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In life data analysis (also called “Weibull analysis”), the practitioner attempts to make predictions about the life of all products in the population by fitting a statistical distribution to life data from a representative sample of units. The parameterized distribution for the data set can then be used to estimate important life characteristics of the product such as reliability or probability of failure at a specific time, the mean life and the failure rate. Life data analysis requires the practitioner to:
This document presents an overview of basic concepts in life data analysis (Weibull analysis) and some suggestions for additional research. ReliaSoft’s Weibull++ software provides a complete array of life data analysis tools.
The term “life data” refers to measurements of product life. Product life can be measured in hours, miles, cycles or any other metric that applies to the period of successful operation of a particular product. Since time is a common measure of life, life data points are often called “times-to-failure” and product life will be described in terms of time throughout the rest of this guide. There are different types of life data and because each type provides different information about the life of the product, the analysis method will vary depending on the data type. With “complete data,” the exact time-to-failure for the unit is known (e.g., the unit failed at 100 hours of operation). With “suspended” or “right censored” data, the unit operated successfully for a known period of time and then continued (or could have continued) to operate for an additional unknown period of time (e.g., the unit was still operating at 100 hours of operation). With “interval” and “left censored” data, the exact time-to-failure is unknown but it falls within a known time range. For example, the unit failed between 100 hours and 150 hours (interval censored) or between 0 hours and 100 hours (left censored).
Statistical distributions have been formulated by statisticians, mathematicians and engineers to mathematically model or represent certain behavior. The probability density function (pdf) is a mathematical function that describes the distribution. The pdf can be represented mathematically or on a plot where the x-axis represents time, as shown next.
The equation below gives the pdf for the 3-parameter Weibull distribution. Some distributions, such as the Weibull and lognormal, tend to better represent life data and are commonly called “lifetime distributions” or “life distributions.” In fact, life data analysis is sometimes called “Weibull analysis” because the Weibull distribution, formulated by Professor Waloddi Weibull, is a popular distribution for analyzing life data. The Weibull model can be applied in a variety of forms (including 1-parameter, 2-parameter, 3-parameter or mixed Weibull). Other commonly used life distributions include the exponential, lognormal and normal distributions. The analyst chooses the life distribution that is most appropriate to model each particular data set based on past experience and goodness-of-fit tests.
In order to fit a statistical model to a life data set, the analyst estimates the parameters of the life distribution that will make the function most closely fit the data. The parameters control the scale, shape and location of the pdf function. For example, in the 3-parameter Weibull model (shown above), the scale parameter, η, defines where the bulk of the distribution lies. The shape parameter, β, defines the shape of the distribution and the location parameter, γ, defines the location of the distribution in time. [View a visual demonstration of the effect of the parameters on the probability density function…]
Several methods have been devised to estimate the parameters that will fit a lifetime distribution to a particular data set. Some available parameter estimation methods include probability plotting, rank regression on x (RRX), rank regression on y (RRY) and maximum likelihood estimation (MLE). The appropriate analysis method will vary depending on the data set and, in some cases, on the life distribution selected.
Once you have calculated the parameters to fit a life distribution to a particular data set, you can obtain a variety of plots and calculated results from the analysis, including:
Because life data analysis results are estimates based on the observed lifetimes of a sampling of units, there is uncertainty in the results due to the limited sample sizes. “Confidence bounds” (also called “confidence intervals”) are used to quantify this uncertainty due to sampling error by expressing the confidence that a specific interval contains the quantity of interest. Whether or not a specific interval contains the quantity of interest is unknown.
Confidence bounds can be expressed as two-sided or one-sided. Two-sided bounds are used to indicate that the quantity of interest is contained within the bounds with a specific confidence. One-sided bounds are used to indicate that the quantity of interest is above the lower bound or below the upper bound with a specific confidence. The appropriate type of bounds depends on the application. For example, the analyst would use a one-sided lower bound on reliability, a one-sided upper bound for percent failing under warranty and two-sided bounds on the parameters of the distribution. (Note that one-sided and two-sided bounds are related. For example, the 90% lower two-sided bound is the 95% lower one-sided bound and the 90% upper two-sided bounds is the 95% upper one-sided bound.)
Two-Sided Confidence Bounds
 Lower One-Sided Confidence Bounds |
 Upper One-Sided Confidence Bounds |
