Chapter 6
Probability Plotting
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
6-1
Copyright 1998-2008 W. Q. Meeker and L. A. Escobar.
Based on the authors’ text Statistical Methods for Reliability
Data, John Wiley & Sons Inc. 1998.
December 14, 2015
8h 9min
Purposes of Probability Plots
Probability plots are used to:
• Assess the adequacy of a particular distributional model.
• To detect multiple failure modes or mixture of different
populations.
• Displaying the results of a parametric maximum likelihood
fit along with the data.
• Obtain, by drawing a smooth curve through the points, a
semiparametric estimate of failure probabilities and distributional quantiles.
i
p = F (t; θ, γ) = 1 − exp − (t−γ)
,
θ
h
Linearizing the Exponential CDF
t ≥ γ.
6-3
• Obtain graphical estimates of model parameters (e.g., by
fitting a straight line through the points on a probability
plot).
CDF:
Chapter 6
Probability Plotting
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 Fb 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.
6-2
Probability Plotting Scales: Linearizing a CDF
Main Idea: For a given cdf, F (t), one can linearize the
{ t versus F (t) } plot by:
• Finding transformations of F (t) and t such that the relationship between the transformed variables is linear.
• The transformed axes can be relabeled in terms of the original probability and time variables.
6-4
The resulting probability axis is generally nonlinear and is
called the probability scale. The data axis is usually a linear
axis or a log axis.
0
θ, γ = 50, 0
200
600
200, 0
tp = γ − θ log(1 − p)
50, 200
200, 200
400
800
1000
0
1
2
3
4
Plot with Exponential Distribution Probability Scales
Showing Exponential cdfs as Straight Lines for
Combinations of Parameters θ = 50, 200 and γ = 0, 200
.99
.98
.95
.9
.8
.7
.6
.4
.2
.01
Time
6-6
Standard exponential quantile
Quantiles : tp = γ − θ log(1 − p).
tp on the horizontal axis and p on the vertical axis.
intercept on the time axis and 1/θ is equal to the
the cdf line.
Conclusion:
The { tp versus − log(1 − p) } plot is a straight line.
We plot
γ is the
slope of
Note:
Changing θ changes the slope of the line and changing γ
changes the position of the line.
6-5
Proportion Failing
CDF:
−∞ < y < ∞.
Linearizing the Normal CDF
p = F (y; µ, σ) = Φnor y−µ
,
σ
−1
(p).
Quantiles : yp = µ + σΦnor
−1
Φnor
(p) is the p quantile of the standard normal distribution.
Conclusion:
−1
(p) } will plot as a straight line.
{ yp versus Φnor
µ is the point at the time axis where the cdf intersects the
Φ−1(p) = 0 line (i.e., p = .5). The slope of the cdf line on
the graph is 1/σ.
Linearizing the Lognormal CDF
t > 0.
6-7
Note:
Any normal cdf plots as a straight line with positive slope.
Also, any straight line with positive slope corresponds to a
normal cdf.
CDF:
i
i
h
,
p = F (t; µ, σ) = Φnor log(t)−µ
σ
h
0
µ, σ = 40, 10
20
80, 10
60
80, 5
−1
(p)
yp = µ + σΦnor
40, 5
40
80
100
-2
-1
0
1
2
Plot with Normal Distribution Probability Scales
Showing Normal cdfs as Straight Lines for
Combinations of Parameters µ = 40, 80 and σ = 5, 10
.99
.98
.95
.9
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
.99
.98
.95
.9
.8
.7
.6
.5
.4
.3
.2
.1
.05
.02
.01
0.0
2
0.5
exp(µ), σ = 50, 2
1
1.0
10
20
1.5
50
2.0
500, 2
500, 1
100
200
−1
(p)σ
log(tp) = µ + Φnor
50, 1
5
Time
2.5
3.0
1000
2
1
0
-1
-2
6-8
6 - 10
6 - 12
• Exponential cdfs plot as straight lines with slopes equal to 1.
• Any Weibull cdf plots as a straight line with positive slope.
And any straight line with positive slope corresponds to a
Weibull cdf.
• The slope of the cdf line on the graph is β = 1/σ (but in
the computations use base e logarithms for the times rather
than the base 10 logarithms used for the figures).
• η = exp(µ) can be read from the time axis at the point
where the cdf intersects the log[− log(1−p)] = 0 line, which
corresponds to p ≈ 0.632.
Comments:
Linearizing the Weibull CDF-Continued
500
Plot with Lognormal Distribution Probability Scales
Showing Lognormal cdfs as Straight Lines for
Combinations of exp(µ) = 50, 500 and σ = 1, 2
Time
Standard quantile
Standard quantile
−1
Quantiles : tp = exp µ + σΦnor
(p) .
−1
(p)σ
Then log(tp) = µ + Φnor
Conclusion:
−1
{ log(tp) versus Φnor
(p) } will plot as a straight line.
6-9
exp(µ) can be read from the time axis at the point where
−1
the cdf intersects the Φnor
(p) = 0 line. The slope of the cdf
line on the graph is 1/σ (but in the computations use base e
logarithms for the times rather than the base 10 logarithms
used for the figures).
i
Linearizing the Weibull CDF
h
,
p = F (t; µ, σ) = Φsev log(t)−µ
σ
i
t > 0.
Note:
Any given lognormal cdf plots as a straight line with positive
slope. Also, any straight line with positive slope corresponds
to a lognormal distribution.
CDF:
h
−1
Quantiles : tp = exp µ + σΦsev
(p) = η[− log(1 − p)]1/β ,
−1
(p) = log[− log(1 − p)], η = exp(µ), β = 1/σ.
where Φsev
This leads to
1
log(tp) = µ + log[− log(1 − p)]σ = log(η) + log[− log(1 − p)]
β
Conclusion:
{ log(tp) versus log[− log(1 − p)] } will plot as a straight line.
6 - 11
Proportion Failing
Proportion Failing
0.0
1
1.0
500, 1
10
20
1.5
50
2.0
50, 1
500, .5
100
200
2.5
500
3.0
1000
log(tp) = log(η) + log[− log(1 − p)] β1
0.5
5
η, β = 50, .5
2
-4
-3
-2
-1
0
1
Plot with Weibull Distribution Probability Scales
Showing Weibull cdfs as Straight Lines for
Combinations of η = 50, 500 and β = .5, 1
.98
.9
.7
.5
.3
.2
.1
.05
.03
.02
.01
Time
Criteria for Choosing Plotting Positions
6 - 13
Standard quantile
Some applications that suggest criteria:
• Checking distributional assumptions.
• Estimation of parameters.
1.0
0.8
0.6
0.4
0.2
0.0
0
5000
10000
15000
Kilometers
20000
25000
6 - 17
Nonparametric Estimate of F (t) for the Shock
Absorbers. Simultaneous Approximate 95%
Confidence Bands for F (t)
6 - 15
Criteria for choosing plotting positions should depend on the
application or purpose for constructing the probability plot.
Proportion Failing
• Display of maximum likelihood results with data.
Proportion Failing
Choosing Plotting Positions to
Plot the Nonparametric Estimate of F
• The discontinuity and randomness of Fb (t) make it difficult to choose a definition for pairs of points (t, Fb ) to plot.
• With times reported as exact, it is has been traditional to
plot { ti versus Fb (ti) } at the observed failure times.
General Idea: Plot an estimate of F at some specified set
of points in time and define plotting positions consisting of
a corresponding estimate of F at these points in time.
6 - 14
Plotting Positions: Continuous Inspection Data
and Multiple Censoring
Fb (t) is a step function until the last reported failure time,
but the step increases may be different than 1/n.
i
h
io
1nbh
F t(i) + ∆ + Fb t(i) − ∆ .
2
Plotting Positions: { t(i) versus pi} with
pi =
Justification: This is consistent with the definition for single
censoring.
6 - 16
Plotting Positions: Continuous Inspection Data and
Single Censoring
n
.
ti versus i−.5
n
o
Let t(1), t(2), . . . be the ordered failure times. When there is
not ties, Fb (t) is a step function increasing by an amount 1/n
until the last reported failure time.
Plotting Positions:
• Justification:
i
h
io
1nbh
i − .5
F t(i) + ∆ + Fb t(i) − ∆
=
n
2
h
i
i
−
.5
E t(i) ≈ F −1
.
n
where ∆ is positive and small.
• When the model fits well, the ML line approximately goes
through the points.
• Need to adjust these plotting positions when there are ties.
6 - 18
4.0
•
• •
4.2
15000
.8
.4
.1
.01
.0005
.98
.8
.3
.05
.005
.0005
5000
5000
•
••
20000
•
4.4
••
25000
15000
Weibull Probability Plot
10000
15000
Lognormal Probability Plot
10000
15000
Loglogistic Probability Plot
10000
200
200
200
0
-2
-4
-6
25000
25000
25000
6 - 19
6 - 21
300
300
300
.98
.8
.9
.95
.7
.6
.4
.3
.2
.1
.05
.02
.005
.002
.0005
1.0
0.8
0.6
0.4
0.2
0.0
.999
.98
.9
.7
.5
.3
.2
.1
.05
.02
.01
.005
.003
.001
.0005
50
3.8
1.8
150
4.0
10000
•
• •
4.2
15000
Kilometers
200
•
250
•
•
20000
Thousands of Cycles
150
200
•
2.4
250
4.4
•
25000
•
300
300
• •• •
••• ••
•••••••
••••
•• ••
••••
••• •
•••
2.2
Log Thousands of Cycles
••
2.0
•
•
•
100
Thousands of Cycles
350
2
1
0
-1
-2
-3
6 - 20
6 - 22
350
-6
-4
-2
0
2
Weibull Probability Plot for the Alloy T7987 Fatigue
Life and Simultaneous Approximate 95% Confidence
Bands for F (t)
100
Plot of Nonparametric Estimate of F (t) for the Alloy
T7987 Fatigue Life and Simultaneous Approximate
95% Confidence Bands for F (t)
•
Lognormal Probability Plot of the Shock Absorber
Data. Also Shown are Simultaneous Approximate 95%
Confidence Bands for F (t)
•
30000
30000
150
Weibull Probability Plot
100
150
Lognormal Probability Plot
100
150
Loglogistic Probability Plot
100
•
Weibull Probability Plot of the Shock Absorber Data.
Also Shown are Simultaneous Approximate 95%
Confidence Bands for F (t)
3.8
•
10000
25000
25000
50
50
50
5000
Log Kilometers
5000
Log Kilometers
.98
.9
.7
.5
.3
.2
.1
.05
.02
.01
.005
.003
.001
.0005
Kilometers
Six-Distribution Probability Plots
of the Shock Absorber Data
20000
20000
5000
Shock Absorber Data (Both Failure Modes)
Probability Plots and Simultaneous 95% Confidence Intervals
Smallest Extreme Value Probability Plot
15000
15000
30000
.98
.5
25000
.1
.02
20000
.98
.5
15000
.1
.02
10000
Logistic Probability Plot
10000
Normal Probability Plot
10000
.003
.0005
5000
5000
5000
.98
.003
.0005
.98
.8
.4
.1
.01
.0005
.98
.8
.3
.05
.005
.0005
Six-Distribution Probability Plots
Alloy T7987 Fatigue Life
300
Alloy T7987 Fatigue Data
Probability Plots and Simultaneous 95% Confidence Intervals
Smallest Extreme Value Probability Plot
250
.98
.5
.1
.02
.003
200
.98
.5
.1
.02
.003
150
300
.0003
100
250
.0003
50
Normal Probability Plot
200
.95
.6
.1
.01
.001
.0001
.98
.8
.4
.1
150
200
300
.98
.8
.4
.1
100
150
Logistic Probability Plot
100
250
.005
.0001
50
50
.005
.0001
.95
.6
.1
.01
.001
.0001
Standard quantile
6 - 24
Standard quantile
Standard quantile
6 - 23
Proportion Failing
Proportion Failing
Proportion Failing
Proportion Failing
Proportion Failing
Proportion Failing
.98
.995
.95
50
1.8
2.0
150
200
2.4
250
300
••
• ••
•• •
•••
••••
•••
•• •
•••
• ••••
•••
•••
2.2
Log Thousands of Cycles
••
•
•
100
Weeks
350
100
2
1
0
-1
-2
-3
6 - 25
•
1.0
1.5
•
2.0
Years
2.5
3.0
•
0.00
0.05
0.10
0.15
0.20
Exponential Distribution Probability Plot of the
Heat-Exchanger Tube Crack Data and Simultaneous
Approximate 95% Confidence Bands for F (t)
.2
.1
.001
95
90
80
70
60
50
40
30
20
5
10
2
1
.5
.2
5
10
Weeks
100
R Proc Reliability
SAS
Nonparametric Lognormal Probability Plot
of the IUD Data
6 - 26
6 - 30
6 - 28
R Proc Reliability probability plots
Here we show some SAS
for the IUD data.
Biomedical Examples
Standard quantile
.9
.8
.7
.5
.3
.2
.1
.05
.02
.005
.002
.0005
Let (t0, t1], . . . , (tm−1, tm] be the inspection times.
The upper endpoints of the inspection intervals ti, i = 1, 2, . . .,
are convenient plotting times.
Plotting Positions: { ti versus pi }, with
pi = Fb (ti)
When there are no censored observations beyond tm,
F (tm) = 1 and this point cannot be plotted on probability
paper.
E[Fb (ti)] = F (ti).
10
6 - 27
Justification: with no losses, from standard binomial theory,
95
90
80
70
60
50
40
30
20
10
5
2
1
.5
.2
5
Proportion Failing
Lognormal Probability Plot for the Alloy T7987
Fatigue Life and Simultaneous Approximate 95%
Confidence Bands for F (t)
Proportion Failing
Plotting Positions: Interval Censored Inspection Data
Thousands of Cycles
Standard quantile
6 - 29
Percent
R Proc Reliability
SAS
Weibull Probability Plot of the IUD Data
Percent
Probability Plots with Specified Shape Parameters
The probability plotting techniques can be extended to construct probability plots for:
• Distributions that are not members of the location-scale
family.
6 - 31
• To help identify, graphically, the need for non-zero threshold
parameter.
• Estimate graphically a shape parameter.
Distributions with a Threshold Parameter
• The lognormal, Weibull, gamma, and other similar distributions can be generalized by the addition of a threshold
parameter, γ, to shift the beginning of the distribution away
from 0.
• These distributions are particularly useful for fitting skewed
distributions that are shifted far to the right of 0.
"
#
log(t − γ) − µ
,
σ
t>γ
6 - 33
• For example, the cdf and quantiles of the 3-parameter lognormal distribution can be expressed as
p = F (t; µ, σ, γ) = Φnor
.99
.98
.9
.95
50
50
150
••
κ = .8
200
••
•••
••••
••••
••••
••••••
•••••••••••••••
100
κ=2
200
••
300
• •
••
••
250
250
300
•
••
•• •
Thousands of Cycles
150
••••
•••
•••••
••••
•••••
•••••
•••••••
•••••
100
350
350
Proportion Failing
.8
.5
.001
.99
.98
.95
.9
.7
.5
.2
.001
Thousands of Cycles
Proportion Failing
.99
.98
.9
.95
.7
.4
.001
.98
.9
.8
.6
.3
.1
.001
50
50
150
κ = 1.2
200
•
•• •
•••
••••
••••
••••
•••••
•••••••
•••••••••
100
κ=5
200
300
•
••
•• •
250
250
300
•
••
••
Thousands of Cycles
150
••
••••
•••••
••••
••••
•••••
••••••
•••
••••
••
100
Thousands of Cycles
6 - 35
350
350
Gamma Probability Plot with κ = .8, 1.2, 2, 5 for the
Alloy T7987 Fatigue Life with Simultaneous
Approximate 95% Confidence Bands for F (t)
Proportion Failing
Proportion Failing
γ=1
2
γ=2
3
4
t
5
γ=3
6
7
8
t > γ.
6 - 32
Pdf for three-parameter lognormal distributions
for µ = 0 and σ = .5 with γ = 1,2,3
0.8
0.6
0.4
f(t)
0.2
0.0
1
p = F (t; θ, κ, γ) = ΓI t−γ
θ ;κ ,
R
Linearizing the 3-Parameter Gamma CDF
CDF:
R
Quantiles : tp = γ + ΓI−1(p; κ)θ.
where ΓI(z; κ) = 0z xκ−1e−xdx/Γ(κ) and Γ(κ) = 0∞ xκ−1e−xdx.
Conclusion:
{ tp versus ΓI−1(p; κ) } will plot as a straight line.
The probability axis depends on specification of the shape
parameter κ.
γ is the intercept on the time axis (because ΓI−1(p; κ) = 0
when p = 0). The slope of the cdf line is equal to 1/θ.
6 - 34
Note:
Changing θ changes the slope of the line and changing γ
changes the position of the line.
i
p = F (t; µ, σ) = Φsev log(t−γ)−µ
,
σ
h
t > γ.
Linearizing the 3-Parameter Weibull CDF Using Linear
Time Axis and Specified Shape Parameter
CDF:
Quantiles : tp = γ + η[− log(1 − p)]1/β ,
where Φsev (z) = 1 − exp[− exp(z)], η = exp(µ), β = 1/σ.
Conclusion:
{ tp versus [− log(1 − p)]1/β } will plot as a straight line.
• The probability axis for this linear-time-axis Weibull probability plot requires specification of the shape parameter β.
• γ is the intercept on the time axis. The slope of the cdf
line is equal to 1/η.
6 - 36
• The plot allows graphical estimation the threshold parameter γ.
50
100
150
•
•• •
200
•
•••
••
••
••
••
•• •
••
•• •••
•••••
•••
•••
Thousands of Cycles
• •
••
••
250
300
350
6 - 37
Linear-Scale Weibull Plot with β = 1.4 for the Alloy
T7987 Fatigue Life with Simultaneous Approximate
95% Confidence Bands for F (t)
.99
.98
.95
.9
.8
.7
.6
.5
.4
.3
.2
.1
.001
.7
.4
.2
10
•
20
•
•
100
200
•
200
•
• •
• • ••
••
•••
••
••
•
50
100
•
•
••
••
• •
••
••
••
••
••
50
0
-10
-20
-30
-40
2.0
1.5
1.0
0.5
0.0
Proportion Failing
Proportion Failing
κ = .1
•
•
κ=4
Millions of Cycles
•
20
Millions of Cycles
.95
.7
.4
.2
.1
.01
.98
.9
.7
.5
.3
.1
.01
10
10
•
20
•
•
50
100
••
••
• ••
••
•• •
••
••
••
κ=1
•
•
50
100
•
•
••
••
• •
••
••
••
••
••
κ = 20
Millions of Cycles
•
20
Millions of Cycles
•
200
•
200
1
0
-1
-2
-3
-4
3.4
3.2
3.0
2.8
2.6
2.4
Standard quantile
6 - 39
Standard quantile
.1
.01
.98
.9
.7
.4
.2
.1
.01
10
Standard quantile
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
•
•
0
•
••
••
1
•
2
2
3
3
•
•••
••
•••
1.0
••
•• •
•••
••
••
•
•
0.0
1
••
••••
••
••
•••
••
•
0
•
•
4
•
5
•
•
• •
4
•
5
•
•
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
•
•
•
••
• •
1.0
1.0
••• •
•••
• ••
1.0
•
••
••
•• •
•••
••
••
•
0.0
•
0.0
•••
•••
••
•••
••
•
0.0
•
2.0
•
2.0
••
••
•
•
•
•
•
2.0
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
••
• •
•
1
•
•
2
3
•
Sample Size= 10
•
•
0
•
••
•• •
1.0
• •
•
2.0
•
•
Sample Size= 20
•• •
••
••
•
•
0.0
3
• •
• ••
•• • •
•• •
2
4
5
•
Sample Size= 40
1
•••
•••
••
••
•••
••
•
0
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
••
•
•
•
0.0
1.0
1
•
•••
•
••
•••
••
••
•
•
0
•
2.0
2
••
••
1.0
3
2.0
3.0
•
•
•
••
• ••
••
• •••
•••••
••
••
•••
••
0.0
•
•
•
•
4
•
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
•
•
0.5
1
0.4
••
••
0.8
3
1.5
2
•
••
•••
•••
•••••
••
••
•••
••
•
0
•
•
•
2.5
4
1.2
•
•
•
•
•
6
6 - 41
5
• •
• •
••
• •
•••
•• •
••
••
•
•
0.0
•
Random Exponential Variates Plotted on Normal
Probability Plots with Sample Sizes of n=10, 20, and
40. Five Replications of Each Probability Plot
Standard quantile
GENG Probability Plots of the Ball Bearing Fatigue
Data with Specified κ= .1, 1, 4, and 20
Proportion Failing
Proportion Failing
Proportion Failing
p = F (t; θ, β, κ) = ΓI
t β ;κ .
θ
Linearizing the Generalized Gamma CDF
CDF:
i1/β
h
.
Quantiles : tp = θ ΓI−1(p; κ)
Then log(tp) = log(θ) + log[ΓI−1(p; κ)] β1 .
Conclusion:
{ log(tp) versus log[ΓI−1(p; κ)] } will plot as a straight line.
The scale parameter θ is the intercept on the time scale, corresponding to the time where the cdf crosses the horizontal
line at log[ΓI−1(p; κ)] = 0.
The slope of the line on the graph with time on the horizontal
axis is β.
6 - 38
Note: The probability scale for the GENG probability plot
requires a given value of the shape parameter κ.
-2
• •
•
•
•
-0.5
•
•
-2
•
• •
0.5
0
•••
•••
••
-1
0
•
1.5
•
•
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1
1
•
••
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• • ••
••
•••
•••
••
-1
•
••
•
•
•
•
2
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
•
-2
•
•
•
0
0
1
1
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1
•
2
•
•
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0
•
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-1
•
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-1
•
-2
• ••
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-2
•
•
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
-1
•
•
••
•
•
• •
0
1
2
•
Sample Size= 10
•
0
1
••
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••
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• ••
-1
••
••
•
2
•
•
Sample Size= 20
•
-2
0
1
•
2
•
Sample Size= 40
-1
••
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• ••
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•••
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•
-2
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
•
-1
•
-1.5
•
•
-2
•
•
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•
1
•
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0
2
•
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1
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2
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0
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•
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-2
•
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
•
•
•
•
-0.5
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•
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0.0
••
• ••
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•
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0
0.5
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•
•
•
2
•
•
•
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1.0
1
•
••
• ••
•••
•••
•••
••
• ••
• •
•
-1.0
•
Random Normal Variates Plotted on Normal
Probability Plots with Sample Sizes of n=10, 20, and
40. Five Replications of Each Probability Plot
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
0.99
0.90
0.70
0.30
0.05
0.01
6 - 40
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.
6 - 42
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
0
Hours
1000
1500
Bleed Failure Data
Bleed System (All Bases)
Event Plot
500
100
DBase
OtherBase
50
200
Hours
500
1000
2000
2000
Bleed Failure Data
With Individual Weibull Distribution ML Estimates
Weibull Probability Plot
Bleed System
Separate Weibull Probability Plots
for Base D and Other Bases
20
6 - 43
Count
39
52
46
31
48
102
158
2
312
101
2
101
2
100
9
100
23
2
56
27
55
20
56
22
55
22
56
11
53
11
55
20
55
8
55
4
55
2
152
3
152
3
6 - 45
5000
6 - 47
Proportion Failing
0.020
0.015
0.010
200
All Bases
Hours
600
•
•
•
1000
Only Base D
500
Hours
200
••
••
••
2000
1400
.02
.01
.003
.001
.0005
.0002
.02
.01
.003
.001
.0005
.0002
50
50
•
•
200
Hours
500
1000
•
All Bases
•
500
Hours
200
•
Hours
200
500
•
Omitting Base D
•
2000
•
•
•
2000
•
2000
6 - 44
6 - 46
5000
•
•
•
•• •
•••
••
••
•
Bleed System
Failure Data Analysis
CDF plot and Weibull Probability Plots
0
50
Bleed System (All Bases)
Weibull Probability Plot
100
Bleed Failure Data
with Nonparametric Simultaneous 95% Confidence Bands
Weibull Probability Plot
50
Bleed System
Failure Data Analysis-Conclusions
20
Proportion Failing
Proportion Failing
0.005
0.0
.2
.1
.03
.01
.003
.00001
.00002
.00005
.0001
.0002
.0005
.001
.003
.005
.01
.03
.02
.05
.0005
Proportion Failing
6 - 48
• The problem at base D was caused by salt air. A change in
maintenance procedures there solved the main part of the
reliability problem with the bleed systems.
• The relatively small slope for the other bases (β ≈ .85)
suggested infant mortality or accidental failures.
• The large slope (β ≈ 5) for Base D indicated strong wearout.
• Separate analyses of the Base D data and the data from
the other bases indicated different failure distributions.
Fraction Failing
Bleed System Failure Data
(Abernethy, Breneman, Medlin, and Reinman 1983)
• Failure and running times for 2256 bleed systems.
.0001
.0003
.0002
.0005
.001
.003
.005
.01
.03
.02
.05
.1
• The Weibull probability plot suggest changes in the failure
distribution after 600 hours. The data shows that 9 of the
19 failures had occurred at Base D.
Fraction Failing
Transmitter Vacuum Tube Data
(Davis 1952)
Number
Failing
109
42
17
7
13
6 - 49
• Life data for a certain kind of transmitter vacuum tube used
in the output stage of high-power transmitters.
Days
Interval Endpoint
Lower
Upper
0
25
25
50
50
75
75
100
100
∞
• The data are read-out (interval censored) data.
F(t)
.98
1.3
1.4
•
1.5
•
1.7
Log Days
1.6
Days
1.8
60
•
1.9
80
2.0
•
100
-1.0
-0.5
0.0
0.5
1.0
Weibull Probability Plot of the V7 Transmitter Tube
Failure Data with Simultaneous Approximate 95%
Confidence Bands for F (t)
β
.9
.7
.5
40
6 - 51
6 - 53
σ
0
?
|
|
?
40
|
|
Days
?
60
|
|
80
Transmitting Tube Time-to-Failure Data
20
?
100
|
V7 Transmitter Tube Failure Data
Event Plot
Row
1
2
3
4
5
F(t)
.98
.95
.9
.8
.7
1.3
1.4
•
1.5
Days
•
1.7
Log Days
1.6
40
1.8
60
•
1.9
80
2.0
•
100
13
7
17
42
109
Count
2.0
1.5
1.0
0.5
0.0
-0.5
6 - 50
6 - 52
Lognormal Probability Plot of the V7 Transmitter
Tube Failure Data with Simultaneous Approximate
95% Confidence Bands for F (t)
3.0
2.0
1.5
1.0
0.5
.6
.5
.4
.3
20
Standard quantile
0.5
1.0
1.5
2.0
2.5
3.0
.3
20
Other Topics in Chapter 6
Probability plotting for arbitrarily censored data.
Standard quantile