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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