Section 5.3
Probability Plots
Probability plots are used to
 Decide if data potentially follow some distribution
 Approximate parameter values can be estimated from the slope and x intercept
o These can even be used if some of the data have not yet been observed
 “Censored” data
 For example only 7 of 10 engines have failed at this point.
 We can’t use X to estimate mean lifetime
o Maximum likelihood estimates of parameter values are better.
o The graphical estimates
 Give a check on estimates derived in other ways.
 Give quick values to use for rough calculations
The distributions we will use
 Normal
 Exponential
 Weibull
In all cases
 Fairly straight means that the distribution could work for those data.
 The slope and x-intercept of the linear trend tell us about values for parameters for the
distributions
o andfor normal distributions
o and threshold for exponential distribution
o and for Weibull distributions
Normal Probability Plots



Done in Chapter 3
Plot standard normal quantiles versus ordered data
o Normal quantiles either from
 Table of quantiles
 Excel using NORMINV function
The new information in chapter 5 are graphical estimates of the mean and standard
deviation.
Plot
Y Axis : quantiles of Z 
X 

Y Axis : Ordered X values
1

Slope
X  intercept: Z=0  X=
Slope 
1

From data
 X intercept ≈ 
 1/slope ≈ 

X




Fairly similar slopes  fairly similar standard deviations.
Lower stress = 900 has larger X intercept indicating a larger mean.
Spring last longer with less stress.
Exponential Probability Plots
P  X  x   F  x   p for x > 0.
1  e x /  p
 ln 1  p  
x

 i  0.5  x
 ln 1 

n  

Plot
X axis : Ordered x
 i  0.5 
Y axis :  ln 1 

n 

 i  0.5 
Plotting  ln 1 
 versus ordered x values should have
n 

 Slope ≈ 1/
o ≈ 1/slope
 Intercept ≈ 0
o A nonzero x-intercept is the threshold which no values are below.
 Threshold ≈ x-intercept
 Essentially Q(0).
o The threshold is like 0 for the usual exponential distribution, the lower bound on
possible values.
o mean time after threshold
o Mean of X = threshold + 
n
65
i
1
2
3
4
5
6
7
8
9
10
11
12
(i-0.5)/n
0.008
0.023
0.038
0.054
0.069
0.085
0.100
0.115
0.131
0.146
0.162
0.177
-LN(1-(i-0.5)/n)
0.008
0.023
0.039
0.055
0.072
0.088
0.105
0.123
0.140
0.158
0.176
0.195
Service
Time
8.00
8.00
8.00
9.00
9.00
10.00
10.00
10.00
10.00
10.00
12.00
12.00
Etc….
-Ln( 1 - (i-0.5)/n)
5.000
y = 0.0641x - 0.566
4.000
3.000
2.000
1.000
0.000
0.00
20.00
40.00
Service Time
60.00
80.00
100.00
Weibull Probability Plot
P  X  x   p  1  e( x / )
1  p  e ( x /  )


ln 1  p   ( x /  ) 
ln   ln 1  p     ln  x    ln  

 i  0.5  
ln   ln 1 
   ln  x    ln  
n  


Plot
X axis : Ordered ln  x 

 i  0.5  
Y axis : ln   ln 1 

n  


Slope  
X intercept  ln( )
  e x intercept
Weibull Probability Plot
Figure 5.25
n
20
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
(i-0.5)/n
0.025
0.075
0.125
0.175
0.225
0.275
0.325
0.375
0.425
0.475
0.525
0.575
0.625
0.675
0.725
0.775
0.825
0.875
Ln(-Ln(1-(i-0.5)/n)))
-3.676
-2.552
-2.013
-1.648
-1.367
-1.134
-0.934
-0.755
-0.592
-0.440
-0.295
-0.156
-0.019
0.117
0.255
0.400
0.556
0.732
Page
276
Voltage
39.4
45.3
49.2
49.4
51.3
52.0
53.2
53.2
54.9
55.5
57.1
57.2
57.5
59.2
61.0
62.4
63.8
64.3
Ln(Voltage)
3.674
3.813
3.896
3.900
3.938
3.951
3.974
3.974
4.006
4.016
4.045
4.047
4.052
4.081
4.111
4.134
4.156
4.164
19
20
0.925
0.975
0.952
1.305
67.3
67.7
Ln(-Ln(1-(i-0.5)/n)))
2.000
y = 9.1269x - 37.232
1.000
0.000
3.500
3.700
3.900
4.100
-1.000
4.300
4.500
Ln(Voltage)
-2.000
-3.000
-4.000
Slope
Guess at 
9.13
X Intercept ≈ 37.232/ 9.1269
Exp(X Intercept)
Maximum likelihood would be better for estimating these
parameters.
4.08
59.11
Guess at Ln()
Guess at 
4.209
4.215