2DS00

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2DS00

• Statistics 1 for Chemical Engineering

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Lecturers

• Dr. A. Di Bucchianico
• Department of Mathematics,
• Statistics group
• HG 9.24
• phone [phone]
• a.d.bucchianico_at_tue.nl
• Ir. G.D. Mooiweer,
• Department of Mathematics
• ICTOO
• HG 9.12
• phone [phone] (Thursdays)
• g.d.mooiweer_at_tue.nl

• Dr. R.W. van der Hofstad
• Department of Mathematics,
• Statistics group
• HG 9.04
• phone [phone]
• rhofstad_at_win.tue.nl

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Goals of this course

• to prepare students for (first-year) laboratory
  assignments
• to learn students how to perform basic
  statistical analyses of experiments
• to learn students how to use software for data
  analysis
• to learn students how to avoid pitfalls in
  analysing measurements

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Important to remember

• Web site for this course www.win.tue.nl/sandro/
  2DS00/
• No textbook, but handouts (Word) Powerpoint
  sheets through web site
• Bring notebook to both lectures and self-study
• (Optional) buy lecture notes 2256 Statgraphics
  voor regulier onderwijs
• (Optional) buy lectures notes 2218 Statistisch
  Compendium

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How to study

• read lecture notes briefly before lecture
• ask questions during lecture
• study lecture notes carefully after lecture
• make excercises during guided self-study
• reread lecture notes after guided self-study
• try out previous examinations shortly before the
  examination
• N.B. Lecture notes (pdf documents) ? PowerPoint
  files

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Week schedule

• Week 1 Measurement and statistics
• Week 2 Error propagation
• Week 3 Simple linear regression analysis
• Week 4 Multiple linear regression analysis
• Week 5 Nonlinear regression analysis

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Detailed contents of week 1

• measurement errors
• graphical displays of data
• summary statistics
• normal distribution
• confidence intervals
• hypothesis testing

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Measurements and statistics

• perfect measurements do not exist
• possible sources of measurement errors
• reading
• environment
• temperature
• humidity
• ...
• impurities
• ...

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Necessity of good measurement system
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Three experiments
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Types of measurement errors

• Random errors
• always present
• reduce influence by averaging repeated
  measurements
• Systematic errors
• requires adjustment/repair of measuring devices
• Outliers
• recording errors
• mistakes in applying procedures

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Illustration of measurement concepts
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Accuracy
difference between average of measured values and
true value
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Accuracy

• relates to systematic errors
• absolute error
• relative error

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Location statistics

• mean
• median
• trimmed means

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Precision

• the degree in which consistent results are
  obtained

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Accurate and precise
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Statistics for precision standard deviation co

• standard deviation
• standard error
• variation coefficient
• variance
• range

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Robust statistics for precision

• robust statistics
• less sensitive to outliers
• difficult mathematical theory
• requires use of statistical software
• interquartile range
• IQR 75 quantile 25 quantile 3rd quartile
  1st quartile
• mean absolute deviation

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Graphical displays

• always make graphical displays for first
  impression
• one picture says more than 1000 words

2 3.1 4 1.9 2.8
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Basic graphical displays

• scatter plot
• watch out for scale (automatic resizing)
• time sequence plot
• for detecting time effects like warming up
• Box-and-Whisker plot
• outliers
• quartiles
• skewness

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Time sequence plot
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Box-and-Whisker plot
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(No Transcript)
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Probability theory

• (cumulative) distribution function
• density
• density to distribution function

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The concept of probability density
density function
a
b
area denotes probability that observation falls
between a and b
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Normal distribution
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Normal distribution

• bell shaped curve
• Important because of Central Limit Theorem
• Normal distribution
• symmetric around µ (location of centre)
• spread parametrised by ?2
• http//www.win.tue.nl/marko/statApplets/function
  Plots.html
• http//www-stat.stanford.edu/naras/jsm/NormalDen
  sity/NormalDensity.html
• µ0 and ?21 standard normal distribution Z

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More on normal distribution

• Area between
• ? ? 0,67? is 0,500
• ? ? 1,00? is 0,683? ? 1,645? is 0,975
• ? ? 1,96? is 0,950
• ? ? 2,00? is 0,954
• ? ? 2,33? is 0,980? ? 2,58? is 0,990
• ? ? 3,00? is 0,997

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Standardisation

• X normally distributed with parameters ? en ?2,
  then (X-?)/? standard normal
• suppose
• ?3
• ?24

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Testing normality

• many statistical procedures implicitly assume
  normality
• if data are not normally distributed, then
  outcome of procedure may be completely wrong
• user is always responsible for checking
  assumptions of statistical procedures
• Graphical checks
• normal probability plot
• density trace
• Formal check
• Shapiro-Wilks test

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Estimation of density function histogram
curve normal distribution with sample mean and
variance as parameters
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Drawbacks of the histogram

• misused for investigating normality
• time ordering of data is lost
• shape depends heavily on bin width bin location

Histogram for strength
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4
same data set
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frequency
2
1
0
24
29
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39
44
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strength

• shape is stable for data sets of size 75 or
  larger
• optimal number of bins ??n

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Alternative to histogram Density Trace

• Density Trace (also called naive density
  estimator)
• use moving bins instead of fixed bins
• choose bin width (automatically in Statgraphics)
• count number of observations in bin at each
  point
• divide by length of bin

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Density Trace

• Example dataset

4/9
3/9
2/9
1/9
1
2
3
4
5
6

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Choice of bin widths in density trace

• too small bin width yields too fluctuating curve
• too large bin width yields too smooth curve

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Patterns in distribution normal curve

• Depicted by a bell-shaped curve
• Indicates that measurement process is running
  normally

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Patterns in distribution bi-modal curve

• Distribution appears to have two peaks
• May indicate that data from more than process
  are mixed together

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Patterns in distribution saw-toothed

• Also commonly referred to as a comb distribution,
  appears as an alternating jagged pattern
• Often indicates a measuring problem
• improper gauge readings
• gauge not sensitive enough for readings

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Testing normality
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Normal Probability Plot
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Normally distributed?
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Normal Probability Plot of not normally
distributed data
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Test for Normality Shapiro-Wilks

• statistical test for Normality Shapiro-Wilks
• idea sophisticated regression analysis in the
  spirit of normal probability plot
• makes Normal Probability Plot objective
• check outliers (measurement error? normality
  sometimes disturbed by single observation)
• analyse if not normally distributed

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Statgraphics Shapiro Wilks
Tests for Normality for width Computed
Chi-Square goodness-of-fit statistic
254.667 P-Value 0.0 Shapiro-Wilks W statistic
0.921395 P-Value 0.000722338

• Interpretation
• value statistic itself cannot need be
  interpreted
• P-value indicates how likely normal distribution
  is
• use ? 0.01 as critical value in order to avoid
  too strict rejections of normality

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Dixons test

• Box-and-Whisker plot graphical test of outliers
• if data are normally distributed, then formal
  test may be used

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Disadvantages of point estimators
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Confidence intervals

• 95 confidence interval for µ probability 0.95
  that interval contains true value µ
• more observations ? narrower interval (effect in
  particular for n lt 20)
• higher confidence ? wider interval
• example ?0,05 ?

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Confidence intervals example
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Hypothesis testing

• example test whether there is a systematic
  error

Hypothesis Tests for meting Sample mean
4.994 Sample median 5.01 t-test ------ Null
hypothesis mean 5.0 Alternative not
equal Computed t statistic -0.155011 P-Value
0.880233 Do not reject the null hypothesis for
alpha 0.05.