Construct and interpret scatter plots to describe patterns of association

Learning goal: Construct a scatter plot for bivariate measurement data, and describe the pattern of association between the two quantities using the words clustering, outlier, positive association, negative association, no association, linear association and nonlinear association. This is the analyze-and-interpret part of the GAISE model: the question has been asked and the data collected, and the work now is to see what the data say.

Bivariate measurement data means that two measured quantities have been recorded for each subject. The pair of numbers belongs together because it came from the same person, plant, cup or car. A list of practice hours for one group of students and a separate list of free-throw scores for another group is not bivariate data; nothing links a particular number of hours to a particular score. A scatter plot needs pairs.

A scatter plot places one dot for each pair. The horizontal axis carries the first quantity and the vertical axis carries the second. Nothing is connected, nothing is stacked and no dot is left out, because every dot is one subject. The plot is the graph a statistician looks at first when asking whether two measurements tend to move together.

Here is the hypothetical data set used through this lesson. Thirteen students on a school basketball team recorded how many hours they practiced in one week and how many free throws out of 20 they made at Friday's test. The numbers are invented for teaching, not measured from a real team.

Student A: 1 hour, 6 made. B: 1 hour, 8 made. C: 2 hours, 7 made. D: 2 hours, 9 made. E: 2 hours, 18 made. F: 4 hours, 12 made. G: 4 hours, 11 made. H: 4 hours, 13 made. I: 5 hours, 14 made. J: 6 hours, 15 made. K: 6 hours, 17 made. L: 7 hours, 18 made. M: 8 hours, 19 made.

To construct the plot, choose the axes and their scales first. Hours run from 1 to 8, so a horizontal axis from 0 to 9 with a tick every hour works. Free throws run from 6 to 19, so a vertical axis from 0 to 20 works. Then plot each pair: student A is the dot at (1, 6), student M is the dot at (8, 19). Two students with the same hours, such as C and D at 2 hours, produce two different dots at different heights; they are not combined.

Reading the pattern. Look at the twelve dots other than student E. As hours increase, made shots increase. That is a positive association: larger values of one quantity tend to go with larger values of the other. If larger values of one tended to go with smaller values of the other, the association would be negative. If the dots formed a shapeless cloud with no tendency either way, there would be no association.

Those twelve dots also lie close to an imaginary straight line running from the lower left to the upper right. That makes the association linear. The word does not mean the dots sit exactly on a line; real measurements usually scatter around it. It means a straight line describes the trend well. An association is nonlinear when the dots follow a curve, so that a straight line would be systematically wrong in some region.

Student E, at (2, 18), sits far above the trend of the other twelve. A dot that lies well away from the overall pattern is an outlier. Notice what the word does and does not say. It says E's pair does not follow the pattern the others follow. It does not say the data point is a mistake, and it does not say it should be deleted. A sensible response is to check the record and then think about a reason: perhaps E has played for years and needs little practice. The outlier stays on the plot and is described in the interpretation.

Clustering describes dots bunched together in one region of the plot, close in both quantities at once. Four of the thirteen dots, students A to D, sit together at 1 to 2 hours and 6 to 9 made shots. No student practiced 3 hours, so a gap separates these four from the rest of the rising pattern: that is a cluster at the low end of the plot. Student E also has 2 hours but sits far above the four, so E is not part of the cluster: sharing a value on one axis is not the same as being close on the plot. Clusters can suggest a subgroup, for instance students who share a busy schedule, and they matter because a pattern seen in one cluster might not hold in another.

A nonlinear negative example. A second hypothetical data set records the temperature of a cup of cocoa, in degrees Celsius, at several times after it was poured: 0 minutes, 90°; 5 minutes, 70°; 10 minutes, 56°; 15 minutes, 46°; 20 minutes, 39°; 30 minutes, 30°; 40 minutes, 26°; 60 minutes, 23°. Temperature falls as time increases, so the association is negative. But the drop in the first five minutes is 20 degrees, from 10 to 15 minutes it is 10 degrees, and from 40 to 60 minutes it is only 3 degrees. The dots fall steeply and then flatten toward room temperature. A straight line would predict the cocoa freezing eventually, which is wrong. This association is negative and nonlinear.

A no-association example. Ten hypothetical students recorded shoe size and a quiz score out of 100: (5, 80), (6, 72), (6, 95), (7, 68), (7, 88), (8, 75), (8, 91), (9, 70), (9, 84), (10, 79). Scores at size 6 are 72 and 95; scores at size 9 are 70 and 84. Larger shoe sizes do not tend to bring higher or lower scores. The plot is a cloud, and the honest description is no association.

The misconception to avoid. Association is not causation. The basketball plot shows that hours and made shots tend to rise together in this group; it does not show that practice produced the scores. Students who are already skilled may enjoy practicing more, or both may depend on something unmeasured. A scatter plot can describe a pattern of association. Explaining why the pattern exists needs more than the plot.

Interpreting in a sentence. A complete description names the direction, the form, any outliers and any clusters, and stays inside the data. For the basketball set: "For these 13 students, more practice hours are associated with more free throws made; the association is positive and roughly linear, one student at 2 hours and 18 made is an outlier above the pattern, and four students cluster at 1 to 2 hours with 6 to 9 made." Every word of that sentence can be checked against the plot.

Worked plotting check: Student H recorded 4 hours and 13 made. On axes with hours horizontal and made shots vertical, the dot goes 4 units right of the origin and 13 units up. Students F and G also recorded 4 hours, with 12 and 11 made, so two more dots sit directly below H, at (4, 12) and (4, 11). All three dots stay on the plot, because three students are three subjects. A plot that shows one dot at 4 hours has lost two data points.

Worked pattern description: In the cocoa data, every later reading is lower than the earlier one, so the direction is negative. The sizes of the drops are 20, 14, 10, 7, 9, 4 and 3 degrees over the successive intervals, which are not equal per minute: 20 degrees in the first 5 minutes but only 3 degrees in the last 20 minutes. The drop per minute shrinks steadily from one interval to the next (4 degrees per minute at first, under 1 by the 20-to-30-minute interval, and less than 0.2 at the end), a systematic flattening across the whole record rather than the up-and-down scatter that dots show around a straight trend, so the dots bend and the association is nonlinear. A description reads: "temperature falls quickly at first and then levels off; negative, nonlinear association, no outliers."

Worked outlier judgement: Suppose student E's record is checked and the 18 is correct. The pair (2, 18) is still an outlier, because it is far from the pattern of the other twelve dots, not because it is wrong. If instead the scorer had typed 18 in place of 8, the corrected pair (2, 8) would fall inside the pattern and the outlier would disappear. Whether a point is an outlier depends on its distance from the pattern; whether it stays in the data depends on whether the measurement was valid.

Construct and annotate: Draw the basketball scatter plot on graph paper with hours from 0 to 9 across and made shots from 0 to 20 up. Plot all 13 pairs, label the outlier and circle the cluster of four dots at 1 to 2 hours and 6 to 9 made. Then plot the cocoa data on a second grid and the shoe-size data on a third. Under each plot write one sentence naming direction, form, outliers and clusters.

Independent performance: Collect bivariate measurement data from at least ten subjects on a question you choose, for example arm span and height, or minutes of reading and pages finished. Record both measurements for each subject in a table. Construct the scatter plot with labeled, scaled axes. Write an interpretation that (1) names the direction as positive, negative or none, or describes it separately over parts of the range if it changes direction, (2) names the form as linear or nonlinear with a reason, or says no form can be seen when there is no association, (3) identifies any outlier and what you did about checking it, or says that none appears, (4) identifies any cluster, or says that none appears, and (5) states that the plot shows association, not cause. Success means every claim in the interpretation can be pointed to on the plot and every dot on the plot can be found in the table.