Ch2.3 & 2.4 More Graphical Displays & Misleading Graphs
Ch 2.3 More Graphical Display
Objective A : Polygon
A1. Class Midpoints
- A class midpoint is the sum of consecutive lower class limits divided by 2.
Example 1: The following frequency distribution represents a random sample of 40 car speeds for
crossing an intersection in Sylmar. Find the class midpoints of each class.
42-45 / 20
46-49 / 15
50-53 / 8
54-57 / 5
58-61 / 2
Speed (mph) / Frequency (f) / Class Midpoints
42-45 / 20
46-49 / 15
50-53 / 8
54-57 / 5
58-61 / 2
A2. Frequency of Polygons
A frequency polygon is a graph that uses line segments connect to points directly above the class
midpoint value. The heights of the points are the class frequencies. Two additional line segments are
drawn connecting each end of the graph with the horizontal axis.
Example 2 : Construct a frequency polygon for example 1.
From the table of Example 1:
Class Midpoint / Frequency (f)20
15
8
5
2
Example 3 : (p.105 #6)
Deaths by Legal Intervention Deaths by legal intervention refers to injuries inflicted by law-
enforcement agents in the course of arresting or attempting to arrest lawbreakers,
suppressing disturbances, maintaining order, and other legal action (including legal
execution). In 2006, 174 such deaths occurred in 16 states in the United States. The
following frequency polygon represents these deaths by age.
(a) What is the class width? How many classes are represented in the graph ?
(b) What is the midpoint of the first class? What are the lower and upper limits of the first
class?
(c) What is the midpoint of the last class? What are the lower and upper limits of the last
class?
(d) Which age group had 35 deaths due to legal intervention?
(e) Which two age groups have the highest number of deaths due to legal intervention? Estimate the
number of deaths for these age groups.
(f) Estimate the relative frequency for the class 20 – 29.
Objective B : Cumulative Frequency Table and Ogive
B1. Cumulative Frequency/Cumulative Relative Frequency Table
- For continuous data, a cumulative frequency table displays the total number of observations less than or
equal to the upper class limit of a class.
- For continuous data, a cumulative frequency table displays the percentage of observations less than or
equal to the upper class limit of a class.
Example 1 : Construct a cumulative frequency table for the data summarized below.
Speed (mph) / Frequency ()42-45.99 / 20
46-49.99 / 15
50-53.99 / 8
54-57.99 / 5
58-61.99 / 2
Cumulative frequency table:
Speed (mph) / Cumulative Frequency ()B2. Ogive
An ogive 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. And 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).
Example 1 : Construct a ogive for the previous example.
Cumulative Frequency Table :
Speed (mph) / Cumulative Frequency ()Cumulative frequency ogive:
Objective C : Time Series Graphs
- A time series graph represents the values of a variable that have been collected over a specified
period of time. The horizontal axis is the time and the vertical axis is the value of the variable.
Line segments are drawn by connective consecutive points of time and corresponding value of
the variable.
Example 1: The following time-series graph shows the annual U.S. motor vehicle
production from 1990 through 2008.
(a) Estimate the number of motor vehicles produced in the United States in 1991.
(b) Estimate the number of motor vehicles produced in the United States in 1999.
(c) Use the results from (a) and (b) to estimate the percent increase in the number of
motor vehicles produced from 1991 to 1999.
(d) Estimate the percent decrease in the number of motor vehicles produced from
1999 to 2008.
Ch 2.4 Graphical Misrepresentations of Data
- The most common graphical misinterpretation of data is accomplished through manipulation of the
scale of the graph.
Example 1 :
Union Membership The following relative frequency histogram represents the proportion of
employed people aged 25 to 64 years old who were members of a union.
(a) Describe how this graph is misleading. What might a reader conclude from the graph?
(b) Redraw the histogram so that it is not misleading
Example 2 :
Inauguration Cost The following is a USA Today-type graph. Explain how it is misleading.