Learning Outcomes
- Use a graph to describe the distribution for a set of qualitative (categorical) data
Qualitative Data Discussion
Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.
| De Anza College | Foothill College | ||||
|---|---|---|---|---|---|
| Number | Percent | Number | Percent | ||
| Full-time | [latex]9,200[/latex] | [latex]40.9[/latex]% | Full-time | [latex]4,059[/latex] | [latex]28.6[/latex]% |
| Part-time | [latex]13,296[/latex] | [latex]59.1[/latex]% | Part-time | [latex]10,124[/latex] | [latex]71.4[/latex]% |
| Total | [latex]22,496[/latex] | [latex]100[/latex]% | Total | [latex]14,183[/latex] | [latex]100[/latex]% |
Table 1.2 Fall Term 2007 (Census day)
Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning which graphs to use. Two graphs that are used to display qualitative data are pie charts and bar graphs.
In a pie chart, categories of data are represented by wedges in a circle and are proportional in size to the percent of individuals in each category.
In a bar graph, the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal.
A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).
Look at Figure 1.5 and Figure 1.6 and determine which graph (pie or bar) you think displays the comparisons better.
It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the “best” graph depending on the data and the context. Our choice also depends on the purpose for which we are using the data.
Figure 1.5
Figure 1.6
Percentages That Add to More (or Less) Than [latex]100[/latex]%
Sometimes percentages add up to be more than [latex]100[/latex]% (or less than [latex]100[/latex]%). In the graph, the percentages add to more than [latex]100[/latex]% because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than [latex]100[/latex]%.
| Characteristic/Category | Percent |
|---|---|
| Full-Time Students | [latex]40.9[/latex]% |
| Students who intend to transfer to a 4-year educational institution | [latex]48.6[/latex]% |
| Students under age 25 | [latex]61.0[/latex]% |
| TOTAL | [latex]150.5[/latex]% |
Table 1.3 De Anza College Spring 2010
Figure 1.7
Omitting Categories and Missing Data
The table displays Ethnicity of Students but is missing the “Other/Unknown” category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart.
| Frequency | Percent | |
|---|---|---|
| Asian | [latex]8,794[/latex] | [latex]36.1[/latex]% |
| Black | [latex]1,412[/latex] | [latex]5.8[/latex]% |
| Filipino | [latex]1,298[/latex] | [latex]5.3[/latex]% |
| Hispanic | [latex]4,180[/latex] | [latex]17.1[/latex]% |
| Native American | [latex]146[/latex] | [latex]0.6[/latex]% |
| Pacific Islander | [latex]236[/latex] | [latex]1.0[/latex]% |
| White | [latex]5,978[/latex] | [latex]24.5[/latex]% |
| TOTAL | [latex]22,044[/latex] out of [latex]24,382[/latex] | [latex]90.4[/latex]% out of [latex]100[/latex]% |
Table 1.4 Ethnicity of Students at De Anza College Fall Term 2007 (Census Day)
Figure 1.8 Ethnicity of Students
The following graph is the same as the previous graph but the “Other/Unknown” percent ([latex]9.6[/latex]%) has been included. The “Other/Unknown” category is large compared to some of the other categories (Native American, [latex]0.6[/latex]%, Pacific Islander [latex]1.0[/latex]%). This is important to know when we think about what the data are telling us.
This particular bar graph in Figure 1.9 can be difficult to understand visually. The graph in Figure 1.10 is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.
Figure 1.9 Bar Graph with Other/Unknown Category
Figure 1.10 Pareto Chart with Bars Sorted by Size
Recall: Percents
Percents are widely used to describe how something has changed. A percent is a ratio (fraction) whose denominator is 100.
Recall: Conversion
Since a percent is a ratio, they can easily be expressed as both fractions and decimals.
Percent to a Fraction
- Write the percent as a ratio with the denominator 100.
- Simplify the fraction if possible.
Percent to a Decimal
- Write the percent as a ratio with the denominator 100.
- Convert the fraction to a decimal by dividing the numerator by the denominator.
Pie Charts: No Missing Data
The following pie charts have the “Other/Unknown” category included (since the percentages must add to [latex]100[/latex]%). The chart in Figure 1.11 is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in Figure 1.9.
Figure 1.11
