How Tightly Do the Points Follow the Pattern?
In “Recognizing Linear and Nonlinear Form,” you learned to identify whether a scatterplot has a roughly straight pattern, a curve, or clusters. Once you have described the form, you can ask a different question: how closely do the points follow that overall pattern? This is the strength of the association.
A strong association has points that stay relatively close to a clear pattern. A weak association has more scatter: knowing the explanatory variable gives less of a consistent indication of where the response values will fall. A moderate association lies between those descriptions. These are visual judgments, not categories determined by a universal cutoff.
To judge strength, first identify the overall pattern the points appear to follow. For a roughly linear pattern, look at how far the points tend to lie from the straight path. For a curved pattern, consider how closely the points trace the curve. The question is not whether every point lies exactly on the pattern—real data usually have some scatter—but whether the pattern remains clear despite that scatter.
Strength does not describe whether the pattern rises or falls. Direction answers that: as the explanatory variable increases, does the response tend to increase or decrease? A strong association may be positive or negative. Strength also does not describe form: a strong pattern can be linear or nonlinear. Keep these features separate when you describe a plot.
A Practical Way to Compare Strength
When comparing two or more scatterplots, focus on how consistently the points follow each plot’s overall pattern. If the plots have the same form and use comparable scales, this comparison is usually straightforward: the plot with the tighter, clearer band of points shows the stronger association. The plot with a more diffuse cloud shows the weaker association.
A useful check is to imagine the general path followed by the points, without drawing a precise line or curve. Then ask: do most points stay close to that path, or are they spread widely around it? Also ask whether the pattern is clear across the range of the explanatory variable, rather than being suggested by only a few observations.
Decide whether the pattern is roughly linear or curved, and whether it tends to rise or fall from left to right.
Notice whether the points stay close to the overall path or spread far away from it.
Check that the plots show comparable variables and scales. A visual comparison can be misleading if the axes or ranges differ substantially.
Use “weak,” “moderate,” or “strong” to describe the scatter around the pattern, then state the form and direction as appropriate.
Words such as “weak,” “moderate,” and “strong” summarize a visual judgment. There is no single AP Statistics rule that assigns every plot to one category based on a specific amount of spread. In a written response, support the label: say that the points lie close to the overall pattern, or that they show considerable scatter around it.
Worked Example: Comparing Three Positive Patterns
A fictional sports science class records weekly training hours, \(x\), and the number of successful repetitions in a timed drill, \(y\), for six athletes. These invented observations are shown below. Imagine plotting each set of paired values on the same axes.
| Athlete | Training hours, \(x\) | Set A, repetitions | Set B, repetitions | Set C, repetitions |
|---|---|---|---|---|
| 1 | 1 | 3 | 2 | 7 |
| 2 | 2 | 4 | 7 | 3 |
| 3 | 3 | 6 | 4 | 8 |
| 4 | 4 | 7 | 8 | 4 |
| 5 | 5 | 9 | 5 | 6 |
| 6 | 6 | 10 | 10 | 5 |
Set A. The repetitions generally increase as training hours increase, and the points stay close to a rising, roughly straight path. There is some variation, but the upward pattern remains clear. This is a strong positive linear association.
Set B. The overall tendency is also upward: the values at the lower training hours are generally lower than those at the higher hours. However, the points fall farther from a straight rising path than in Set A. The association is positive but more scattered, so it is reasonable to describe it as moderate.
Set C. The response values move up and down as training hours increase, without a clear overall rising or falling path. The points do not closely follow a consistent pattern. The association is weak, or there may be no apparent association in this small plot.
Compare the sets. With the same explanatory-variable values and comparable axes, the visual ranking is Set A as strongest, Set B as intermediate, and Set C as weakest. The comparison is about how tightly each set of points follows a pattern—not about the size of the response values themselves.
Describe the first two in context. One suitable statement is: “Training hours and repetitions show a strong positive linear association in Set A; the association is still positive in Set B but is more moderate because its points are more scattered around the upward pattern.” This gives both the direction and the comparative strength.
Strength Is Not Direction, Steepness, or Form
Some features of a scatterplot can look important but do not, by themselves, tell you how strong an association is. A steep upward pattern is not automatically stronger than a shallow upward pattern. Steepness describes how much the response changes for a given change in the explanatory variable; strength describes how consistently the points follow the pattern. Likewise, a downward pattern is not weaker just because it is negative.
The form matters to the description, but it is distinct from strength. If the points follow a curve closely, the association can be strong even though it is not linear. As noted in the earlier tutorial on form, a curved pattern should not be treated as no association just because it does not follow a straight path. Look at how tightly the points trace the curve.
When plots use very different axis ranges, be cautious about comparing apparent scatter. A narrow-looking cloud can look broad if the graph is stretched in one direction; the same points can look different under different scales. Before making a comparison, read the axes and consider how much of each variable’s displayed range the scatter occupies. When possible, compare plots shown with consistent scales.
Worked Example: A Strong Negative Pattern and a Moderate Positive Pattern
A fictional community garden records the number of hours of afternoon shade and the number of seedlings flowering in a week for two different garden beds. The measurements are invented. Each set uses the same axes: shade hours on the horizontal axis and flowering seedlings on the vertical axis.
| Observation | Shade hours | Bed A: flowering seedlings | Bed B: flowering seedlings |
|---|---|---|---|
| 1 | 1 | 18 | 16 |
| 2 | 2 | 16 | 19 |
| 3 | 3 | 13 | 13 |
| 4 | 4 | 11 | 17 |
| 5 | 5 | 8 | 12 |
| 6 | 6 | 6 | 15 |
Assess Bed A. As shade hours increase, the number of flowering seedlings generally decreases. The values stay near a descending straight path, with relatively little scatter. This is a strong negative linear association.
Assess Bed B. There is a general tendency for the number of flowering seedlings to be lower at greater shade hours, but the values vary more around that downward tendency. This is a negative association with more scatter than Bed A’s, so a moderate description is reasonable.
Compare without confusing direction and strength. Bed A shows the stronger association even though its direction is negative. Bed B’s direction is also negative, but its points do not follow the downward pattern as consistently. A suitable comparison is: “Both beds show negative associations between shade hours and flowering seedlings, but Bed A’s association is stronger because its points lie more tightly around a descending linear pattern.”
The comparison does not establish that shade caused the differences in flowering. The plots describe the observed associations; other features of the beds could also be related to flowering.
Worked Example: Strength Around a Curved Pattern
A fictional technician records the air temperature, \(x\), in degrees Celsius and the time, \(y\), in minutes, required for a cooling device to reach its target. The invented values form a U-shaped pattern.
| Trial | Temperature, \(x\) (°C) | Cooling time, \(y\) (minutes) |
|---|---|---|
| 1 | 0 | 20 |
| 2 | 5 | 12 |
| 3 | 10 | 7 |
| 4 | 15 | 5 |
| 5 | 20 | 8 |
| 6 | 25 | 13 |
| 7 | 30 | 21 |
Describe form and direction. Cooling time decreases as temperature rises toward 15°C, then increases as temperature rises beyond about 15°C. The overall form is curved, so describing the entire plot with only one direction would leave out the change in behavior.
Judge strength around the right pattern. The points lie close to a clear U-shaped path. The association is strong in the visual sense: the curved pattern is consistent, even though the pattern is not linear. Strength is judged around the overall curve, not around an imagined straight line.
Write the conclusion carefully. “The scatterplot shows a strong curved association between temperature and cooling time: cooling time falls and then rises as temperature increases.” This sentence names the strength and form, and explains the changing direction in context. It does not incorrectly call the pattern linear or say there is no association.
Common Mistakes and AP Exam Tips
- Using direction as a strength label: “Positive” and “negative” describe whether the pattern rises or falls, not whether it is strong. State direction and strength separately.
- Assuming steep means strong: Steepness is not tightness. A shallow pattern can be strong if the points stay close to it; a steep pattern can be weak if the points are widely scattered.
- Calling a curved pattern weak because it is not straight: First recognize form, then judge how closely the points follow that form. A clear curve can show a strong association.
- Giving a label without visual evidence: “The association is moderate” is less informative than “The association is moderate because the points show a clear upward tendency but are fairly scattered around it.”
- Comparing plots with different scales as if they were identical: Read the axes and check the displayed ranges. Different scales can change the visual impression of spread.
- Claiming causation from strength: A strong association means the plotted variables follow a consistent pattern; it does not, by itself, show that one variable causes changes in the other.
A full-credit AP description typically identifies the direction and form when they are visible, then supports its strength judgment with how tightly the points follow the pattern. For example: “There is a strong negative linear association because the points lie close to a downward-sloping straight pattern.” Avoid treating the label alone as the explanation.
Check Your Understanding
Use the scatterplot descriptions to judge strength and explain your reasoning.
- Two plots show positive, roughly linear patterns on the same scales. In Plot A, points stay close to an upward path; in Plot B, they are much more spread out. Which association is stronger, and what visual feature supports your answer?
- A plot shows a clear downward trend, with points close to a straight path. State its direction and strength separately.
- A scatterplot has a clear U-shaped pattern, and the points stay close to the curve. How should you describe its form and strength?
- Why is it incorrect to call a steep pattern strong without considering how scattered its points are?
- What should you check before comparing the apparent strength of two scatterplots displayed with different axis scales?