Transcript Quality Control Tools

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
• Make inference about populations
• Use statistical package/Minitab
Text Book and References
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

“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
Quizzes
Exam1
Exam II
Final Exam
15%
15%
20%
20%
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
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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

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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
2) – {[( x ) 2]/n}

(x
i
i
also =
n -1
• Standard deviation = S
S = Square root (V)
• Coefficient of variation = S/ x
• Inter-quartile range (IQR)
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
Time Frequency Plot
15
14
13
12
11
y 10
9
8
7
6
5
0
10
20
30
Observation number
40
50
Time Frequency Plot
15
14
13
12
11
y 10
9
8
7
6
5
0
10
20
30
Observation number
40
50
Control Charts
105
Concentration
Upper control limit = 100.5
95
x = 91.50
85
Lower control limit = 82.54
75
0
10
20
Observation number
30
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


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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 into two 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

Data Table 1.1 Compressive Strength of 80
Aluminum Lithium Alloy
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
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
6
7
7
5
5
1
4
2
4
3
8
0
9
7
8
1
7
1
0 8
0 3
1 3 5 3 5
9 5 8 3 1
7 1 3 4 0
0 7 3 0 5
5 4 4 1 6
3 6 14 1
6 0 9 3 4
1 0 8
8 9
frequency
6
8
0
2
0
9
8 6 8 0 8
8 7 9
1 0 6
1
1
1
2
3
3
6
8
12
10
10
7
6
4
1
3
1
Number of Stems Considerations
Stem
6
7
8
9
Leaf
1 3 4 5 5 6
0 1 1 3 5 7 8 8 9
1 3 4 4 7 8 8
2 3 5
Stem number considerations
Stem
6L
6U
7L
7U
8L
8U
9L
9U
leaf
1 3
5 5
0 1
5 7
1 3
7 8
2 3
5
4
6
1 3
8 8
4 4
8
Number of Stems
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Between 20 and 5
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Roughly n where n number of data
points
Percentiles
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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 =
Cumulative
Relative Frequency
(Frequency/ n)
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
25
Frequency
20
15
10
5
0
70 90 110 130 150 170 190 210 230 250
Compressive Strength (psi)
Cumulative Frequency
90
80
70
60
50
40
30
20
10
0
1
Strength
100
150
200
Histogram: is the graph of the frequency distribution table that
shows class intervals V.S. freq. or (Cumulative) Relative freq.
250
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
First Quartile
Extreme Outliers
1.5 IQR
Second Quartile
Outliers
1.5 IQR
Third Quartile
Outliers
IQR
1.5 IQR
Extreme Outliers
1.5 IQR
100
150
Strength
200
250
120
110
Quantity Index
100
90
80
70
1
2
Plant
3