maths

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maths statistics Dr William
Megill 2E2.31 enswmm_at_bath.ac.uk
ME10305
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• textbooks resources
• Engineering
• Mitchell and Jolley 1988, Winston Jackson
  1995 (both in library at 300.001.5)
• Montgomery et al. 2006 Engineering Statistics.
  4th Ed. Wiley Sons (Waterstones or Amazon)
• Biology
• Zar (1984) Biostatistical Analysis
• Sokal Rohlf (1973) Introduction to
  Biostatistics
• Dickson Massey available in library
• HELM
• Helping Engineers Learn Mathematics

• textbooks resources
• Engineering
• Mitchell and Jolley 1988, Winston Jackson
  1995 (both in library at 300.001.5)
• Montgomery et al. 2006 Engineering Statistics.
  4th Ed. Wiley Sons (Waterstones or Amazon)
• Biology
• Zar (1984) Biostatistical Analysis
• Sokal Rohlf (1973) Introduction to
  Biostatistics
• Dickson Massey available in library
• HELM
• Helping Engineers Learn Mathematics

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Role of Stats in Engineering

• or Why are you here?

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• Scientific Method
• Observe
• Generalise
• Hypothesise
• Test
• Learn
• Report

(auto eng/paint quality) (evaporation due to
heat) (sun is the problem) (paint
thickness) (cars in various places) (statistics) (
it was/wasnt the sun)

• Research Programme
• Context / Field
• Theory
• Hypothesis
• Variables
• Experiment
• Analysis
• Conclusions

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• Scientific Method
• Observe
• Generalise
• Hypothesise
• Test
• Learn
• Report

• Engineering Method
• Description of the problem
• Identify important factors
• Propose/refine model
• Collect data
• Manipulate the model
• Confirm the solution
• Conclusions

• Research Programme
• Context / Field
• Theory
• Hypothesis
• Variables
• Experiment
• Analysis
• Conclusions

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Statistics

• What is Stats?
• Collection
• Presentation
• Analysis
• of data to make decisions/solve problems
• Why do Engineers need Statistics?
• Take Real World into account
• Variability (not measurement error)

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Examples

• Gas mileage
• Is it always the same?
• What influences your measurement?

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Examples

• Production processes
• e.g. Rubber o-rings
• Made from synthetic rubber following recipe
• Measure e.g. tensile strength
• 1030, 1035, 1020, 1049, 1028, 1026, 1019, 1010
  (psi)
• Describe as an average /- scatter
• Stats calls this a model X m e
• Plot somehow dot diagram

1010
1020
1030
1040
1050
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Examples

• Not good enough, so modify the recipe
• Looks good, but
• Will it always work this well?
• Are 8 samples enough to be reliable?
• What risk are we taking?
• e.g. is it possible the difference was all simply
  chance?
• Hence need for statistical thinking

1060
1070
1010
1020
1030
1040
1050
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Collecting Engineering Data

• Three types of experiment
• Retrospective study
• (historical data)
• Observational study
• (watch what happens)
• Designed experiment
• (muck with it
• and then see what happens)

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Example - Paint on cars

• Observation paint fades on cars in Kansas
• Retrospective study
• Look at cars of a particular age
• measure paint thickness
• use sunshine data
• Advantages
• Data already exists, just have to analyse it
• Problems
• No control of variation
• Cant test specific factors (get the whole
  package)
• Data quality questionable
• Biases can sneak in
• Often huge datasets, full of non-random sampling,
  bias, etc.
• Need a really spectacular grasp of Stats
• Frustrating!

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Example Paint on cars

• Observation paint fades on cars in Kansas
• Observational study
• Look at cars of a particular age
• Measure paint thickness over time
• Measure sun exposure
• Advantages
• Easy to do, not expensive, non-destructive,
  non-interfering
• Problems
• Range of parameters limited by external factors
• Cant tease out confounding effects
• Large datasets
• Often how its done! ends up needing Stats
  consulting

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Example Paint on Cars

• Observation paint fades on cars in Kansas
• Designed experiment
• Make deliberate purposeful changes to
  controllable variables (factors)
• Observe system output
• Decision or inference about which variables
  responsible for the changes.
• What are factors in Kansas paint example?
• .

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Experimental Design
Test matrix

• Multiple factors (independent variables)
• Heat (evaporation)
• Rain (dissolution)
• Oxidation
• Response variable
• Paint thickness (microns)
• Factorial design
• This can get crazy
• So go for fractional factorial design

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Randomness

• Stats based on samples of a population
• Population can be
• real this months bearings
• conceptual set of all possible measurements
• Samples must be randomly selected
• Otherwise get biases false conclusions
• e.g. maths skills of undergrads
• Select using random number tables, dice, computer
• Each member of the population has an equal chance
  of being selected
• Not always easy
• Particularly in retrospective or observational
  studies
• Be careful interpreting data, even from
  respectable sources.

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Statistical models

• Mechanistic
• built from underlying understanding of a
  phenomenon
• e.g. Ohms law I E/R
• might need to include scatter, so I E/R e
• e term includes effects of all unmodeled sources
  of variability
• Empirical
• built from functional combination of observed
  variables
• e.g. t t0 b1T b2R b3O e

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Statistical models

• Empirical
• Model to combine two effects
• (Pull strength) b0 b1 (wire length) b2 (die
  height) e
• even if dont quite know why, we can make
  predictions...