Weibull Distribution - Probability Plotting
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Weibull Distribution - Probability Plotting |
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Probability Plotting is tedious and computer routines are often employed. Commercial software is available for these calculations, such as the Reliability & Maintenance Analyst. The following section describes probability plotting for the Weibull distribution using the Reliability & Maintenance Analyst. The manual method is located here. Probability plotting supports the 2-parameter and 3-parameter Weibull distribution, and is an excellent method for determining goodness-of-fit. To determine the goodness-of-fit, select the "Transformed" option in the Plot Type frame, and click the "Plot" button. If the plotted points form a straight line, the distribution provides a good time to fail model for the data. The "R-Squared" value is a measure of how well the data forms a straight line. An R-Squared value of 1.0 indicates a perfectly straight line. R-Squared is also known as the coefficient of determination. To estimate the parameters of the Weibull distribution using probability plotting, follow these steps:
The figure below shows the Weibull probability plotting screen using the data in the file "Demo2.dat".
The default for this operation is a location parameter of zero. In the Location Parameter frame, the software will determine the best value for the location parameter if "Software Estimate" is selected. The user is also given the option of entering a location parameter in the text box below the "User Entered" option. Be cautious when using a non-zero location parameter. A positive location parameter indicates a zero probability of failure for time less than the value of the location parameter, and a negative location parameter means that the population had failures before testing began. Unless there is a sound engineering for one of these conditions, it is best to use a location parameter equal to zero. Clicking the "Plot" button gives a probability plot with 2 options - transformed data and untransformed data. The transformed data produced a plot of the data that has been transformed to produce a straight line with a 95% confidence interval surrounding it. If the plotted points do not follow a straight line, the Weibull distribution with the estimated parameters does not provide an adequate time to fail model. The untransformed option plots the cumulative distribution function against time. This is useful for determining the probability of failure at a given time. The title of the graphs can be changed by editing the text in the Graph Title frame. To check the spelling of the title, click the "Spell Check" button. To predict reliability or time-to-fail using the estimated parameters use the Predicting Module. By taking the logarithm of the Weibull cumulative distribution function twice and rearranging,
 versus
ln x, and fitting a straight line to the points, the parameters of the Weibull
distribution can be estimated. The slope of the plot provides an estimate of b , and the y-intercept can be used to estimate q .
The cumulative distribution function, F(x), is usually estimated from the median rank, but other estimates such as the mean rank and the Kaplan-Meier product limit estimator are also used. Specialized probability paper is available for probability plotting. Using probability paper eliminates the need to transform the data prior to plotting. Example: Determine the parameters of the Weibull distribution using probability plotting for the data given below. Solution: The table below is constructed to obtain the necessary plotting data.
Click here to download this solution in Microsoft Excel. Confidence limits can be added to this plot using 5% and 95% ranks. Plotting position for the 5% and 95% ranks are found from the expression
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