Download Engineering Fundamentals and Problem Solving, 6e

Engineering
Fundamentals and Problem Solving, 6e
Chapter 10
Statistics
Chapter Objectives
• Analyze a wide variety of data sets using
descriptive techniques (mean, mode, variance,
standard deviation, and correlation)
• Learn to apply the appropriate descriptive
statistical techniques in a variety of situations
• Create graphical representations of individual
and grouped data points with graphs and
histograms
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
2
The Need for Statistical Methods
 “Quality is job one”
 “…..The basic concept of using statistical signals to
improve performance can be applied to any area
where output exhibits variation, such as component
dimensions, bookkeeping error rates, performance
characteristics of a computer information system, or
transit times for incoming materials…..”
Continuous Process Control Manual
Ford Motor Company
The Need for Statistical Methods
“…..As world competition intensifies, understanding
and applying statistical concepts and tools is
becoming a requirement for all employees. Those
individuals who get these skills in school will have a
real advantage when they apply for their first job.”
Paul H. O’Neill
CEO, Aluminum company of America
The Need for Statistical Methods
“ The competitive position of industry in the US
demands that we greatly increase the knowledge of
statistics among our engineering graduates…….. The
economic survival in today’s world cannot be ensured
without access to modern productivity tools, notably
applications of statistical methods.”
Arno Penzias
VP at AT&T Bell laboratories
A Model for Problem Solving
 State the problem or question
 Collect and analyze data
 Interpret the data and make decisions
 Implement and verify the decisions
 Plan next actions
From Data Tables to Probability
Goal: Improving the quality of any process
Solution: Using tools of statistics to make decisions from
data in an organized way.
How do we obtain good data on which to base
these decisions?
Most good plans for collecting data make use of
randomization which is tied to probability
Descriptive Statistics
• Used to summarize or describe important
features of a data set
• Parameters are calculated from available
observations
• Engineers generally contend with samples rather
than entire populations of data
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
8
Numerical Summaries of Data
• Data are the numeric observations of a
phenomenon of interest
• The totality of all observations is called a
population (finite or infinite)
• A portion used for analysis is a random
sample from the population
• The collection described in terms of shape,
outliers, center, and spread (SOCS)
• Center  mean; Spread  variance
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
Population vs. Sample
Population described by
its parameters (, )
Sample described by its
statistics ( x , s)
The statistics are used to estimate the parameters.
Measures of Central Tendency
• MEDIAN – “Middle” value in a sample
• MODE – The most common value in the sample
(there may be more than one mode)
• MEAN – Arithmetic average
– Geometric average
– Harmonic average
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
11
Mean
 If the n observations in a random sample are
denoted by x1, x2, . . ., xn, the sample mean is
x1  x 2    xn
1 n
x 
  xi
n
n i 1
 For a finite population with
N equally likely
values, the population mean is
1
   x i fX (x i ) 
N
i 1
n
N
x
i 1
i
Measures of Variation
• Represent the amount of disparity (dispersal,
scatter) between the data points and the mean
• Variance
2
(
x

x
)
 i
s2 
n 1
• Standard Deviation
s s
2
Note: n-1 is the number of degrees of freedom left
after calculating n
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
13
Variance
 If the n observations in a random sample are
denoted by x1, x2, . . ., xn, the sample variance
is
n
n
1
1

2
2
2
2
s 
(x i  x ) 
 x i  n x 

n  1 i 1
n  1  i 1

 For a finite population with
N equally likely
values, the population variance is
2
1

N
N
2
(
x


)
 i
i 1
Frequency Distributions
 A frequency distribution is a compact summary




of data, expressed as a table, graph, or function.
The data is gathered into bins or cells, defined
by class intervals.
The number of classes, multiplied by the class
interval, should exceed the range of the data.
Number of bins approximately equal to square
root of the sample size
The boundaries of the class intervals should be
convenient values, as should the class width.
Frequency Distribution Table
Histograms
 A histogram is a visual display of a frequency
distribution, similar to a bar chart or a stem-andleaf diagram.
 Steps to build one with equal bin widths:
1. Label the bin boundaries on the horizontal
scale.
2. Mark & label the vertical scale with the
frequencies or relative frequencies.
3. Above each bin, draw a rectangle whose
height = the frequency or relative frequency.
Shape of Frequency Distribution
Example 10.2
• Interstate Safety Corridors are established on certain
roadways with a propensity for strong cross winds, blowing
dust, and frequent fatal accidents.
• A driver is expected to turn on the headlights and pay special
attention to the posted speed limit in these corridors.
• In one such Safety Corridor in northern New Mexico, the
posted speed limit is 75 miles per hour.
• The Department of Public Safety set up a radar checkpoint
and the actual speed of 36 vehicles that passed the checkpoint
is shown in the Table below.
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
20
Example – cont’d
Actual speeds of cars in Safety Corridor
70
85
71
75
65
66
78
69
82
76
90
78
70
68
69
85
77
91
80
61
71
72
89
69
86
81
62
63
76
80
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
76
80
71
72
70
92
21
Example – cont’d
a.) Make a frequency distribution table using 5 as a class
width (e.g. 60.0 – 64.9)
b.) Construct a histogram
Highway Speed
9
Interval
Frequency
60-64.9
3
65-69.9
6
70-74.9
8
75-79.9
7
80-84.9
5
85-89.9
4
2
90-94.5
3
1
8
7
Frequency
6
5
4
3
0
60-64.9
65-69.9
70-74.9
75-79.9
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
80-84.9
85-89.9
90-94.5
22
Example – cont’d
c) The mean
x

x 
i
n
2716

 75.44
36
d) The standard deviation
 n( xi )  ( xi )
 
n(n  1)

2
2



1/ 2
 36(207360)  (2716) 


36
(
35
)


2
1/ 2
 8.3715
e) The variance
 2  70.0825
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
23