Chapter 6
Probability Plotting
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
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.
January 13, 2014
3h 41min
6-1
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
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.
• Obtain graphical estimates of model parameters (e.g., by
fitting a straight line through the points on a probability
plot).
6-3
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.
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.
6-4
Linearizing the Exponential CDF
CDF:
i
(t−γ)
p = F (t; θ, γ) = 1 − exp − θ
,
Quantiles : tp = γ − θ log(1 − p).
h
t ≥ γ.
Conclusion:
The { tp versus − log(1 − p) } plot is a straight line.
We plot tp on the horizontal axis and p on the vertical axis.
γ is the intercept on the time axis and 1/θ is equal to the
slope of the cdf line.
Note:
Changing θ changes the slope of the line and changing γ
changes the position of the line.
6-5
Plot with Exponential Distribution Probability Scales
Showing Exponential cdfs as Straight Lines for
Combinations of Parameters θ = 50, 200 and γ = 0, 200
tp = γ − θ log(1 − p)
.99
4
.98
200, 0
.95
3
50, 200
.9
2
200, 200
.8
.7
1
.6
Standard exponential quantile
Proportion Failing
θ, γ = 50, 0
.4
.2
.01
0
0
200
400
600
800
1000
Time
6-6
Linearizing the Normal CDF
CDF:
y−µ
p = F (y; µ, σ) = Φnor σ ,
Quantiles : yp = µ + σΦ−1
nor (p).
−∞ < y < ∞.
Φ−1
nor (p) is the p quantile of the standard normal distribution.
Conclusion:
{ yp versus Φ−1
nor (p) } will plot as a straight line.
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.
Note:
Any normal cdf plots as a straight line with positive slope.
Also, any straight line with positive slope corresponds to a
normal cdf.
6-7
Plot with Normal Distribution Probability Scales
Showing Normal cdfs as Straight Lines for
Combinations of Parameters µ = 40, 80 and σ = 5, 10
yp = µ + σΦ−1
nor (p)
.99
.98
2
.95
.9
Proportion Failing
.7
.6
.5
0
.4
.3
.2
Standard quantile
1
.8
-1
80, 10
.1
.05
µ, σ = 40, 10
.02
40, 5
-2
80, 5
.01
0
20
40
60
80
100
Time
6-8
Linearizing the Lognormal CDF
h
i
log(t)−µ
CDF:
p = F (t; µ, σ) = Φnor
,
σ
i
h
−1
Quantiles : tp = exp µ + σΦnor (p) .
t > 0.
Then log(tp) = µ + Φ−1
nor (p)σ
Conclusion:
{ log(tp) versus Φ−1
nor (p) } will plot as a straight line.
exp(µ) can be read from the time axis at the point where
the cdf intersects the Φ−1
nor (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).
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.
6-9
Plot with Lognormal Distribution Probability Scales
Showing Lognormal cdfs as Straight Lines for
Combinations of exp(µ) = 50, 500 and σ = 1, 2
log(tp) = µ + Φ−1
nor (p)σ
0.0
0.5
1.0
1.5
2.0
2.5
3.0
.99
.98
2
.95
.9
Proportion Failing
.7
.6
.5
0
.4
.3
500, 2
.2
Standard quantile
1
.8
-1
.1
exp(µ), σ = 50, 2
.05
500, 1
50, 1
-2
.02
.01
1
2
5
10
20
50
100
200
500
1000
Time
6 - 10
Linearizing the Weibull CDF
i
log(t)−µ
, t > 0.
CDF:
p = F (t; µ, σ) = Φsev
σ
i
h
−1
Quantiles : tp = exp µ + σΦsev (p) = η[− log(1 − p)]1/β ,
h
where Φ−1
sev (p) = log[− log(1 − p)], η = exp(µ), β = 1/σ.
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
Linearizing the Weibull CDF-Continued
Comments:
• 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.
• 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).
• Any Weibull cdf plots as a straight line with positive slope.
And any straight line with positive slope corresponds to a
Weibull cdf.
• Exponential cdfs plot as straight lines with slopes equal to 1.
6 - 12
Plot with Weibull Distribution Probability Scales
Showing Weibull cdfs as Straight Lines for
Combinations of η = 50, 500 and β = .5, 1
log(tp) = log(η) + log[− log(1 − p)] β1
0.0
0.5
1.0
1.5
2.0
.98
2.5
3.0
50, 1
1
.9
.7
0
500, .5
η, β = 50, .5
.3
-1
.2
-2
.1
.05
Standard quantile
Proportion Failing
.5
-3
.03
.02
-4
500, 1
.01
1
2
5
10
20
50
100
200
500
1000
Time
6 - 13
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
Criteria for Choosing Plotting Positions
Criteria for choosing plotting positions should depend on the
application or purpose for constructing the probability plot.
Some applications that suggest criteria:
• Checking distributional assumptions.
• Estimation of parameters.
• Display of maximum likelihood results with data.
6 - 15
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.
Plotting Positions: { t(i) versus pi} with
i
h
io
1nbh
pi =
F t(i) + ∆ + Fb t(i) − ∆ .
2
Justification: This is consistent with the definition for single
censoring.
6 - 16
Nonparametric Estimate of F (t) for the Shock
Absorbers. Simultaneous Approximate 95%
Confidence Bands for F (t)
1.0
Proportion Failing
0.8
0.6
0.4
0.2
0.0
0
5000
10000
15000
20000
25000
Kilometers
6 - 17
Plotting Positions: Continuous Inspection Data and
Single Censoring
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:
n
ti versus i−.5
n
o
.
• Justification:
i
h
io
1nbh
i − .5
b
=
F t(i) + ∆ + F 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
Weibull Probability Plot of the Shock Absorber Data.
Also Shown are Simultaneous Approximate 95%
Confidence Bands for F (t)
Log Kilometers
3.8
4.0
4.2
4.4
.98
.9
Proportion Failing
.5
.3
.2
.1
.05
•
•
.02
• •
•
•
•
•
••
-2
-4
•
.01
0
.005
.003
Standard quantile
.7
-6
.001
.0005
5000
10000
15000
20000
25000
Kilometers
6 - 19
Lognormal Probability Plot of the Shock Absorber
Data. Also Shown are Simultaneous Approximate 95%
Confidence Bands for F (t)
Log Kilometers
3.8
4.0
4.2
4.4
.98
2
.95
.9
Proportion Failing
.7
.6
.4
.3
.2
.1
.05
•
•
.02
• •
•
•
•
•
•
•
0
-1
Standard quantile
1
.8
-2
•
.005
.002
-3
.0005
5000
10000
15000
20000
25000
Kilometers
6 - 20
Six-Distribution Probability Plots
of the Shock Absorber Data
Shock Absorber Data (Both Failure Modes)
Probability Plots and Simultaneous 95% Confidence Intervals
.98
.5
Smallest Extreme Value Probability Plot
.1
.02
.1
.02
.003
.0005
.003
.0005
Proportion Failing
5000
10000
15000
20000
25000
30000
Normal Probability Plot
.98
Weibull Probability Plot
.98
.5
5000
15000
25000
Lognormal Probability Plot
.98
.8
.4
.1
.01
.0005
10000
.8
.4
.1
.01
.0005
5000
10000
15000
20000
25000
30000
Logistic Probability Plot
.98
5000
15000
25000
Loglogistic Probability Plot
.98
.8
.3
.05
.005
.0005
10000
.8
.3
.05
.005
.0005
5000
10000
15000
20000
25000
30000
5000
10000
15000
25000
6 - 21
Plot of Nonparametric Estimate of F (t) for the Alloy
T7987 Fatigue Life and Simultaneous Approximate
95% Confidence Bands for F (t)
1.0
Proportion Failing
0.8
0.6
0.4
0.2
0.0
100
150
200
250
300
350
Thousands of Cycles
6 - 22
Six-Distribution Probability Plots
Alloy T7987 Fatigue Life
Alloy T7987 Fatigue Data
Probability Plots and Simultaneous 95% Confidence Intervals
Smallest Extreme Value Probability Plot
.98
.5
.1
.02
.003
.0003
.0003
50
Proportion Failing
Weibull Probability Plot
.98
.5
.1
.02
.003
100
150
200
250
300
50
Normal Probability Plot
.98
.8
.4
.1
.005
.0001
.005
.0001
100
150
200
150
200
300
Lognormal Probability Plot
.98
.8
.4
.1
50
100
250
300
50
Logistic Probability Plot
100
150
200
300
Loglogistic Probability Plot
.95
.6
.1
.01
.001
.0001
.95
.6
.1
.01
.001
.0001
50
100
150
200
250
300
50
100
150
200
300
6 - 23
Weibull Probability Plot for the Alloy T7987 Fatigue
Life and Simultaneous Approximate 95% Confidence
Bands for F (t)
Log Thousands of Cycles
1.8
2.0
2.2
2.4
2
.999
Proportion Failing
.7
.5
.3
.2
.1
.05
•
•
•
.02
.01
.005
.003
•
••
•
•
•
•
•
•
•
••
•
•
••
•• •
•
•
•
• ••
•
•
•
•
•
•
••••
0
-2
-4
Standard quantile
.98
.9
-6
.001
.0005
50
100
150
200
250
300
350
Thousands of Cycles
6 - 24
Lognormal Probability Plot for the Alloy T7987
Fatigue Life and Simultaneous Approximate 95%
Confidence Bands for F (t)
Log Thousands of Cycles
1.8
2.0
2.2
2.4
.995
.98
2
Proportion Failing
.9
.8
.7
.5
.3
.2
.1
•
•
•
.05
.02
.005
.002
••
•
•
•
• ••
•
•
•
• •
•
•
••
•
•
•••
•
•
•••
•
•
•
•
•
•••
1
0
-1
Standard quantile
.95
-2
-3
.0005
50
100
150
200
250
300
350
Thousands of Cycles
6 - 25
Exponential Distribution Probability Plot of the
Heat-Exchanger Tube Crack Data and Simultaneous
Approximate 95% Confidence Bands for F (t)
.2
0.15
.1
0.10
•
•
.001
Standard quantile
Proportion Failing
0.20
0.05
•
0.00
1.0
1.5
2.0
2.5
3.0
Years
6 - 26
Plotting Positions: Interval Censored Inspection Data
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.
Justification: with no losses, from standard binomial theory,
E[Fb (ti)] = F (ti).
6 - 27
Biomedical Examples
R Proc Reliability probability plots
Here we show some SAS
for the IUD data.
6 - 28
R Proc Reliability
SAS
Weibull Probability Plot of the IUD Data
95
90
80
70
60
50
40
30
Percent
20
10
5
2
1
.5
.2
5
10
100
Weeks
6 - 29
R Proc Reliability
SAS
Nonparametric Lognormal Probability Plot
of the IUD Data
95
90
80
Percent
70
60
50
40
30
20
10
5
2
1
.5
.2
5
10
100
Weeks
6 - 30
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.
• To help identify, graphically, the need for non-zero threshold
parameter.
• Estimate graphically a shape parameter.
6 - 31
Pdf for three-parameter lognormal distributions
for µ = 0 and σ = .5 with γ = 1,2,3
0.8
0.6
f(t)
0.4
0.2
γ=1
γ=3
γ=2
0.0
1
2
3
4
5
6
7
8
t
6 - 32
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.
• For example, the cdf and quantiles of the 3-parameter lognormal distribution can be expressed as
"
#
log(t − γ) − µ
p = F (t; µ, σ, γ) = Φnor
,
σ
t>γ
6 - 33
Linearizing the 3-Parameter Gamma CDF
t−γ
p = F (t; θ, κ, γ) = ΓI θ ; κ ,
CDF:
Quantiles : tp = γ + Γ−1
I (p; κ)θ.
R
t > γ.
R
where ΓI(z; κ) = 0z xκ−1e−xdx/Γ(κ) and Γ(κ) = 0∞ xκ−1e−xdx.
Conclusion:
{ tp versus Γ−1
I (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 Γ−1
I (p; κ) = 0
when p = 0). The slope of the cdf line is equal to 1/θ.
Note:
Changing θ changes the slope of the line and changing γ
changes the position of the line.
6 - 34
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)
κ = .8
κ = 1.2
.98
.95
.9
••
••••
•
•
•••
••••
•
•
••••
•••••••••••••••
.8
.5
.001
50
Proportion Failing
Proportion Failing
.99
100
150
200
•
••
•• •
250
300
.9
•
••• •
•
•
•••
•••
•
•
•
•••
••••••
••
•
•
•
•
•
•
•••
.7
.4
.001
50
100
150
200
•
••
•• •
250
300
Thousands of Cycles
κ=2
κ=5
.95
.9
•
••• •
•
•
•••
••••
•
•
••••
•••••
•
•
•
•
•••
••••
50
.95
Thousands of Cycles
.99
.98
.7
.5
.2
.001
.98
350
100
150
200
Proportion Failing
Proportion Failing
.99
• •
•• ••
.98
.9
.8
.6
••
••••
•
•
•••
•••
•
•
•
••••
••••
•
•
•
•
••••
•••
.3
.1
.001
250
300
Thousands of Cycles
350
350
50
100
150
200
• •
• ••
250
300
350
Thousands of Cycles
6 - 35
Linearizing the 3-Parameter Weibull CDF Using Linear
Time Axis and Specified Shape Parameter
CDF:
h
i
log(t−γ)−µ
p = F (t; µ, σ) = Φsev
,
σ
t > γ.
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.
• The plot allows graphical estimation the threshold parameter γ.
6 - 36
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
Proportion Failing
.95
.9
.8
.7
.6
.5
••
•
•
•••
•
•
•
•
•
•
•
•
•
•••
.4
.3
.2
.1
.001
50
100
150
•
•
•
••
•
•
•
•
••
•
•
••
•
••
200
•
•
•
•
••
250
300
350
Thousands of Cycles
6 - 37
Linearizing the Generalized Gamma CDF
CDF:
β
p = F (t; θ, β, κ) = ΓI θt ; κ .
h
i1/β
−1
Quantiles : tp = θ ΓI (p; κ)
.
1
Then log(tp) = log(θ) + log[Γ−1
I (p; κ)] β .
Conclusion:
{ log(tp) versus log[Γ−1
I (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[Γ−1
I (p; κ)] = 0.
The slope of the line on the graph with time on the horizontal
axis is β.
Note: The probability scale for the GENG probability plot
requires a given value of the shape parameter κ.
6 - 38
GENG Probability Plots of the Ball Bearing Fatigue
Data with Specified κ= .1, 1, 4, and 20
•
.1
•
0
-10
-20
-30
•
-40
.4
.2
.1
•
•
••
••••
••
••
•
1
0
-1
-2
•
-3
•
-4
.01
20
50
100
200
20
50
100
Millions of Cycles
Millions of Cycles
κ=4
κ = 20
.7
.4
.2
.1
•
•
•
••
•• •
•
••
•••
•
• •
•
••
•
1.5
1.0
0.5
•
0.0
.01
20
2.0
50
100
Millions of Cycles
200
.98
Proportion Failing
.98
.9
10
10
Standard quantile
10
Proportion Failing
.7
••
••
•
•
•
200
•
.9
.7
.5
.3
.1
•
•
••
•• •
•
••
•
••
•
•• •
•
••
3.4
3.2
3.0
2.8
2.6
•
.01
Standard quantile
.01
.95
Standard quantile
.2
• • •
•• • ••
•
••••
•
••
•
Proportion Failing
.7
.4
κ=1
Standard quantile
Proportion Failing
κ = .1
2.4
10
20
50
100
200
Millions of Cycles
6 - 39
Random Normal Variates Plotted on Normal
Probability Plots with Sample Sizes of n=10, 20, and
40. Five Replications of Each Probability Plot
Sample Size= 10
0.99
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
0.5
1.5
•
•
•
•
••
•
•
0.30
•
0.05
0.01
-2
-1
0.99
•
0.90
0.70
0
1
•
•
•
•
•
•
••
0.30
•
2
0.05
0.01
-1
0.99
•
0.90
0.70
0
1
2
•
•
•
••
•
•
•
0.30
•
0.05
0.01
-1.5
•
0.90
0.70
-0.5
•
•
•
•
••
•
•
•
0.5
-0.5
0.5
Sample Size= 20
0.99
•
0.90
0.70
0.30
0.05
0.01
••
•
•
-2
•
•••
• ••
•
•
•
•
••
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0.99
0.90
0.70
0.30
0.05
0.01
-1
0
1
•
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••
•
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•
•
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•
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0.90
0.70
0.05
0.01
-1
•
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•
-2
0.99
0
1
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-1
0.30
0.05
0.01
0
•
0.90
0.70
•
-2
0.99
1
2
•
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-1
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0.30
0.05
0.01
0
•
0.90
0.70
•
-2
0.99
1
2
••
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•
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-1.0
0.0
1.0
Sample Size= 40
0.99
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•
•
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•
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•
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0.90
0.70
0.30
0.05
0.01
•
-2
-1
0
1
•
0.99
0.90
0.70
0.30
0.05
0.01
2
•
•
•
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-1
0
1
0.99
0.90
0.70
0.30
0.05
0.01
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-1
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•
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0.70
0.30
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0
1
2
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2
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0
1
2
6 - 40
Random Exponential Variates Plotted on Normal
Probability Plots with Sample Sizes of n=10, 20, and
40. Five Replications of Each Probability Plot
Sample Size= 10
0.99
0.99
•
0.90
0.70
0.30
0.05
0.01
•
•
0
•
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0.30
0.05
0.01
1
2
3
4
0.99
•
0.90
0.70
5
•
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0.0
•
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0.01
1.0
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•
0.90
0.70
2.0
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0.70
0.30
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1
2
3
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0.70
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1.0
2.0
3.0
•
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0.0
0.4
0.8
1.2
Sample Size= 20
0.99
0.90
0.70
0.30
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1
2
3
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4
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6
6 - 41
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
Bleed System Failure Data
(Abernethy, Breneman, Medlin, and Reinman 1983)
• Failure and running times for 2256 bleed systems.
• 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.
6 - 43
Bleed System
Failure Data Analysis
CDF plot and Weibull Probability Plots
All Bases
Proportion Failing
0.020
0.015
0.010
.02
.01
.003
•
.001
.0005
0.005
•
•
• ••
•••
•
••
•
•
•
•
•
.0002
0.0
200
600
1000
1400
50
200
500
Hours
Hours
Only Base D
Omitting Base D
.2
.1
.03
•
•
.01
.003
Proportion Failing
0
Proportion Failing
Proportion Failing
All Bases
••
•• •
•
.02
.01
.0005
•
•
.003
•
.001
.0005
•
2000
•
•
•
•
•
.0002
50
200
500
Hours
2000
50
200
500
2000
Hours
6 - 44
Bleed System (All Bases)
Event Plot
Bleed Failure Data
Count
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
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
0
500
1000
1500
2000
Hours
6 - 45
Bleed System (All Bases)
Weibull Probability Plot
Bleed Failure Data
with Nonparametric Simultaneous 95% Confidence Bands
Weibull Probability Plot
.05
.03
.02
.01
.005
Fraction Failing
.003
.001
.0005
.0002
.0001
.00005
.00002
.00001
20
50
100
200
500
1000
2000
5000
Hours
6 - 46
Bleed System
Separate Weibull Probability Plots
for Base D and Other Bases
Bleed Failure Data
With Individual Weibull Distribution ML Estimates
Weibull Probability Plot
.1
DBase
OtherBase
.05
.03
.02
Fraction Failing
.01
.005
.003
.001
.0005
.0003
.0002
.0001
20
50
100
200
500
1000
2000
5000
Hours
6 - 47
Bleed System
Failure Data Analysis-Conclusions
• Separate analyses of the Base D data and the data from
the other bases indicated different failure distributions.
• The large slope (β ≈ 5) for Base D indicated strong wearout.
• The relatively small slope for the other bases (β ≈ .85)
suggested infant mortality or accidental failures.
• 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.
6 - 48
Transmitter Vacuum Tube Data
(Davis 1952)
• Life data for a certain kind of transmitter vacuum tube used
in the output stage of high-power transmitters.
• The data are read-out (interval censored) data.
Days
Interval Endpoint
Lower
Upper
0
25
25
50
50
75
75
100
100
∞
Number
Failing
109
42
17
7
13
6 - 49
V7 Transmitter Tube Failure Data
Event Plot
Transmitting Tube Time-to-Failure Data
Count
Row
1
?
|
|
2
109
?
|
|
3
42
?
4
|
17
|
?
|
5
7
13
0
20
40
60
80
100
Days
6 - 50
Weibull Probability Plot of the V7 Transmitter Tube
Failure Data with Simultaneous Approximate 95%
Confidence Bands for F (t)
Log Days
0.5
1.0
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
F(t)
.98
•
1.0
•
.9
•
1.5
0.5
.7
2.0
0.0
•
2.5
Standard quantile
β
.5
-0.5
3.0
-1.0
.3
20
40
60
80
100
Days
6 - 51
Lognormal Probability Plot of the V7 Transmitter
Tube Failure Data with Simultaneous Approximate
95% Confidence Bands for F (t)
Log Days
1.0
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
F(t)
.98
2.0
.95
•
.9
1.5
•
1.0
0.5
•
.8
.7
0.5
.6
•
.5
Standard quantile
3.0
2.0
1.5
σ
0.0
.4
-0.5
.3
20
40
60
80
100
Days
6 - 52
Other Topics in Chapter 6
Probability plotting for arbitrarily censored data.
6 - 53