Figure 3-5 describes the relation between the number of hours students spent studying and the students’ grades on a statistics exam. In this example, the independent variable (x, on the horizontal axis) is the number of hours spent studying, and the dependent variable (y, on the vertical axis) is the grade on the statistics exam.

The scatterplot in Figure 3-5 suggests that more hours studying is associated with higher grades; it includes each participant’s two scores (one for hours spent studying and the other for grade received) that reveal the overall pattern of scores. In this scatterplot, the values on both axes go down to 0, but they don’t have to. Sometimes the scores are clustered and the pattern in the data might be clearer by adjusting the range on one or both axes. (If it’s not practical for the scores to go down to 0, be sure to indicate this with cut marks.)

52

Edward R. Tufte’s (1997/2005, 2001/2006b, 2006a) beautiful books demonstrate simple ways to create clearer graphs. One guideline is to increase the “data-ink ratio”—display more data with less ink. For example, a range-frame is a scatterplot or related graph that indicates the range of the data on each axis to the minimum and maximum scores. Eliminating the ends of the axes in Figure 3-6 frames the same data from Figure 3-5 within its minimum and maximum values, and increases the data-to-ink ratio.

To create a scatterplot:

53

A scatterplot between two scale variables can tell three possible stories. First, there may be no relation at all; in this case, the scatterplot looks like a jumble of random dots. This is an important scientific story if we previously believed that there was a systematic pattern between the two variables.

Second, a linear relation between variables means that the relation between variables is best described by a straight line. When the linear relation is positive, the pattern of data points flows upward and to the right. When the linear relation is negative, the pattern of data points flows downward and to the right. The data story about hours studying and statistics grades in Figures 3-5 and 3-6 indicates a positive, linear relation.

A nonlinear relation between variables means that the relation between variables is best described by a line that breaks or curves in some way. Nonlinear simply means “not straight,” so there are many possible nonlinear relations between variables. For example, the Yerkes–Dodson law described in Figure 3-7 predicts the relation between level of arousal and test performance. As professors, we don’t want you so relaxed that you don’t even show up for the test, but we also don’t want you so stressed out that you have a panic attack. You will maximize your performance somewhere in the happy middle described by a nonlinear relation (in this case, an upside-down U-curve).

The first type of line graph, based on a scatterplot, is especially useful because the line of best fit minimizes the distances between all the data points from that line. That allows us to use the x value to predict the y value and make predictions based on only one piece of information. For example, we can use the line of best fit in Figure 3-8 to predict that a student will earn a test score of about 62 if she studies for 2 hours; if she studies for 13 hours, she will earn a score of about 100. For now, we can simply eyeball the scatterplot and draw a line of best fit; in Chapter 14, you will learn how to calculate a line of best fit.

54

Here is a recap of the steps to create a scatterplot with a line of best fit:

A second situation in which a line graph is more useful than just a scatterplot involves time-related data. A time plot, or time series plot, is a graph that plots a scale variable on the y-axis as it changes over an increment of time (for example, hour, day, century) labeled on the x-axis. As with a scatterplot, marks are placed above each value on the x-axis (for example, at a given hour) at the value for that particular time on the y-axis (i.e., the score on the dependent variable). These marks are then connected with a line. It is possible to graph several lines on the same graph in a time plot, as long as the lines use the same scale variable on the y-axis. With multiple lines, the viewer can compare the trends for different levels of another variable.

Figure 3-9, for example, shows positive attitudes and negative attitudes around the world, as expressed on Twitter. The researchers analyzed more than half a billion tweets over the course of 24 hours (Golder & Macy, 2011) and plotted separate lines for each day of the week. These fascinating data tell many stories. For example, people tend to express more positive attitudes and fewer negative attitudes in the morning than later in the day; people express more positive attitudes on the weekends than during the week; and the weekend morning peak in positive attitudes is later than during the week, perhaps an indication that people are sleeping in.

Here is a recap of the steps to create a time plot:

3-3: Bar graphs depict data for two or more categories. They tell a data story more precisely than do either pictorial graphs or pie charts.

Figure 3-10 shows two different ways of depicting the percentage of Internet users in a given country who visited Twitter.com in June 2010. One graph is an alphabetized bar graph; the other is a Pareto chart. Where does Canada’s usage fit relative to that of other countries? Which graph makes it is easier to answer that question?

Bar graphs can help us understand the answers to interesting questions. For example, researchers wondered whether piercings and tattoos, once seen as indicators of a “deviant” worldview, had become mainstream (Koch, Roberts, Armstrong, & Owen, 2010). They surveyed 1753 college students with respect to numbers of piercings and tattoos, as well as about a range of destructive behaviors including academic cheating, illegal drug use, and number of arrests (aside from traffic arrests). The bar graph in Figure 3-11 depicts one finding: The likelihood of having been arrested was fairly similar among all groups, except among those with four or more tattoos, 70.6% of whom reported having been arrested at least once. A magazine article about this research advised parents, “So, that butterfly on your sophomore’s ankle is not a sign she is hanging out with the wrong crowd. But if she comes home for spring break covered from head to toe, start worrying” (Jacobs, 2010).

57

Liars’ Alert! The small differences among the students with no tattoos, one tattoo, and two or three tattoos could be exaggerated if a reporter wanted to scare parents. Compare Figure 3-12 to the first three bars of Figure 3-11. Notice what happens when the fourth bar for four or more tattoos is eliminated: The values on the y-axis do not begin at 0, the intervals change from 10 to 2, and the y-axis ends at 20%. The exact same data leave a very different impression. (Note: If the data are very far from 0, and it does not make sense to have the axis go down to 0, indicate this on the graph by including double slashes—called cut marks—like those shown in Figure 3-12.)

Here is a recap of the steps to create a bar graph. The critical choice for you, the graph creator, is in step 2.

Tufte (2001) has a plan for better bar graphs. In Figure 3-13, Tufte (a) eliminated the vertical axis; (b) kept the data labels on the y-axis; and (c) replaced the horizontal tick marks with thin white lines through the bars—another increase in the data–ink ratio.

Solutions to these Check Your Learning questions can be found in Appendix D.