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Unit 6: Probability Plotting

Notes largely based on “Statistical Methods for Reliability Data”

by W.Q. Meeker and L. A. Escobar, Wiley, 1998 and on their class notes.

Ramón V. León

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Unit 6 Objectives• Describe applications for probability plots• Explain the basic concepts of probability plotting• Show how to linearize a cdf on special plotting scales• Explain how to plot a nonparametric estimate of the cdf

to judge the adequacy of a particular parametric distribution

• Explain methods of separating useful information from noise when interpreting a probability plot

• Use a probability plot to obtain graphical estimates of reliability characteristics like failure probabilities and quantiles

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Purpose of Probability Plots

• Asses the adequacy of a particular distributional model• To detect multiple failure modes or mixtures of different

populations• Obtain graphical estimates of model parameters (e.g., by

fitting a straight line through the points on a probability plot)

• Displaying the results of a parametric likelihood fit along with the data

• Obtain, by drawing a smooth curve through the points, a nonparametric estimate of failure probabilities and distributional quantiles

Probability plots are used to:

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Probability Plotting Scales: Linearizing a CDF

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Linearizing the Exponential CDF

log(1 ) ptpγ

θ−

− − =

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Linearizing the Normal CDF

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Linearizing the Lognormal CDF

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Linearizing the Weibull CDF

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Linearizing the Weibull CDF - Continued

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Choosing Plotting Positions to Plot the Nonparametric Estimate of the CDF

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Criteria for Choosing Plotting Positions

• Checking distributional assumptions• Estimation of parameters• Display of maximum likelihood results

with data.

Criteria for choosing plotting positions should depend on the application or purpose for constructing the probability plot

Some applications that suggest criteria:

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Plotting Positions: Continuous Inspection Data and Single Censoring

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Plotting Positions: Continuous Inspection Data and Multiple Censoring

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Six Distribution Probability Plots of the

Shock Absorber Data

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Six Distribution Probability

PlotsAlloy

T7987 Fatigue Life

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Plotting Positions: Interval Censored Inspection Data

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Probability Plots with Specified Shape Parameters

• Distributions that are not members of the location-scale family

• To help identify, graphically, the need for non-zero threshold parameters

• Estimate graphically a shape parameter

The probability plotting techniques can be extended to construct probability plots for:

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Distributions with a Threshold Parameter

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Linearizing the 3-Parameter Gamma CDF

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Linearizing the 3-Parameter Weibull CDF Using Linear Time Axis and Specified Shape Parameter

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Linearizing the Generalized Gamma CDF

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Notes on the Application of Probability Plotting

• Using simulation to help interpret probability plots– Try different assumed distributions and compare the

results– Assess linearity; allowing for more variability in the

tails• Use simultaneous nonparametric confidence bands• Use simulation or bootstrap to calibrate

• Possible reason for a bend in a probability plot– Sharp bend or change in slope generally indicates an

abrupt change in a failure process

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