One common way to change the impression conveyed by a graph is to alter the scale on the vertical axis, the horizontal axis, or both. For example, consider the data on collisions of large marine vessels operating in European waters over a certain five-year period, summarized in Table 2.10. Figure 2.38 is a MINITAB bar graph showing the frequency of collisions for each of the three locations listed in the table. The graph shows that in-port collisions occur more often than collisions at sea or collisions in restricted waters.

Now, suppose you want to use the same data to exaggerate the difference between the number of in-port collisions and the number of collisions in restricted waters. One way to do this is to increase the distance between successive units on the vertical axis—that is, stretch the vertical axis by graphing only a few units per inch. A telltale sign of stretching is a long vertical axis, but this indication is often hidden by starting the vertical axis at some point above the origin, 0. Such a graph is shown in the SPSS printout in Figure 2.39. By starting the bar chart at 250 collisions (instead of 0), it appears that the frequency of in-port collisions is many times greater than the frequency of collisions in restricted waters.

Another method of achieving visual distortion with bar graphs is by making the width of the bars proportional to their height. For example, look at the bar chart in Figure 2.40a, which depicts the percentage of the total number of motor vehicle deaths in a year that occurred on each of four major highways. Now suppose we make both the width and the height grow as the percentage of fatal accidents grows. This change is shown in Figure 2.40b. The distortion is that the reader may tend to equate the area of the bars with the percentage of deaths occurring at each highway when, in fact, the true relative frequency of fatal accidents is proportional only to the height of the bars.

Although we’ve discussed only a few of the ways that graphs can be used to convey misleading pictures of phenomena, the lesson is clear: Look at all graphical descriptions of data with a critical eye. In particular, check the axes and the size of the units on each axis. Ignore the visual changes, and concentrate on the actual numerical changes indicated by the graph or chart.

The information in a data set can also be distorted by using numerical descriptive measures, as Example 2.20 shows.

Another distortion of information in a sample occurs when only a measure of central tendency is reported. Both a measure of central tendency and a measure of variability are needed to obtain an accurate mental image of a data set.

Suppose, for instance, that you want to buy a new car and are trying to decide which of two models to purchase. Since energy and economy are both important issues, you decide to purchase model A because its EPA mileage rating is 32 miles per gallon in the city, whereas the mileage rating for model B is only 30 miles per gallon in the city.

However, you may have acted too quickly. How much variability is associated with the ratings? As an extreme example, suppose that further investigation reveals that the standard deviation for model A mileages is 5 miles per gallon, whereas that for model B is only 1 mile per gallon. If the mileages form a mound-shaped distribution, they might appear as shown in Figure 2.41. Note that the larger amount of variability associated with model A implies that more risk is involved in purchasing that model. That is, the particular car you purchase is more likely to have a mileage rating that will differ greatly from the EPA rating of 32 miles per gallon if you purchase model A, while a model B car is not likely to vary from the 30-miles-per-gallon rating by more than 2 miles per gallon.

We conclude this section with another example on distorting the truth with numerical descriptive measures.