1. Frequency Distributions and Graphs
Chapter 2
Frequency Distributions and Graphs

2. Objectives
Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives.

3. 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).

4. 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.

5. 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

6. Example of a Histogram
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
134.5 1
25.5
16.5
07.5
98.5
89.5 60 50 40 30 20 10
Number of college applicants F r e q u n c y

7. 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.

8. 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.

9. Example of a Frequency Polygon
For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct a frequency polygon.
Frequency Polygon
Class Limits
Frequency
90 – 98 6
99 – 107 22
108 – 116 43
117 – 125 28
126 – 134 9 60 50 y c n 40 e u q 30 e r F 20 10 94
103
112
121
130
Number of college applicants

10. 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.

11. 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.

12. Example of an Ogive
For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct an ogive.
Ogive
125.5
116.5
107.5
98.5
89.5
120
100 80 60 40 20
Number of college applicants
Cumulative Frequency
134.5
Class Limits
Frequency
90 – 98 6
99 – 107 22
108 – 116 43
117 – 125 28
126 – 134 9

13. 2-4 Other Types of Graphs
Pareto charts - a Pareto chart is used to represent a frequency distribution for a categorical variable.

14. Example of a Pareto Charti c d e R b r y a p
Assault 1 3 2 9 4 6 5 . 8 P a r e t o C h a r t f o r t h e n u m b e r o f C r i m e s I n v e s t i g a t e d b y L a w D e f c t C o u n P r m % E
ment Officers in U.S. National Parks During 1995.

15. 2-4 Other Types of Graphs
Time series graph - A time series graph represents data that occur over a specific period of time.

16. 2-4 Other Types of Graphs - Time Series GraphU T H O R I T Y T R A N S I T R I D E R S H I P 8 9 ) s 8 7 n o i i l l 8 5 m n 8 3 ( i p 8 1 i h s r 7 9 e d i R 7 7 7 5 1 9 9 1 9 9 1 1 9 9 2 1 9 9 3 1 9 9 4 Y e a r

17. 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.

18. 2-4 Other Types of Graphs - Pie Graph
Robbery (29, 12.1%)
Pie Chart of the Number of Crimes Investigated by Law Enforcement Officers In U.S. National Parks During 1995
Rape (34, 14.2%)
Homicide (13, 5.4%)
Assaults (164, 68.3%)