Chapter 1
Picturing Distributions with Graphs
BPS - 3rd Ed.
Chapter 1
1
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.
BPS - 3rd Ed.
Chapter 1
2
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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Chapter 1
3
Variables
 Categorical
– Places an individual into one of several
groups or categories
 Quantitative
(Numerical)
– Takes numerical values for which
arithmetic operations such as adding and
averaging make sense
BPS - 3rd Ed.
Chapter 1
4
Case Study
The Effect of Hypnosis
on the
Immune System
reported in Science News, Sept. 4, 1993, p. 153
BPS - 3rd Ed.
Chapter 1
5
Case Study
The Effect of Hypnosis
on the
Immune System
Objective:
To determine if hypnosis strengthens the
disease-fighting capacity of immune cells.
BPS - 3rd Ed.
Chapter 1
6
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.
BPS - 3rd Ed.
Chapter 1
7
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)
BPS - 3rd Ed.
Chapter 1
8
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
BPS - 3rd Ed.
Chapter 1
9
Case Study
Variables measured
categorical
quantitative
BPS - 3rd Ed.
 Easy
or difficult to achieve
hypnotic trance
 Group assignment
 Pre-study white blood cell
count
 Post-study white blood cell
count
Chapter 1
10
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)
BPS - 3rd Ed.
Chapter 1
11
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.
BPS - 3rd Ed.
Chapter 1
12
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.
BPS - 3rd Ed.
Chapter 1
13
Case Study
 The
cohort of women were followed for
14 years.
 The number of CHD (fatal and nonfatal)
cases were counted (1292 cases).
BPS - 3rd Ed.
Chapter 1
14
Case Study
Variables measured
quantitative
categorical
BPS - 3rd Ed.
 Age
(in 1976)
 Weight in 1976
 Weight at age 18
 Incidence of coronary heart
disease
 Smoker or nonsmoker
 Family history of heart disease
Chapter 1
15
Distribution
 Tells
what values a variable takes and
how often it takes these values
 Can
be a table, graph, or function
BPS - 3rd Ed.
Chapter 1
16
Displaying Distributions
 Categorical
variables
– Pie charts
– Bar graphs
 Quantitative
variables
– Histograms
– Stemplots (stem-and-leaf plots)
BPS - 3rd Ed.
Chapter 1
17
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%
BPS - 3rd Ed.
Chapter 1
18
Class Make-up on First Day
Pie Chart
Senior
20.9%
Freshman
41.9%
Junior
14.0%
Sophomore
23.3%
BPS - 3rd Ed.
Chapter 1
19
Class Make-up on First Day
45.0%
41.9%
Bar Graph
40.0%
35.0%
Percent
30.0%
23.3%
25.0%
20.9%
20.0%
14.0%
15.0%
10.0%
5.0%
0.0%
Freshman
Sophomore
Junior
Senior
Year in School
BPS - 3rd Ed.
Chapter 1
20
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 %
BPS - 3rd Ed.
Chapter 1
21
Example: U.S. Solid Waste (2000)
Pie Chart
BPS - 3rd Ed.
Chapter 1
22
Example: U.S. Solid Waste (2000)
Bar Graph
BPS - 3rd Ed.
Chapter 1
23
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
BPS - 3rd Ed.
Chapter 1
24
Shape of the Data
 Symmetric
– bell shaped
– other symmetric shapes
 Asymmetric
– right skewed
– left skewed
 Unimodal,
BPS - 3rd Ed.
bimodal
Chapter 1
25
Symmetric
Bell-Shaped
BPS - 3rd Ed.
Chapter 1
26
Symmetric
Mound-Shaped
BPS - 3rd Ed.
Chapter 1
27
Symmetric
Uniform
BPS - 3rd Ed.
Chapter 1
28
Asymmetric
Skewed to the Left
BPS - 3rd Ed.
Chapter 1
29
Asymmetric
Skewed to the Right
BPS - 3rd Ed.
Chapter 1
30
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
BPS - 3rd Ed.
Chapter 1
31
Histograms
 For
quantitative variables that take
many values
 Divide the possible values into class
intervals (we will only consider equal widths)
 Count how many observations fall in
each interval (may change to percents)
 Draw picture representing distribution
BPS - 3rd Ed.
Chapter 1
32
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)
BPS - 3rd Ed.
Chapter 1
33
Case Study
Weight Data
Introductory Statistics class
Spring, 1997
Virginia Commonwealth University
BPS - 3rd Ed.
Chapter 1
34
Weight Data
192
152
135
110
128
180
260
170
165
150
BPS - 3rd Ed.
110
120
185
165
212
119
165
210
186
100
195
170
120
185
175
203
185
123
139
106
Chapter 1
180
130
155
220
140
157
150
172
175
133
170
130
101
180
187
148
106
180
127
124
215
125
194
35
Weight Data: Frequency Table
Weight Group
100 - <120
120 - <140
140 - <160
160 - <180
180 - <200
200 - <220
220 - <240
240 - <260
260 - <280
Count
7
12
7
8
12
4
1
0
1
sqrt(53) = 7.2, or 8 intervals; range (260100=160) / 8 = 20 = class width
BPS - 3rd Ed.
Chapter 1
36
Weight Data: Histogram
14
Number of students
12
10
8
6
Frequency
4
2
0
100
120
140
160
180 200
Weight
220 240
260
280
* Left endpoint is included in the group, right endpoint is not.
BPS - 3rd Ed.
Chapter 1
37
Stemplots
(Stem-and-Leaf Plots)
 For
quantitative variables
 Separate each observation into a stem (first
part of the number) and a leaf (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
BPS - 3rd Ed.
Chapter 1
38
Weight Data
192
152
135
110
128
180
260
170
165
150
BPS - 3rd Ed.
110
120
185
165
212
119
165
210
186
100
195
170
120
185
175
203
185
123
139
106
Chapter 1
180
130
155
220
140
157
150
172
175
133
170
130
101
180
187
148
106
180
127
124
215
125
194
39
Weight Data:
Stemplot
(Stem & Leaf Plot)
Key
20|3 means
203 pounds
Stems = 10’s
Leaves = 1’s
BPS - 3rd Ed.
Chapter 1
10
11
12
13 5
14
15 2
16
17
18
19 2
20
21
22
23
24
25
26
192
152
135
40
Weight Data:
Stemplot
(Stem & Leaf Plot)
Key
20|3 means
203 pounds
Stems = 10’s
Leaves = 1’s
BPS - 3rd Ed.
Chapter 1
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
0166
009
0034578
00359
08
00257
555
000255
000055567
245
3
025
0
0
41
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.
BPS - 3rd Ed.
Chapter 1
42
Extended Stem-and-Leaf Plots
Example: 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
BPS - 3rd Ed.
Leaves 0-4 would go on each
upper stem (first “15”), and leaves
5-9 would go on each lower stem
(second “15”).
Chapter 1
43
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).
BPS - 3rd Ed.
Chapter 1
44
Class Make-up on First Day
(Fall Semesters: 1985-1993)
Class Make-up On First Day
70%
60%
Percent of Class
That Are Freshman
50%
40%
30%
20%
10%
0%
1985
1986
1987
1988
1989
1990
1991
1992
1993
Year of Fall Semester
BPS - 3rd Ed.
Chapter 1
45
Average Tuition (Public vs. Private)
BPS - 3rd Ed.
Chapter 1
46