Objectives (BPS chapter 1)
Picturing Distributions with Graphs

What is Statistics ?

Individuals and variables

Two types of data: categorical and quantitative

Ways to chart categorical data: bar graphs and pie charts

Ways to chart quantitative data: histograms and stemplots

Interpreting histograms

Time plots
Statistics ?

Statistics is the science of data
Individuals and Variables

Individuals are the objects described by a set of
data. Individuals may be people, but they may
also be animals or things.

A variable is any characteristic of an individual. A
variable can take different values for different
individuals.
Example: A data set layout
Two types of variables
A variable can be either

A categorical variable places an individual into one of
several groups or categories. What can be counted is
the count or proportion of individuals in each category.
or

A quantitative variable takes numerical values for
which arithmetic operations such as adding and
averaging make sense.

The distribution of a variable tells us what values it
takes and how often it takes these values.
Example

Data from a medical study contain values of
many variables for each of the people who
were the subjects of the study. Which of the
following variables are categorical and which
are quantitative?
a)
Gender (female or male) --- Categorical
Age (years) --- Quantitative
Race (black, white or other) --- Categorical
Smoker (yes or no) --- Categorical
Systolic blood pressure --- Quantitative
Level of calcium in the blood --- Quantitative
b)
c)
d)
e)
f)
Ways to chart categorical data
Because the variable is categorical, the data in the graph can be
ordered any way we want (alphabetical, by increasing value, by year,
by personal preference, etc.).

Bar graphs
Each category is
represented by
a bar.

Pie charts
Peculiarity: The slices must
represent the parts of one whole.
Example: Top 10 causes of death in the United States, 2001
Rank Causes of death
Counts
Percent of
top 10s
Percent of
total
deaths
1 Heart disease
700,142
37%
29%
2 Cancer
553,768
29%
23%
3 Cerebrovascular
163,538
9%
7%
4 Chronic respiratory
123,013
6%
5%
5 Accidents
101,537
5%
4%
6 Diabetes mellitus
71,372
4%
3%
7 Flu and pneumonia
62,034
3%
3%
8 Alzheimer’s disease
53,852
3%
2%
9 Kidney disorders
39,480
2%
2%
32,238
2%
1%
10 Septicemia
All other causes
629,967
26%
For each individual who died in the United States in 2001, we record what was
the cause of death. The table above is a summary of that information.
Bar graphs
Each category is represented by one bar. The bar’s height shows the count (or
sometimes the percentage) for that particular category.
Top 10 causes of death in the U.S., 2001
The number of individuals
who died of an accident in
2001 is approximately
100,000.
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Counts (x1000)
800
700
600
500
400
300
200
100
0
Pie charts
Each slice represents a piece of one whole.
The size of a slice depends on what percent of the whole this category represents.
Percent of people dying from
top 10 causes of death in the U.S., 2000
Ways to chart quantitative data

Histograms and stemplots
These are summary graphs for a single variable. They are very useful to
understand the pattern of variability in the data.

Line graphs: time plots
Use when there is a meaningful sequence, like time. The line connecting
the points helps emphasize any change over time.

Other graphs to reflect numerical summaries (see chapter 2)
Example
Histograms
The range of values that a
variable can take is divided
into equal-size intervals.
The histogram shows the
number of individual data
points that fall in each
interval.
The first column represents all states with a percent Hispanic in their
population between 0% and 4.99%. The height of the column shows how
many states (27) have a percent Hispanic in this range.
The last column represents all states with a percent Hispanic between 40%
and 44.99%. There is only one such state: New Mexico, at 42.1% Hispanic.
Interpreting histograms
When describing a quantitative variable, we look for the overall pattern and for
striking deviations from that pattern. We can describe the overall pattern of a
histogram by its shape, center, and spread.
Histogram with a line connecting
each column  too detailed
Histogram with a smoothed curve
highlighting the overall pattern of
the distribution
Most common distribution shapes

Symmetric
distribution
A distribution is symmetric if the right and left
sides of the histogram are approximately mirror
images of each other.

A distribution is skewed to the right if the right
side of the histogram (side with larger values)
extends much farther out than the left side. It is
skewed to the left if the left side of the histogram
Skewed
distribution
extends much farther out than the right side.
Complex,
multimodal
distribution

Not all distributions have a simple overall shape,
especially when there are few observations.
Outliers
An important kind of deviation is an outlier. Outliers are observations
that lie outside the overall pattern of a distribution. Always look for
outliers and try to explain them.
The overall pattern is fairly
symmetrical except for two
states clearly not belonging
to the main trend. Alaska
and Florida have unusual
representation of the
elderly in their population.
A large gap in the
distribution is typically a
sign of an outlier.
Alaska
Florida
Stem plots
How to make a stem plots:
1) Separate each observation into a stem,
consisting of all but the final (rightmost) digit,
and a leaf, which is that remaining final digit.
Stems may have as many digits as needed, but
each leaf contains only a single digit.
2) Write the stems in a vertical column with the
smallest value at the top, and draw a vertical
line at the right of this column.
3) Write each leaf in the row to the right of its
stem, in increasing order out from the stem.
Original data: 9, 9, 22, 32, 33, 39, 39, 42, 49, 52, 58, 70
STEM
LEAVES
State
Percent
Alabama
Alaska
Arizona
Arkansas
California
Colorado
Connecticut
Delaware
Florida
Georgia
Hawaii
Idaho
Illinois
Indiana
Iowa
Kansas
Kentucky
Louisiana
Maine
Maryland
Massachusetts
Michigan
Minnesota
Mississippi
Missouri
Montana
Nebraska
Nevada
NewHampshire
NewJersey
NewMexico
NewYork
NorthCarolina
NorthDakota
Ohio
Oklahoma
Oregon
Pennsylvania
RhodeIsland
SouthCarolina
SouthDakota
Tennessee
Texas
Utah
Vermont
Virginia
W ashington
W estVirginia
W isconsin
W yoming
1.5
4.1
25.3
2.8
32.4
17.1
9.4
4.8
16.8
5.3
7.2
7.9
10.7
3.5
2.8
7
1.5
2.4
0.7
4.3
6.8
3.3
2.9
1.3
2.1
2
5.5
19.7
1.7
13.3
42.1
15.1
4.7
1.2
1.9
5.2
8
3.2
8.7
2.4
1.4
2
32
9
0.9
4.7
7.2
0.7
3.6
6.4
Step 1:
Sort the
data
State
Percent
Maine
W estVirginia
Vermont
NorthDakota
Mississippi
SouthDakota
Alabama
Kentucky
NewHampshire
Ohio
Montana
Tennessee
Missouri
Louisiana
SouthCarolina
Arkansas
Iowa
Minnesota
Pennsylvania
Michigan
Indiana
W isconsin
Alaska
Maryland
NorthCarolina
Virginia
Delaware
Oklahoma
Georgia
Nebraska
W yoming
Massachusetts
Kansas
Hawaii
W ashington
Idaho
Oregon
RhodeIsland
Utah
Connecticut
Illinois
NewJersey
NewYork
Florida
Colorado
Nevada
Arizona
Texas
California
NewMexico
0.7
0.7
0.9
1.2
1.3
1.4
1.5
1.5
1.7
1.9
2
2
2.1
2.4
2.4
2.8
2.8
2.9
3.2
3.3
3.5
3.6
4.1
4.3
4.7
4.7
4.8
5.2
5.3
5.5
6.4
6.8
7
7.2
7.2
7.9
8
8.7
9
9.4
10.7
13.3
15.1
16.8
17.1
19.7
25.3
32
32.4
42.1
Percent of Hispanic residents
in each of the 50 states
Step 2:
Assign the
values to
stems and
leaves
Time Plots


Time plots are for variables measured over
time and it is all about changes over time.
A time plot of a variable plots each observation
against the time at which it was measured.
Always put time on the horizontal scale of your
plot and the variable you are measuring on the
vertical scale. Connecting the data points by
line helps emphasize any change over time
Example

How have
college tuition
and fees
changed over
time?
Time Plot
Time plot of the average
tuition paid by students
at public and private
college foe academic
year 1970-1971 to 20012002