Chapter 2: Organizing and
Summarizing Data
2.3 Additional Displays of Quantitative Data
Additional Displays of Quantitative Data
• Stem and Leaf Plot
• Cumulative Frequency and Relative Frequency Tables and Ogives
A
(also called
) is an effective way to
summarize univariate numerical data when the data set is not too large.
Each number in the data set is broken into two pieces, a stem and a leaf
• The stem consists of all but the rightmost digit, and the leaf consists of
the rightmost digit.
• An ordered stem plot has the leaf digits in each row ordered from
smallest to largest.
• Some drawbacks to stem-and-leaf diagrams:
Not useful with very large data sets, or large range of data
Awkward with data containing many digits
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To construct a stem plot:
Think of each observation as a stem, consisting of all but the rightmost
digit, which is the leaf
Identify the stem and leaf unit, i.e. if the rightmost digit is ones, then the leaf unit is
ones and the stem unit is tens (even for 3-digit numbers).
Write the stems from smallest to largest in a vertical column to the left a
vertical line
List a stem value in the range of the data even if there are no leaves for that stem.
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To construct a stem plot (continued):
Write each leaf to the right of the vertical line in the row that contains
the appropriate stem, i.e. 36 will be in the 3 stem row and the leaf written is 6
List the leaves, using all data values, even those that are repeated.
Leaves are separated by spaces not commas.
Leaves in each row should be arranged in ascending order.
Write title and labels
Give your graph a title.
Label above each set of leaves the name of the group.
Indicate the units for stems and leaves somewhere in the display. The leaf unit is
determined by the place value of the rightmost digit in the leaf.
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RVCC Associate Degree Graduates Survey May 2014
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Example: Draw a stem and leaf plot using one line per stem to
organize the data.
Example of
plot using two line per stem to organize the data.
The first stem line has leaf values 0-4
The second stem line has leaf values 5-9
:
Draw a stem and leaf
Frequency Polygons
• A frequency polygon is a graph that uses points, connected by line
segments, to represent the frequencies for the classes. It is constructed
by plotting a point above each class midpoint on a horizontal axis at a
height equal to the frequency of the class. Next, line segments are
drawn connecting consecutive points. Two additional line segments
are drawn connecting each end of the graph with the horizontal axis.
• Reminder: A class midpoint is the sum of consecutive lower class
limits divided by 2.
To construct a frequency polygon of the data in Table
13, used previously, calculate the class midpoints of
each class, as shown.
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Now plot points whose x-coordinates are the class midpoints and
y-coordinates are the frequencies; connect them with line
segments. Next determine the midpoint of the class preceding the
first class to be 37.5 and the midpoint of the class following
the last class to be 337.5.
Finally, connect
each end of the
graph with the
horizontal axis at
(37.5, 0) and
(337.5, 0),
respectively, and
obtain the figure.
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Ogive for cumulative Frequency and Relative
Frequency Display
• A cumulative frequency distribution displays the aggregate
frequency of the category. In other words, for discrete data, it displays
the total number of observations less than or equal to the category. For
continuous data, it displays the total number of observations less than
or equal to the upper class limit of a class.
• A cumulative relative frequency distribution displays the proportion
(or percentage) of observations less than or equal to the category for
discrete data and the proportion (or percentage) of observations less
than or equal to the upper class limit for continuous data.
Table 17 displays the cumulative frequency and cumulative
relative frequency of the data summarized in Table 13, used
previously. Table 17 shows that 54% of the fines were under
$200.
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An ogive (read as “oh jive”) is a graph that
represents the cumulative frequency or cumulative
relative frequency for the class. It is constructed by
plotting points whose x-coordinates are the upper
class limits and whose y-coordinates are the
cumulative frequencies or cumulative relative
frequencies of the class. Then line segments are
drawn connecting consecutive points. An additional
line segment is drawn connecting the first point to
the horizontal axis at a location representing the
upper limit of the class that would precede the first
class (if it existed).
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Construct a relative frequency ogive using the data in Table 17
by plotting points whose x-coordinates are the upper class
limits and whose y-coordinates are the cumulative relative
frequencies of the classes. Then connect the points with line
segments.
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If the value of a variable is measured at different
points in time, the data are referred to as time
series data.
A time-series plot is obtained by plotting the
time in which a variable is measured on the
horizontal axis and the corresponding value of
the variable on the vertical axis. Line segments
are then drawn connecting the points.
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EXAMPLE
(1 of 2)
Drawing a Time-Series Plot
The Partisan Conflict Index (PCI) tracks
the degree of political disagreement among
U.S. politicians in the federal government.
It is found by measuring the frequency of
newspaper articles reporting disagreement
in a given month. Higher values of the
index suggest greater conflict among
political parties, Congress, and the
President. The data in Table 18 represents
the PCI in December from 2004 to 2022.
Construct a time-series plot of the data. In
what year was the index highest?
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EXAMPLE
(2 of 2)
Drawing a Time-Series Plot
The figure shows the
time-series plot.
Notice the jump in the
PCI following the 2010
midterm elections. The
PCI was highest in
2016. In November of
2016 Donald Trump
was elected President
of the United States.
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