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/(chi2 (2r +2; (1-phi)/2))

Upper Confidence Limit = 2T/(chi2 (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 # FailuresONE 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)

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