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issue: October 2003 APPLIANCE Magazine

Engineering: Reliability Testing
Making the Most of Life Test Data

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by Tedd Karr, owner, Compass Training & Consulting

Whether it's complex assemblies or individual components, the main question an appliance industry designer has to face is: "Will this product perform reliably?" However, how can it be determined if a product is reliable or meets or exceeds the life of the competition?

There always seems to be some type of constraint, whether it is too few prototypes, lack of test criteria, tight schedules, high costs, long lead times, or limited testing facilities to tackle such a problem. This following will discuss one way an designer can investigate reliability using life test data.

Switch Failure Data
Unit #1
34,907 cycles
Unit #2
18,281 cycles
Unit #3
15,399 cycles
Unit #4
27,063 cycles
Unit #5
25,363 cycles
Table 1. Raw Data
The data in Table 1 shows overload cycle tests on a sample of five electrical switches conducted until all units failed. Having limited test data such as this is not an unusual situation. Often, the designer must determine product acceptability based on minimal data combined with education, experience, and luck. If marketing specifications require a life of a minimum of 12,000 cycles, it appears logical that a product is acceptable, considering that all samples surpassed the required minimum. However, with only five samples, can there be any confidence in a conclusion? The answer is yes; there is a valid method for extracting maximum information from as few as five life test results by using the Weibull Distribution.

Unlike the commonly used Normal or bell-shaped curve, the Weibull distribution is actually a family of curves. Mathematically, it is defined as:

is a scale parameter (affects if the curve is peaked or flatter);
is a shape parameter (affects overall curve shape); and
is a location parameter (and the smallest value of X, which is normally set to zero and greatly simplifies the equation).

The curve of the Weibull can vary greatly as these parameters change. The most important parameter is , which can change the overall shape of the curve. In general, values vary from 1/3 to 5. When = 1, the shape becomes the exponential curve, which has been used to explain the decay rate of electronic components. When approaches 3.5, is 1, and is 0, then the Weibull distribution looks like the Normal or bell-shaped distribution. For most other statistical distributions, the data is checked against criteria so that the proper distribution is fit to the data. The Weibull works somewhat in reverse since it is a family of distributions where the data defines the shape of the distribution.

Failure Order (i)
Switch Failure Data
15,399 cycles
18,281 cycles
25,363 cycles
27,063 cycles
34,907 cycles
Table 2. Ranked Data
A simple method exists using Weibull distribution paper to plot life test data and to generate estimates r egarding failure rates. Weibull probability plotting paper is a version of log-log paper and can be downloaded or printed from the Internet. Although the vertical or Y-axis is listed as a "percent," these values are generated from a statistical formula called Median Ranks. The term median is the middle value of a dataset such that 50 percent of all values are higher than the median and 50 percent are lower than the median. The term rank means to put the data into some type of order or priority.

Table 2 shows the switch cycle data after it is placed into ascending order. In order to generate a Weibull graph, a Y-axis (percent) and X-axis point (number of cycles) will be required for each data point. The Y-axis or Median Rank values are estimated by the following equation:
is the failure order number (failure #1, #2, etc.), and
is the total number of samples.

A table of Median Ranks can be found in some college textbooks, especially those teaching reliability concepts. A chart of these values can also be found on the Internet. Calculations done using software typically use the exact Median Ranks formula:


Table 3 shows the Median Rank for the failure data, based on the simplified approximation equation.

Failure Order (i) Switch Failure Data Median Ranks
#1 15,399 cycles 0.12963 (12.96 percent)
#2 18,281 cycles 0.31482 (31.48 percent)
#3 25,363 cycles 0.50 (50 percent)
#4 27,063 cycles 0.68519 (68.52 percent)
#5 34,907 cycles 0.87037 (87.04 percent)
Table 3. Failure Order and Median Ranks
To manually complete a Weibull plot from the switch data is not a difficult process and takes the following basic steps:

1. Rank each failure in ascending order from 1 through N, where N is the total number of failed samples. (As shown in Table 2.)

2. Calculate the Median Ranks for each sample. (As shown in Table 3.)

3. Plot Y-axis points for the Median Ranks and corresponding X-axis points for the number of cycles until failure.

Graph 1. Switch Life Cycle Failures. CLICK for a larger graphic.
4. Draw a best-fit line through these data points. (As shown in Graph 1.)

When data points are plotted on log paper, they should all be nearly linear. Any "outliers" or extreme values should be studied to determine if they are erroneous. Also, a noticeable curvature to the plotted line indicates that the Weibull may not be the appropriate distribution.

The following three underlying factors are most likely to affect plotted data:

1. Failure data by its nature and in itself is extreme data.

2. Sampling error (probability) may produce a higher or lower data point than expected, but the datapoint still may be valid.

3. Human errors, test set errors, or other failure modes may cause inaccuracy in the datapoint.


Graph 2. Switch Life Cycle Failures. CLICK for a larger graphic.
Although the Y-axis is labeled as "percent," it can also be referred to as "unreliability," which is defined as 100 percent - reliability. For instance, in Graph 2 at the 50-percent level, the expected number of cycles until failure is approximately 25,000. This indicates that one-half of all switches built to this design will be expected to fail at less than 25,000 cycles.

Two other percentage levels are also of interest, the 10-percent level and the 90-percent level. For this switch example, the 10-percent level correlates to approximately 15,000 cycles. The prediction at this level indicates 1 in 10 switches will not live past 15,000 cycles. At the other end of the scale, the 90-percent level correlates to a value of approximately 33,000 cycles - only 1 switch from 10 is expected to survive past 33,000 cycles.


Graph 3. Switch Life Cycle Failures CLICK for a larger graphic.
Are these values acceptable? As stated previously, the minimum life specification was 12,000 cycles, and all the raw data in Table 1 exceeded that minimum specification. On Graph 3, the 12,000-cycle specification minimum correlates to an "unreliability" of just more than 4.2 percent. Based on this prediction, if switch failures happen within the warranty period, then a warranty return rate of approximately 1 in 20 units is possible. Whether this potential return rate is acceptable then becomes a business decision rather than an engineering one.

The Weibull distribution is a rare case in which minimal data can be effectively used for decision making with confidence. Although it is simple to generate Weibull graphs manually, most companies have some type of quality or reliability software that contains Weibull calculations. In addition to an increased level of accuracy, software can also quickly generate values of and and full color charts. For companies that don't use Weibull graphs and data, the problem is usually one of not understanding this powerful method rather than a lack of software.
About the Author

Ted Karr is the owner of Compass Training & Consulting located in Greenville, NC, U.S. He specializes in engineering statistics to optimize product designs and manufacturing processes. Mr. Karr has an engineering degree from NC State University, an MBA from East Carolina University, and holds numerous professional certifications.



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