EngStats Wk1 Descriptive Stats PDF

8/8/2019 EngStats Wk1 Descriptive Stats PDF

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Recall

Variability

DataStatistics

Engineering method


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RecallPopulation

Sample


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Engineering Statistics

Descri tive Statistics(chapter 6 montg.)


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Motivation (bioelectronics)

Large set of data. Highly-dimensional data How to make sense of such data?


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Motivation

, ..,

Large set of data. Highly-dimensional data How to make sense of such data?

Aircraft 1, , Aircraft 1000


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Outline Descriptive vs. Inferential Statistics Numerical summaries of data

Data display Stem-and-Leaf diagrams

Freq. distributions & histograms Box plots Probability Plots


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Descriptive vs. Inferential StatisticsStatistics

Descriptive Inferential

Numerical summary Graphical display Confidence Interval Hypothesis tests


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Numerical Summaries of Data


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Numerical Summaries of Data

Measures

Tendency Dispersion

Mode Median Mean

Range Variance Standard Deviation


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Data Summary

Descriptive statistics Measures of central tendency

Mean: weighted average Mode: most common observation Median: half the sample is larger

mode: most frequentvalue

median: value s.t. 50% of observationsabove/below

mean: value s.t. sum of deviationsweighted by frequency same oneither side


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Data Summary

Descriptive statistics Measures of scatter

variance (Standard deviation) ran e Coefficient of variation


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We will be focussing on the followings:

Mean

Variance / Standard deviation

Data Summary

Proportion

There exist differences between population &sample measurements


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Numerical Summaries of DataPopulation

p

x

2

s

2

Similarly forvs. s


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Formulas:

Mean

Data Summary

Variance

Standard deviation


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Example

Consider the 8 observations on pull-off forcecollected from prototype engine connectors. Theeight observations are:

1 . , 2 . , 3 . , 4 . , 5 . , x 6 = 13.5, x 7 = 12.6, and x 8 = 13.1

Q: Find the sample mean, sample mode, samplemedian, sample variance and sample standard deviation


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Measures


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Example: Solution

Sample mean,

n

x x x x n

..21 +++=v

pounds

xi

i

138

1.13..9.126.12 8

1

=

+++=

=

=


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Example: Solution

Sample variance,

x xi )(

82

v

pounds s

poundsn

s

48.02886.0

2886.0181

==

=

=

==

What does this figure means?


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Use of calculatorSet the calculator (Casio 570MS) to the following:

(1) Clear screenPress Shift, Press CLR, Choose 1 (for clear screen, Scl),

Press =ress .

(2) Choosing SD modePress MODE, MODE,

Choose 1 (for standard deviation, SD),Press = . (note: SD should appear on the display screen)


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Use of calculator(3) Entering data: eg. 1,2,3,4

Press 1; Press M + ; Press 2; Press M + ; Press 3; PressM + ; Press 4; Press M + .

Shift 2; choose 1, gives the sample mean x = 2:5. Shift 2; choose 3, gives the sample standard

deviation s = 1:29. Shift 1; choose 1, gives = 30. Shift 1; choose 3, gives n = 4.

2 x


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ExampleCompressive strength of 80 Al-Li alloy specimens

105 221 183 186 121 181 180 143

97 154 153 174 120 168 167 141

245 229 174 199 181 158 176 110

163 131 154 115 160 208 158 133

207 180 190 193 194 133 156 123

134 178 76 167 184 135 229 146

218 157 101 171 165 172 158 169199 151 142 163 145 171 148 158

160 175 149 87 160 237 150 135

196 201 200 176 150 170 118 149


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Use: informative general visual display of data(each with at least 2 digits)

- shape of distribution

- central tendency

Data display: stem-and-leaf diag.

n x x ,...,1

- sprea o a a

Works well especially for small sample size,

eg. 20 observations.


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Steps to construct

Stem-and-Leaf diag.

(1) Divide each number x i into two parts:a stem , consisting of one or more of the

leading digitsa leaf , consisting of the remaining digit.

(2) List the stem values in a vertical column.(3) Record the leaf for each observation beside itsstem.(4) Write the units for stems and leaves on thedisplay.


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Stem-and-Leaf Diagram

105 221 183 186 121 181 180 143

97 154 153 174 120 168 167 141

Compressive strength of 80 Al-Li alloy specimensStem Leaf Frequency

7 6 1

8 7 1

9 7 1

10 5 1 2

11 5 8 0 3

105 221 183 186 121 181 180 143

97 154 153 174 120 168 167 141

Stem Leaf Frequency

7 6 1

8 7 1

9 7 1

10 5 1 2

11 5 8 0 3

105 221 183 186 121 181 180 143

97 154 153 174 120 168 167 141

Stem Leaf Frequency

7 6 1

8 7 1

9 7 1

10 5 1 2

11 5 8 0 3

105 221 183 186 121 181 180 143

97 154 153 174 120 168 167 141

Stem Leaf Frequency

7 6 1

8 7 1

9 7 1

10 5 1 2

11 5 8 0 3

105 221 183 186 121 181 180 143

97 154 153 174 120 168 167 141

Stem Leaf Frequency

7 6 1

8 7 1

9 7 1

10 5 1 2

11 5 8 0 3

105 221 183 186 121 181 180 143

97 154 153 174 120 168 167 141

Stem Leaf Frequency

7 6 1

8 7 1

9 7 1

10 5 1 2

11 5 8 0 3

Stem-and-leaf diag.

245 229 174 199 181 158 176 110

163 131 154 115 160 208 158 133

207 180 190 193 194 133 156 123

134 178 76 167 184 135 229 146

218 157 101 171 165 172 158 169

199 151 142 163 145 171 148 158

160 175 149 87 160 237 150 135

196 201 200 176 150 170 118 149

12 1 0 3 3

13 4 1 3 5 3 5 6

14 2 9 5 8 3 1 6 9 8

15 4 7 1 3 4 9 8 8 6 8 0 8 12

16 3 9 7 3 9 5 9 8 7 9 10

17 8 5 4 4 1 6 2 1 0 6 10

18 0 3 6 1 4 1 0 7

19 9 6 0 9 3 4 6

20 7 1 0 8 4

21 8 1

22 1 8 9 3

23 7 1

24 5 1

245 229 174 199 181 158 176 110

163 131 154 115 160 208 158 133

207 180 190 193 194 133 156 123

134 178 76 167 184 135 229 146

218 157 101 171 165 172 158 169

199 151 142 163 145 171 148 158

160 175 149 87 160 237 150 135

196 201 200 176 150 170 118 149

12 1 0 3 3

13 4 1 3 5 3 5 6

14 2 9 5 8 3 1 6 9 8

15 4 7 1 3 4 9 8 8 6 8 0 8 12

16 3 9 7 3 9 5 9 8 7 9 10

17 8 5 4 4 1 6 2 1 0 6 10

18 0 3 6 1 4 1 0 7

19 9 6 0 9 3 4 6

20 7 1 0 8 4

21 8 1

22 1 8 9 3

23 7 1

24 5 1

245 229 174 199 181 158 176 110

163 131 154 115 160 208 158 133

207 180 190 193 194 133 156 123

134 178 76 167 184 135 229 146

218 157 101 171 165 172 158 169

199 151 142 163 145 171 148 158

160 175 149 87 160 237 150 135

196 201 200 176 150 170 118 149

12 1 0 3 3

13 4 1 3 5 3 5 6

14 2 9 5 8 3 1 6 9 8

15 4 7 1 3 4 9 8 8 6 8 0 8 12

16 3 9 7 3 9 5 9 8 7 9 10

17 8 5 4 4 1 6 2 1 0 6 10

18 0 3 6 1 4 1 0 7

19 9 6 0 9 3 4 6

20 7 1 0 8 4

21 8 1

22 1 8 9 3

23 7 1

24 5 1

245 229 174 199 181 158 176 110

163 131 154 115 160 208 158 133

207 180 190 193 194 133 156 123

134 178 76 167 184 135 229 146

218 157 101 171 165 172 158 169

199 151 142 163 145 171 148 158

160 175 149 87 160 237 150 135

196 201 200 176 150 170 118 149

12 1 0 3 3

13 4 1 3 5 3 5 6

14 2 9 5 8 3 1 6 9 8

15 4 7 1 3 4 9 8 8 6 8 0 8 12

16 3 9 7 3 9 5 9 8 7 9 10

17 8 5 4 4 1 6 2 1 0 6 10

18 0 3 6 1 4 1 0 7

19 9 6 0 9 3 4 6

20 7 1 0 8 4

21 8 1

22 1 8 9 3

23 7 1

24 5 1

245 229 174 199 181 158 176 110

163 131 154 115 160 208 158 133

207 180 190 193 194 133 156 123

134 178 76 167 184 135 229 146

218 157 101 171 165 172 158 169

199 151 142 163 145 171 148 158

160 175 149 87 160 237 150 135

196 201 200 176 150 170 118 149

12 1 0 3 3

13 4 1 3 5 3 5 6

14 2 9 5 8 3 1 6 9 8

15 4 7 1 3 4 9 8 8 6 8 0 8 12

16 3 9 7 3 9 5 9 8 7 9 10

17 8 5 4 4 1 6 2 1 0 6 10

18 0 3 6 1 4 1 0 7

19 9 6 0 9 3 4 6

20 7 1 0 8 4

21 8 1

22 1 8 9 3

23 7 1

24 5 1

245 229 174 199 181 158 176 110

163 131 154 115 160 208 158 133

207 180 190 193 194 133 156 123

134 178 76 167 184 135 229 146

218 157 101 171 165 172 158 169

199 151 142 163 145 171 148 158

160 175 149 87 160 237 150 135

196 201 200 176 150 170 118 149

12 sps 3

13 4 1 3 5 3 5 6

14 2 9 5 8 3 1 6 9 8

15 4 7 1 3 4 9 8 8 6 8 0 8 12

16 3 9 7 3 9 5 9 8 7 9 10

17 8 5 4 4 1 6 2 1 0 6 10

18 0 3 6 1 4 1 0 7

19 9 6 0 9 3 4 6

20 7 1 0 8 4

21 8 1

22 1 8 9 3

23 7 1

24 5 1


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Ordered Stem-and-Leaf Stem Leaf Frequency

1 7 6 1

2 8 7 1

3 9 7 1

5 10 1 5 2

>= stem

median

Stem-and-Leaf diag.

8 11 0 5 8 3

11 12 0 1 3 3

17 13 1 3 3 4 5 5 6

25 14 1 2 3 5 6 8 9 9 8

37 15 0 0 1 3 4 4 6 7 8 8 8 8 12

(10) 16 0 0 0 3 3 5 7 7 8 9 10

33 17 0 1 1 2 4 4 5 6 6 8 10

23 18 0 0 1 1 3 4 6 7

16 19 0 3 4 6 9 9 6

10 20 0 1 7 8 4

6 21 8 1

5 22 1 8 9 3

2 23 7 1

1 24 5 1


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more compact summary than stem-and-leaf diagram

range of data divided into intervals: bins, class

interval

Data display:Frequency distributions & Histograms

Histogram- visual display of frequency dn.

- shape of distribution

- central tendency- spread of data

more stable for larger datasets, eg. 75- 100++


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Histogram

appear to be normally distributed relatively sensitive to changes in number of

bins/band width (esp. Small datasets)

Histogram for compression strength data


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Histogram:cumulative distribution plot

Cumulative distribution plots Position of mean and median change based on

the general shape of distribution


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Data display: Box Plot

also known as box-and-whisker plots

of data.

Centre, spread, deviation from symmetry& outliers


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Box Plot


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Box PlotStem Leaf Frequency

1 7 6 1

2 8 7 1

3 9 7 1

5 10 1 5 2

8 11 0 5 8 3

11 12 0 1 3 317 13 1 3 3 4 5 5 6

25 14 1 2 3 5 6 8 9 9 8

37 150 0 1 3 4 4 6 7 8 8 88 12

(10) 16 0 0 0 3 3 5 7 7 8 9 10

33 17 0 1 1 2 4 4 5 6 6 8 10

23 18 0 0 1 1 3 4 6 7

16 19 0 3 4 6 9 9 6

10 20 0 1 7 8 4

6 21 8 1

5 22 1 8 9 3

2 23 7 1

1 24 5 1


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Data display: Probability Plots

Use: visual examination to check datadistribution*

normal, lognormal, Weibull distribution etc.

Focus: Normal Probability plots


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Normal Probability Plot

Plot *standardized normal scores z j vs. x j Checking normality of data Plotted points fall approx. on straight line


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Normal Probability Plot

Deviation from normality(indication of non-normal distribution)(a) Light-tailed dn. (b) heavy-tailed dn. (c) positiveskewed dn.


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Next class Read chapter 2. Probability

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EngStats Wk1 Descriptive Stats PDF