Look at the Shape Before Summarizing It
In “Describing Direction in a Scatterplot,” you learned to move from left to right and describe how the response tends to change as the explanatory variable increases. A related question is about form: what overall shape do the points make? The points may follow a roughly straight pattern, trace a curve, or gather into distinct groups called clusters.
Form matters because a numerical summary can compress many points into one number and hide the shape that is visible in the plot. In particular, a numerical summary designed for straight-line association may not describe a curved pattern well. Before calculating or interpreting a summary, inspect the scatterplot and identify its form.
Linear Form: A Roughly Straight Pattern
A scatterplot has linear form when its points tend to lie around a straight path. In real data, they usually do not fall exactly on a line. The question is whether a straight path gives a reasonable description of the overall shape, allowing for some scatter around it.
Linear form is not the same as positive or negative direction. Form describes the shape; direction describes whether the response generally rises or falls as the explanatory variable increases. A roughly straight pattern can slope upward or downward. It can also be weak or strong, depending on how closely the points follow the straight path. Those are separate features to describe.
When a plot is roughly linear, a numerical summary designed to describe straight-line association may be useful. But first check the graph: a single number cannot tell you whether the points follow a straight path or a different shape.
Nonlinear Form: A Systematic Curve
A scatterplot has nonlinear form when its points follow a clear pattern that bends rather than staying around a straight path. The curve may bend upward or downward, or it may change direction. A curved pattern is still an association; it is not the same as random scatter or no apparent association.
A changing direction is one clue to look for. For example, the response might decrease over part of the explanatory-variable range and then increase. Calling such a pattern simply positive or negative would leave out its shape. As in the earlier tutorial on direction, describe what the response tends to do across the range, but now explain that the pattern is curved.
Clusters: Separate Groups Within the Plot
A cluster is a group of points that lies relatively close together and is separated from another group or from the rest of the data. Clusters may appear as separate clouds, bands, or concentrations of points. They are a feature of the plotted data that should be described, not automatically treated as stray points to ignore.
When you see clusters, look for information about the observational units. The groups might correspond to different varieties, locations, age groups, machines, or other categories. If a relevant group label is available, compare the patterns within groups as well as the combined pattern. A single summary of all the points may conceal differences between groups or suggest an overall pattern that does not describe either group well.
A plot can have more than one feature. For instance, separate clusters might each have a roughly straight pattern, while the overall collection has gaps between groups. A careful description names both the clustering and any within-group pattern that is visible.
A Reliable Order for Describing Form
Recall which variable is explanatory and which is the response, as in “Explanatory and Response Variables in Scatterplots.”
Decide whether the points follow a roughly straight path, a systematic curve, or distinct clusters. Do not decide from a numerical summary alone.
Note direction, changes in direction, gaps, and groups when relevant. Keep these descriptions separate from the form label.
A summary of linear association cannot, by itself, reveal curvature or explain why separate groups occur.
Worked Example: Practice Sessions and Endurance
A fictional coach records the number of practice sessions completed by each athlete and the time each athlete can maintain a steady cycling pace. Practice sessions are the explanatory variable, measured in sessions; endurance time is the response variable, measured in minutes. The invented observations are shown below.
| Athlete | Practice sessions | Endurance time (minutes) |
|---|---|---|
| A | 1 | 4 |
| B | 2 | 5 |
| C | 3 | 7 |
| D | 4 | 8 |
| E | 5 | 10 |
| F | 6 | 11 |
Assess the form. From left to right, the endurance times generally rise, and the plotted points would lie near an upward straight path. They are not all exactly on one line, but there is no obvious bend or separation into groups. The form is roughly linear.
Describe it in context. A suitable description is: “Among these athletes, more practice sessions tend to be associated with longer endurance times, in a roughly linear pattern.” This statement names the variables, gives the direction, and describes the form without claiming that every athlete follows the same pattern.
Keep the conclusion limited. These observations show an association in the recorded data. The scatterplot alone does not establish that practice sessions caused the longer endurance times. Other differences among athletes could be involved.
Worked Example: Outdoor Temperature and Energy Use
A fictional building manager records outdoor temperature and the building’s daily energy use. Temperature is the explanatory variable, in degrees Celsius; energy use is the response variable, in kilowatt-hours. These invented values form a clear curve: energy use is higher at the cold and warm ends of the temperature range and lower around the middle.
| Day | Temperature (°C) | Energy use (kWh) |
|---|---|---|
| 1 | 0 | 42 |
| 2 | 5 | 32 |
| 3 | 10 | 24 |
| 4 | 15 | 20 |
| 5 | 20 | 22 |
| 6 | 25 | 30 |
| 7 | 30 | 42 |
Judge the form first. Energy use falls as temperature increases from 0°C to about 15°C, then rises as temperature increases further. The points trace a clear U-shaped pattern. This is nonlinear form, not a roughly straight pattern and not a plot with no apparent association.
Describe the pattern in context. A suitable description is: “Daily energy use is highest at the colder and warmer temperatures in these observations and lowest around 15°C; the association between temperature and energy use is curved.” This explains the change in direction instead of forcing the whole plot into one positive or negative description.
See why a number can mislead. The correlation \(r\) is a numerical summary of linear association. For these seven points, the means are \(\bar{x}=15\) and \(\bar{y}=212/7\). The sum of the paired centered products is \(-30\), not zero. For example, the pairs at 5°C and 25°C contribute \(-20\), those at 10°C and 20°C contribute \(-10\), and the pair at 0°C and 30°C contributes 0. The sum of squared deviations for temperature is 700, and the sum for energy use is \(3440/7\). Therefore:
The correlation is close to zero, but it is not exactly zero. More importantly, the plot has a clear U-shaped pattern. A value of \(r\) close to zero does not show that these variables have no association; it indicates that their overall linear association is close to zero. The scatterplot reveals the curvature that the single number does not describe.
Worked Example: Seedling Growth in Two Varieties
A fictional greenhouse compares nutrient concentration with seedling height for two plant varieties. Nutrient concentration is the explanatory variable, measured in units per liter; height is the response variable, measured in centimeters. The invented observations are labeled by variety so that possible clusters can be seen.
| Variety | Nutrient concentration | Height (cm) |
|---|---|---|
| A | 1 | 8 |
| A | 2 | 9 |
| A | 3 | 11 |
| A | 4 | 12 |
| B | 7 | 20 |
| B | 8 | 22 |
| B | 9 | 23 |
| B | 10 | 25 |
Identify the groups. The four Variety A observations form a lower-left group, while the four Variety B observations form an upper-right group. There is a noticeable gap between the groups in both concentration and height. The combined plot is clustered.
Look within the groups. Within each variety, the points tend to rise as concentration increases, and each group appears to have a roughly straight upward pattern. The overall plot therefore has distinct clusters and visible within-group positive patterns. Saying only “the points form a straight increasing pattern” would omit the separation between varieties.
Explain why the labels matter. Variety is a plausible feature to consider because the clusters line up with the variety labels. The plot does not establish why the groups differ, but it does show that describing all eight observations as if they formed one undifferentiated cloud would lose useful information. A careful analysis reports the clusters and considers the groups separately before relying on one overall summary.
Common Mistakes and AP Exam Tips
- Calling every association linear: A general rise or fall does not guarantee a straight form. Check whether the points bend systematically. Full-credit wording identifies the curve when one is visible.
- Calling a curve “no association” because it changes direction: A U-shaped pattern is structured, not random. Describe its bend and the response’s change across the explanatory-variable range.
- Treating a correlation near zero as proof of no relationship: Correlation summarizes linear association and can be close to zero when a strong curved pattern bends in opposite directions over the range. Refer to the scatterplot’s shape.
- Ignoring clusters: Separate clouds may correspond to meaningful categories. Mention the clusters and, when labels are given, describe what happens within each group as well as across the full plot.
- Mixing form with direction or strength: “Linear” describes shape, “positive” or “negative” describes direction, and strength describes how closely points follow a pattern. A complete description keeps these features distinct.
- Claiming that a visual pattern proves causation: A scatterplot shows association. Unless the study design supports a causal conclusion, do not say that one variable caused the other to change.
On an AP response, name the visible form and support it with what the points do. For example: “The scatterplot shows a curved association: energy use falls and then rises as temperature increases.” If there are separate groups, name them. Do not let a numerical summary replace that description.
Check Your Understanding
For each situation, identify the form and explain what the plot supports.
- A scatterplot of hours of daylight and daily electricity use falls and then rises in a clear U shape. What is its form, and why would “no apparent association” be misleading?
- A plot of distance traveled and travel time has points scattered around an upward straight path. Describe its form and direction separately.
- A scatterplot contains two separate clouds, each labeled with a different equipment model. What feature should you report, and what should you examine within each group?
- For a clear curved pattern, a numerical summary of linear association is close to zero. What can you conclude from that number, and what must you check in the plot?
- Why should you describe clusters or curvature before relying on a single numerical summary?