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Chapter 8

Probability Plotting and

Hazard Plotting

Introduction

• When sampling from a population, a probability plot of

data can often yield a better understanding of the

population than traditional statements made only

about the mean and standard deviation.

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8.1 S4/IEE Application Examples:

Probability Plotting

• One random paid invoice was selected each day from last

year’s invoices, where the number of days beyond the due

date was measured and reported (i.e., days sales

outstanding [DSO]). The DSO for each sample was

plotted in sequence of occurrence on a control chart. No

special causes were identified in the control chart. A

normal probability plot of the sample data had a null

hypothesis test of population normality being rejected at a

level of 0.05. A lognormal probability plot tits the data well.

From the lognormal probability plot, we can estimate the

percentage of invoices from the current process that are

beyond a 30-day criterion.

8.1 S4/IEE Application Examples:

Probability Plotting

• One random sample of a manufactured part was selected

each day over the last year, where the diameter of the

parts were measured and reported. The diameter for each

sample was plotted in sequence of occurrence on a control

chart. No special causes were identified in the control

chart. A normal probability plot of the sample data failed to

reject the null hypothesis of population normality at a level

of 0.05. From a normal probability plot we can estimate

the percentage of parts that the customer receives from

the current process that are beyond a specification limit.

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8.2 Description

• Percent characteristics of a population can be

determined from the cumulative distribution function

CDF, which is the integration of the probability density

function (PDF).

• Probability and hazard plots are useful in visually

assessing how well data follow distributions and in

estimating from data the unknown parameters of a

PDF/CDF.

• These plots can also be used to estimate percent less

than (or greater than) characteristics of a population.

8.2 Description

A basic concept behind probability plotting is that if data plotted on a probability distribution scale follow a straight line, then the population from which the samples are drawn can be represented by that distribution.

• Manual plotting: probability papers • Table Q1 Normal probability paper

• Table Q2 Lognormal probability paper

• Table Q3 Weibull probability paper

• Table R1 Normal hazard paper

• Table R2 Lognormal hazard paper

• Table R3 Weibull hazard paper

• http://www.weibull.com/GPaper/index.htm

• Computer programs • Minitab

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8.2 Description

• Probability and hazard plots of the same data are

interchangeable for practice purposes (Nelson 1982).

• The book uses probability plotting when data are not

censored (e.g., all component failure times are

available from a reliability test.)

• The book uses hazard plotting when there are

censored data (e.g., not all the components failed

during a reliability test.)

8.3 Probability Plotting

• The axes are transformed on probability paper

such that a particular CDF shape will appear as a

straight line if the data are from that distribution.

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8.4 Example 8.1: PDF, CDF, and

Probability Plot

5.7

5.8

5.9

6.0

6.1

6.1

6.3

6.3

6.4

6.5

6.6

6.8

7.0

7.4

7.6

3.8

4.6

4.6

4.9

5.2

5.3

5.3

5.4

5.6

5.6

Minitab: Graph Histogram

8.4 Example 8.1: PDF, CDF, and

Probability Plot

Minitab:

Graph

Empirical

CDFs

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8.4 Example 8.1: PDF, CDF, and

Probability Plot

Minitab:

Graph

Probability

Plots

8.4 Example 8.1: PDF, CDF, and

Probability Plot

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Z(i)

i X(i) F(i) Z(i)

1 3.8 0.02 -2.05375

2 4.6 0.06 -1.55477

3 4.6 0.10 -1.28155

4 4.9 0.14 -1.08032

5 5.2 0.18 -0.91537

6 5.3 0.22 -0.77219

7 5.3 0.26 -0.64335

8 5.4 0.30 -0.5244

9 5.6 0.34 -0.41246

10 5.6 0.38 -0.30548

11 5.7 0.42 -0.20189

12 5.8 0.46 -0.10043

13 5.9 0.50 0

14 6.0 0.54 0.100434

15 6.1 0.58 0.201893

16 6.1 0.62 0.305481

17 6.3 0.66 0.412463

18 6.3 0.70 0.524401

19 6.4 0.74 0.643345

20 6.5 0.78 0.772193

21 6.6 0.82 0.915365

22 6.8 0.86 1.080319

23 7.0 0.90 1.281552

24 7.4 0.94 1.554774

25 7.6 0.98 2.053749

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8.5 Probability Plot Positions and

Interpretation of Plots

• Probability plot has one axis that describes

percentage of the population, while the other axis

describes the variable of concern.

• With uncensored data, a simple generic form

commonly used to determine the percentage value

for ranked data, Fi, is 𝐹𝑖 =100(𝑖−0.5)

𝑛 for i=1, 2, …, n.

(Table P)

8.5 Probability Plot Positions and

Interpretation of Plots

• Data from a distribution follow a straight line when

plotted on a probability paper created from that

distribution (normal, lognormal, Weibull, etc.)

• Probability plots have many applications:

• A “knee” indicates that the data are from two or

more distributions.

• Outliers

• Relative to measured data

• Reliability of a device

• Relative to DOE analyses

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8.6 Hazard Plots

• Most nonlife data are complete, i.e., not censored.

• Reliability test of life data may also be complete

when the time to failure of each testing unit is noted.

• Reliability tests commonly contain testing units that

have not experienced failure.

• Multiple time censoring: failure time are noted and

units may be removed from test at any time.

8.6 Hazard Plots

Time

Unit

Under

test

Failure

Continuing test

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8.6 Hazard Plots

1. Ranked data are assigned a reverse rank number (j), which

is independent of whether the data points were from

censoring or failure. A “+” sign indicates that the device

has not yet failed at the noted time.

2. A hazard value (100/ j) for each failure point is determined,

where j is the reverse ranking.

3. Cumulative hazard values are determined for these failure

points. This value is the sum of the current hazard value

and the previous failed cumulative hazard value. These

hazard values may exceed 100%.

4. The cumulative hazards are plotted with the failure times

on hazard plotting paper.

8.7 Example 8.2:

Hazard Plotting X Cen(0

) j Hazard Cum

Haz. X

3.8 1 25 0.0400 0.0400 3.8

4.5 1 24 0.0417 0.0817 4.5

4.6 1 23 0.0435 0.1251 4.6

4.9 1 22 0.0455 0.1706 4.9

5.2 0 21 0.0000 0.1706

5.3 1 20 0.0500 0.2206 5.3

5.3 1 19 0.0526 0.2732 5.3

5.4 0 18 0.0000 0.2732

5.6 1 17 0.0588 0.3321 5.6

5.6 1 16 0.0625 0.3946 5.6

5.7 1 15 0.0667 0.4612 5.7

5.8 0 14 0.0000 0.4612

5.9 1 13 0.0769 0.5381 5.9

6.0 1 12 0.0833 0.6215 6.0

6.1 1 11 0.0909 0.7124 6.1

6.1 1 10 0.1000 0.8124 6.1

6.3 0 9 0.0000 0.8124

6.3 1 8 0.1250 0.9374 6.3

6.4 1 7 0.1429 1.0802 6.4

6.5 1 6 0.1667 1.2469 6.5

6.6 1 5 0.2000 1.4469 6.6

6.8 0 4 0.0000 1.4469

7.0 1 3 0.3333 1.7802 7.0

7.4 1 2 0.5000 2.2802 7.4

7.6 0 1 0.0000 2.2802

1.0

10.0

0.0100 0.1000 1.0000 10.0000

Tim

e (

years

)

Cumulative hazard

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Minitab:

Stat

Reliability

Dist. Analysis

(right censored)

8.7 Example 8.2:

Hazard Plotting

8.7 Example 8.2:

Hazard Plotting

Minitab:

Stat

Reliability

Dist. Analysis

(right censored)

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8.8 Summarizing the Creation of

Probability and Hazard Plots

1. A histogram is a graphical representation of a frequency

distribution determined from sample data. The empirical

cumulative frequency distribution can also be plotted. Both

distributions are estimates of the actual population

distributions. (PDF and CDF)

2. The procedure used to determine a PDF is first to assume

that the data follow certain distribution, and then observe

whether the data can be modeled by that distribution. This

observation of data fit to a distribution can be evaluated

using a probability plot or hazard plot.