Making Use of Reliability Statistics

Fred Schenkelberg

Everything Varies

Decisions to Make

Questions to Answer

Experiments to Analyze

Convincing Evidence

How do you do treat data?

EXPLORATORY DATA ANALYSIS

Fun with Plotting

Given Some Data

Rural MaleRural FemaleUrban MaleUrban Female

Rural MaleRural FemaleUrban MaleUrban Female

Rural MaleRural FemaleUrban MaleUrban Female

Rural MaleRural FemaleUrban MaleUrban Female

Rural MaleRural FemaleUrban MaleUrban Female

50-54

55-59

60-64

65-69

70-74

0 20 40 60 80 100

Death Rates in Virginia - 1940

Dot Plot

A B C D E F G H

25

1020

50100Box Plot

Histogram 1990 - 2010 California Temperatures (°C)

Celisus

Density

-10 0 10 20 30 40

0.00

0.01

0.02

0.03

0.04

Q-Q Plot

0 5 10 15

05

1015

x

qchi

sq(p

poin

ts(x

), df

= 4

)

Basic View of Dataset 0 20 40 60 80 100 120

020

040

060

0

x

020

4060

8010

0

x

0 50 100 150

0.0

0.01

00.

020

Quantiles of Standard Normal

x

-2 0 2

020

4060

8010

0

Scatter or Run Plot

0 10 20 30 40 50

-2-1

01

2

Simple Use of Color In a Plot

Just a Whisper of a Label

Scatter Plots in Matrix

Sepal.Length

2.0 2.5 3.0 3.5 4.0 0.5 1.0 1.5 2.0 2.5

4.5

5.5

6.5

7.5

2.02.53.03.54.0

Sepal.Width

Petal.Length

12

34

56

7

4.5 5.5 6.5 7.5

0.51.01.52.02.5

1 2 3 4 5 6 7

Petal.Width

Edgar Anderson's Iris Data

With the right tool this is easy

FIELD DATA Let’s explore some data

3 Months of Field Data Concentrator Field Data 2p Weibull vs WeiBayes

Folio1\Concentrator 1: m=1.5000, s=0.1000, Rb=2.1122, h=16.4576, Z=0.999817465905239Folio1\Concentrator: b=2.9361, h=9.0133, Z=0.999817465905239

T ime, (t)

Un

relia

bili

ty,

F(t

)

0 . 100 10.0001.0000.100

0.500

1.000

5.000

10.000

50.000

90.000

99.000

0.100

x 137

x 140

x 3

x 137

x 140

x 3

Probability-W eibull

Folio1\ConcentratorW eibull-2PMLE RRM MED FMF=280/S=38062

Data PointsProbability Line

Folio1\Concentrator 1W eibull-Bayesian-2PMLE MED MED BSNF=280/S=38062

Data PointsProbability Line

Fred Schenke lbergConsult ing8/21/200710:12:25 AM

8 Months by System

1.00 100.0010.000.01

0.05

0.10

0.50

1.00

5.00

10.00

50.00

90.00

99.00

0.01

ReliaSoft's Weibull++ 6.0 - www.Weibull.com

Probability - Weibull

Time, (t)

Unr

elia

bilit

y, F

(t)

9/30/2005 10:06Fred Schenkelberg ConsultingFred Schenkelberg

WeibullCompressor

W2 RRX - RRM MEDF=849 / S=153493

β1=2.2557, η1=50.7130, ρ=0.9999

Plastics

W2 RRX - RRM MEDF=1314 / S=153028

β2=1.3281, η2=162.3354, ρ=0.9994

Sieve

W2 RRX - RRM MEDF=360 / S=143343

β3=2.0955, η3=89.5413, ρ=0.9998

Solenoid

W2 RRX - RRM MEDF=550 / S=153792

β4=2.4939, η4=48.1558, ρ=0.9999

System

W2 RRX - RRM MEDF=29930 / S=311217CB[FM]@90.00%2-Sided-B [T2]

β5=1.5656, η5=46.3456, ρ=0.9992

Time to Repair Data ReliaSoft W eibull++ 7 - www.Re liaSoft. com

Probability - Lognormal

µ =−1 .4 6 2 5 , σ= 1 .6 5 0 8 , ρ= 0 .9 6 1 1

T ime, (t)

Un

relia

bili

ty,

F(t

)

0 . 010 100.0000.100 1.000 10.0000.010

0.0500.100

0.500

1.000

5.000

10.000

50.000

99.990

0.010

Probability-Lognormal

Line 4 Depa lle t izer Single Serving MTTRLognormal-2PRRX SRM MED FMF=1245/S=0

Data PointsProbability Line

Fred Schenke lbergConsult ing11/23/20072:19:26 PM

Count of Failures over Cut

Depth

DEPTH CUT

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2000 4000 6000 8000 10000 12000 14000 16000

Frac

tion

Failin

g

Thu May 11 17:05:37 PDT 2006

RSS DEPTH CUT data Nonparametric CDF Estimate

with Nonparametric pointwise 95% Confidence Bands

Better Questions

Questions?

EXPERIMENTS Asking better questions

Setting Priorities

R t( ) = e−tη( )β

Comparison (hypothesis test)

1 2-1

01

23

45

group

extra

Welch Two Sample t-test data: extra by group t = -1.8608, df = 17.776, p-value = 0.07939 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -3.3654832 0.2054832 sample estimates: mean in group 1 mean in group 2 0.75 2.33

Regression

Average Children Height in centimeters versus Age in Months

Regression

height = 0.635 age + 64.928

Response Surface

Design of Experiments

Questions?

Sample Size? Just how many do we need?

Minimum Samples n = ln 1−C( )

m ln R( )

Talk about the Risk

Options with Limited Samples

How do you extract value from limited samples?

Next Steps on Your Journey 1.  Gather failure data 2.  Plot the data 3.  Ask questions 4.  Embrace Statistics 5.  Enjoy!

DESIGNING EXPERIMENTS

Consider objectives and possible outcomes

What is the Objective?

What are the possible

outcomes?

What is the decision point?

Questions?

FINDING VALUE

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