Transcript Lecture2

MER301: Engineering
Reliability
LECTURE 2:
Chapter 1:
Role of Statistics in Engineering
Chapter 2:
Data Summary and Presentation
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
1
Summary of Lecture 2 Topics
 Summary of Chapter 1 Topics




Engineering Method
Statistics in Engineering
Collection of Engineering Data
Observing Processes over Time


Populations and Samples
Data Displays
 Summary of Chapter 2 Topics





Central Point and Spread


L Berkley Davis
Copyright 2009
Dot Diagrams
Histograms
Box and Whisker Plots
Scatter plots
Median,Quartiles,Interquartile range
Means, Variances and Standard Deviations
MER301: Engineering Reliability
Lecture 2
2
Engineering Method
L Berkley Davis
Copyright 2009

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
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Lecture 2
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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 Capabilityto ensure product CTQs
are met in light of manufacturing capability
Validation of product performance
L Berkley
Davis
Union College
Copyright
2009Engineering
Mechanical
MER301: Engineering Reliability
Lecture 2
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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)


Y  fn( x1 , x2 ,...xn )
L Berkley Davis
Copyright 2009
The Designer
must understand
MER301: Engineering Reliability
Product and Process CTQ’s
Lecture 2
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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
L Berkley Davis
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MER301: Engineering Reliability
Lecture 2
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Experimental Gage R&R Precision and Accuracy
Not Accurate, Not Precise
Not Accurate, Precise
L Berkley Davis
Copyright 2009
Accurate, Not Precise
Accurate, Precise
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Lecture 2
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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

Accounts for variation in x’s, by letting each x be described
by a mean value and a variation
 Probabilistic/Statistical
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 2
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Engineering Models
Physics Based Models
Physics Based Fluid Mechanics Models





Conservation of Mass, Momentum,
 Continuity
and Energy

Fluid Mechanics/Heat Transfer
   V  0
t

Continuity,Navier-Stokes,
 Momentum
Energy, Acoustics, Lubrication,
Turbulence
DV
 V


 
 V  V  P  F   2 V
Elasticity
Dt
 t


Stress/ Strain,isotropic media,
 Energy
Beam/Column Theory
Electromagnetic Theory

Maxwell’s Laws, Ohm’s Law,
Q
De
   k 2 T    q r  
Wave equations, Plasma

t
Dt
dynamics
Dynamics

Kinematics,Inertia, Rigid
Bodies
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
L Berkley Davis
Copyright 2009
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…
L Berkley Davis
Copyright 2009
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 2
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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
L Berkley Davis
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MER301: Engineering Reliability
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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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MER301: Engineering Reliability
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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
L Berkley Davis
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MER301: Engineering Reliability
Lecture 2
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Collection of Engineering Data
Retrospective Study
 Uses existing data to model existing
processes/designs in order to make
predictions about future performance
Observational
Study
 Quality of data often
an issue with
this kind of study
 Process or phenomenon is
recorded
 Insufficient data set(too few x’s or too narrow a
range of variation of x’s) watched and data is
 Not enough samples for statistical
validity variables
 All relevant
 Validity of measurements in question
are measured
Designed
 Measurements are made with the
 Retrospective Studies often
used
in
required
rigor
failure RCA’s
Union College
Mechanical Engineering
Experiment
 System Output (big Y’s)observed
 There is no intervention in the under controlled conditions
process/phenomenon on the part
 Y=fn(control x’s, noise x’s)
of those making the study
 Control variables are manipulated
MER301: Engineering Reliability
Lecture 1
Union College
Mechanical Engineering
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MER301: Engineering Reliability
Lecture 1
 Noise variables must be identified
 Study environment is regulated
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 Used to establish “cause and
effect” between x’s and Y’s
L Berkley Davis
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MER301: Engineering Reliability
Lecture 2
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 1
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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
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
Process Variation over Time
monitor/correct process
- Run or Control Charts
performance
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 1
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Union College
Mechanical Engineering
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 2
MER301: Engineering Reliability
Lecture 1
27
15
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
MER301: Engineering Reliability
Lecture 2
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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 aboutSample data are what we get…..
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 2
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Data Displays
 Dot Diagrams
 Histograms
Dotplot for Weight
Frequency
15
100
110
120
130
140
150
160
170
180
10
5
190
Weight
0
100 110 120 130 140 150 160 170 180 190 200
Weight
 Box and Whisker
L Berkley Davis
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 Scatter Plots
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Lecture 2
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Pareto Charts
 Widely used in process analysis to
identify the most frequent failures
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Copyright 2009
MER301: Engineering Reliability
Lecture 2
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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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MER301: Engineering Reliability
Lecture 2
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Central point-Population Mean
 For a population of
size N….
Descriptive Statistics
Variable: L2MeanEx
Anderson-Darling Normality Test
A-Squared:
P-Value:
21.45 22.20 22.95 23.70 24.45 25.20 25.95 26.70 27.45 28.20
Mean
StDev
Variance
Skewness
Kurtosis
N
24.9898
1.0143
1.02887
5.96E-02
-5.6E-02
5000
95% Confidence Interval for Mu
Minimum
1st Quartile
Median
3rd Quartile
Maximum
21.1995
24.2996
24.9634
25.6760
28.4057
N

x
i 1
0.801
0.038
i
95% Confidence Interval for Mu
N
24.9616
24.94
24.96
24.98
25.00
25.02
25.0179
95% Confidence Interval for Sigma
0.9948
1.0346
95% Confidence Interval for Median
95% Confidence Interval for Median
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 2
24.9317
25.0001
21
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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 2
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Spread-Population Variance
 Measure of variation
in the population
Descriptive Statistics
Variable: L2MeanEx
Anderson-Darling Normality Test
N

2
 ( xi   )
 i 1
A-Squared:
P-Value:
2
0.801
0.038
21.45 22.20 22.95 23.70 24.45 25.20 25.95 26.70 27.45 28.20
Mean
StDev
Variance
Skewness
Kurtosis
N
24.9898
1.0143
1.02887
5.96E-02
-5.6E-02
5000
95% Confidence Interval for Mu
Minimum
1st Quartile
Median
3rd Quartile
Maximum
21.1995
24.2996
24.9634
25.6760
28.4057
N
95% Confidence Interval for Mu
24.9616
24.94
24.96
24.98
25.00
25.02
25.0179
95% Confidence Interval for Sigma
0.9948
1.0346
95% Confidence Interval for Median
95% Confidence Interval for Median
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 2
24.9317
25.0001
23
Data
Central Point-Sample Mean
Yi
68.4
66.4
69.5
71.6
71.4
72.5
64.6
68.5
71.2
66.8
67.6
65.6
65.3
67.1
67.5
64.8
67.9
68.2
69.1
67.8
67.4
68.3
71.7
68.8
68.1
 n observations in a
sample are denoted
by x1, x2, …, xn,
n
x
L Berkley Davis
Copyright 2009
 xi
n  25
Descriptive Statistics
Mean
Standard Error
68.244
0.432707754
Median
68.1
Mode
#N/A
Standard Deviation
2.163538768
Sample Variance
Kurtosis
Skewness
4.6809
-0.395792379
0.316647157
Range
i 1
7.9
Minimum
64.6
Maximum
72.5
n
Sum
n  25
1706.1
Count
25
Largest(1)
72.5
Smallest(1)
64.6
Confidence Level(95.0%)
MER301: Engineering Reliability
Lecture 2
0.893064904
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Data
Central Point-Sample Median
Yi
68.4
66.4
69.5
71.6
71.4
72.5
64.6
68.5
71.2
66.8
67.6
65.6
65.3
67.1
67.5
64.8
67.9
68.2
69.1
67.8
67.4
68.3
71.7
68.8
68.1

n observations in a sample
are denoted by x1, x2, …, xn,
n
 xi
x  i 1
n
Descriptive Statistics
Mean
68.244
Standard Error
0.432707754
Median
68.1
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%
#N/A
22
68.3
11
58.30%
2.163538768
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%
7.9
15
67.5
17
33.30%
Minimum
64.6
21
67.4
18
29.10%
Maximum
72.5
14
67.1
19
25.00%
1706.1
10
66.8
20
20.80%
Mode
Standard Deviation
Sample Variance
4.6809
Kurtosis
-0.395792379
Skewness
0.316647157
Range
Sum
Count
Largest(1)
n  25
25
72.5
Smallest(1)
64.6
Confidence Level(95.0%)
L Berkley Davis
Copyright 2009
Point
0.893064904
MER301: Engineering Reliability
Lecture 2
2
66.4
21
16.60%
12
65.6
22
12.50%
13
65.3
23
8.30%
16
7
64.8
64.6
24
25
4.10%
0.00%
25
Spread-Sample Variance
 Measure of variation
in the sample
 Note n-1 rather than
N as divisor
n
2
 ( xi  x )
2
i 1
s 
n 1
Descriptive Statistics
Mean
Standard Error
68.244
0.432707754
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
68.1
#N/A
2.163538768
4.6809
-0.395792379
0.316647157
Range
7.9
Minimum
64.6
Maximum
72.5
Sum
Count
25
Largest(1)
72.5
Smallest(1)
64.6
Confidence Level(95.0%)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 2
1706.1
0.893064904
26
Sample Mean and Variance…Rank Order
Median..Histogram and Box Plot…
n
x
 xi
i 1
n
s 
2
n
 ( xi  x )
i 1
n 1
Descriptive Statistics
Mean
Standard Error
68.244
0.432707754
Median
68.1
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%
1706.1
2
66.4
21
16.60%
Count
25
12
65.6
22
12.50%
Largest(1)
72.5
13
65.3
23
8.30%
64.6
16
7
64.8
64.6
24
25
4.10%
0.00%
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
#N/A
2.163538768
4.6809
-0.395792379
0.316647157
Range
7.9
Minimum
64.6
Maximum
72.5
Sum
Smallest(1)
Confidence Level(95.0%)
L Berkley Davis
Copyright 2009
2
0.893064904
MER301: Engineering Reliability
Lecture 2
27
Summary of Lecture 2 Topics
 Summary of Chapter 1 Topics




Engineering Method
Statistics in Engineering
Collection of Engineering Data
Observing Processes over Time


Populations and Samples
Data Displays
 Summary of Chapter 2 Topics





Central Point and Spread


L Berkley Davis
Copyright 2009
Dot Diagrams
Histograms
Box and Whisker Plots
Scatter plots
Median,Quartiles,Interquartile range
Means, Variances and Standard Deviations
MER301: Engineering Reliability
Lecture 2
28