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The Role of Statisticsin Engineering
ENM 500
Chapter 1
The adventure begins…
A look ahead
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1-1 The Engineering Method and Statistical Thinking
Figure 1.1 The engineering method
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The Engineering or Scientific Method
• Figure 1-1 Describes the Scientific or Engineering Method.
• Several steps rely on statistical methods– Conduct experiments – how are efficient experiments
designed?– Identify the important factors – how do we account for
variability when we measure these factors?– Confirm the solution – how do we accept or reject a
solution/hypothesis based on measurements?
• Variability complicates the task.• Statistical methods help us understand and deal with
variability.
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• Statistical techniques are useful for describing and understanding variability.
• By variability, we mean successive observations of a system or phenomenon do not produce exactly the same result.
• Statistics gives us a framework for describing this variability and for learning about potential sources of variability.
1-1 The Engineering Method and Statistical Thinking
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Why is variability important to us?
• We want to predict results and control results with accuracy. Variability makes predictions and control more difficult and less accurate.
• If a particular part was required to be 1” + 0.010” and the actual standard deviation was 0.010”, almost one-third of the parts would be out of tolerance, even if their mean was exactly 1.000”!
• Would you rather work in a room that had a constant temperature of 70o or one where the temperature alternated between 50o and 90o every 30 minutes?
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Why Do We Study Probability & Statistics?
• Statistics – deals with the collection, presentation, analysis, and use of data to make decisions, solve problems, and design products & processes. – use statistics to draw inferences. Examples: quality, performance, or
durability of a product, weather forecasts, utilization or loading of system.
• Probability – allows us to use information & data to make intelligent statements & forecasts about future events. – Probability helps quantify the risks associated with statistical
inferences
• Prob & Stat are foundations for other coursework, e.g. reliability and quality courses, robust design, simulation, design of experiments, decision analysis, forecasting, time-series analysis, and operations research.
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What do we want to know about our data?
A measure of central tendency: Average or mean -
A measure of variability: Sample variance –
Sample Standard Deviation -
1 2
1
.... 1 nn
ii
x x xx x
n n
2 2
1
1( )
1
n
ii
s x xn
2s s
We build models to explain this variability
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An ExampleSample 1 Sample 2
17 23
21 16
23 17
20 21
18 25
22 18
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1 2
1 2
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19 23
22 18
21 24
x 20 x 20
2.16 3.62s s
10
20
30
X
13X s
13X s
23X s
23X s
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Sample vs. Population Measures – Statistical Inference
• The sample mean ( ) estimates the population mean ( )
• The sample variance ( ) estimates the population variance ( )
SAMPLE POPULATION
MEAN:
VARIANCE:
x
2s2
2 2
1
1( )
1
n
ii
s x xn
2 2
1
1( )
N
ii
x xN
1
1 N
ii
xN
1
1 n
ii
x xn
The population can sometimes be conceptual and essentially have infinite
size.
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Sample vs. Population Measures
We use sample measures ( ) to draw conclusions about the population measures ( ).
• The sample will be a (random) subset of the population
• The population may not yet exist, so the sample may be from a small set of prototypes (analytic)– There is an issue of stability – do the prototypes accurately
reflect the prospective population?
2, x s2,
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Sample Data – May be obtained from:
• Observational Study – sample is drawn randomly from current process or system
• Designed experiment – deliberate changes are made to the controllable variables of a process or system. The system output is observed & inferences made about the effects of controlling the input.
• Retrospective Study – Historical observations. Were you fortunate enough that the needed variables were actually collected accurately!?
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Concept of Models
• Common engineering/physical models: – F = ma – I = E/R – d = vt
• Mechanistic models: used when we understand the physical mechanism relating these variables.
• Empirical models: use our engineering & scientific knowledge of the phenomena, but are not built on first-principle understanding of the underlying mechanism. They are data driven.
Let the data do the talking, right?
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1-3 Mechanistic and Empirical Models
A mechanistic model is built from our underlying knowledge of the basic physical mechanism that relates several variables.
Example: Ohm’s Law
Current = voltage/resistance
I = E/R
I = E/R +
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1-3 Mechanistic and Empirical Models
An empirical model is built from our engineering and scientific knowledge of the phenomenon, but is not directly developed from our theoretical or first-principles understanding of the underlying mechanism.
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1-3 Mechanistic and Empirical Models
Example
Suppose we are interested in the average molecular weight (Mn) of a polymer. Now we know that Mn is related to the viscosity of the material (V), and it also depends on the amount of catalyst (C) and the temperature (T ) in the polymerization reactor when the material is manufactured. The relationship between Mn and these variables is
Mn = f(V,C,T)say, where the form of the function f is unknown.
where the ’s are unknown parameters.
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1-3 Mechanistic and Empirical Models
In general, this type of empirical model is called a regression model.
The estimated regression line is given by
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Figure 1-15 Three-dimensional plot of the wire and pull strength data.
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Figure 1-16 Plot of the predicted values of pull strength from the empirical model.
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Designing Engineering Experiments
• Experiments are often used to confirm theory or to evaluate various design options– Often, several factors may be important– Each factor may have more than one level of concern
• Full factorial design – considers all factors at all levels of interest– For K factors, each having two levels, a total of 2K
experiments are required– For K = 4, N = 16– For K = 8, N = 256
• Fractional factorial design – only a subset of factor combinations are actually tested
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Design of Experiments (DOE)
• Assume you want to investigate the impact of three factors on the pull-off force of a connector:– Wall thickness (3/32” and 1/8”)– Cure times (1 hour and 24 hours)– Cure temperature (70o F and 100o F)
• We can now conduct an experiment to assess the impact of each of these variables (separately & interacting), each variable being assessed at two different levels
• Since other sources of variability may be present, we would do multiple experiments (replicate) at each design point.
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Full Factorial Design
Figure S1-1 The factorial experiment for the connector wall thickness problem.
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Importance of Factor Interactions
Figure S1-2 The two-factor interaction between cure time and cure temperature.
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The Key Distinction
• The key difference between observational studies and experimental designs is this:
– In a proper experiment you can eliminate confounding factors and isolate effects of interest.
– In an observational study you take existing data. This may make it impossible to distinguish the effects of two factors that appear to explain observations equally well.
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Time Series
• The correct analysis and interpretation of data collected over time is very important in assessing & controlling the performance of a system or process.
– When is performance normal & when is it out of control?– What factors are driving a system out of control?– What corrections should be applied to regain control? – When has a change occurred – a fundamental shift in the
process behavior?
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1-2.5 Observing Processes Over Time
Figure 1-11 Adjustments applied to random disturbances over control the process and increase the deviations from the target.
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1-2.5 Observing Processes Over Time
Figure 1-12 Process mean shift is detected at observation number 57, and one adjustment (a decrease of two units) reduces the deviations from target.
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1-2.6 Observing Processes Over Time
Figure 1-13 A control chart for the chemical process concentration data.
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1-4 Probability and Probability Models
• Probability models help quantify the risks involved in statistical inference, that is, risks involved in decisions made every day.
• Probability provides the framework for the study and application of statistics.
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Let’s Toss a Coin
• There are 1000 coins one of which contains two heads; the others are fair. A coin is selected at random and tossed 10 times. If heads appear on all ten tosses, what is the probability that the coin selected is the two-headed coin?
P(two-headed is selected) = .001P(toss 10 heads in a row – fair coin) = (1/2)10 = 1/1024 .001Therefore P(two-headed coin selected given 10 heads observed) .5
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