A Scatterplot Can Hide How Its Points Were Collected
In “Common Mistakes Describing Scatterplots,” you practiced describing direction, unusual features, form, and strength without leaving out context or claiming more than the plot supports. A further question is whether the points belong to distinct groups or occur in a meaningful time order. If that information is hidden, a single scatterplot can suggest a pattern that does not describe what is happening within each group or over time.
A lurking variable is a variable not included in the displayed scatterplot that may help explain an observed association. Time and group membership are two possible lurking variables. For example, two quantities may both increase as a business grows, creating an upward pattern even if increases in one are not connected to increases in the other. Or two groups may occupy different regions of a plot, so the pattern between the groups differs from the patterns within them.
This does not mean the scatterplot is wrong. It means that a description of the combined data may leave out an important feature of how the observations are arranged. As in “Adding a Categorical Variable to a Scatterplot,” labels or symbols can help show group membership. When time is involved, the order in which observations occurred can also matter, even if the scatterplot itself does not show that order.
Check the Order of Observations Over Time
A scatterplot displays paired values of two quantitative variables, but it does not automatically show when each pair was recorded. If both variables change over time, the points may form an apparent association because they share a time trend. In that situation, time may be related to both plotted variables, while the scatterplot alone cannot show whether one variable is related to the other after accounting for time.
When measurements were collected in sequence, find out whether observations represent days, months, production runs, or another ordered series. A time plot, which places time on the horizontal axis and one measured variable on the vertical axis, can help reveal a trend, cycle, or sudden change. You can also mark the order on a scatterplot, for example with arrows, if that makes the sequence clear. These displays provide context; they do not by themselves establish a cause.
Worked Example: Website Visits and Support Requests
Suppose a fictional small business records weekly website visits, in hundreds, and customer support requests per day during six consecutive weeks. The paired measurements are listed in time order:
| Week | Website visits, hundreds | Support requests per day |
|---|---|---|
| 1 | 2 | 12 |
| 2 | 3 | 13 |
| 3 | 4 | 12 |
| 4 | 5 | 15 |
| 5 | 6 | 14 |
| 6 | 7 | 16 |
Describe what the scatterplot suggests. From the first week to the last, website visits rise from 2 hundred to 7 hundred, and support requests rise from 12 to 16 per day. The plotted pairs would show an overall positive association, with some week-to-week variation in support requests.
Restore the time order. The observations are consecutive weeks, and both measurements generally increase across those weeks. A time plot would show that upward movement. The scatterplot by itself does not show that the pairs were recorded in sequence, so it can encourage a reader to treat the association as if it were simply a relationship between visits and requests.
State a cautious conclusion. A suitable description is: “Across these six weeks, website visits and customer support requests show a positive association, and both generally increase over time. The pattern may reflect business growth or another change over time; these data do not establish that more visits caused more requests.”
The key observation is that calendar time provides another possible explanation for the joint increase. For instance, a business expansion could increase both visits and the number of customers contacting support. The data shown do not tell us which explanation is correct. They also do not establish whether more visits directly lead to more requests.
A time pattern can be more complicated than a steady increase. Measurements may rise and fall in a cycle, or a process may change suddenly after a policy or equipment change. A scatterplot may compress these stages into one cloud of points. If the order is relevant, examine it rather than relying only on a left-to-right description of the scatterplot.
When Two Groups Are Combined
Group membership can change the way an association appears. The group labels might represent different course levels, neighborhoods, product types, or other categories. If the groups occupy separate regions of the scatterplot, the pattern between their centers may be different from the pattern among observations within either group.
One important possibility is a reversal: each group shows an association in one direction, while the combined data show an association in the opposite direction. This can happen when group membership is related to both plotted variables. The label “positive” or “negative” for the combined data may therefore hide contrasting subgroup patterns. A useful response is to describe the groups separately and then explain how their combined arrangement differs.
Worked Example: Study Hours and Scores in Two Course Levels
Imagine a fictional school recording weekly study hours and quiz scores for students in two course levels. Each point represents one student. The values are:
| Course level | Study hours, \(x\) | Quiz score, \(y\), points |
|---|---|---|
| Foundation | 1 | 58 |
| Foundation | 2 | 54 |
| Foundation | 3 | 50 |
| Foundation | 4 | 46 |
| Advanced | 6 | 92 |
| Advanced | 7 | 88 |
| Advanced | 8 | 84 |
| Advanced | 9 | 80 |
Describe each group first. In the Foundation group, study hours increase from 1 to 4 while quiz scores decrease from 58 to 46 points. The four values form a decreasing straight-line pattern. In the Advanced group, study hours increase from 6 to 9 while scores decrease from 92 to 80 points. That group also shows a decreasing straight-line pattern.
Compare the groups’ locations. The Foundation students’ mean study time is \((1+2+3+4)/4=2.5\) hours and their mean quiz score is \((58+54+50+46)/4=52\) points. The Advanced students’ mean study time is \((6+7+8+9)/4=7.5\) hours and their mean score is \((92+88+84+80)/4=86\) points. The Advanced group is located farther to the right and higher on the graph.
Describe the pooled pattern accurately. If the course-level labels are ignored, the group centers are arranged in an upward direction: the group with higher study time also has higher scores. The combined data therefore show an overall positive linear association, even though the association within each course level is negative. The two groups form distinct clusters, and each cluster slopes downward.
Write the conclusion in context. “For these students, the pooled scatterplot shows a positive association between weekly study hours and quiz score, but the pattern within each course level is negative. The Advanced students have both higher study times and higher scores overall, so the pooled association does not describe the within-level patterns.”
This reversal warns against concluding that, within a course level, students who study more tend to score higher based on the combined plot. It also does not show that studying lowers scores within either level. The example describes associations in a small fictional data set; it does not establish why scores differ or how they would change if a student studied more.
The combined plot can be described, but it should not be treated as a complete summary when visible groups have different patterns. As in “Identifying Clusters and Gaps,” note the clusters and their locations. Then ask whether direction and strength are similar within each cluster. If a group-coded plot is available, use its labels rather than mentally treating every point as part of one undifferentiated cloud.
A Group Difference Can Create a Pattern
A second subgroup issue is that an association in the pooled data may mostly reflect separation between the groups, while neither group shows a clear association on its own. The following example uses deliberately small values to make the visual comparison clear. The point is not to calculate a correlation; it is to notice what changes when the group labels are restored.
Worked Example: Screen Time and Sleep in Two Age Bands
Consider a fictional set of observations of daily leisure screen time, in hours, and nightly sleep, in hours. The observations are separated into two age bands:
| Age band | Screen time, \(x\), hours | Sleep, \(y\), hours |
|---|---|---|
| 11–12 years | 1 | 8 |
| 11–12 years | 2 | 9 |
| 11–12 years | 3 | 8 |
| 11–12 years | 4 | 9 |
| 16–17 years | 5 | 6 |
| 16–17 years | 6 | 7 |
| 16–17 years | 7 | 6 |
| 16–17 years | 8 | 7 |
Look within each age band. For ages 11–12, sleep alternates between 8 and 9 hours as screen time increases; there is no clear consistent linear pattern in these four values. For ages 16–17, sleep alternates between 6 and 7 hours as screen time increases; again, there is no clear consistent linear pattern within the group.
Now consider the pooled observations. The younger group has screen times from 1 to 4 hours and sleep values from 8 to 9 hours. The older group has screen times from 5 to 8 hours and sleep values from 6 to 7 hours. Combining the groups produces an overall downward arrangement: observations with more screen time also tend to have fewer hours of sleep. Much of that separation occurs between the age bands, rather than as a clear pattern within either band.
Describe the limits. “In the combined data, daily leisure screen time and nightly sleep have a negative association, but there is little clear linear association within either age band in these observations. The pooled pattern is closely tied to differences between the age groups.” The data do not show that screen time caused the age-group difference or that increasing screen time would reduce a particular student’s sleep.
When groups are separated, an apparent pooled association can become weaker or less clear. That observation is useful: it tells you that the combined direction alone does not capture the whole structure. It does not prove that there is no relationship within any group in a larger population; it describes the pattern in the displayed data.
A Practical Check for Time and Subgroups
Before finalizing a scatterplot description, ask how the observations were collected and whether they have a meaningful order or group label. This check complements DUFS; it does not replace describing direction, unusual features, form, and strength.
As in “Writing Descriptions in Context,” name what each point represents and what the axes measure, including units.
If observations were recorded over time, identify the sequence and consider whether both variables change as time passes.
Check whether a categorical variable, such as age band or course level, distinguishes parts of the plot. Use color, symbols, or labels if available.
Describe direction and form within visible groups, then compare those patterns with the arrangement of all observations together.
State what the plot shows, but do not claim that the explanatory variable causes the response or that the same pattern applies within every group.
Common Mistakes and AP Exam Tips
The aim is not to invent a hidden explanation whenever a scatterplot has an association. Instead, use information supplied in the question or visible in the display to check whether time order or subgroup membership matters.
- Do not ignore group labels. If groups have different patterns, a full-credit description says what happens within them and how their combined arrangement differs.
- Do not describe only the group centers. In the course-level example, the groups are arranged upward overall, but each group slopes downward. Mentioning only the positive pooled pattern misses a central feature.
- Do not treat a time sequence as interchangeable observations. If measurements were made in order, say so and check whether both variables change over time. The scatterplot may not preserve that information.
- Do not claim that a lurking variable has been proven to cause the pattern. Time or group membership may help account for an association, but a plot alone does not establish a causal explanation.
- Do not assert that a subgroup has no association beyond the evidence. If a small group shows no clear pattern, say that there is no clear association in the displayed observations, rather than declaring that no relationship exists.
- Do not confuse a pooled description with a within-group description. Name which one you mean. “Overall in the combined data” and “within each age band” describe different views of the same observations.
A careful AP response is specific about the scope of its claim. For example: “The combined data show a positive association, but each course level shows a negative pattern; the higher-scoring Advanced group also tends to have more study hours.” This wording distinguishes the pooled association from the subgroup associations and avoids claiming that study time caused the scores.
Check Your Understanding
Use time order and subgroup membership to decide what a careful scatterplot description should include.
- Two measurements both rise across several consecutive months. What additional display or information could help you assess whether time contributes to their apparent association?
- In the study-hours example, what is the direction within each course level, and how does it compare with the pooled direction?
- Why is it incomplete to describe the screen-time and sleep data only as a negative association?
- A plot has two clusters marked with different symbols, but no group names are provided. What should you say about the clusters, and what should you avoid guessing?
- Write one cautious sentence describing a positive pooled association that reverses within two subgroups.