Statistics Chapter

1
Statistics Chapter

• Measures of Central Tendency

2
Introduction

• Statistics
• Helps us better understand the data we work with.

• From www.bls.gov
• Earnings
• Median hourly earnings of welders, cutters,
  solderers, and brazers were 14.72 in May 2004.
  The middle 50 percent earned between 11.90 and
  18.05. The lowest 10 percent had earnings of
  less than 9.79, while the top 10 percent earned
  over 22.20.
• Median hourly earnings of machinists were 16.33
  in May 2004. The middle 50 percent earned between
  12.84 and 20.33. The lowest 10 percent earned
  less than 10.08, while the top 10 percent earned
  more than 24.34.

3
Introduction

• Statistics
• Helps us better understand the data we work with.

Thermal eye watches weld quality     A steel
maker and a research centre have developed an
automated system for monitoring the quality of
welds in industrial seam-welding. An optical
pyrometer senses the temperature of the weld as
it emerges from the machine. The weld is
acceptable if the average and standard deviation
of the temperatures along its length both fall
within set values. Studies show excellent
correlation with traditional inspection
techniques. The system can also detect any
gradual deterioration in the weld quality,
indicating the need for preventative maintenance
of the welding machine.
4
Introduction

• Statistics
• Helps us better understand the data we work with.

• What single number best represents this data?
• How spread out is the data? (How similar are
  the s similar in size?)
• What is the likelihood of a particular result to
  happen based on this data?

5
Central Tendency

• What 1 single number best represents a collection
  of numbers (data)?

6
Central Tendency Analogy
7
Central Tendency
Machinists Pay
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8
Central Tendency
Machinists Pay
(May 2005) Mean annual earnings 35,350
(May 2005) Median Hourly Salary 16.51
Machinists Pay Data
9
Central Tendency

• Mean

10
Mean

• Mean is the average of a set of numbers.

11
Mean Sample Problem 1

• Average 1-way distance driven from home to SWTC

mean 25.0
12
Mean Sample Problem 2

• Weekly Production Report

mean 883 bolts/day
13
Statistics

• Median

14
Median

• Middle

15
Median

• Median tunes-out flyers.

mean 137,929
mean 83.8
16
Median

• Even amount of data

Median 77.5
88
72
71
92
60
83
17
Median

• Odd amount of data

88
72
71
92
60
83
75
18
Sample Problem

• Number of defects per shift (2 weeks of data)

Mean ? Median ?
4, 4, 3, 12, 8, 5, 5, 4, 4, 3
Mean 5.2 Median 4
19
Statistics

• Mode

20
Mode

• Most common
• Most frequently occurring
• Most popular

Soft Drinks
0.75
21
Mode
19 yrs
22
Mode
19 yrs
22 yrs
23
Homework

• Practice Set 4
• Pages 11 -12 1-3

24
Statistics

• Measures of Variability

25
Introduction
Ave depth 4.2 ft
26
Introduction
Introduction
Ave depth 4.2 ft
27
(No Transcript)
28
Data Set 1 1, 3, 5, 6, 6, 8, 8, 10, 10 mean
6.33
Data Set 2 5, 5, 5, 6, 7, 7, 7, 7, 8 Mean 6.33
29
Range

• Range Highest - Lowest
• A small range means
• A large range means

30
Range
Data 1, 3, 5, 6, 6, 8, 8, 10, 10
31
Range
Data 5, 5, 5, 6, 7, 7, 7, 7, 8
32
Data Set 1 1, 3, 5, 6, 6, 8, 8, 10, 10 mean
6.33 range 9
Data Set 2 5, 5, 5, 6, 7, 7, 7, 7, 8 mean
6.33 range 3
33
Range Our Class

• Age of students

range 32
34
Range Our Class

• Hours per week studying

range 13.5 hrs/wk
35
Statistics

• Measures of Variability Standard Deviation

36
Standard Deviation

• Standard deviation is another way to measure
  variation.
• Standard Deviation tells you the average
  distance a piece of data is from that groups
  mean.

37
5.1 2 3.1

• Hours per week studying
• Mean 5.1 hrs/wk

5.1 4 1.1
7 5.1 1.9
15 5.1 9.9
etc
Standard Deviation 3.5 hrs
2
0
4
6
8
10
12
14
16
38
Compute Standard Deviation

• Calculator
• Computer
• Excel

Sx
stdev(A1A7)
8.55
39
Review

• Measures of variability tell you
• how similar a collection of numbers are to one
  another
• how different they are from one another.

Range 7 Standard Deviation 2.5
89, 90, 92, 95, 88, 90
Range 11.7 Standard Deviation 31
90, 71, 80, 93, 62, 84
40
Comparison

• Range

Pro easy to compute Con only looks at the
high and low values nothing else. Can be
affected by flyers.
41
Comparison

• Standard Deviation

Pro gives a more accurate picture of
variability since it involves all of the data in
the computation. Con takes longer to
compute
42
In-class Problems

• Question 1 Which group of data will have a
  large standard deviation? Which will have a
  small standard deviation?
• Question 2 For which group of data will the
  computed range be deceptive?

Set 1 7.85, 8.21, 7.15, 6.82, 5.95, 8.05, 8.72
Set 2 7.15, 12.98, 13.25, 12.01, 12.37, 11.95
43
Questions (1 of 3)

• Traffic speeds of all vehicles moving along a
  stretch of freeway are monitored for several
  days. The standard deviation was found to be
  relatively low, that means
• A) the vehicles are moving at a relatively
  uniform speed

44
Questions (1 of 3)

• Traffic speeds of all vehicles moving along a
  stretch of freeway are monitored for several
  days. The standard deviation was found to be
  relatively low, that means
• A) the vehicles are moving at a relatively
  uniform speed
• B) the vehicles are moving at very dissimilar
  speedssome are moving quite slow and others are
  moving quite fast.

45
Questions (2 of 3)

• Which machining shop has a wider range of ages in
  its force?
• A) Acme Metals
• Standard Deviation 5 yrs
• B) Emerald City Machining
• Standard Deviation 7.8 years

46
Questions (3 of 3)

• Which measure range or standard deviation,
  would give the most realistic indication of
  variability for this data set?

Number of off-site welding jobs performed per
week.
11, 15, 12, 14, 18, 12, 12, 13, 1, 2, 10, 15, 13,
12, 16
47
Homework

• Practice Set 6
• Pages 22 - 23 1-4

48
Statistics

• Normal Distributions

49
Normal Distributions

• Data gets collected about a certain topic
• IQ Scores
• Heights
• Number of Arrests
• Age
• Size of Manufactured Parts

50
Normal Distributions

• As you record the data, you note that numbers
  tend to pile up in certain areas.

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red blue yellow green orange
51
Normal Distributions

• As you record the data, you note that numbers
  tend to pile up in certain areas.

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red blue yellow green orange
52
Normal Distributions

• Curve

red blue yellow green orange
53
Normal Distribution

• IQ
• mean 100
• st. dev 15

X X X X X
X X X X
X X X X
X X
X X
X
X
100
115
85
130
70
55
145
54
IQ Info

• Genius IQ is generally considered to begin around
  140 to 145, representing 0.25 of the population
  (1 in 400).  Here's a rough guide
• 115-124 - Above average (e.g., university
  students)
• 125-134 - Gifted (e.g., post-graduate students)
• 135-144 - Highly gifted (e.g., intellectuals)
• 145-154 - Genius (e.g., professors)
• 155-164 - Genius (e.g., Nobel Prize winners)
• 165-179 - High genius
• 180-200 - Highest genius
• gt200 - "Unmeasurable genius"

55
Normal Distribution

• Heights of U.S Women Age 18-25
• mean 65.5 inches
• standard deviation 2.5 inches

65.5
68
63
70.5
60.5
58
73
56
Standard Deviation Characteristics
Weight of Candy Bars Produced
IQ Scores
What can be learned from the different curve
shapes?
Traffic Speeds
Age of Auto Thieves
57
Standard Deviation Characteristics

• IQ

15
15
100
115
85
58
Standard Deviation Characteristics

• IQ

15 15
15 15
100
115
85
130
70
59
Standard Deviation Characteristics

• IQ

15 15 15
15 15 15
100
115
85
145
55
70
130
60
Normal Distributions

• For any data that is normally distributed
• 68 of the data is located within 1 standard
  deviation from the mean.

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61
Normal Distributions

• For any data that is normally distributed
• 95 of the data is located within 2 standard
  deviations from the mean.

? ?
? ?
62
Normal Distributions

• For any data that is normally distributed
• 99.7 of the data is located within 3 standard
  deviations from the mean.

? ? ?
? ? ?
63
Practice

• Daily attendance at the annual county fair for
  the past ten years.

What is the mean for this data?
What is the standard deviation for this data?
4500
4750
4250
5000
4000
3750
5250
64
Homework

• Practice Set 7
• Page 28 1 - 5

65
Statistics

• Normal Distribution Applications

66
Using Normally Distributed Data

• Data on the Heights of U.S. Women

65.5
68
63
70.5
60.5
58
73
67
Applications
68
Applications
69
Using Normally Distributed Data

• Car Battery Service Life

Service Life (Months) of SureStart Car Batteries

• How long does the average SureStart Battery last?

• Would it be better for this company to set a 44
  month warranty or a 56 month warranty?

52
54
48
56
46
44
58
70
Applications
Warranties
71
Sample Problems

• Data Lifespan of 60w Lightbulbs

As a manufacturer, what would be the safest claim
to make about the bulbs? a) Our bulbs are
guaranteed to last 2000 hrs. b) Our bulbs are
guaranteed to last 2300 hrs. c) Our bulbs are
guaranteed to last 1500 hours.
2000
1850
1600
2450
2150
1550
2300
72
Sample Problems

• Establish ad campaigns to target products.

a) What is the average age of your customers?
b) What percent of your customers are between
the ages of 15 and 23?
c) What percent of your customers are over 25
years old?
19
21
17
23
15
13
25
73
Sample Problems

• Establish ad campaigns to target products.

d) What percent of your customers are 15 years
old or less?
e) If you sold to 15,000 customers last year,
how many were between the ages of 17 and 21?
19
21
17
23
15
13
25
74
Sample Problems

• IQ Data

a) In a classroom of 20 people, how many will
have IQs between 85 and 115?
100
115
85
130
70
55
145
75
Sample Problems
b) At SWTC, assuming around 1400 full time
students, how many will have an IQ below 85?
How many below 70?

• IQ Data

100
115
85
130
70
55
145
76
Sample Problems
c) In Platteville, assuming a population of
about 10,000 people, how many will have an IQ
above 130?

• IQ Data

100
115
85
130
70
55
145
77
Homework

• Practice Set 8
• Pages 34-35 1-3

78
Statistics

• Quality Control and SPC

79
Quality Control

• Helps to understand what is happening with the
  products that are being produced.

80
Statistics

• Process Capability

81
Process Capability
This machines job is to fill 50 gallon
drums. What kind of results (gallons/barrel)
should I expect from this process?
If I know what is typical for this process I can
more easily spot when problems show up.
Drum-filling
FillingMachine
82
Production
Goal Describe what is happening with the
drum-filling process. The intent is to fill each
drum with 50 gallons of fluid.
Drum-filling
FillingMachine
Mean 50.1 gallons St. Dev. 0.05 gallons
83
Data from drum filling
Mean 50.1 gallons St. Dev. 0.05 gallons
99.7
50.1
50.15
50.05
50.2
50.0
49.95
50.25
Upper Control Limit 50.25 gal
UCL 50.25 gal
Lower Control Limit 49.95 gal
LCL 49.95 gal
84
Once the UCL and LCL are computed, I now know the
capabilities of this process.
Drum-filling
FillingMachine
85
Statistics

• Tolerance

86
Tolerance

• Tolerance information tells you how to judge
  quality.

• Weight
• Length
• Volume
• Color

87
Tolerance
Tolerance for Volume
50.0 gal ? 0.2 gal
Minimum Volume 50 0.2 49.8 gal Maximum
Volume 50 0.2 50.2 gal
Drum-filling
50.18 gal
49.75 gal
NO
OK
88
Capability vs Tolerance

• Capability
• UCL
• LCL

• Tolerance
• Minimum
• Maximum

89
Capability vs Tolerance

• Capability
• UCL
• LCL

• Tolerance
• Minimum
• Maximum

90
Capability vs Tolerance

• Capability
• UCL 50.25 gal
• LCL 49.95 gal

• Tolerance
• Minimum 49.8 gal
• Maximum 50.2 gal

91
Sample Problem
Tolerance for diameter 2.5000 ? 0.0015
Process mean diameter of 2.5003 standard
deviation of 0.0006
92
Control Limits and Tolerance

• Compute each

• Control limits for this process
• mean 2.5003
• st. dev. 0.0006
• UCL __________
• LCL __________

• Tolerance limits
• 2.5000 ? 0.0015
• max dia __________
• min dia __________

93
Capability vs Tolerance

• Capability
• UCL 2.5021
• LCL 2.4985

• Tolerance
• Minimum 2.4985
• Maximum 2.5015

94
Control Charts

• Show what is happening with a process throughout
  the day

---- g
240 g
242 g
239 g
240 g
238 g
95
Control Chart

UCL
12.5 fl oz

12.0 fl oz

11.5 fl oz
LCL
7 am 12.4 12.5 12.3 12.3 12.5 62 12.4 0.2
8 am 12.5 12.4 12.5 12.3 12.3 62 12.4 0.2
9am etc etc etc etc etc etc 12.6 0.1
10am etc etc etc etc etc etc 12.4 0.4
11am etc etc etc etc etc etc 12.2 0.2
12 pm etc etc etc etc etc etc 12.4 0.5
1 pm etc etc etc etc etc etc 11.8 0.1
2 pm etc etc etc etc etc etc 12.0 0.3
3 pm
Measurements
96
Control Chart
UCL

0.5

LCL
0
7 am 12.4 12.5 12.3 12.3 12.5 62 12.4 0.2
8 am 12.5 12.4 12.5 12.3 12.3 62 12.4 0.2
9am etc etc etc etc etc etc 12.6 0.1
10am etc etc etc etc etc etc 12.4 0.4
11am etc etc etc etc etc etc 12.2 0.2
12 pm etc etc etc etc etc etc 12.4 0.5
1 pm etc etc etc etc etc etc 11.8 0.1
2 pm etc etc etc etc etc etc 12.0 0.3
3 pm
Measurements
97
Trends
UCL

0.5

LCL
0
98
Statistics

• Correlation

99
Introduction

• Correlation is there a relationship between two
  sets of data?
• Engine Size versus Fuel Economy
• Temperature versus Crime Rate
• Average temperature versus Home Heating Costs

100
Procedure - Step 1

• Step 1 Create a scatter graph

101
Procedure Step 2
Engine Size 4.0 L Mileage 20.5 mpg
102
Procedure Step 2
Engine Size 2.0 L Mileage 23.5 mpg
103
Procedure Step 3
104
Procedure Step 3
I
II
IV
III
105
Procedure Step 4

• Quadrant I _____
• Quadrant II _____
• Quadrant III _____
• Quadrant IV _____

I
II
IV
III
106
Procedure Step 5

• Quadrant I Quadrant III _____
• Quadrant II Quadrant IV _____

I
II

• Positive Correlation
• Negative Correlation
• No Correlation

IV
III
107
Procedure Step 6

• Positive Correlation

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Crime Rate
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Temperature
108
Procedure Step 6

• Negative Correlation

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Mileage
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Engine Size
109
Procedure Step 6

• No Correlation

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110
Procedure Step 6

• Quadrant I Quadrant III __2___
• Quadrant II Quadrant IV __12___

I
II

• Positive Correlation
• Negative Correlation
• No Correlation

IV
III
111
Correlation Criteria

• Correlation Rules (p. 55)
• Positive
• Sum of Quadrants I and III more than twice the
  sum of Quadrants II and IV.
• Negative
• Sum of Quadrants II and IV more than twice the
  sum of Quadrants I and III.
• No Correlation
• When neither of the above occur.

112
Homework

• Practice Set 11
• Pages 59 60, 1 and 2