Categorical Data Charts

Bar Graphs

Bar graphs are used to compare categories. The horizontal axis is usually used to show the categories while the vertical axis usually shows the frequency of each category. However, the axes can be reversed if a horizontal rather than a vertical bar graph is desired. The bars on a bar graph do not touch because the categories are completely separate from each other, and since the categories have no order, they can be arranged at will.

Figure 1, illustrates an example of a horizontal bar graph. The title tells us that the graph is showing the water required to produce certain foods and consumer goods. The vertical axis shows the independent variable of foods and consumer goods (the categories), while the horizontal axis shows the water needed in liters. The water needed is the dependent variable (it depends on which product is selected) and is a continuous measure. Consequently, the horizontal axis has marks at equal intervals that represent 2500 liters. The length of each bar next to a category illustrates the water needed to produce that product. For example, 1 computer takes 20,000 liters of water to produce it, while 1 slice of wheat bread needs 40 liters. The order of the categories are listed top to bottom from the highest to lowest water needs. This makes the graph look organized, but the categories could be reorganized in any order.

Figure 1.

A bar chart is made up of paired data. In figure 1, the independent variable is product, while the dependent variable is water footprint. Consequently, the ordered pairs are of the form (product, water footprint). For example, the ordered pair (apple, 70) tells us that an apple has a water footprint of 70 liters. Notice the scale on the horizontal axis. By scale we mean the units into which the axis is divided. Numbers should always start at zero and go up in a linear fashion: the distance between two numbers on the axis should be constant.

Pie Charts (Circle Graphs)

Pie charts show the composition of data, or the parts of a whole. Each piece of the pie represents a category, and each category is represented by an appropriately sized segment. None of the segments overlap as they represent different categories, and the sum of the percents in each category must add to 100%. A pie chart must also include the total number, [latex]n[/latex], of the frequencies, and it must have a title explaining the contents of the chart. In addition, each segment is labeled with the category and the percent of the whole it represents. 

Figure 2. An example of a pie chart

In Figure 2, the whole pie (the whole circle) represents the total products sold at a store in the 3rd quarter (Q3); the pieces of that pie show us the percentage of sales each department made. A pie chart like this makes it very easy to see that the clothing department makes up the largest section of sales, and fragrances the smallest as they have the largest and smallest segments of the whole. Consequently, a pie chart is appropriate to use if comparing different categories. The one thing that is missing from figure 2, is the total number of products sold, [latex]n[/latex]. Without this it is impossible to determine the number of products sold in any given category. For example, if we knew that there were 3,457,940 total items sold, then we could calculate the number of items sold in “Accessories” by multiplying the percent by the total items sold. i.e. [latex]\text{number of items sold in Accessories}=24\text{% of }3,457,940=829,905.6[/latex]. Of course, this answer would be rounded to the nearest item, [latex]829,906[/latex], as accessories are counted in whole numbers.

ASIDE: A pie chart uses a complete circle to represent 100% of the data (1 whole), then appropriately sized segments are drawn to represent the percent of the whole represented by each category. The angle of the segment at the center of the circle can be found by multiplying [latex]360°[/latex] (the total number of degrees at the center of any circle) by the percentage of the whole being represented. For example, in figure 2, the angle for the “Clothing” segment is [latex]35\text{% of }360°=126°[/latex].

Pie charts are not always good for illustrating categorical data. If there are too many categories, it becomes impossible to tell them apart or assess their relative size. Pie charts should only be used when there is a small number (≤ 6) of categories.

Figure 3. An obviously problematic pie chart with too many categories.

Discrete Data Charts

Histogram

Histograms are particularly useful for large data sets. Histograms are similar to bar graphs, but are used for discrete (or grouped continuous) data. A histogram consists of contiguous (adjoining) bars. The horizontal axis is labeled with what the data represents, and the vertical axis is labeled frequency (or percent frequency). The graph will have the same shape with either label. The difference between a histogram and a bar graph lies in the way the bars are constructed. In a bar graph, the bars do not touch and the categories can be arranged in any order. In a histogram, the independent variable is discrete so has a defined order to it. Consequently, the bars must be ordered numerically along a continuously scaled axis.

Suppose that in a quick campus study, college students were asked how many books they had purchased for the semester. The number of books is discrete data, since books are counted and the number of books purchased has an obvious order.

The frequency table summarizes the data collected:

The histogram in figure 4 displays the number of books on the [latex]x[/latex]-axis and the frequency of purchase on the [latex]y[/latex]-axis.

Figure 4. An example of a histogram

Notice the values along the horizontal axis. The whole numbers for the number of books purchased lie in the center of each bar. This causes the bars, which must touch due to the continuous scale on the horizontal axis, to be half-way between the integer values.

Although the table is easy to read, the graph effectively illustrates the shape, the center, and the spread of the data. The minimum, maximum, and mode are easy to pick out. The minimum is [latex]1[/latex] (the middle value of the box to the far left), the maximum is [latex]6[/latex] (the middle value of the box to the far right), which means the range is [latex]6-1=5[/latex]. The mode, which is the most frequent value is found by identifying the category with the tallest bar, so the mode is [latex]3[/latex]. This tells us that most of the students sampled bought three books that semester.