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CHAPTER 2
Frequency Distributions and Graphs
Objectives
Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives.
2-3 Histograms, Frequency Polygons, and Ogives
The three most commonly used graphs in research are:
The histogram. The frequency polygon. The cumulative frequency graph, or
ogive (pronounced o-jive).
2-3 Histograms, Frequency Polygons, and Ogives
The histogram is a graph that displays the data by using vertical bars of various heights to represent the frequencies.
Procedure for Drawing a Histogram
Draw and label the x and y axes Represent the frequency on the y axis
and the class boundaries on the x axis Using the frequencies as heights, draw
vertical bars for each class
Example of a Histogram
134.5125.5116.5107.598.589.5
60
50
40
30
20
10
0
Number of college applicants
Fre
quen
cy
For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct a histogram.
Class Limits Frequency
90 – 98 6
99 – 107 22
108 – 116 43
117 – 125 28
126 – 134 9
2-3 Histograms, Frequency Polygons, and Ogives
A frequency polygon is a graph that displays the data by using lines that connect points plotted for frequencies at the midpoint of classes. The frequencies represent the heights of the midpoints.
Procedure for Drawing a Frequency Polygon
Find the midpoints for each class Draw the x and y axes. Label the frequency on
the y axis and the midpoints on the x axis. Using the midpoints for the x values and the
frequencies as the y values, plot the points. Connect the adjacent points with line
segments. Draw a line back to the x axis at the beginning and end of the graph.
Example of a Frequency Polygon
Frequency Polygon
For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct a frequency polygon.
Class Limits Frequency
90 – 98 6
99 – 107 22
108 – 116 43
117 – 125 28
126 – 134 9
13012111210394
60
50
40
30
20
10
0
Number of college applicants
Fre
qu
en
cy
2-3 Histograms, Frequency Polygons, and Ogives
A cumulative frequency graph or ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.
Procedure for Drawing a Cumulative Frequency Graph
Find the cumulative frequency for each class Draw the x and y axes. Label the cumulative
frequencies on the y axis and the class boundaries on the x axis.
Plot the cumulative frequency at each upper boundary.
Starting with the first upper boundary; connect the adjacent points with straight lines. Then extend the graph to the first lower class boundary on the x axis.
Example of an Ogive
For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct an ogive.
Class Limits Frequency
90 – 98 6
99 – 107 22
108 – 116 43
117 – 125 28
126 – 134 9
Ogive
125.5116.5107.598.589.5
120
100
80
60
40
20
0
Number of college applicants
Cu
mu
lativ
e F
req
ue
ncy
134.5
2-4 Other Types of Graphs
Pareto charts - a Pareto chart is used to represent a frequency distribution for a categorical variable.
Example of a Pareto Chart
Homicide
RobberyRape
Assault
13 29 34164 5.412.114.268.3
100.0 94.6 82.5 68.3
250
200
150
100
50
0
100
80
60
40
20
0
Defect
CountPercentCum %
Pe r
cent
Co u
nt
Enforcement Officers in U.S. National Parks During 1995. Pareto Chart for the number of Crimes Investigated by Law
2-4 Other Types of Graphs
Time series graph - A time series graph represents data that occur over a specific period of time.
2-4 Other Types of Graphs - Time Series Graph
19941993199219911990
89
87
85
83
81
79
77
75
Year
Rid
ersh
ip (
in m
illio
ns)PORT AUTHORITY TRANSIT RIDERSHIP
2-4 Other Types of Graphs
Pie graph - A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.
2-4 Other Types of Graphs - Pie Graph
Robbery (29, 12.1%)
Rape (34, 14.2%)
Assaults (164, 68.3%)
Homicide (13, 5.4%)
Pie Chart of the Number of Crimes Investigated byLaw Enforcement Officers In U.S. National Parks During 1995
