MER301: Engineering Reliability

L Berkley DavisCopyright 2009

MER301: Engineering Reliability 1

MER301: Engineering Reliability

LECTURE 2:

Chapter 1: Role of Statistics in EngineeringChapter 2: Data Summary and Presentation


MER301: Engineering ReliabilityLecture 2

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Summary of Lecture 2 Topics

Summary of Chapter 1 Topics Engineering Method Statistics in Engineering Collection of Engineering Data Observing Processes over Time

Summary of Chapter 2 Topics Populations and Samples Data Displays

Dot Diagrams Histograms Box and Whisker Plots Scatter plots

Central Point and Spread Median,Quartiles,Interquartile range Means, Variances and Standard Deviations

L Berkley Davis Copyright 2009


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Engineering Method Successful design and

introduction of a new product is dependent on a rigorous engineering process that is executed with discipline and attention to detail

Design for Six Sigma is one such process that allows the designer to explicitly account for the effects of variation



4Union CollegeMechanical Engineering

Elements of Design for Six Sigma Flowdown of Customer Requirements(CTQ’s) to Engineering

Measurement System Analysis(Gage R&R)

Statistical Design Methods(Probabilistic Analyses) rather than Deterministic(Mathematical) Analysis

Quantitative Transfer Functions linking CTQ’s(Y’s) to x’s

Disciplined Risk Assessment Process

Design Optimization and Robust Design allow products to be minimally sensitive to design, operating and manufacturing variation

Design for Manufacturability/Process Capability to ensure product CTQs are met in light of manufacturing capability

Validation of product performance



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Critical to Quality Variables(CTQ’s) Products/Processes have measures of performance,

operational flexibility, reliability, and cost that are directly seen by the end customer These are called CTQ variables(Big Y’s) and are the

ultimate measurement of an engineered product or process Big Y’s are functions of other variables that the engineer

must control in the design(control variables) or allow to be uncontrolled(noise)

The Designer must understand

Product and Process CTQ’s

),...,( 21 nxxxfnY

L B e r k l e y D a v i sC o p y r i g h t 2 0 0 9

M E R 3 0 1 : E n g i n e e r i n g R e l i a b i l i t yL e c t u r e 2

5U n i o n C o l l e g eM e c h a n i c a l E n g i n e e r i n g

E l e m e n t s o f D e s i g n f o r S i x S i g m a F l o w d o w n o f C u s t o m e r R e q u i r e m e n t s ( C T Q ’ s ) t o E n g i n e e r i n g

M e a s u r e m e n t S y s t e m A n a l y s i s ( G a g e R & R )

S t a t i s t i c a l D e s i g n M e t h o d s ( P r o b a b i l i s t i c A n a l y s e s ) r a t h e r t h a n D e t e r m i n i s t i c ( M a t h e m a t i c a l ) A n a l y s i s

Q u a n t i t a t i v e T r a n s f e r F u n c t i o n s l i n k i n g C T Q ’ s ( Y ’ s ) t o x ’ s

D i s c i p l i n e d R i s k A s s e s s m e n t P r o c e s s

D e s i g n O p t i m i z a t i o n a n d R o b u s t D e s i g n a l l o w p r o d u c t s t o b e m i n i m a l l y s e n s i t i v e t o d e s i g n , o p e r a t i n g a n d m a n u f a c t u r i n g v a r i a t i o n

D e s i g n f o r M a n u f a c t u r a b i l i t y / P r o c e s s C a p a b i l i t y t o e n s u r e p r o d u c t C T Q s a r e m e t i n l i g h t o f m a n u f a c t u r i n g c a p a b i l i t y

V a l i d a t i o n o f p r o d u c t p e r f o r m a n c e



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Measurement System Errors… Total Error in a measurement is defined as the difference

between the True Value and the Measured Value of Y Accuracy of Measurement System is defined as the

difference between a Standard Reference and the Average Observed Measurement

Two general categories of error – Bias or Accuracy Error and Precision Error (excluding gross blunders)

Total Error = Bias Error + Precision Error for independent random variables

Measurement System Error is described by Average Bias Error (Mean Shift)and a statistical estimate of the Precision Error (Variance)

Measurement System Analysis is a Fundamental Part of Every Experiment














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Not Accurate, Not Precise Accurate, Not Precise

Not Accurate, Precise Accurate, Precise

Experimental Gage R&R -Precision and Accuracy



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Engineering Models Mathematical Model:Quantitative description of a

system/event with descriptive equations Physics Based(Mechanistic) Models built from first principles Empirical Models built from Data and Engineering Knowledge Both Physics Based and Empirical Models can be either

Deterministic or Statistical/Probabilistic Deterministic

For Y=fn(x’s) , model does not explicitly account for variation Probabilistic/Statistical

Accounts for variation in x’s, by letting each x be described by a mean value and a variation













Engineering Models


Physics Based Models Conservation of Mass, Momentum,

and Energy Fluid Mechanics/Heat Transfer

Continuity,Navier-Stokes, Energy, Acoustics, Lubrication, Turbulence

Elastic ity Stress/ Strain,isotropic media,

Beam/Column Theory Electromagnetic Theory

Maxwell’s Laws, Ohm’s Law, Wave equations, Plasma dynamics

Dynamics K inematics,Inertia, Rigid

Bodies

Unio n Col l eg eMec ha nic al Engi ne eri ng

Physics Based Fluid Mechanics Models

Continuity

Momentum

Energy

0

Vt

VFVVVV

2

PtDt

D

Dt

DTk

t

Q e

rq2



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Empirical Modeling- Regression Analysis

The Big Y is the Pull Strength.. Wire Length and Die Height are the independent variables

The goal here is to use the data to create an empirical model that relates the value of Y to the values of the x’s

The methodology is to conduct a regression analysis…



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Statistics in Engineering… Engineers work with data sets and need methods and

tools to summarize data and draw conclusions Descriptive statistics to present data in an understandable manner Measures of central points and variation to characterize and data

Engineers deal with variation in all of their work. Variation arises from: Real variation caused by parts tolerance, materials property variations or

operational differences Apparent or Gage R&R variation from measurement system error

A consequence of variation is that engineers must deal with probability in product assembly, product performance, and product reliability

Statistical Design Methods are needed to deal with probabilistic design



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Statistical Methods/Tools… Probability –The Laws of Chance Descriptive Statistics- Analytical and graphical

methods that allow us to describe or picture a data set

Inferential Statistics- Methods by which conclusions can be drawn about a large group of objects based on observing only a portion of the objects

Model Building- Development of prediction equations(transfer functions) from experimental data



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Uses of Statistical Tools Establishing design targets from CTQ’s Data collection(sampling,gage R&R,DOE)

Sampling strategy Analysis of data(means,variances, generation of transfer

functions, descriptive statistics) Statistical Inference/hypothesis testing

Model Building/Optimization/Validation

Statistical Design/Process Control



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Collection of Engineering Data

Union CollegeMechanical Engineering


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Retrospective Study Uses existing data to model existing

processes/designs in order to make predictions about future performance

Quality of data often an issue with this kind of study Insufficient data set(too few x’s or too narrow a

range of variation of x’s) Not enough samples for statistical validity Validity of measurements in question

Retrospective Studies often used in failure RCA’s



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Observational Study

Process or phenomenon is watched and data is recorded All relevant variables are measured Measurements are made with the

required rigor

There is no intervention in the process/phenomenon on the part of those making the study



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Designed Experiment

System Output (big Y ’s)observed under controlled conditions Y =fn(control x’s, noise x’s) Control variables are manipulated Noise variables must be identified Study environment is regulated

Used to establish “cause and effect” between x’s and Y ’s



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Designed Factorial Experiments

Several process variables(factors) and their ranges are identified as being significant in a Factorial Study



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Observing Processes over Time

All processes exhibit variation over time…variation may be caused by random factors or by system degradation(wear)

Control Charts can be used to monitor/correct process performance



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Process Variation over Time - Run or Control Charts

Observing Processes over Time



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Summary of Chapter 2 Topics

Populations and Samples Data Displays





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Populations and Samples Population- entire group of objects being studied Sample- collection of objects from which data are actually gathered

Sample may be all or part of the entire population Sample Data are used to make predictions about the Population Validity of the predictions depends on how the Sample is taken and how big

it is… Both Populations and Samples are characterized by the Central Point

and the Spread of the variables being studied

Populations are what we want to know about- Sample data are what we get…..


Data Displays Dot Diagrams Histograms

Box and Whisker Scatter Plots


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100 110 120 130 140 150 160 170 180 190

Weight

Dotplot for Weight

100 110 120 130 140 150 160 170 180 190 200

0

5

10

15

Weight

Fre

que

ncy



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Pareto Charts Widely used in process analysis to identify

the most frequent failures



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Measures of Central Point and Spread

Percentile Ordered ranking of Data

Median – measure of central tendency Not sensitive to Outliers

Quartiles – divides data into 4 equal parts First or lower, second, third or upper

Interquartile Range – measure of Spread



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Central point-Population Mean

For a population of size N….

N

xN

ii

1

21.45 22.20 22.95 23.70 24.45 25.20 25.95 26.70 27.45 28.20

95% Confidence Interval for Mu

24.94 24.96 24.98 25.00 25.02

95% Confidence Interval for Median

Variable: L2MeanEx

A-Squared:P-Value:

MeanStDevVarianceSkewnessKurtosisN

Minimum1st QuartileMedian3rd QuartileMaximum

24.9616

0.9948

24.9317

0.8010.038

24.9898 1.01431.02887

5.96E-02-5.6E-02

5000

21.199524.299624.963425.676028.4057

25.0179

1.0346

25.0001

Anderson-Darling Normality Test


95% Confidence Interval for Sigma


Descriptive Statistics



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What is Variance?

Variance is a quantitative measure of the square of the difference between each measurement in a sample and the mean of the sample.

Comparison of the(square root of)variance to the mean gives information as to how well a process is controlled



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Spread-Population Variance

Measure of variation in the population

N

xN

ii

1

2

2)(

21.45 22.20 22.95 23.70 24.45 25.20 25.95 26.70 27.45 28.20


24.94 24.96 24.98 25.00 25.02


Variable: L2MeanEx

A-Squared:P-Value:

MeanStDevVarianceSkewnessKurtosisN

Minimum1st QuartileMedian3rd QuartileMaximum

24.9616

0.9948

24.9317

0.8010.038

24.9898 1.01431.02887

5.96E-02-5.6E-02

5000

21.199524.299624.963425.676028.4057

25.0179

1.0346

25.0001

Anderson-Darling Normality Test


95% Confidence Interval for Sigma





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Central Point-Sample Mean

n observations in a sample are denoted by x1, x2, …, xn,

n

xx

n

ii

1

DataYi

68.466.469.571.671.472.564.668.571.266.867.665.665.367.167.564.867.968.269.167.867.468.371.768.868.1


Mean 68.244Standard Error 0.432707754

Median 68.1

Mode #N/A

Standard Deviation 2.163538768

Sample Variance 4.6809

Kurtosis -0.395792379

Skewness 0.316647157

Range 7.9

Minimum 64.6

Maximum 72.5

Sum 1706.1

Count 25Largest(1) 72.5

Smallest(1) 64.6

Confidence Level(95.0%) 0.893064904

25n

25n



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Central Point-Sample Median

n observations in a sample are denoted by x1, x2, …, xn,

n

xx

n

ii

1

DataYi

68.466.469.571.671.472.564.668.571.266.867.665.665.367.167.564.867.968.269.167.867.468.371.768.868.1

Point Data Rank Percent

6 72.5 1 100.00%

23 71.7 2 95.80%

4 71.6 3 91.60%

5 71.4 4 87.50%

9 71.2 5 83.30%

3 69.5 6 79.10%

19 69.1 7 75.00%24 68.8 8 70.80%

8 68.5 9 66.60%

1 68.4 10 62.50%

22 68.3 11 58.30%

18 68.2 12 54.10%

25 68.1 13 50.00%17 67.9 14 45.80%

20 67.8 15 41.60%

11 67.6 16 37.50%

15 67.5 17 33.30%

21 67.4 18 29.10%

14 67.1 19 25.00%10 66.8 20 20.80%

2 66.4 21 16.60%

12 65.6 22 12.50%

13 65.3 23 8.30%

16 64.8 24 4.10%7 64.6 25 0.00%



Median 68.1Mode #N/A

Standard Deviation 2.163538768

Sample Variance 4.6809

Kurtosis -0.395792379


Range 7.9

Minimum 64.6

Maximum 72.5

Sum 1706.1


Smallest(1) 64.6


25n



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Spread-Sample Variance

Measure of variation in the sample

Note n-1 rather than N as divisor

1

)(1

2

2

n

xxs

n

ii



Median 68.1

Mode #N/A

Standard Deviation 2.163538768Sample Variance 4.6809

Kurtosis -0.395792379


Range 7.9

Minimum 64.6

Maximum 72.5

Sum 1706.1


Smallest(1) 64.6



Point Data Rank Percent

6 72.5 1 100.00%

23 71.7 2 95.80%

4 71.6 3 91.60%

5 71.4 4 87.50%

9 71.2 5 83.30%

3 69.5 6 79.10%

19 69.1 7 75.00%24 68.8 8 70.80%

8 68.5 9 66.60%

1 68.4 10 62.50%

22 68.3 11 58.30%

18 68.2 12 54.10%

25 68.1 13 50.00%17 67.9 14 45.80%

20 67.8 15 41.60%

11 67.6 16 37.50%

15 67.5 17 33.30%

21 67.4 18 29.10%

14 67.1 19 25.00%10 66.8 20 20.80%

2 66.4 21 16.60%

12 65.6 22 12.50%

13 65.3 23 8.30%

16 64.8 24 4.10%7 64.6 25 0.00%


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Sample Mean and Variance…Rank OrderMedian..Histogram and Box Plot…

1

)(1

2

2

n

xxs

n

ii



Median 68.1

Mode #N/A

Standard Deviation 2.163538768Sample Variance 4.6809

Kurtosis -0.395792379


Range 7.9

Minimum 64.6

Maximum 72.5

Sum 1706.1


Smallest(1) 64.6


n

xx

n

ii

1



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Summary of Lecture 2 Topics

Summary of Chapter 1 Topics Engineering Method Statistics in Engineering Collection of Engineering Data Observing Processes over Time

Summary of Chapter 2 Topics Populations and Samples Data Displays



MER301: Engineering Reliability

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