76 Chapter 2 Graphical Summaries of Data baseline is at zero, so an accurate impression is given. In Figure 2.22, the baseline is larger than zero, so the rate of increase is exaggerated. When a graph or plot represents how much or how many of something, check the baseline. If it isn’t at zero, the graph may be misleading. Objective 2 Understand the area principle for constructing statistical graphs The Area Principle We often use images to compare amounts. Larger images correspond to greater amounts. To use images properly in this way, we must follow a rule known as the area principle. The Area Principle When amounts are compared by constructing an image for each amount, the areas of the images must be proportional to the amounts. For example, if one amount is twice as much as another, its image should have twice as much area as the other image. When the area principle is violated, the images give a misleading impression of the data. Bar graphs, when constructed properly, follow the area principle. The reason is that all the bars have the same width; only their height varies. Therefore, the areas of the bars are proportional to the amounts. For example, Figure 2.23 presents a bar graph that illustrates a comparison of the cost of jet fuel in 2000 versus 2008. In 2000, the cost of jet fuel was $0.90 per gallon, and in 2008 it had risen to $3.16 per gallon. 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 Cost per gallon ($) Cost of Jet Fuel 2000 2008 Source: www.airlines.org Figure 2.23 Price per gallon of jet fuel in 2000 and 2008. The bar graph accurately represents the difference. The bars in the bar graph differ in only one dimension — their height. The widths are the same. For this reason, the bar graph presents an accurate comparison of the two prices. The price in 2008 is about 3.5 times as much as the price in 2000, and the area of the bar for 2008 is about 3.5 times as large as the area of the bar for 2000. Unfortunately, people often mistakenly vary both dimensions of an image when making a comparison. This exaggerates the difference. Following is a comparison of the cost of jet fuel in the years 2000 and 2008 that uses a picture of an airplane to illustrate the difference. Cost of Jet Fuel 2000 2008 The pictures of the planes make the difference appear much larger than the correctly drawn bar graph does. The reason is that both the height and the width of the airplane have been increased by a factor of 3.5. Thus, the area of the larger plane is more than 12 times the area of the smaller plane. This graph violates the area principle, and gives a misleading impression of the comparison.
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