Example 6: A labor union might produce this graph to show the difference between the average manager salary and the average worker salary. The average manager salary is twice as high as the average worker salary as in a bar graph, but the image is also twice as wide. That makes it look like the manager salary is 4 times as large as the worker salary. The area needs to accurately portray the relationship, otherwise we will have a perceptual distortion. Misleading Scale Another type of distortion in bar charts results from setting the baseline to a value other than zero. The baseline is the bottom of the vertical axis, representing the least number of cases that could have occurred in a category. Normally, this number should be zero. Example 7: Compare the two graphs below showing support for same-sex marriage rights from a poll taken in December, 2008. At a glance, the two graphs suggest very different stories. The second graph makes it look like more than three times as many people oppose marriage rights as support them. But when we look at the scale we can see that the difference is about 12%. By not starting at zero the difference looks enlarged. Stacked Bar Graphs Another type of graph that can be hard to read and sometimes misleading is a stacked bar graph. In a stacked bar graph, the values we are comparing are stacked on top of each other vertically. Example 8: The lists college expenses for two different students and we want to compare them. A stacked bar graph shows the expenses stacked vertically, but we are interested in the differences, not the totals. Expense Student 1 Student 2 Rent $500 $650 Food $125 $125 Tuition $1750 $1450 Books $325 $275 Misc $100 $175 It would be much easier to interpret the differences in a side-by-side bar chart.
Presenting Quantitative Data With categorical data, the horizontal axis is the category, but with quantitative, or numerical, data we have numbers. If we have repeated values we can also make a frequency table. Example 9: A teacher records scores on a 20-point quiz for the 30 students in their class. The scores are: 19, 20, 18, 18, 17, 18, 19, 17, 20, 18, 20, 16, 20, 15, 17, 12, 18, 19, 18, 19, 17, 20, 18, 16, 15, 18, 20, 5, 0 and 0. Here is a frequency table with the scores grouped and put in order. Quiz Score Frequency of Students 0 2 5 1 12 1 15 2 16 2 17 4 18 8 19 4 20 6 Using this table, it would be possible to create a standard bar chart from this summary, like we did for categorical data. However, since the scores are numerical values, this chart wouldn’t make sense; the first and second bars would be five values apart, while the later bars would only be one value apart. Instead, we will to treat the horizontal axis as a number line. This type of graph is called a histogram. Histograms A histogram is like a bar graph, but the horizontal axis is a number line. Unlike a bar graph, there are no spaces between the bars. Here is a histogram for the data given above. Notice that in this histogram, the two scores of 15 are to the right of 15, or between 15 and 16. The horizontal scales on histograms can be confusing for this reason. Some people choose to have bars at ½ values to avoid this ambiguity, as in this second histogram. If we have a large number of different data values, a frequency table listing every possible value would be way too long. There would be too many bars on the histogram to reveal any patterns. For this reason, it is common with quantitative data to group data into class intervals. Class Intervals Class intervals are groupings of the data. In general, we define class intervals so that: Each interval is equal in size. For example, if the first class contains values from the second class should include values from 135-149. We typically have somewhere between 5 and 20 classes, depending on the number of data values we’re working with.