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Statistics for the Behavioral Sciences |
Chapter 3 Outline Graphs |
Roger N. Morrissette, PhD |
Graphs are used to show a relationship between the independent variable and the dependent variable. The independent variable is typically on the x-axis (horizontal line or abscissa) of a graph and the dependent variable is typically on the y-axis (vertical line or ordinate) of a graph. Caution should be taken when drawing a graph due to the horizontal-vertical illusion. This illusion makes vertical lines appear longer than horizontal lines. Graphs can be used to display information that has been summarized in a frequency distribution. Graphs should make the data easier to understand. In this case, the dependent variable is placed on the x-axis and the frequency is on the y-axis. Graphs are used to illustrate trends and to help predict the future.
Graphs should always contain a descriptive Title that informs us what kind of information is being conveyed. Labels on both axes tell us what is being measured. The numbers along the vertical axis tell us in what increments the measurements are being reported. A graph Key is used to identify two different dependent of independent variables. The most common graphs are histograms and polygons. Both are discussed below.
A Frequency Histogram consists of a number of bars placed side by side. The width of each bar indicates the interval size. The height of each bar indicates the frequency of the interval. To create a Frequency Histogram you must first determine the frequency of the intervals of the axes. Then draw the bars of the histogram. The bars are drawn from the lower limits to the upper limits along the x-axis. Since the intervals are connected the bars of the histogram should also be connected. The graph below is an example of a Frequency Histogram:
Polygons consists of points on a graph with lines connecting them. A Frequency Polygon uses a single point rather than a bar to represent an interval on a graph. They use the midpoint of the interval as the single point plotted. The frequency polygon should begin and end at the abscissa. The steps to generating a Frequency Polygon and an example of a Frequency Polygon (without a title) are listed below:
step 1: draw and label the axes
step 2: add two extra intervals: one below the lowest interval and one above the highest interval
step 3: determine the midpoint for each interval
step 4: plot the frequency for each of the midpoints on the graph
step 5: connect the dots with a straight line
Relative frequency is used to compare two distributions that have different numbers of subjects. Relative frequency can be graphed as a Relative Frequency Polygon. Relative frequency polygons are created in the same manner as the frequency polygon. The only difference being that you use relative frequency instead of frequency values. The graph below is an example of a Relative Frequency Polygon:
Cumulative frequency polygons graph the number of subjects in a distribution that fall below a particular score. Cumulative Frequency Polygons are created in the same manner as the frequency polygon. The only difference being that you use cumulative frequency values and the upper real limits of the intervals are used instead of the midpoints. The s-shaped curve of the graph below is called an “ogive”, pronounced “oh-jive.” The graph below is an example of a Cumulative Frequency Polygon:
Cumulative Relative Frequency Polygons are created in the same manner as the cumulative frequency polygon with the only difference being that you use cumulative relative frequency values instead of cumulative frequency on the y-axis. The graph below is an example of a Cumulative Relative Frequency Polygon:
Stem and Leaf Diagrams allow you to display raw data visually. Each raw score is divided into a stem and a leaf. The leaf is typically the last digit of the raw value. The stem is the remaining digits of the raw value. To generate a stem and leaf diagram you must first create a vertical column that contains all of the stems. Then list each leaf next to the corresponding stem. In these diagrams, all of the scores are represented in the diagram without the loss of any information. The graph below is an example of a Stem and Leaf Diagram:
stem leaf
2 2 3 5 5 5 7
3 1 1 4 6
4 0 0 4 5 6 7 7 8 9
5 0 1 1
6 3 3 5 6 7
7 7 8 9
8 1 1 2 6 9 9 9
9 5
VIII. Changing the Shape of a Graph
Any two people can take the same data and make dramatically different graphs with it. You must take care in reading all the labels and axes of graphs so as not to be misled by the representation of the data. The following three graphs were all generated with the same data:
