Engineering Probability and Statistics - SE-205 -Chap 1

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Engineering Probability and Statistics - SE-205 -Chap 1. By S. O. Duffuaa. Course Objectives. Introduce the students to basic probability and statistics and demonstrate its wide application in the area of Systems Engineering. Main Course Outcomes. Students should be able to perform :

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Engineering Probability and Statistics - SE-205 -Chap 1

By

S. O. Duffuaa

Course Objectives
• Introduce the students to basic probability and statistics and demonstrate its wide application in the area of Systems Engineering.
Main Course Outcomes
• Students should be able to perform:
• Summarize and present data
• Describe probability distributions
• Compute probabilities using density/mass functions
• Conduct interval estimation
• Use statistical package/Minitab
Text Book and References
• “Applied Statistics and Probability for Engineers “ by D. C. Montgomery and Runger, 1994.
• “Probability and Statistics for Engineers and Scientists” 5th by Walpole and Mayers.
• Statistics by Murry Speigel
Course Policy
• Home-works and attendance 15%
• Quizzes 15%
• Exam1 20%
• Exam II 20%
• Final Exam 30%
SE- 205 Place in SE Curriculum
• Central Course
• Prerequisite for 7 SE courses
• SE 303, SE 320, SE 323, SE 325,

SE 447, SE 480, SE 463 and may be others. See SE Curriculum Tree

Engineering Problem Solving
• Develop clear and concise problem description
• Identify the important factors in the problem.
• Propose a model for the problem
• Conduct appropriate experimentation
• Refine the model
Engineering Problem Solving
• Validate the solution
• Conclusion and recommendations
Statistics
• Science of data collection, summarization, presentation and analysis for better decision making.
• How to collect data ?
• How to summarize it ?
• How to present it ?
• How do you analyze it and make conclusions and correct decisions ?
Role of Statistics
• Many aspects of Engineering deals with data – Product and process design
• Identify sources of variability
• Essential for decision making
Data Collection
• Observational study
• Observe the system
• Historical data
• The objective is to build a system model usually called empirical models
• Design of experiment
• Plays key role in engineering design
Data Collection
• Sources of data collection:
• Observation of something you can’t control (observe a system or historical data)
• Conduct an experiment
• Surveying opinions of people in social problem
Statistics
• Divided into:
• Descriptive Statistics
• Inferential Statistics
Forms of Data Description
• Point summary
• Tabular format
• Graphical format
• Diagrams
Point Summary

1) Central tendency measures

• Sample Mean x =  xi/n
• Population Mean(µ)
• Median --- Middle value
• Mode --- Most frequent value
• Percentile
Point Summary

2) Variability measures

• Range = Max xi - Min xi
• Variance = V = S 2 =  (xi – x )2/ n-1

also =

• Standard deviation = S

S = Square root (V)

• Coefficient of variation = S/ x
• Inter-quartile range (IQR)

 (xi 2) – {[( xi) 2]/n}

n -1

Diagrams: Dot Diagram
• A diagram that has on the x-axis the points plotted : Given the following grades of a class:

50, 23, 40, 90, 95, 10, 80, 50, 75, 55, 60, 40.

.

.

.

.

0

50

100

Dot Diagram
• A diagram that has on the x-axis the points plotted : Given the following grades of a class:

50, 23, 40, 90, 95, 10, 80, 50, 75, 55, 60, 40.

.

.

.

.

0

50

100

Graphical Format

• Time Frequency Plot
• The Time Frequency Plot tells the following :
• 1) The Center of Data
• 2) The Variability
• 3) The Trends or Shifts in the data
• Control Chart

Control Charts

Upper control limit = 100.5

x = 91.50

Lower control limit = 82.54

Control Charts

Central Line = Average ( X )

Lower Control Limit (LCL)= X – 3S

Upper Control Limit (UCL)= X + 3S

Lecture Objectives
• Sample and population
• Random sample
• Present the following:
• Stem-leaf diagram
• The frequency distribution
• Histogram
Population and Sample
• Population is the totality of observations we are concerned with.
• Example: All Engineers in the Kingdom,

All SE students etc.

• Sample : Subset of the population

50 Engineers selected at random, 10 SE students selected at random.

Mean and Variance
• Sample mean X-bar
• Population mean µ
• Sample variance S2
• Population variance σ2
Stem-And –Leaf Diagram
• Each number xi is divided intotwo parts the stem consisting of one or two leading digits
• The rest of the digits constitute the leaf.
• Example if the data is 126 then 12 is stem

and 6 is the leaf.

What is the stem and leaf for 76

105 221 183 186 121 181 180 143

97 154 153 174 120 168 167 141

245 228 174 199 181 158 176 110

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

Stem-And-Leaf f

Stem leaf frequency

7 6 1

8 7 1

9 7 1

10 5 1 2

11 5 0 8 3

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 0 8 8 6 8 0 8 12

16 3 0 7 3 0 5 0 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

Number of Stems Considerations

Stem Leaf

6 1 3 4 5 5 6

7 0 1 1 3 5 7 8 8 9

8 1 3 4 4 7 8 8

9 2 3 5

Stem number considerations

Stem leaf

6L 1 3 4

6U 5 5 6

7L 0 1 1 3

7U 5 7 8 8

8L 1 3 4 4

8U 7 8 8

9L 2 3

9U 5

Number of Stems
• Between 20 and 5
• Roughly n where n number of data points
Percentiles
• Pth percentile of the data is a value where at least P% of the data takes on this value or less and at least (1-P)% of the data takes on this value or more.
• Median is 50th percentile. ( Q2)
• First quartile Q1 is the 25th percentile.
• Third quartile Q3 is the 75th percentile.
Percentile Computation : Example

Data : 5, 7, 25, 10, 22, 13, 15, 27, 45, 18, 3, 30

Compute 90th percentile.

1. Sort the data from smallest to largest

3, 5, 7, 10, 13, 15, 18, 22, 25, 27, 30, 45

2. Multiply 90/100 x 12 = 10.8 round it to to the next integer which is 11.

Therefore the 90th percentile is point # 11 which is 30.

Percentile Computation : Example
• If the product of the percent with the number of the data came out to be a number. Then the percentile is the average of the data point corresponding to this number and the data point corresponding to the next number.
• Quartiles computation is similar to the percentiles.
Pth percentile = (P/ 100)*n = r

double (round it up & take its rank)

(r)

integer (take Avg. of its rank & # after)

• Inter-quartile range = Q3 – Q1
• Frequency Distribution Table :

1) # class intervals (k) = 5 < k < 20

k ~ n

2) The width of the intervals (W) = Range/k

= (Max-Min) /n

Class Interval

(psi)

Tally

(# data in this interval)

Frequency

Relative Frequency =

(Frequency/ n)

Cumulative Relative Frequency

70 ≤ x < 90

||

2

0.0250

0.0250

90 ≤ x < 110

|||

3

0.0375

0.0625

110 ≤ x < 130

|||| |

6

0.0750

0.1375

130 ≤ x < 150

|||| |||| ||||

14

0.1750

0.3125

150 ≤ x < 170

|||| |||| |||| |||| ||

22

0.2750

0.5875

170 ≤ x <1 90

|||| |||| |||| ||

17

0.2125

0.8000

190 ≤ x < 210

|||| ||||

10

0.1250

0.9250

210 ≤ x < 230

||||

4

0.0500

0.9750

230 ≤ x < 250

||

2

0.0250

1.0000

Histogram

2

5

20

15

Frequency

10

5

0

70

90

110 130

150 170

190

210

230

250

Compressive Strength (

psi

)

Histogram: is the graph of the frequency distribution table that shows class intervals V.S. freq. or (Cumulative) Relative freq.

Box Plot

Whisker extends to largest data point within 1.5 interquartile ranges from third quartile

Whisker extends to smallest data point within 1.5 interquartile ranges from first quartile

Third Quartile

First Quartile

Second Quartile

Extreme Outliers

Outliers

Outliers

Extreme Outliers

1.5 IQR

1.5 IQR

IQR

1.5 IQR

1.5 IQR

Strength

120

110

100

Quantity Index

90

80

70

1

3

2

Plant