Reliability HotWire |
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Reliability Basics | |||||||||||||||

In
last month's Reliability Basics, we examined the reliability function
- what it is and how it can be used. The concept of the lifetime
distribution was introduced, as was the probability density function ( In this issue, we will look at how we can begin to determine estimates of the parameters for each lifetime distribution, based on test data. These estimates can then be used to construct reliability functions and plots, as well as other life data statistics, such as the MTBF. The simplest and longest-used method for parameter estimation is that of probability plotting. This methodology involves plotting the failure times on a specially-constructed plotting paper to determine the fit of the data to a given distribution and, if applicable, estimates of the distribution's parameters.
A distribution's
probability plotting paper is constructed by linearizing the cumulative
density function ( where
and
are parameters. We now need to linearize this function into the form If we now set: and: the This is now a linear
equation, with a slope of
and an intercept of
ln( ). Now
the x- and y-axes of the Weibull probability plotting paper can be
constructed. The x-axis is simply logarithmic, since where Note that since the
mathematical expression for the
The question now arises
of how to plot our failure times on the plotting paper. We can see that
the x-axis values will correspond to our failure times, since we see that the
y-coordinate is based on Median ranks are based
on a solution for the cumulative binomial distribution, based on sample
size and failure number. The median ranks represent the 50% confidence
level ("best guess") estimate for the true unreliability for a failure,
based on the total number of failures and the order number (first, second,
etc.) of the failure in question. There is also an approximation that can
be used to estimate median ranks, called where Based on Benard's approximation, we can now calculate unreliability estimates for each of our failure times. These are shown in the following table:
Now that we have y-coordinate values to go with the x-coordinate failure times, we can now plot our failure data on a Weibull probability plot: The failure times plotted on Weibull probability paper fall in a fairly linear fashion, indicating that our choice of the two-parameter Weibull distribution was valid. If the points did not seem to follow a straight line, we might want to consider using another lifetime distribution to analyze the data. We can now draw a best-fit line through the points. This line represents
the model of the unreliability, as expressed by the linearized
unreliability function, or
pdf. Determination of ,
or the Weibull slope, is relatively easy. As we saw when we were
discussing the linearization of the two-parameter Weibull By drawing a line parallel to the best-fit model line through the slope scale, we can see that the estimate for for this data set is approximately 1.4. Mathematical
manipulation of the Weibull We want to be able to
read the value of
from the x-axis time scale, which can be expressed mathematically as Hence, is where our best-fit unreliability model line intersects with a horizontal line extended from the 63.2% level of the unreliability, or y-axis scale: As the graphic shows, the best-fit model line intersects the 63.2% unreliability line at approximately 44 hours. Therefore, the estimate for for our data is 44 hours. This illustrates the basics of probability plotting for complete data using a two-parameter Weibull example. The methodology can be more difficult for other types of analysis. For example, if the data set contained suspensions, we would have to be able to account for them. This is dealt with by modifying the median rank values for the failure times, although that particular methodology exceeds the scope of this article. (For a more detailed discussion of this methodology, click here.) There are also
shortfalls to this method of parameter estimation. Besides the most
obvious, which is the amount of effort required, manual probability
plotting is not always consistent in the results. Two people plotting a
straight line through a set of points will not always draw this line the
same way, thus coming up with slightly different results. In addition,
probability plotting can be very difficult for analyzing large sample
sizes (
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