ENGINEERING STATISTICS

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ENGINEERING STATISTICS
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Engineering System Analysis

• Engineering systems analysis is the process of
  using observations to qualitatively and
  quantitatively understand a system.
• The use of mathematics to determine how a set of
  interconnected components whose individual
  characteristics are known will behave in response
  to a given input or set of inputs.

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• What is meant by understanding a system?
• The ability to predict future outcomes from the
  system based on hypothetical inputs.
• How do we go about formalizing an understanding
  of a System?
• Our understanding of a system is formalized by a
  model that maps input signals to output signals.
• Why is this important?
• A system model is a key component in the systems
  engineering design cycle.

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Systems Engineering Cycle
System Analysis Cycle
System Design Cycle
Simulate model response
Conceptual design
MODEL
Problem
Optimize design parameters
Evaluate prototype
Build Prototype
Final Design
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• A technician is involved in the implementation of
  engineering designs. An experienced technician
  can extrapolate from previous designs to obtain
  effective solutions to similar problems
• An engineer uses the tools of modeling and
  optimization to generate system designs. An
  experienced engineer should be able to tackle
  problems that are completely novel and should
  provide solutions that are optimal.

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How not to solve a design problem
System Analysis Cycle
System Design Cycle
Simulate model response
Conceptual design
Problem
MODEL
Optimize design parameters
Evaluate prototype
Build Prototype
Final Design
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The Importance of models

• Dictionary definition of model
• A representation of something (usually on a
  smaller scale)
• A simplified description of a complex entity or
  process.
• A representative form or pattern.

A comprehensible simplified description of a real
world system that captures its most significant
patterns or form.
The key property of a model for
systems engineering design is the ability to
predict outcomes from the system.
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Models

• Engineering systems analysis can be thought of as
  the process of using observations to identify a
  model of a system.
• The process of modeling a system is one of
  finding correlations or patterns in the observed
  signals.

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Statistical framework

• Measuring real signals is a statistical process.
• Observed signals will be noisy and this noise
  must be included in the modeling process. Thus,
  all modeling is inherently a statistical process.
• Identified models of systems are uncertain
  approximations of the real world. The modeling
  error itself is interpreted as a statistical
  process.

A systems engineer should have a
good understanding of statistical modeling
and statistical decision methodology
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What is statistics?

• Statistics is the scientific application of
  mathematical principles to the collection,
  analysis, and presentation of data
• at the foundation of all of statistics is data.

Collection Presentation Analysis Use
make decisions and solve problems
deals with
to
Statistics
data
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Engineering statistics

• Engineering statistics is the study of how best
  to
• Collect engineering data
• Summarize or describe engineering data
• Draw formal inferences and practical conclusions
  on the basis of engineering data all the while
  recognizing the reality of variation

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Engineering Statistics

• is the branch of statistics that has three
  subtopics which are particular to engineering.
• Design of experiments (DOE)
• use statistical techniques to test and construct
  model of engineering components and systems.
• Quality control and process control
• use statistics as a tool to manage conformance to
  specifications of manufacturing processes and
  their products.
• Time and method engineering
• use statistics to study repetitive operations in
  manufacturing in order to set standards and find
  optimum (in some sense) manufacturing procedures.

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Data collection methods

• Observational Study
• Experimental Study
• Opposite ends of a continuum where the scale is
  in terms of the degree to which an investigator
  manages important variables in the study

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Types of data

• Qualitative Data (Categorical)
• Non-numerical characteristics associated with
  items in a sample
• Examples
• Eye color (blue, brown, green, etc)
• Engine status (working, not working fixable,
  not working not fixable)
• Quantitative Data (numerical)
• Numerical characteristics associated with items
  in a sample
• Typically counts of occurrences of a phenomenon
  of interest or measurements of some physical
  property
• Can be further broken down into discrete
  (countable) and continuous (uncountable)

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Collection of quantitative data (Measurement)

• If you cant measure, you cant do statistics or
  engineering for that matter!
• Issues
• Validity
• Precision
• Accuracy (unbiasedness)

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Measurement issues

• Validity faithfully representing the aspect of
  interest i.e. usefully or appropriately
  represents the feature of an object or system
• Precision small variation in repeated
  measurements
• Accuracy (unbiasedness) producing the true
  value on average

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Precision and accuracy
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Statistical thinking

• Statistical methods are used to help us describe
  and understand variability.
• By variability, we mean that successive
  observations of a system or phenomenon do not
  produce exactly the same result.

Are these gears produced exactly the same size?
NO!
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Sources of variability
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Example

• An engineer is developing a rubber compound for
  use in O-rings.
• The engineer uses the standard rubber compound to
  produce eight O-rings in a development laboratory
  and measures the tensile strength of each
  specimen.
• The tensile strengths (in psi) of the eight
  O-rings are

1030,1035,1020, 1049, 1028, 1026, 1019, and 1010.
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Variability

• There is variability in the tensile strength
  measurements.
• The variability may even arise from the
  measurement errors
• Tensile Strength can be modeled as a random
  variable.
• Tests on the initial specimens show that the
  average tensile strength is 1027.1 psi.
• The engineer thinks that this may be too low for
  the intended applications.
• He decides to consider a modified formulation of
  rubber in which a Teflon additive is included.

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Random sampling

• Assume that X is a measurable quantity related to
  a product (tensile strength of rubber). We model
  X as a random variable
• Occur frequently in engineering applications
• Random sampling
• Obtain samples from a population
• All outcomes must be equally likely to be sampled
• Replacement necessary for small populations
• Meaningful statistics can be obtained from samples

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Point Estimation

• The probability density function f(x) of the
  random variable X is assumed to be known.
• Generally it is taken as Gaussian distribution
  basing on the central limit theorem.
• Our purpose is to estimate certain parameters of
  f(x), (mean, variance) from observation of the
  samples.

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Sample Mean Variance
M is a point estimator of m S is a point
estimator of s
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Point estimates as random variables

• Since the sample mean and variance depend on the
  random sample chosen, the values of M and S both
  depend on the sample set.
• As such, they also can be considered as random
  variables.

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Examples
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Quality of Estimators

• If y u(x1,x2,...,xN) is a point estimator of a
  parameter q of the population, we want
• Ey q (unbiased)
• Vy should be as small as possible (minimum
  variance)
• Such an estimator is called an unbiased minimum
  variance estimator.

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PDF of sample mean
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For larger sample sizes (N) the probability that
the mean estimate is closer to the mean is
higher.
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Confidence interval
We want to determine an interval I for the actual
mean m so that
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• Given that X is a Gaussian random variable with
  mean m and variance s2.

has distribution N(0,1)
has a chi-square distribution with N-1 degrees
of freedom.
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Define
Then the pdf of t is given by
This distribution is known as Students
t-distribution with k degrees of freedom.
The distribution is named after the English
statistician W.S. Gosset, who published his
research under the pseudonym Student.
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hk(t)
(1-a)/2
(1-a)/2
tk,a
-tk,a
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• Thus if we obtain the estimates M and S from the
  sample set, the actual value of the population
  mean m will lie in the interval

with probability a. This is called a a100
percent confidence interval.

• The values for Students t-distribution are
  tabulated.

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Confidence coefficient
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Example

• Ten measurements were made on the resistance of a
  certain type of wire. Suppose that M10.48 W and
  S1.36 W. We want to obtain a confidence interval
  for m with confidence coefficient 0.90. From the
  table

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Example
The voltage measured at the output of a system
V, Volt
t, msec
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Statistics with 500 measurements
V exp(-t/t)
t 3 msec
t, msec
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Statistical Hypothesis Testing (experiment design)
H0 Mean and variance are not changed (null
hypothesis) H1 Mean and variance are changed
(alternative hypothesis)
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Statistical Hypothesis Testing (process
optimization)
H0 MTBF 3 months (null hypothesis) H1 MTBF
gt 3 months (alternative hypothesis)
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• To guess is cheap. To guess wrongly is expensive
  - Chinese Proverb
• There are three kinds of lies lies, damned
  lies, and statistics - Benjamin Disraeli (?),
  British PM
• First get your facts, then you can distort them
  at your leisure - Mark Twain
• Statistical Thinking will one day be as necessary
  for efficient citizenship as the ability to read
  and write - H. G. Wells