## Displaying Data Graphically and Interpreting the Graph

Graphs are also used to display data or evidence. Graphs are a method of presenting numerical patterns. They condense detailed numerical information into a visual form in which relationships and numerical patterns can be seen more easily. For example, which countries have larger or smaller populations? A careful reader could examine a long list of numbers representing the populations of many countries, but with over 200 nations in the world, searching through such a list would take concentration and time. Putting these same numbers on a graph can quickly reveal population patterns. Economists use graphs both for a compact and readable presentation of groups of numbers and for building an intuitive grasp of relationships and connections.

Three types of graphs are used in this book: line graphs, pie graphs, and bar graphs. Each is discussed below. We also provide warnings about how graphs can be manipulated to alter viewers’ perceptions of the relationships in the data.

**Line Graphs**

The graphs we have discussed so far are called **line graphs**, because they show a relationship between two variables: one measured on the horizontal axis and the other measured on the vertical axis.

Sometimes it is useful to show more than one set of data on the same axes. The data in Table is displayed in Figure which shows the relationship between two variables: length and median weight for American baby boys and girls during the first three years of life. (The **median** means that half of all babies weigh more than this and half weigh less.) The line graph measures length in inches on the horizontal axis and weight in pounds on the vertical axis. For example, point A on the figure shows that a boy who is 28 inches long will have a median weight of about 19 pounds. One line on the graph shows the length-weight relationship for boys and the other line shows the relationship for girls. This kind of graph is widely used by healthcare providers to check whether a child’s physical development is roughly on track.

Boys from Birth to 36 Months | Girls from Birth to 36 Months | ||
---|---|---|---|

Length (inches) | Weight (pounds) | Length (inches) | Weight (pounds) |

20.0 | 8.0 | 20.0 | 7.9 |

22.0 | 10.5 | 22.0 | 10.5 |

24.0 | 13.5 | 24.0 | 13.2 |

26.0 | 16.4 | 26.0 | 16.0 |

28.0 | 19.0 | 28.0 | 18.8 |

30.0 | 21.8 | 30.0 | 21.2 |

32.0 | 24.3 | 32.0 | 24.0 |

34.0 | 27.0 | 34.0 | 26.2 |

36.0 | 29.3 | 36.0 | 28.9 |

38.0 | 32.0 | 38.0 | 31.3 |

Not all relationships in economics are linear. Sometimes they are curves. Figure presents another example of a line graph, representing the data from Table. In this case, the line graph shows how thin the air becomes when you climb a mountain. The horizontal axis of the figure shows altitude, measured in meters above sea level. The vertical axis measures the density of the air at each altitude. Air density is measured by the weight of the air in a cubic meter of space (that is, a box measuring one meter in height, width, and depth). As the graph shows, air pressure is heaviest at ground level and becomes lighter as you climb. Figure shows that a cubic meter of air at an altitude of 500 meters weighs approximately one kilogram (about 2.2 pounds). However, as the altitude increases, air density decreases. A cubic meter of air at the top of Mount Everest, at about 8,828 meters, would weigh only 0.023 kilograms. The thin air at high altitudes explains why many mountain climbers need to use oxygen tanks as they reach the top of a mountain.

Altitude (meters) | Air Density (kg/cubic meters) |
---|---|

0 | 1.200 |

500 | 1.093 |

1,000 | 0.831 |

1,500 | 0.678 |

2,000 | 0.569 |

2,500 | 0.484 |

3,000 | 0.415 |

3,500 | 0.357 |

4,000 | 0.307 |

4,500 | 0.231 |

5,000 | 0.182 |

5,500 | 0.142 |

6,000 | 0.100 |

6,500 | 0.085 |

7,000 | 0.066 |

7,500 | 0.051 |

8,000 | 0.041 |

8,500 | 0.025 |

9,000 | 0.022 |

9,500 | 0.019 |

10,000 | 0.014 |

The length-weight relationship and the altitude-air density relationships in these two figures represent averages. If you were to collect actual data on air pressure at different altitudes, the same altitude in different geographic locations will have slightly different air density, depending on factors like how far you are from the equator, local weather conditions, and the humidity in the air. Similarly, in measuring the height and weight of children for the previous line graph, children of a particular height would have a range of different weights, some above average and some below. In the real world, this sort of variation in data is common. The task of a researcher is to organize that data in a way that helps to understand typical patterns. The study of statistics, especially when combined with computer statistics and spreadsheet programs, is a great help in organizing this kind of data, plotting line graphs, and looking for typical underlying relationships. For most economics and social science majors, a statistics course will be required at some point.

One common line graph is called a **time series**, in which the horizontal axis shows time and the vertical axis displays another variable. Thus, a time series graph shows how a variable changes over time. Figure shows the unemployment rate in the United States since 1975, where unemployment is defined as the percentage of adults who want jobs and are looking for a job, but cannot find one. The points for the unemployment rate in each year are plotted on the graph, and a line then connects the points, showing how the unemployment rate has moved up and down since 1975. The line graph makes it easy to see, for example, that the highest unemployment rate during this time period was slightly less than 10% in the early 1980s and 2010, while the unemployment rate declined from the early 1990s to the end of the 1990s, before rising and then falling back in the early 2000s, and then rising sharply during the recession from 2008–2009.

**Pie Graphs**

A **pie graph** (sometimes called a **pie chart**) is used to show how an overall total is divided into parts. A circle represents a group as a whole. The slices of this circular “pie” show the relative sizes of subgroups.

Figure shows how the U.S. population was divided among children, working age adults, and the elderly in 1970, 2000, and what is projected for 2030. The information is first conveyed with numbers in Table, and then in three pie charts. The first column of Table shows the total U.S. population for each of the three years. Columns 2–4 categorize the total in terms of age groups—from birth to 18 years, from 19 to 64 years, and 65 years and above. In columns 2–4, the first number shows the actual number of people in each age category, while the number in parentheses shows the percentage of the total population comprised by that age group.

Year | Total Population | 19 and Under | 20–64 years | Over 65 |
---|---|---|---|---|

1970 | 205.0 million | 77.2 (37.6%) | 107.7 (52.5%) | 20.1 (9.8%) |

2000 | 275.4 million | 78.4 (28.5%) | 162.2 (58.9%) | 34.8 (12.6%) |

2030 | 351.1 million | 92.6 (26.4%) | 188.2 (53.6%) | 70.3 (20.0%) |

In a pie graph, each slice of the pie represents a share of the total, or a percentage. For example, 50% would be half of the pie and 20% would be one-fifth of the pie. The three pie graphs in Figure show that the share of the U.S. population 65 and over is growing. The pie graphs allow you to get a feel for the relative size of the different age groups from 1970 to 2000 to 2030, without requiring you to slog through the specific numbers and percentages in the table. Some common examples of how pie graphs are used include dividing the population into groups by age, income level, ethnicity, religion, occupation; dividing different firms into categories by size, industry, number of employees; and dividing up government spending or taxes into its main categories.

**Bar Graphs**

A **bar graph** uses the height of different bars to compare quantities. Table lists the 12 most populous countries in the world. Figure provides this same data in a bar graph. The height of the bars corresponds to the population of each country. Although you may know that China and India are the most populous countries in the world, seeing how the bars on the graph tower over the other countries helps illustrate the magnitude of the difference between the sizes of national populations.

Country | Population |
---|---|

China | 1,369 |

India | 1,270 |

United States | 321 |

Indonesia | 255 |

Brazil | 204 |

Pakistan | 190 |

Nigeria | 184 |

Bangladesh | 158 |

Russia | 146 |

Japan | 127 |

Mexico | 121 |

Philippines | 101 |

Bar graphs can be subdivided in a way that reveals information similar to that we can get from pie charts. Figure offers three bar graphs based on the information from Figure about the U.S. age distribution in 1970, 2000, and 2030. Figure (a) shows three bars for each year, representing the total number of persons in each age bracket for each year. Figure (b) shows just one bar for each year, but the different age groups are now shaded inside the bar. In Figure (c), still based on the same data, the vertical axis measures percentages rather than the number of persons. In this case, all three bar graphs are the same height, representing 100% of the population, with each bar divided according to the percentage of population in each age group. It is sometimes easier for a reader to run his or her eyes across several bar graphs, comparing the shaded areas, rather than trying to compare several pie graphs.

Figure and Figure show how the bars can represent countries or years, and how the vertical axis can represent a numerical or a percentage value. Bar graphs can also compare size, quantity, rates, distances, and other quantitative categories.

**Comparing Line Graphs with Pie Charts and Bar Graphs**

Now that you are familiar with pie graphs, bar graphs, and line graphs, how do you know which graph to use for your data? Pie graphs are often better than line graphs at showing how an overall group is divided. However, if a pie graph has too many slices, it can become difficult to interpret.

Bar graphs are especially useful when comparing quantities. For example, if you are studying the populations of different countries, as in Figure, bar graphs can show the relationships between the population sizes of multiple countries. Not only can it show these relationships, but it can also show breakdowns of different groups within the population.

A line graph is often the most effective format for illustrating a relationship between two variables that are both changing. For example, time series graphs can show patterns as time changes, like the unemployment rate over time. Line graphs are widely used in economics to present continuous data about prices, wages, quantities bought and sold, the size of the economy.

**How Graphs Can Be Misleading**

Graphs not only reveal patterns; they can also alter how patterns are perceived. To see some of the ways this can be done, consider the line graphs of Figure, Figure, and Figure. These graphs all illustrate the unemployment rate—but from different perspectives.

Suppose you wanted a graph which gives the impression that the rise in unemployment in 2009 was not all that large, or all that extraordinary by historical standards. You might choose to present your data as in Figure (a). Figure (a) includes much of the same data presented earlier in Figure, but stretches the horizontal axis out longer relative to the vertical axis. By spreading the graph wide and flat, the visual appearance is that the rise in unemployment is not so large, and is similar to some past rises in unemployment. Now imagine you wanted to emphasize how unemployment spiked substantially higher in 2009. In this case, using the same data, you can stretch the vertical axis out relative to the horizontal axis, as in Figure (b), which makes all rises and falls in unemployment appear larger.

A similar effect can be accomplished without changing the length of the axes, but by changing the scale on the vertical axis. In Figure (c), the scale on the vertical axis runs from 0% to 30%, while in Figure (d), the vertical axis runs from 3% to 10%. Compared to Figure, where the vertical scale runs from 0% to 12%, Figure (c) makes the fluctuation in unemployment look smaller, while Figure (d) makes it look larger.

Another way to alter the perception of the graph is to reduce the amount of variation by changing the number of points plotted on the graph. Figure (e) shows the unemployment rate according to five-year averages. By averaging out some of the year-to-year changes, the line appears smoother and with fewer highs and lows. In reality, the unemployment rate is reported monthly, and Figure (f) shows the monthly figures since 1960, which fluctuate more than the five-year average. Figure (f) is also a vivid illustration of how graphs can compress lots of data. The graph includes monthly data since 1960, which over almost 50 years, works out to nearly 600 data points. Reading that list of 600 data points in numerical form would be hypnotic. You can, however, get a good intuitive sense of these 600 data points very quickly from the graph.

A final trick in manipulating the perception of graphical information is that, by choosing the starting and ending points carefully, you can influence the perception of whether the variable is rising or falling. The original data show a general pattern with unemployment low in the 1960s, but spiking up in the mid-1970s, early 1980s, early 1990s, early 2000s, and late 2000s. Figure (g), however, shows a graph that goes back only to 1975, which gives an impression that unemployment was more-or-less gradually falling over time until the 2009 recession pushed it back up to its “original” level—which is a plausible interpretation if one starts at the high point around 1975.

These kinds of tricks—or shall we just call them “presentation choices”— are not limited to line graphs. In a pie chart with many small slices and one large slice, someone must decided what categories should be used to produce these slices in the first place, thus making some slices appear bigger than others. If you are making a bar graph, you can make the vertical axis either taller or shorter, which will tend to make variations in the height of the bars appear more or less.

Being able to read graphs is an essential skill, both in economics and in life. A graph is just one perspective or point of view, shaped by choices such as those discussed in this section. Do not always believe the first quick impression from a graph. View with caution.