Chapter 1
Picturing Distributions with Graphs
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 Question: What is the average height of students in this class?
 Answer: Easy, ask everyone the height, and then find the
average (mean).
 Question: What is the average height of students in
University of Utah?
 Answer: Seems complicated. Too many people, it will be
difficult to ask everyone. Ask a sufficiently large number of
people (but still small compared to the total size of students)
and make an estimate using tools from STATISTICS.
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 1. How do we select the people to ask? How many people
should we ask?
 2. How do we make an estimate ?
 3. Our estimate will be probably be a little different from the
true value. Can we say how close it is to the true value?
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Statistics
Statistics is a science that involves the extraction of
information from numerical data obtained during an
experiment or from a sample. It involves the design
of the experiment or sampling procedure, the
collection and analysis of the data, and making
inferences (statements) about the population based
upon information in a sample.
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5
name
major
megan
ryan
jacob
antony
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guillermo
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nicos
susan
bs
egr
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bs
egr
bs
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ba
ba
score
gpa
25
35
40
50
22
24
31
35
24
3
2.5
3.5
2.5
2.6
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Individuals and Variables
 Individuals
 the objects described by a set of data
 may be people, animals, or things
 Variable
 any characteristic of an individual
 can take different values for different individuals
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Variables
 Places an individual into one of several groups or categories
 Takes numerical values for which arithmetic operations such as
adding and averaging make sense
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Case Study
The Effect of Hypnosis
on the
Immune System
reported in Science News, Sept. 4, 1993, p. 153
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Case Study
The Effect of Hypnosis
on the
Immune System
Objective:
To determine if hypnosis strengthens the
disease-fighting capacity of immune cells.
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Case Study
 65 college students.
 33 easily hypnotized
 32 not easily hypnotized
 white blood cell counts measured
 all students viewed a brief video about the immune system.
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Case Study
 Students randomly assigned to one of three conditions
 subjects hypnotized, given mental exercise
 subjects relaxed in sensory deprivation tank
 control group (no treatment)
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Case Study
 white blood cell counts re-measured after one week
 the two white blood cell counts are compared for each
group
 results
 hypnotized group showed larger jump in white blood cells
 “easily hypnotized” group showed largest immune enhancement
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Case Study
Variables measured
quantitative
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 Pre-study white blood cell count
 Post-study white blood cell count
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Case Study
Weight Gain Spells
Heart Risk for Women
“Weight, weight change, and coronary heart disease
in women.” W.C. Willett, et. al., vol. 273(6), Journal
of the American Medical Association, Feb. 8, 1995.
(Reported in Science News, Feb. 4, 1995, p. 108)
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Case Study
Weight Gain Spells
Heart Risk for Women
Objective:
To recommend a range of body mass index
(a function of weight and height) in terms of
coronary heart disease (CHD) risk in women.
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Case Study
 Study started in 1976 with 115,818 women aged 30 to 55
years and without a history of previous CHD.
 Each woman’s weight (body mass) was determined.
 Each woman was asked her weight at age 18.
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Case Study
 The cohort of women were followed for 14 years.
 The number of CHD (fatal and nonfatal) cases were counted
(1292 cases).
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Case Study
Variables measured
categorical
 Incidence of coronary heart disease
 Smoker or nonsmoker
 Family history of heart disease
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Distribution
 Tells what values a variable takes and how often it takes these
values
 Can be a table, graph, or function
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Displaying Distributions
 Categorical variables
 Pie charts
 Bar graphs
 Quantitative variables
 Histograms
 Stemplots (stem-and-leaf plots)
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Class Make-up on First Day
Data Table
Year
Count
Percent
Freshman
18
41.9%
Sophomore
10
23.3%
Junior
6
14.0%
Senior
9
20.9%
Total
43
100.1%
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Class Make-up on First Day
Pie Chart
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Class Make-up on First Day
Bar Graph
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Example: U.S. Solid Waste (2000)
Data Table
Material
Weight (million tons)
Percent of total
Food scraps
25.9
11.2 %
Glass
12.8
5.5 %
Metals
18.0
7.8 %
Paper, paperboard
86.7
37.4 %
Plastics
24.7
10.7 %
Rubber, leather, textiles
15.8
6.8 %
Wood
12.7
5.5 %
Yard trimmings
27.7
11.9 %
Other
7.5
3.2 %
Total
231.9
100.0 %
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Example: U.S. Solid Waste (2000)
Pie Chart
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Example: U.S. Solid Waste (2000)
Bar Graph
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Histograms
 For quantitative variables that take many values
 Divide the possible values into
(we will only
consider equal widths)
 Count how many observations fall in each interval (may
change to percents)
 Draw picture representing distribution
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Histograms: Class Intervals
 How many intervals?
 One rule is to calculate the square root of the sample size, and
round up.
 Size of intervals?
 Divide range of data (maxmin) by number of intervals desired,
and round to convenient number
 Pick intervals so each observation can only fall in exactly
one interval (no overlap)
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Case Study
Weight Data
Introductory Statistics class
Spring, 1997
Virginia Commonwealth University
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Weight Data
192
152
135
110
128
180
260
170
165
150
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120
185
165
212
119
165
210
186
100
195
170
120
185
175
203
185
123
139
106
180
130
155
220
140
157
150
172
175
133
170
130
101
180
187
148
106
180
127
124
215
125
194
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Weight Data: Frequency Table
Weight Group
100 - <120
120 - <140
140 - <160
160 - <180
180 - <200
200 - <220
220 - <240
240 -<260
260 -<280
Weight Group
100 - <120
120 - <140
140 - <160
160 - <180
180 - <200
200 - <220
220 - <240
240 - <260
260 - <280
sqrt(53) = 7.2,
BPS - 5th Ed.
Count
Count
7
12
7
8
12
4
1
0
1
7
12
7
8
12
4
1
0
1
; range (260100=160) / 8 =
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Number of students
Weight Data: Histogram
100
120
140
160
180 200
Weight
220 240
260
280
* Left endpoint is included in the group, right endpoint is not.
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Examining the Distribution of
Quantitative Data
 Overall pattern of graph
 Deviations from overall pattern
 Shape of the data
 Center of the data
 Spread of the data (Variation)
 Outliers
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Shape of the Data
 Symmetric
 bell shaped
 other symmetric shapes
 Asymmetric
 right skewed
 left skewed
 Unimodal, bimodal
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Symmetric
Bell-Shaped
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Symmetric
Mound-Shaped
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Symmetric
Uniform
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Asymmetric
Skewed to the Left
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Asymmetric
Skewed to the Right
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Outliers
 Extreme values that fall outside the overall pattern
 May occur naturally
 May occur due to error in recording
 May occur due to error in measuring
 Observational unit may be fundamentally different
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Stemplots
(Stem-and-Leaf Plots)
 For quantitative variables
 Separate each observation into a
(first part of the
number) and a
(the remaining part of the number)
 Write the stems in a vertical column; draw a vertical line
to the right of the stems
 Write each leaf in the row to the right of its stem; order
leaves if desired
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1
2
Weight Data
192
152
135
110
128
180
260
170
165
150
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110
120
185
165
212
119
165
210
186
100
195
170
120
185
175
203
185
123
139
106
180
130
155
220
140
157
150
172
175
133
170
130
101
180
187
148
106
180
127
124
215
125
194
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Weight Data:
Stemplot
(Stem & Leaf Plot)
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16 2
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192
152
135
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Weight Data:
Stemplot
(Stem & Leaf Plot)
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0166
009
0034578
00359
08
00257
555
000255
000055567
245
3
025
0
0
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Extended Stem-and-Leaf Plots
If there are very few stems (when the data cover only a
very small range of values), then we may want to create
more stems by splitting the original stems.
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Extended Stem-and-Leaf Plots
: if all of the data values were between 150 and 179,
then we may choose to use the following stems:
15
15
16
16
17
17
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Leaves 0-4 would go on each
upper stem (first “15”), and leaves
5-9 would go on each lower stem
(second “15”).
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Time Plots
 A time plot shows behavior over time.
 Time is always on the horizontal axis, and the variable being
measured is on the vertical axis.
 Look for an overall pattern (trend), and deviations from this
trend. Connecting the data points by lines may emphasize
this trend.
 Look for patterns that repeat at known regular intervals
(seasonal variations).
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Class Make-up on First Day
(Fall Semesters: 1985-1993)
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Average Tuition (Public vs. Private)
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