Organizing and Summarizing
Data Graphically
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When data is collected from a survey or designed
experiment, they must be organized into a
manageable form. Data that is not organized is
referred to as raw data.
Ways to Organize Data
• Tables
• Graphs
• Numerical Summaries (Chapter 3)
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Working with categorical
variables/Data
Categorical data is a result of observing a categorical variable. Each observation
is placed into one of many possible categories or levels determined by the
variable.
Frequency is the count of observations in each category of the variable.
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A frequency distribution lists each category of
data and the number of occurrences for each
category of data.
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EXAMPLE
Organizing Qualitative Data into a Frequency
Distribution
The data on the next slide represent the color
of M&Ms in a bag of plain M&Ms.
Construct a frequency distribution of the color
of plain M&Ms.
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Frequency table
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The relative frequency is the proportion (or percent) of
observations within a category and is found using the
formula:
frequency
relative frequency 
sum of all frequencies
A relative frequency distribution lists the relative
frequency of each category of data.
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Why use the relative
frequency?
The relative frequency makes it easier for us to compare
two or more groups when the number of observations
within each of those groups are not the same.
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EXAMPLE
Organizing Qualitative Data into a Relative
Frequency Distribution
Use the frequency distribution obtained in the
prior example to construct a relative frequency
distribution of the color of plain M&Ms.
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Relative Frequency
12
 0.2667
45
0.2222
0.2
0.1333
0.0667
0.1111
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Bar Graphs
A bar graph is constructed by labeling each category of data on
either the horizontal or vertical axis and the frequency or relative
frequency of the category on the other axis.
In many cases since the variable has set categories, the bars from
each category often not touch each other in a bar graph.
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EXAMPLE
Constructing a Frequency and
Relative Frequency Bar Graph
Use the M&M data to construct
(a) a frequency bar graph and
(b) a relative frequency bar graph.
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EXAMPLE
Comparing Two Data Sets
The following data represent the marital status (in millions) of U.S. residents 18 years of
age or older in 1990 and 2006. Draw a side-by-side relative frequency bar graph of the
data.
Marital Status
1990
2006
Never married
40.4
55.3
Married
112.6
127.7
Widowed
13.8
13.9
Divorced
15.1
22.8
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Marital Status in 1990 vs. 2006
0.7
0.6
Relative Frequency
0.5
1990
0.4
2006
0.3
0.2
0.1
0
Never married
Married
Widowed
Divorced
Marital Status
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Pie Charts
A pie chart is a circle divided into sectors. Each
sector represents a category of the variable of
interest The area of each sector is proportional
to the frequency of the category. In other
words each section of the pie represents the
relative frequency for that category.
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EXAMPLE
Constructing a Pie Chart
The following data represent the marital status (in millions) of U.S. residents 18 years of
age or older in 2006. Draw a pie chart of the data.
Marital Status
Frequency
Never married
55.3
Married
127.7
Widowed
13.9
Divorced
22.8
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Working With Numerical
Variables/Data
Numerical data is obtained by measurements from an object of interest or
where we count the number of objects that have a specific trait. Unlike
categorical variables we are interested more in the values/measurements over
the categories themselves.
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The first step in summarizing quantitative data is to
determine whether the data is discrete or continuous.
If the data is discrete and there are relatively few
different values of the variable, the categories of data
will be the observations (as in qualitative data). If the
data is discrete, but there are many different values of
the variable, or if the data is continuous, the categories
of data (called classes) must be created using intervals
of numbers.
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EXAMPLE
Constructing Frequency and Relative
Frequency Distribution from Discrete Data
The following data represent the number of available
cars in a household based on a random sample of 50
households. Construct a frequency and relative
frequency distribution.
3
4
1
3
2
0
2
1
3
3
1
2
3
2
2
2
2
2
1
1
1
1
4
2
2
1
2
1
2
2
1
2
2
0
1
2
0
1
3
1
0
2
2
2
3
2
4
2
2
5
Data based on results reported by the United States Bureau of the Census.
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The Histogram
A histogram is constructed by drawing rectangles
for each class of data whose height is the
frequency or relative frequency of the class. The
width of each rectangle should be the same and
they should touch each other.
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EXAMPLE
Drawing a Histogram for Discrete Data
Draw a frequency and relative frequency histogram for
the “number of cars per household” data.
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EXAMPLE
Organizing Continuous Data into a
Frequency and Relative Frequency Distribution
The following data represent the time between eruptions (in
seconds) for a random sample of 45 eruptions at the Old
Faithful Geyser in California. Construct a frequency and
relative frequency distribution of the data.
Source: Ladonna Hansen, Park Curator
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The smallest data value is 672 and the largest data
value is 738. We will create the classes so that the
lower class limit of the first class is 670 and the class
width is 10 and obtain the following classes:
670 - 679
680 - 689
690 - 699
700 - 709
710 - 719
720 - 729
730 - 739
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EXAMPLE
Constructing a Frequency and Relative
Frequency Histogram for Continuous Data
Using class width of 10:
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EXAMPLE
Constructing a Frequency and Relative
Frequency Histogram for Continuous Data
Using class width of 5:
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Describing the Shape of the Distribution
Shape is the physical appearance
of the distribution
Uni-modal distributions have one peak.
Bi-modal distributions have two peaks.
Multi-modal distributions have multiple peaks.
In this course we will deal more with the uni-modal
distributions since many of our assumptions involve
a normal distribution which is uni-modal.
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Common Uni-modal
Shapes
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EXAMPLE
Identifying the Shape of the Distribution
Identify the shape of the following histogram which represents the time between
eruptions at Old Faithful.
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EXAMPLE
Identifying the Shape of the Distribution
Answer: the shape of the distribution appears to be
skew left or negatively skewed.
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