A scatterplot is a graph that depicts the relation between two scale variables. The values of each variable are marked along the two axes, and a mark is made to indicate the intersection of the two scores for each participant. The mark is above the participant’s score on the x-axis and across from the score on the y-axis.

EXAMPLE 3.1

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.

Figure 3-5

Scatterplot of Hours Spent Studying and Statistics Grades This scatterplot depicts the relation between hours spent studying and grades on a statistics exam. Each dot represents one student’s score on the independent variable along the x-axis and on the dependent variable along the y-axis.

The scatterplot in Figure 3-5 suggests that more hours studying leads to 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.)

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.

Figure 3-6

Range-Frame Improves on a Scatterplot A range-frame is a traditional scatterplot that indicates the minimum and maximum observed values on the axes by erasing all ink beyond these points. This simple alteration increases the ratio of ink dedicated to actual data to overall printed ink in this graph.

To create a scatterplot:

1. Organize the data by participant; each participant will have two scores, one on each scale variable.
2. Label the horizontal x-axis with the name of the independent variable and its possible values, starting with 0 if practical.
3. Label the vertical y-axis with the name of the dependent variable and its possible values, starting with 0 if practical.

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4. Make a mark on the graph above each study participant’s score on the x-axis and next to his or her score on the y-axis.
5. To convert to a range-frame, simply erase the axes below the minimum score and above the maximum score.

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 ∪-curve).

Figure 3-7

Nonlinear Relations The Yerkes–Dodson law predicts that stress/anxiety improves test performance—but only to a point. Too much anxiety leads to an inability to perform at one’s best. This inverted U-curve illustrates the concept, but a scatterplot would better clarify the particular relation between these two variables.

A line graph is used to illustrate the relation between two scale variables.

EXAMPLE 3.2

Figure 3-8

The Line of Best Fit The line of best fit allows us to make predictions for a person’s value on the y variable from his or her value on the x variable.

The first type of line graph, based on a scatterplot, is especially useful because the best-fit line 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 if a student studies for 2 hours, she will earn a test score of about 62; 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 16, you will learn how to calculate a line of best fit.

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

1. Label the x-axis with the name of the independent variable and its possible values, starting with 0 if practical.
2. Label the y-axis with the name of the dependent variable and its possible values, starting with 0 if practical.
3. Make a mark above each study participant’s score on the x-axis and next to his or her score on the y-axis.
4. Visually estimate and sketch the line of best fit through the points on the scatterplot.
5. Maximize the data–ink ratio by converting to a range frame: Erase the axes below the minimum score and above the maximum score.

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, second, 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 minute) 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.

EXAMPLE 3.3

Figure 3-9

Hourly Moods as Seen Through Twitter Researchers tracked positive attitudes and negative attitudes expressed through Twitter over the course of a day and from around the globe (Golder & Macy, 2011). Time plots allow for multiple scale variables on one graph; in this case, there are separate lines for each day of the week, allowing us to see, for example, that the lines for Saturday and Sunday tend to be highest for positive attitudes and lowest for negative attitudes across the day.

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 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:

1. Label the x-axis with the name of the independent variable and its possible values. The independent variable should be an increment of time (e.g., hour, month, year).
2. Label the y-axis with the name of the dependent variable and its possible values, starting with 0 if practical.
3. Make a mark above each value on the x-axis at the value for that time on the y-axis.
4. Connect the dots.

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5. As you did with the scatterplot, maximize the data–ink ratio by converting to a range-frame: Erase the y-axis below the minimum y value and above the maximum y value.

A bar graph is a visual depiction of data in which the independent variable is nominal or ordinal and the dependent variable is scale. The height of each bar typically represents the average value of the dependent variable for each category.

MASTERING THE CONCEPT

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.

A Pareto chart is a type of bar graph in which the categories along the x-axis are ordered from highest bar on the left to lowest bar on the right.

EXAMPLE 3.4

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?

Figure 3-10

The Flexibility of the Bar Graph The standard bar graph provides a comparison of Twitter usage among 14 levels of a nominal dependent variable, country. The Pareto chart, a version of a bar graph, orders the countries from highest to lowest along the horizontal axis, which allows us to more easily pick out the highest and lowest bars. We can more easily know that Canada places in the middle of these countries, and that the United States and the United Kingdom are toward the bottom. We have to do more work to draw these conclusions from the original bar graph.

EXAMPLE 3.5

Figure 3-11

Bar Graphs Highlight Differences Between Averages or Percentages This bar graph depicts the percentages who have been arrested at least once (other than in a traffic arrest) for four groups of U.S. university students: those with no tattoos, one tattoo, two to three tattoos, or four or more tattoos. A bar graph can more vividly depict differences between percentages than just the typed numbers themselves can: 8.5, 18.7, 12.7, and 70.6.

Bar graphs can help us understand the answers to interesting questions. For example, researchers wondered whether piercings and tattoos, once viewed as indicators of a “deviant” worldview, had become mainstream (Koch, Roberts, Armstrong, & Owen, 2010). They surveyed 1753 American 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).

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.)

Figure 3-12

Deceiving with the Scale To exaggerate a difference between means, graphmakers sometimes compress the rating scale that they show on their graphs. When possible, label the axis beginning with 0, and when displaying percentages, include all values up to 100%.

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

1. Label the x-axis with the name and levels (i.e., categories) of the nominal or ordinal independent variable.

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2. Label the y-axis with the name of the scale dependent variable and its possible values, starting with 0 if practical.
3. For every level of the independent variable, draw a bar with the height of that level’s value on the dependent variable.

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.

Figure 3-13

Redesigning the Bar Graph Eliminating the frame and the y-axis and adding thin white lines through the bars, as suggested by Tufte (2001/2006b), makes this bar graph easier to read and increases the data–ink ratio.

A pictorial graph is a visual depiction of data typically used for an independent variable with very few levels (categories) and a scale dependent variable. Each level uses a picture or symbol to represent its value on the scale dependent variable.

A pie chart is a graph in the shape of a circle, with a slice for every level (category) of the independent variable. The size of each slice represents the proportion (or percentage) of each level.

CHECK YOUR LEARNING

Clarifying the Concepts

• 3-4 How are scatterplots and line graphs similar?
• 3-5 Why should we typically avoid using pictorial graphs and pie charts?

Calculating the Statistics

• 3-6 What type of visual display of data allows us to calculate or evaluate how a variable changes over time?

Applying the Concepts

• 3-7 What is the best type of common graph to depict each of the following data sets and research questions? Explain your answers.
  1. Depression severity and amount of stress for 150 university students. Is depression related to stress level?
  2. Number of inpatient mental health facilities in Canada as measured every 10 years between 1890 and 2000. Has the number of facilities declined in recent years?
  3. Number of siblings reported by 100 people. What size family is most common?
  4. Mean years of education for six regions of the United States. Are education levels higher in some regions than in others?
  5. Calories consumed in a day and hours slept that night for 85 people. Does the amount of food a person eats predict how long he or she sleeps at night?