Picturing Distributions with Graphs
Chapter 1
BPS - 5th Ed.
1
Chapter 1
BPS - 5th Ed.
2
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.
Chapter 1
BPS - 5th Ed.
3
 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
Chapter 1
BPS - 5th Ed.
4
 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
Chapter 1
BPS - 5th Ed.
5
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)
Chapter 1
BPS - 5th Ed.
6
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.
Chapter 1
BPS - 5th Ed.
7
 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.
Chapter 1
BPS - 5th Ed.
8
 The cohort of women were followed for 14 years.
 The number of CHD (fatal and nonfatal) cases were
counted (1292 cases).
Chapter 1
BPS - 5th Ed.
9
Variables measured
quantitative
 Age (in 1976)
 Weight in 1976
 Weight at age 18
categorical
 Incidence of coronary heart disease
 Smoker or nonsmoker
 Family history of heart disease
Chapter 1
BPS - 5th Ed.
10
 Tells what values a variable takes and how often it
takes these values
 Can be a table, graph, or function
Chapter 1
BPS - 5th Ed.
11
 Categorical variables
 Pie charts
 Bar graphs
 Quantitative variables
 Histograms
 Stemplots (stem-and-leaf plots)
Chapter 1
BPS - 5th Ed.
12
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%
Chapter 1
BPS - 5th Ed.
13
Pie Chart
Senior
20.9%
Freshman
41.9%
Junior
14.0%
Sophomore
23.3%
Chapter 1
BPS - 5th Ed.
14
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
Chapter 1
BPS - 5th Ed.
15
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 %
Chapter 1
BPS - 5th Ed.
16
Pie Chart
Chapter 1
BPS - 5th Ed.
17
Bar Graph
Chapter 1
BPS - 5th Ed.
18
 Overall pattern of graph
 Deviations from overall pattern
 Shape of the data
 Center of the data
 Spread of the data (Variation)
 Outliers
Chapter 1
BPS - 5th Ed.
19
 Symmetric
 bell shaped
 other symmetric shapes
 Asymmetric
 right skewed
 left skewed
 Unimodal, bimodal
Chapter 1
BPS - 5th Ed.
20
Chapter 1
BPS - 5th Ed.
21
Chapter 1
BPS - 5th Ed.
22
Chapter 1
BPS - 5th Ed.
23
Chapter 1
BPS - 5th Ed.
24
Chapter 1
BPS - 5th Ed.
25
 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
Chapter 1
BPS - 5th Ed.
26
 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
Chapter 1
BPS - 5th Ed.
27
 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)
Chapter 1
BPS - 5th Ed.
28
Weight Data
Introductory Statistics class
Spring, 1997
Virginia Commonwealth University
Chapter 1
BPS - 5th Ed.
29
192
152
135
110
128
180
260
170
165
150
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
Chapter 1
170
130
101
180
187
148
106
180
127
124
BPS - 5th Ed.
215
125
194
30
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
Chapter 1
BPS - 5th Ed.
31
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.
Chapter 1
BPS - 5th Ed.
32
 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
Chapter 1
BPS - 5th Ed.
33
1
2
192
152
135
110
128
180
260
170
165
150
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
Chapter 1
170
130
101
180
187
148
106
180
127
124
BPS - 5th Ed.
215
125
194
34
Key
20|3 means
203 pounds
Stems = 10’s
Leaves = 1’s
10
11
12
13 5
14
15 2
16
17
18
19 2
20
21
22
23
24
25
26
Chapter 1
BPS - 5th Ed.
192
152
135
35
Key
20|3 means
203 pounds
Stems = 10’s
Leaves = 1’s
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Chapter 1
0166
009
0034578
00359
08
00257
555
000255
000055567
245
3
025
0
0
BPS - 5th Ed.
36
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.
Chapter 1
BPS - 5th Ed.
37
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
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
BPS - 5th Ed.
38
 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).
Chapter 1
BPS - 5th Ed.
39
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
Chapter 1
BPS - 5th Ed.
40
Average Tuition (Public vs. Private)
Chapter 1
BPS - 5th Ed.
41