Looking at data: distributions
- Displaying distributions with graphs
IPS section 1.1
© 2006 W.H. Freeman and Company (authored by
Brigitte Baldi, University of California-Irvine; adapted by
Jim Brumbaugh-Smith, Manchester College)
Objectives
Displaying distributions with graphs

Recognize numerical vs. categorical data

Construct graphs representing a distribution of numerical data

Histograms

Stemplots

Describe overall patterns in numerical data

Identify exceptions to overall patterns

Discuss pros and cons of histograms vs. stemplots
Terminology

Individual (or “observation”)

Variable

numerical (or “quantitative”)

categorical (or “qualitative)

Value

Frequency

absolute

relative

Frequency table

Distribution
Terminology (cont’d)

Graphs for quantitative data

Histogram

Stemplot (or “stem-and-leaf diagram”)

Boxplot (section 1.2)

Symmetric

Skewed right (or “positively skewed”)

Skewed left (or “negatively skewed”)

Peak (or “mode”)

Unimodal vs. Bimodal

Outlier
Variables
In a study, we collect data from individuals, more formally known as
observations. Observations can be people, animals, plants, or any
object or process of interest.
A variable is a characteristic that varies among individuals in a
population or in a sample (a subset of a population).
Example: age, height, blood pressure, ethnicity, leaf length, first language
The distribution of a variable tells us what values the variable takes
and how often it takes on these values. A distribution can also be
thought of as the pattern of variation seen in the data.
Two types of variables

Variables can be either numerical (a/k/a quantitative) …

Something that can be counted or measured for each individual and then
added, subtracted, averaged, etc. across individuals in the population.

Example: How tall you are, your age, your blood cholesterol level, the
number of credit cards you own

… or categorical (a/k/a qualitative).

Something that falls into one of several categories. What can be
computed is the count or proportion of individuals in each category.

Example: Your blood type (A, B, AB, O), your hair color, your ethnicity,
whether you paid income tax last tax year or not
How do you know if a variable is categorical or quantitative?
Ask:
 What are the individuals in the sample?
 What is being recorded about those individuals?
 Is that a number (“quantitative”) or a statement (“categorical”)?
Categorical
Quantitative
Each individual is
assigned to one of
several categories.
Each individual is
attributed a
numerical value.
Individuals
in sample
DIAGNOSIS
AGE AT DEATH
Patient A
Heart disease
56
Patient B
Stroke
70
Patient C
Stroke
75
Patient D
Lung cancer
60
Patient E
Heart disease
80
Patient F
Accident
73
Patient G
Diabetes
69
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
% of top
10s
% 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
Top 10 causes of deaths in the United States 2001
The number of individuals
who died of an accident in
2001 is approximately
100,000.
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Each category is represented by one bar. The bar’s height shows the count (or
sometimes the percentage) for that particular category.
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700
600
500
400
300
200
100
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Top 10 causes of deaths in the United States 2001
Bar graph sorted by rank
 Easy to analyze
Sorted alphabetically
 Much less useful
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 United States in 2000
Make sure your
labels match
the data.
Make sure
all percents
add up to 100.
Percent of deaths from top 10 causes
Percent of
deaths from
all causes
Child poverty before and after government
intervention—UNICEF, 1996
What does this chart tell you?
•The United States has the highest rate of child
poverty among developed nations (22% of under 18).
•Its government does the least—through taxes and
subsidies—to remedy the problem (size of orange
bars and percent difference between orange/blue
bars).
Could you transform this bar graph to fit in 1 pie
chart? In two pie charts? Why?
The poverty line is defined as 50% of national median income.
Histograms
Vertical bar chart where horizontal
axis is a numerical scale
corresponding to the data.

Vertical axis represents frequency
(how many) or relative frequency
(what proportion)

“Peaks” correspond to commonly
occurring data values.

“Valleys” and “tails” correspond to
values which do not occur as
frequently.


Most states have between 0 and 10 percent Hispanic residents

A very small number have between 25 and 45 percent.
Histograms
The range of values that a
variable can take is divided
into equal size intervals.
The histogram shows the
number (i.e., frequency) 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 the percent Hispanic residents 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% Hispanics.
Creating a histogram
What “class size” should you use?


Use an appropriate number of classes − usually between 5 and 15 work
well, depending on number of data values being represented.
Either too few or too many classes will obscure the pattern in the data.

Not so detailed that it is no longer summary

Avoid using many classes having frequency of only 0 or 1

Not overly summarized so that you lose all the information
 rule of thumb: start with 5 to 10 classes
Look at the distribution and refine your classes.
(There isn’t a unique or “perfect” histogram.)
Guidelines for histograms

Label the horizontal axis with a consistent numerical scale. (Don’t
leave any gaps in the scale or compress the scale toward the left or
right side of the graph)

Use vertical bars of equal width.

Label the horizontal scale on the class boundaries.

An exception is when each bar corresponds to a single whole
number. Then it is reasonable to label each bar at its midpoint.
Same data set
Not
summarized
enough
Too summarized
Interpreting histograms
When describing the distribution of 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 extends
much farther out than the right.
Skewed
right
distribution
Complex,
multimodal
distribution

Not all distributions have a simple overall shape,
especially when there are few observations.
Histogram of Drydays in 1995
IMPORTANT NOTE:
Your data are the way they are.
Do not try to force them into a
particular shape.
It is a common misconception
that if you have a large enough
data set, the data will eventually
turn out nice and symmetrical.
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
More on Outliers

If outlier is incorrect data (e.g., data entry error, false response)

Correct it if possible.

Discard if absolutely sure it is wrong.

Discarding data might introduce a bias if uncorrectable errors
tend to be mainly high (or mainly low) values.

If outlier is correct data consider effect on analysis

Which statistical technique is most appropriate?

How are conclusions affected by the outlier?

Some Other Strategies:

Refine population definition if unusual responses are not part
of intended study group.

If sample size is quite small collect more data to see if any
gaps fill in.
Stemplots (or “stem-and-leaf ” diagrams)
How to make a stemplot:
1) Truncate (or “trim”) the data to appropriate level of
accuracy.
2) Separate each observation into a stem, consisting of
all but the final (rightmost) digit, and a leaf, which is
the remaining final digit. Stems may have as many
digits as needed, but each leaf contains a single digit.
3) Write the stems in a column with the smallest value
at the top, and draw a vertical line at the right of this
column.
4) Write each leaf in the row to the right of its stem, in
increasing order out from the stem. Leaves should
be aligned in vertical columns. (Why?)
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
Stemplots versus histograms
Stemplots are quick and dirty histograms that can easily be done by
hand, therefore very convenient for back of the envelope calculations.
However, they are rarely found in scientific publications.
Stemplots versus histograms
1) Advantages of Stemplots

Quick to do by hand (no frequency table needed)

Maintains all the numerical data

Good for comparing two distributions (using back-to-back plots)
2) Disadvantages

Not as crisp visually (compared to a graphics-quality histogram)

Most people like numerical scale on horizontal axis.
Variations on stemplots
1) Stem-splitting

Use to increase the number of stems

Create one stem for leaves 0-4

Second stem for leaves 5-9
2) Back-to-back Stemplots

Use to compare two data sets measured on same scale (e.g., male vs.
female heights).

Use single stem with leaves on opposite sides for different data sets.
Ways to chart quantitative 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.

Histograms and stemplots
These are summary graphs for a single variable. They are very useful to
understand the pattern of variability in the data.

Other graphs to reflect numerical summaries (see Chapter 1.2)
Line graphs: time plots
In a time plot, time always goes on the horizontal, x axis.
We describe time series by looking for an overall pattern and for striking
deviations from that pattern. In a time series:
A trend is a rise or fall that
persist over time, despite
small irregularities.
A pattern that repeats itself
at regular intervals of time is
called seasonal variation.
Retail price of fresh oranges
over time
Time is on the horizontal, x axis.
The variable of interest—here
“retail price of fresh oranges”—
goes on the vertical, y axis.
This time plot shows a regular pattern of yearly variations. These are seasonal
variations in fresh orange pricing most likely due to similar seasonal variations in
the production of fresh oranges.
There is also an overall upward trend in pricing over time. It could simply be
reflecting inflation trends or a more fundamental change in this industry.
A time plot can be used to compare two or more
data sets covering the same time period.
1918 influenza epidemic
# Cases # Deaths
Date
800
800 700
10000
9000
10000
8000
9000
7000
8000
6000
7000
5000
6000
4000
5000
3000
4000
2000
3000
2000
1000
1000 0
700 600
600 500
500 400
400 300
300 200
200 100
100 0
0
0
# deaths reported
1918 influenza epidemic
we
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ewk
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130
552
738
414
198
90
56
50
71
137
178
194
290
310
149
Incidence
36
531
4233
8682
7164
2229
600
164
57
722
1517
1828
1539
2416
3148
3465
1440
# cases diagnosed
week 1
week 2
week 3
week 4
week 5
week 6
week 7
week 8
week 9
week 10
week 11
week 12
week 13
week 14
week 15
week 16
week 17
1918 influenza epidemic
# Cases
# Cases
# Deaths
# Deaths
The pattern over time for the number of flu diagnoses closely resembles that for the
number of deaths from the flu, indicating that about 8% to 10% of the people
diagnosed that year died shortly afterward from complications of the flu.
Scales matter
Death rates from cancer (US, 1945-95)
Death rates from cancer (US, 1945-95)
Death rate (per
thousand)
250
200
150
100
250
Death rate (per thousand)
How you stretch the axes and choose your
scales can give a different impression.
200
150
100
50
50
0
1940
1950
1960
1970
1980
1990
0
1940
2000
1960
1980
2000
Years
Years
Death rates from cancer (US, 1945-95)
250
Death rates from cancer (US, 1945-95)
220
Death rate (per thousand)
Death rate (per thousand)
200
150
100
50
0
1940
1960
Years
1980
2000
A picture is worth a
thousand words,
200
BUT
180
160
There is nothing like
hard numbers.
 Look at the scales.
140
120
1940
1960
1980
Years
2000
Why does it matter?
What's wrong with
these graphs?
Cornell’s tuition over time
Careful reading
reveals that:
1. The ranking graph covers an 11-year period, the
tuition graph 35 years, yet they are shown
comparatively on the cover and without a
horizontal time scale.
2. Ranking and tuition have very different units, yet
both graphs are placed on the same page without
a vertical axis to show the units.
Cornell’s ranking over time
3. The impression of a recent sharp “drop” in the
ranking graph actually shows that Cornell’s rank
has IMPROVED from 15th to 6th ...