1.4 Representing a Categorical Variable with Graphs

Bar Graphs

1. Determine the categories you want to include in the graph.
2. Count the number of observations in each category.
3. Mark the frequencies on the vertical axis and the categories on the horizontal axis.
4. Draw the bars, with the height of each bar representing the frequency of the corresponding category.
5. Add a title and axis labels to the graph to help interpret the data.

Source: Prem S. Mann: Introductory Statistics. John Wiley and Sons Inc. 2020

Pie Charts

1. Determine the categories you want to include in the pie chart. (Example: Commuter, non-commuter)
2. Calculate the fraction of the whole that is represented by each category (Example: Out of 50 respondents, 30 commuters would occupy 3/5ths of the pie, while 20 non-commuters would occupy the remaining 2/5ths of the pie).
3. Draw a circle and divide it into slices that are proportional to the fractions calculated in step 2.
4. Label each slice with the corresponding category and the percentage it represents.
5. Add a title to the pie chart to help interpret the data.

• The choice between bar graphs and pie charts will depend on how many categories that variable of your interest assumes and the size of it. Whenever you have many categories or few categories with about the same frequencies, then the bar graph should be your first choice. If the pie has many slices or slices of the same size, it will be hard to compare the groups.
• Be careful of quantity distortions and keeping the area principle.

Contingency Table (Two-Way Table)

1. Determine the variables you want to include in the table.
2. Count the number of observations in each category of each variable.
3. Organize the counts in a table, with each row representing a category of one variable and each column representing a category of the other variable.
4. Add row and column totals to the table. (This step is the easiest to forget!)
5. Analyze the table to identify any patterns or trends in the data. (This is important when establishing context and responding to Multiple Choice and Free Response Questions in the AP exam!)

Key Vocabulary

• Frequency Table
• Relative Frequency Table
• Pie Chart
• Bar Graph
• Two-way Table

Real-Life Applications: To Trust or Not To Trust a Bar/Pie Chart?

• Using bar/pie charts to compare variables on different scales: Charts are best used to compare categories or groups that are on the same scale. If you are comparing variables that are on different scales, it can be difficult to accurately compare the sizes of the bars/pie slices.

Source: Infogram

• Using bar/pie charts to show continuous data: Charts are best used to show categorical data, not continuous data. If you have continuous data, it is usually better to use a different type of graph, such as a line graph or scatterplot.
• Using bar/pie charts to show small differences: Charts are not very effective at showing small differences between categories. If the differences between the categories are small, it may be difficult to accurately interpret the graph.
• Using bar/pie charts to show trends over time: Charts are not well suited for showing trends over time. For this purpose, it is usually better to use a line graph or a time series plot.
• Using bar/pie charts to show more than two variables: Charts are typically used to compare two variables. If you want to show more than two variables, it is usually better to use a different type of graph. The example below compares A, B, and C; here, you can see that it might make more sense to use a bar chart over a pie chart.

Source: Wikipedia

• Using bar/pie charts to show a false impression of size: Truncated bar graphs (bar graphs that don't start at a y-value of 0) can be misleading if the truncation is not clearly labeled or if the truncation is done in a way that distorts the data. For example, if the truncation is done at an arbitrary value, it could give the impression that the data is more evenly distributed than it really is. See the example below and notice how the differences between 2010 and 2011 are more noticeable in a truncated bar graph (left) compared to the usual bar graph (right).

Source: Wikipedia