Reliability HotWire |
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Defining Distributions | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Generally speaking, the
object of performing a life data analysis is to be able to predict the
future performance of a certain product. This prediction is based on the
observed behavior of a relatively small group of these objects that are
considered to be representative of the entire population. Occurrences of
interest are observed, and then statistics is used to model the frequency
and probability of these occurrences. Most frequently, the occurrences of
interest are the failure of the units on test. A number of units are
placed on test and run until failures occur. This data set is then
analyzed and modeled, and predictions regarding the product's failure
behavior are calculated. These predictions are made via the familiar life
data analysis metrics such as reliability, failure rate, etc.
The basis for these metrics is a mathematical function that models how the
failure occurrences are distributed over time. This function is called the In order to get a
better idea of how the
These values could represent failure times, or product dimension variations, or any other information. At this point, what the data set represents is immaterial; we just have a representative sample of data from a process that we want to characterize. One way we can begin to do this is through the use of a histogram. In order to construct a histogram, we separate the data into "bins" or ranges, and count how many of the data points fall into each range. This information can then be plotted in a bar chart. Following is a histogram for the data with a range size of 50.
From this initial histogram, we can begin to characterize where the data set fails with respect to its values. Note that the y-axis is in terms or relative proportion rather than the raw number of data points falling into the range. In other words, we can use the graph to get an idea of how the data set is distributed. This graph illustrates that a relatively high proportion of the data falls between the values of 0 and 50, 47%. The range from 51 to 100 contains 34% of the data, from 101 to 150 contains 14% of the data, and values of 151 or greater comprise 5% of the data. If all of these values are added up, they will sum to 100%, indicating that we have been able to account for all of our data points in this histogram. This is a good start, but the histogram can be improved. Our first histogram contains only four ranges, which may not be sufficient to get a good idea of how the data points are distributed. We can try to refine the histogram by creating more ranges that are smaller in width. The following histogram shows the data in a histogram with a range size of 25. This represents an improvement over the initial histogram in that it gives us a more refined picture of where the data points tend to fall. It shows that the most likely area for the data is in the range from 26 to 50. Note that the scale of the y-axis has dropped from 0.5 to 0.35. This is because as we divide the same number of data points into smaller ranges, the relative values for the bars in the histogram will decrease. However, the values of all of the bars will still add up to 1, or 100%. We can further refine our histogram by again decreasing the size of the ranges. The following chart shows a histogram for the data with a range size of 10. This further refines the picture of where the data points tend to fall, showing a sharp increase to a maximum in the 21 to 30 range, which gradually tails off to the right. Note that once again the y-axis range has decreased, although all of the bars in the chart would still add up to 1. If we had enough data,
we could continue to create histograms with smaller and smaller ranges to
get a more refined picture of the distribution of the data. The ultimate
conclusion of this process would be a histogram that has, in theory, a
range that is infinitesimally small. In other words, we would have a
continuous function. Such functions do exist, and are the Note that the The formal mathematical
definition of a In other words, the In reliability terms,
this function gives us the probability that a failure occurs between time In next month's
Reliability Basics, we'll look at how the
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