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 bellshaped curve, the Weibull distribution is actually a family of curves. Mathematically, it is defined as:



where: 

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 bellshaped 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

#1 
15,399
cycles

#2 
18,281
cycles

#3 
25,363
cycles

#4 
27,063
cycles

#5 
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 loglog paper and can be downloaded or printed
from the Internet. Although the vertical or Yaxis 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 Yaxis (percent) and Xaxis point (number of cycles)
will be required for each data point. The Yaxis or Median Rank values are
estimated by the following equation:



where: 

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 Yaxis points for the Median Ranks and corresponding Xaxis points for the number of cycles until failure.
Graph 1. Switch Life Cycle Failures. CLICK for
a larger graphic. 
4. Draw a bestfit 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 Yaxis 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 50percent
level, the expected number of cycles until failure is approximately 25,000.
This indicates that onehalf 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 10percent level and the
90percent level. For this switch example, the 10percent 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 90percent 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 12,000cycle 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. 