 + 56 + 63 + 70 + 77 + 84 = 546

See Table A on the next page to review the example in table form.
Again, the estimate of Failure Rate in %/Month counts all the failures in the most recent month of all date
codes of the previous 12 months against the same 12 month base of units shipped. After the first 12 months
passes, the base count of units shipped would be a consistent 12,000 per month in our example. Failure
counts do vary month to month, depending on variables within the return process, the ability of Customers
to detect failures, use conditions and other variables. Shipping counts in reality will also vary where for
example, 900 may be shipped in one month and perhaps 1,200 in a peak month. See Table B with failure
numbers randomized.
Some companies use a 3 month aged and 12 month rolling average to smooth the results out. Monthly
estimates will naturally have variation from a 12 month average due to lower quantities and the impact of
age in the field for a given month or date code.
Note: many companies verify returned or replaced units and in some instances find the product operates
and meets all specifications. These units are sometimes labeled as a “No Trouble Found” or “NTF” and
need to be evaluated as well due to other issues including:
 User training not comprehensive
 Vague or unclear operating instructions, etc.
 Customer was interested in credit only
 Poor product ergonomics

Table A Hypothetical Calculation of Monthly Failure Rates in %/month
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Jan Feb Mar
Units Shipped 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
12 month Cum Units in Service 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 12000 12000
Failures 7 7 7 7 7 7 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7 7
7 7 7 7 7 7 7 7 7
7 7 7 7 7 7 7 7
7 7 7 7 7 7 7
7 7 7 7 7 7
7 7 7 7 7
7 7 7 7
7 7 7
7 7
7
Total Failures Within Month 7 14 21 28 35 42 49 56 63 70 77 84 84 84
Failure Rate in % per Month 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
(Total F Within M/Cum Units)
Cum Failures in 1st 12 months 7 21 42 70 105 147 196 252 315 385 462 546

Table B Hypothetical Calculation of Monthly Failure Rates in %/month
RANDOMIZED using Pair of Dice
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Jan Feb Mar
Units Shipped 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
12 month Cum Units in Service 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 12000 12000
Failures 6 7 6 9 6 3 4 6 8 6 4 6
9 12 5 8 6 7 9 10 7 6 11 8
8 6 9 7 4 2 8 8 7 9 11 4
6 6 10 11 7 9 2 9 4 8 6
10 7 8 10 6 6 5 10 7 7
8 4 6 7 9 6 9 6 8
7 7 9 6 6 2 5 6
8 6 5 9 4 5 8
4 10 9 7 8 6
7 6 10 9 6
7 8 4 8
6 10 7
7 8
11
Total Failures Within Month 6 16 26 26 39 41 45 55 67 66 74 86 88 85
Failure Rate in % per Month 0.600 0.800 0.867 0.650 0.780 0.683 0.643 0.688 0.744 0.660 0.673 0.717 0.733 0.708
(Total F Within M/Cum Units) Avg. = 0.710
Cum Failures in 1st 12 months 6 22 48 74 113 154 199 254 321 387 461 547

Section 1.2 Estimating Field Failure Rate and MTBF based on Field %/Month
In order to determine the failure rate based on hours, some knowledge of the actual typical usage per day
and per month is needed. Is the product used 24 hours a day, 7 days a week or some value less than this
100% use cycle? Let’s assume that a company has determined that its product is in use typically half a day
or 12 hours/day and is in use on weekends. This means that the product is used on average 30.33 days per
month and 12 hours per day for a monthly total of 364 hours per month per unit.
It would be very difficult to measure or estimate the actual field product usage for the previous 12 months
unless some type of monitoring or record keeping is maintained. Units in service from January of the
previous year on the following January would have approximately 12 months of time in use. Units from
last February would have 11 months, and so on to December units with one month of usage. A conversion
of the %/month based on knowledge of the usage would be a reasonable way to estimate failure rate or
MTBF:
FR = (0.7%/month) X (1 day/12 hours) X ( 1 month/30.333 days) X (1/100) ≈ .000020
MTBF = 1/FR = 1/.000020 = 50,000 hours
Section 1.3 Accelerated Life Tests
What Are Accelerated Life Tests:
The failure rate of many devices and products are inherently low. As a result, many industries use
accelerated testing or Accelerated Life Tests to assess the reliability of components and products. Elevated
stresses are used to produce, in a shorter period of time, the same failures that would be observed in use by
Customers. Temperature, humidity, power or voltage stresses, vibration, shock, salt spray and other
stresses are commonly used during accelerated testing on electronic products. Mechanical devices may be
cycled repeatedly and may also be exposed to operational or beyond operational environmental extremes to
generate failures.
Why Accelerated Life Testing:
There are several reasons for performing Accelerated Life Testing on Products.
1. Generally it is to identify failure modes and weak areas of a design not anticipated through the use
of other reliability improvement methods such as Design Review, using approved components,
predictions and Engineering Simulation methods. The goal is to eliminate failures prior to the
useful life.
2. Another reason is to estimate product or component failure rates or MTBF using the Field
Equivalent hours generated versus the number of failures that occur. These results can be used to
estimate or predict actual failure rates or MTBF that may occur in the field and allow for planning
of service or repair, required spares and logistical decisions.

Several areas where Accelerated Life Testing can be effective:
 When updating or developing new products, testing can be performed on prototype and pilot run
units to in order to identify failure modes. Once identified, these can be resolved with engineering
corrective action.
 Another important use is in the testing existing product models in order to duplicate or identify
failure modes currently occurring during field use by Customers. These failures generally show up
as warranty replacements or warranty repairs performed by an approved Service function.
Duplicating the failure modes in testing allows redesigned units to be retested and verified for
elimination or reduction in the occurrence of the identified failures.
It is very useful to use test methods on an ongoing basis including previous vintages or models as this
repetition allows the evolution of a consistent predictive procedure. Though some testing programs are
dictated by a Standards or Customer Requirements and are specific as to what stresses need to be applied
and for how long, these may be made more severe in order to generate failures.
Accelerated Life Test Example: a Consumer Product Electronics Manufacturer runs product samples
through a week’s worth of temperature, humidity, vibration, salt spray and performs a drop test. They then
perform a complete set of functional and parametric tests to detect failures. One week of testing is 168
hours. They have determined that their acceleration factor versus actual field conditions is 20 to 1. This
would be equivalent of a year’s worth of failures for the same number of units occurring in 2.6 weeks of
accelerated testing (52 weeks/year / 20). The cycle is repeated two more times for a total of three cycles or
test loops..
Field Equivalent Hours
If the Consumer Products Manufacturer used 25 product test units and performed three sets or loops of
Accelerated Life Testing, they would then be able to determine the Field Equivalent hours. The Field
Equivalent hours is defined as:
T = Field Equivalent Hours = Test Hours x A.F. x n
Where:
Test Hours = 168 hours X 3 loops = 504 hours
A.F. = 20 from past history
n = # of units
Therefore: T = Field Equivalent Hours = 504 X 20 X 25 = 252,000 hours
If 5 failures occur, the MTBF would then be:
MTBF = 252,000/5 = 50,400 Hours
Accelerated Life Test Example:
A manufacturer of communications microphones developed a machine that allowed for the mounting of
multiple microphones and their associated coil cords onto fixtures. The machine had the capability to
stretch the coil cords up and down to their limit, while simultaneously twisting and rocking the microphone
unit. Each cycle incremented each of three counters allowing for a count of the three movements to be

registered. The wiring within the cable was included in the circuit that controlled the motors that provided
the movements of the fixtures. When a wire within the cable became open, the device would stop and the
counters would register how many cycles were completed to that point. Indicator lights identified which
microphone and associated cable had failed.
Failures could then be plotted versus number of cycles so that various designs could be compared. Units
were dissected, analyzed and photographed so that these failures modes could also be compared directly to
units in the field that failed and that were returned by Customers. Improved designs resulted in a
significant reduction in warranty returns for a very slight increase in material costs after changes in the
design. Of course, new designs should be verified and analyzed using this machine and not risk shipping
product to the Customer until improved.
Section 2.1 Chi Square and the MTBF Confidence Limit Formulas
The failure rate or MTBF calculated is a nominal value. Based on the test or field results and a selected
Confidence Level, a Confidence Interval can be calculated around the NOMINAL value within which the
true population value will fall with a probability equal to the stated confidence level, ex: 90%. The size of
the confidence interval varies with the number of failures and the number of field equivalent hours (sample
size). If a low number of failures occur, the confidence interval may vary over a considerable range.
The formulas used to calculate the end points, or Confidence Limits of the confidence interval use the
critical value of the Chi-square distribution.
Lower Confidence Limit = 2T/(chi
2
(2r +2; (1-phi)/2))
Upper Confidence Limit = 2T/(chi
2
(2r; (1+phi)/2))
Where:
T = Actual Field Hours or
T = Field Equivalent Hours (Test Hours x A.F. x n or # of units)
phi = Level of Confidence Selected (90% for most cases)
chi
2
= the Chi Square value from Table 1 or other published tables for 2r+2 or 2r degrees of freedom
where r is the number of failures.i
Note: MTBF values are rounded to the nearest 100.
An example would be if 500,000 field equivalent hours were accumulated on test units and 10 failures
occurred, the MTBF would be 50,000 hours.
The lower confidence limit would then be:
Lower Confidence Limit = (2 X 500,000)/33.924 = 29,500 Hours

In this case 33.924 is a standard chi
2
table value that intersects at 22 (2r + 2) degrees of freedom and at 1-
(0.9/2) or 0.05 (alpha).
The value of 33.924 can also be obtained from Table 1: CHI SQUARE VALUES FOR 90%
CONFIDENCE LIMITS under “Lower Conf. Limits” and “10” failures. This indicates that the true
population MTBF may be as low as 29,500 hours.
The upper confidence limit would be:
Upper Confidence Limit = (2 X 500,000)/10.851 = 92,200 Hours
The 10.851 is a standard chi
2
table value that intersects at 20 (2r) degrees of freedom and at 1+ (0.9/2) or
0.95 (alpha). The value of 10.851 can also be obtained from Table 1: CHI SQUARE VALUES FOR 90%
CONFIDENCE LIMITS under “Upper Conf. Limits” and “10” failures. This indicates that the true MTBF
may be as high as 92,200 hours.
Therefore the interval of 29,500 hours to 92,200 hours will contain the true MTBF 90% of the time. This is
a -40%, + 84% variation.
The lower and upper confidence can more be more easily calculated by multiplying the NOMINAL MTBF
by the “Chi Square” factors in Table 2. The Table 2 “CHI SQUARE FACTORS FOR 90%
CONFIDENCE LIMITS” includes factors for 0 to 50 failures.
Using the example MTBF of 50,000 hours, find the multipliers for the lower and upper confidence limits
for 10 failures in Table 2. These are 0.590 and 1.845 respectively. The calculation of the limits would then
be:
Lower Confidence Limit = 50,000 hours x 0.590 = 29,500 hours
Upper Confidence Limit = 50,000 hours x 1.845 = 92,200 hours
Notice that this is very close to the values calculated using the Confidence Limit formulas.
Section 2.2 Comparing Different Products, Vintages or Competitive Products
If two or more products or the same product produced at different times or with modifications are to be
compared, the NOMINAL MTBF Values cannot be used. The confidence intervals must not overlap for a
difference to be concluded at a 90% confidence level. If the confidence intervals overlap, no difference can
be stated. This is true even though the NOMINAL values are different. See the following Data 1 and
Graph 1:

Data 1, Comparison of Product Vintages:
Graph 1, Comparison of Product Vintages:
Note that Model B Significantly improves in MTBF (no overlap) from its Prototype, Pilot Run to 1st
Production test units. Model B Pilot Run units, our Previous Model A, and the Competitor #1 units are
basically equal due to the overlap in the confidence limits. The Model B 1st
Production units are
significantly higher in MTBF than all other test groups due to no overlap of the confidence limits. To
obtain narrower confidence limits requires that more test units be used, or an increase in the acceleration
factor be made with additional stresses applied to generate more failures.
COMPARISON OF PRODUCTS - VINTAGES, VERSIONS, or COMPETTIVE - 90% CL
Test Accel. Field Eq.
Product and/or Vintage # units Hours Factor Hours Failures MTBF LCL UCL
Model B Prototypes 10 250 15 37,500.00 12 3,125.0 1,928.8 5,415.8
Model B Pilot Run 25 504 15 189,000.00 12 15,750.0 9,720.9 27,295.5
Model B 1st Production 25 504 20 252,000.00 2 126,000.0 40,026.7 709,137.0
Previous Model A 25 504 15 189,000.00 12 15,750.0 9,720.9 27,295.5
Competitor #1 10 504 15 75,600.00 9 8,400.0 4,813.7 16,101.5
100
1,000
10,000
100,000
1,000,000
Model B
Prototypes
Model B
Pilot Run
Model B 1st
Production
Previous
Model A
Competitor
#1
H
O
U
R
S
Product Vintage
Comparison of Products: MTBF (LCL, UCL)

Section 2.3 One Sided Confidence Limits for MTBF
Often, when testing or when field results are obtained, a comparison to a minimum required MTBF is
needed. In this case only the lower confidence limit for MTBF (or upper confidence limit for failure rate)
is of interest. It is not of concern how high the MTBF is and the level of confidence can be used entirely in
the calculation of the lower confidence limit. This is referred to as a one-sided confidence limit. In this
case the formula for the lower confidence limit is the same except for the second term in determining the
standard is 1- phi instead of 1 – (phi/2) or
1–(0.9/2), in our case.
With the same 50,000 hour MTBF example and T of 500,000 hours, the lower confidence limit would then
be:
Lower Confidence Limit = (2 X 500,000)/30.813 = 32,450 Hours
Where 30.813 is the standard chi
2
table value that intersection at 22 (2r + 2) and now (1 – 0.9) or 0.10.
The 32,450 hours should exceed the minimum MTBF established by the requirements. The value of 30.813
can also be obtained from Table 3: CHI SQUARE VALUES FOR ONE SIDED 90% CONFIDENCE
LIMITS under “Lower Conf. Limits” and “10” failures. This indicates that the true population MTBF is at
least 32,450 hours. No estimate or limit on how high the MTBF is obtained.
It is useful to look at the variation in one sided 90% and perhaps 60% Lower Confidence Limits normalized
for any acceleration factor and sample size versus the number of failures. It can be seen that as the number
of failures increases, the confidence limits get closer to the MTBF. Conversely, as the number of failures
decreased the limits widen significantly the gap between the MTBF and the Lower Confidence Limit. See
Graph 2.

Graph 2
Again, with our example of a 50,000 hour MTBF, 10 failures and 500,000 field equivalent hours we can
determine the variation or distance of the one-sided Lower Confidence Limit from the MTBF. Using
Graph 1, on the X-axis find the line for 10 failures and move up vertically until the MTBF line is
intersected. An (MTBF/A.F. * N) of 67 is found. The 90% Lower Confidence Limit value intersects the
10 failures line at approximately 43. These have a ratio of approximately 1.56 which is also the ratio of the
MTBF of 50,000 hours and the calculated one sided Lower Confidence Limit of 32,450 hours. This can be
used for any MTBF, sample size, or acceleration factor. So one could also estimate the One Sided 90%
Confidence Limit dividing the 50,000 hour MTBF by the 1.56 ratio obtained from Graph 2, or an estimate
of 32,050 hours.
Example 1: Say you have an MTBF of 100,000 hours and 15 failures occurred. The vertical line intersects
the normalized MTBF curve at 45 and the Lower 90% Confidence Limit curve at 31. This ratio is 45/31 or
1.45. The 90% Lower Confidence Limit could be then estimated by dividing 100,000/1.45 or 68,965
hours. This LCL for the MTBF needs to equal to or exceed your minimum MTBF product requirement.
0
20
40
60
80
100
120
140
5 10 15 20 30 40 50
M
T
B
F
/
A
F
*
N
Number of Failures
MTBF/AF * N vs # Failures
ONE SIDED CONFIDENCE LIMIT
90% LCL 60% LCL MTBF

Example 2: An extremely reliable product has a historical MTBF of 500,000 hours. How many test units
would be required to generate 5 failures in testing assuming the Accelerated Life Testing procedure has an
acceleration factor of 20 to 1?
Using the Curves in Graph 2, find 5 failures. The vertical line intersects the MTBF curve at 134. Since:
134 = (MTBF/(A.F. X N) = 500,000/(20 X n)
Therefore:
n = MTBF/(A.F. X 134) = 500,000/(20 X 134) = 187 test units
Section 2.4 Failure Rate Confidence Limits, Two Sided
Sometimes it is desired to review results as a Failure Rate (λ), rather than an MTBF. In the case of Failure
Rate, it is desired to minimize this in a component or product. The Lower Confidence Limit for the MTBF
can be used to calculate the Upper Confidence Limit for Failure Rate by taking the reciprocal. Also, the
Upper Confidence Limit for MTBF can be used to obtain the Lower Confidence Limit for Failure Rate the
same way. The 29,500 hour MTBF Lower Confidence Limit would become:
1/29,500 = 0.0000339 failures per hour
The value 0.0000339 is then the Upper Confidence Limit for Failure Rate. This can be converted into a
sometimes more meaningful %/month value if you know the conversion factors based on an empirical
study of effective daily usage. For example, it may have been determined that a product’s usage is 12
hours per day rather than 24 hours. The conversion of Failure Rate to %/month would then be:
FR (%/M) = 0.0000339 X (12 hours/1day) X (33.333 days/month) X 100 = 1.234%/month
See the following Tables 1 through 4, Chi Square Values and Chi Square Factors for MTBF Limits, for
both two sided and one sided:

Tables 1 through 4, Chi Square Values and Chi Square Factors for MTBF Limits, for
both two sided and one sided:
Table 1: CHI SQUARE VALUES FOR 90% CONFIDENCE
LIMITS
0.90
Lower
Conf. Upper Conf.
Lower
Conf. Upper Conf.
# of
Failures Limits Limits # of Failures Limits Limits
0 5.991 na
1 9.488 0.103 26 72.153 36.437
2 12.592 0.711 27 74.468 38.116
3 15.507 1.635 28 76.778 39.801
4 18.307 2.733 29 79.082 41.492
5 21.026 3.940 30 81.381 43.188
6 23.685 5.226 31 83.675 44.889
7 26.296 6.571 32 85.965 46.595
8 28.869 7.962 33 88.250 48.305
9 31.410 9.390 34 90.531 50.020
10 33.924 10.851 35 92.808 51.739
11 36.415 12.338 36 95.081 53.462
12 38.885 13.848 37 97.351 55.189
13 41.337 15.379 38 99.617 56.920
14 43.773 16.928 39 101.879 58.654
15 46.194 18.493 40 104.139 60.391
16 48.602 20.072 41 106.395 62.132
17 50.998 21.664 42 108.648 63.876
18 53.384 23.269 43 110.898 65.623
19 55.758 24.884 44 113.145 67.373
20 58.124 26.509 45 115.390 69.126
21 60.481 28.144 46 117.632 70.882
22 62.830 29.787 47 119.871 72.640
23 65.171 31.439 48 122.108 74.401
24 67.505 33.098 49 124.342 76.164
25 69.832 34.764 50 126.574 77.929
Note: for "0" failures, only a LCL for MTBF can be calculated using 2T/5.991
Field Equivalent Hours (Test Hours x A.F. x # of units)

Table 2: CHI SQUARE FACTORS FOR 90% CONFIDENCE
LIMITS
Multiply FACTORS by the MTBF
0.90
Lower
Conf. Upper Conf.
Lower
Conf. Upper Conf.
# of Failures Limits Limits # of Failures Limits Limits
1 0.211 19.496 26 0.721 1.427
2 0.318 5.628 27 0.725 1.417
3 0.387 3.669 28 0.729 1.407
4 0.437 2.928 29 0.733 1.398
5 0.476 2.538 30 0.737 1.389
6 0.507 2.296 31 0.741 1.381
7 0.532 2.131 32 0.744 1.374
8 0.554 2.010 33 0.748 1.366
9 0.573 1.917 34 0.751 1.359
10 0.590 1.843 35 0.754 1.353
11 0.604 1.783 36 0.757 1.347
12 0.617 1.733 37 0.760 1.341
13 0.629 1.691 38 0.763 1.335
14 0.640 1.654 39 0.766 1.330
15 0.649 1.622 40 0.768 1.325
16 0.658 1.594 41 0.771 1.320
17 0.667 1.569 42 0.773 1.315
18 0.674 1.547 43 0.775 1.311
19 0.682 1.527 44 0.778 1.306
20 0.688 1.509 45 0.780 1.302
21 0.694 1.492 46 0.782 1.298
22 0.700 1.477 47 0.784 1.294
23 0.706 1.463 48 0.786 1.290
24 0.711 1.450 49 0.788 1.287
25 0.716 1.438 50 0.790 1.283

Table 3: CHI SQUARE VALUES FOR ONE SIDED 90% LOWER
CONFIDENCE LIMITS
0.90
Lower
Conf. Upper Conf.
Lower
Conf. Upper Conf.
0 4.605 na
1 7.779 na 26 67.673 na
2 10.645 na 27 69.919 na
3 13.362 na 28 72.160 na
4 15.987 na 29 74.397 na
5 18.549 na 30 76.630 na
6 21.064 na 31 78.860 na
7 23.542 na 32 81.085 na
8 25.989 na 33 83.308 na
9 28.412 na 34 85.527 na
10 30.813 na 35 87.743 na
11 33.196 na 36 89.956 na
12 35.563 na 37 92.166 na
13 37.916 na 38 94.374 na
14 40.256 na 39 96.578 na
15 42.585 na 40 98.780 na
16 44.903 na 41 100.980 na
17 47.212 na 42 103.177 na
18 49.513 na 43 105.372 na
19 51.805 na 44 107.565 na
20 54.090 na 45 109.756 na
21 56.369 na 46 111.944 na
22 58.641 na 47 114.131 na
23 60.907 na 48 116.315 na
24 63.167 na 49 118.498 na
25 65.422 na 50 120.679 na
Note: for "0" failures, only a LCL for MTBF can be calculated using 2T/5.991
Field Equivalent Hours (Test Hours x A.F. x # of units)

Table 4: CHI SQUARE FACTORS FOR One Sided 90% LOWER
CONFIDENCE LIMIT
Multiply FACTOR by the MTBF
0.90
Lower
Conf. Upper Conf.
Lower
Conf. Upper Conf.
1 0.257 na 26 0.768 na
2 0.376 na 27 0.772 na
3 0.449 na 28 0.776 na
4 0.500 na 29 0.780 na
5 0.539 na 30 0.783 na
6 0.570 na 31 0.786 na
7 0.595 na 32 0.789 na
8 0.616 na 33 0.792 na
9 0.634 na 34 0.795 na
10 0.649 na 35 0.798 na
11 0.663 na 36 0.800 na
12 0.675 na 37 0.803 na
13 0.686 na 38 0.805 na
14 0.696 na 39 0.808 na
15 0.704 na 40 0.810 na
16 0.713 na 41 0.812 na
17 0.720 na 42 0.814 na
18 0.727 na 43 0.816 na
19 0.734 na 44 0.818 na
20 0.740 na 45 0.820 na
21 0.745 na 46 0.822 na
22 0.750 na 47 0.824 na
23 0.755 na 48 0.825 na
24 0.760 na 49 0.827 na
25 0.764 na 50 0.829 na
i
Halpern, Siegmund (1978), The Assurance Sciences: an introduction to quality control and reliability, Prentice Hall, New Jersey.

Practical Use of Failure Rates and Mean Time to Failure Data (2)

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

Similaire à Practical Use of Failure Rates and Mean Time to Failure Data (2)

Similaire à Practical Use of Failure Rates and Mean Time to Failure Data (2) (20)

Practical Use of Failure Rates and Mean Time to Failure Data (2)