Which Way Does the Pattern Go?
In “Making a Scatterplot on the TI-84,” you learned to place each observational unit’s paired values at a point \((x,y)\), with the explanatory variable on the horizontal axis and the response variable on the vertical axis. Once the plot is made, a first question is: as \(x\) increases, what tends to happen to \(y\)?
The answer describes the direction of the association. A scatterplot may show a positive association, a negative association, or no apparent association. Describe what the plotted values tend to do; do not suggest that every point must follow the same pattern.
Positive and Negative Association
In a scatterplot, move your attention from left to right: the explanatory-variable values increase in that direction. If the points generally rise as you move right, the association is positive. If they generally fall, the association is negative. The points do not need to lie exactly on a straight path. Direction describes the overall tendency, not a rule that each successive point must obey.
A useful description connects the direction to the situation. Name the explanatory and response variables, and state how the response tends to change as the explanatory variable increases. Include units when they help make the meaning clear. For example, “As daily practice time increases, a player’s serve speed tends to increase” is more informative than “The graph goes up.”
Direction is not the same as strength or form. A plot can rise while its points are tightly grouped or widely scattered around that upward tendency. Those are different features to describe. In this tutorial, focus on whether the response tends to rise, fall, or show no clear pattern as you move from lower to higher explanatory-variable values.
When Is There No Apparent Association?
Use “no apparent association” when the plotted points show no clear pattern. In such a plot, knowing whether an observation has a relatively low or high \(x\)-value does not reveal a consistent tendency for its \(y\)-value to be low, high, or to change in an orderly way. This is a description of what the displayed data show, not proof that the variables can never be related.
Be careful not to use “no association” just because the response does not steadily increase or steadily decrease across the whole graph. A plot might show a clear pattern that changes direction: for example, the response may fall over part of the \(x\)-range and then rise. Such data show an association, even though there is no single overall positive or negative direction. Describe the visible pattern accurately rather than labeling it “no association.”
A quick check is to trace the plotted pattern from left to right. If the points show a clear overall upward tendency, describe a positive association. If they show a clear overall downward tendency, describe a negative association. If they show no clear pattern, describe no apparent association. If they form a clear pattern that changes direction, say that there is an association with no single overall positive or negative direction.
Identify the explanatory variable on the horizontal axis and the response variable on the vertical axis.
As the explanatory-variable values increase, look at what the response values tend to do.
Say positive, negative, or no apparent association when those descriptions fit; if a clear pattern changes direction, describe that instead of calling it no association.
Name the variables and describe the response’s tendency as the explanatory variable increases.
Worked Example: Study Time and Quiz Score
A fictional teacher records the amount of time students spend reviewing a set of practice questions and their scores on a related quiz. Review time is the explanatory variable, measured in minutes; quiz score is the response variable, measured in points out of 20.
| Student | Review time (minutes) | Quiz score (points) |
|---|---|---|
| A | 10 | 8 |
| B | 15 | 10 |
| C | 20 | 11 |
| D | 25 | 13 |
| E | 30 | 14 |
| F | 35 | 17 |
Read from left to right. As review time goes from 10 to 35 minutes, the quiz scores generally rise from 8 to 17 points. In this small set of observations, the pattern has a positive direction.
Describe it in context. A suitable description is: “Among these students, longer review time tends to be associated with higher quiz scores.” The word “tends” is important: it describes the overall pattern without claiming that every student who reviews longer must score higher.
Keep the conclusion within the evidence. The scatterplot describes an association in the recorded data. It does not establish that additional review time caused higher scores; other differences among students could also be involved. The direction statement should not go beyond what the plot shows.
Worked Example: Outdoor Temperature and Hot-Drink Sales
A fictional café tracks the afternoon outdoor temperature and the number of hot drinks sold during that afternoon. Temperature is the explanatory variable, in degrees Celsius; hot-drink sales are the response variable, in cups.
| Afternoon | Temperature (°C) | Hot drinks sold (cups) |
|---|---|---|
| 1 | 4 | 86 |
| 2 | 7 | 79 |
| 3 | 10 | 72 |
| 4 | 13 | 65 |
| 5 | 16 | 61 |
| 6 | 19 | 53 |
Read from left to right. As temperature increases from 4°C to 19°C, the number of hot drinks sold generally decreases from 86 cups to 53 cups. The overall direction is negative.
Describe it in context. A suitable statement is: “In these recorded afternoons, higher outdoor temperatures tend to be associated with fewer hot drinks sold.” This says which variable tends to go down as the other rises and uses the context of the data.
Avoid an absolute claim. Do not say that every warmer afternoon must have lower hot-drink sales. The description concerns the general direction of the plotted observations. Also, the plot alone does not prove that temperature caused the change in sales.
Worked Example: A Plot with No Apparent Association
A fictional community garden records the number of minutes a volunteer spends watering a particular plot and the number of tomatoes harvested from it that week. Watering time is the explanatory variable, in minutes; tomato harvest is the response variable, in tomatoes.
| Garden plot | Watering time (minutes) | Tomatoes harvested |
|---|---|---|
| A | 12 | 18 |
| B | 18 | 11 |
| C | 23 | 21 |
| D | 29 | 14 |
| E | 35 | 19 |
| F | 41 | 12 |
Check for a pattern, not just a direction. As watering time increases, harvest counts go up and down, with no clear overall tendency to rise or fall. These values do not show an obvious pattern relating the two variables.
Describe what is visible. A suitable statement is: “For these garden plots, the scatterplot shows no apparent association between watering time and the number of tomatoes harvested.” This does not mean that the variables must be unrelated in every garden or under every condition. It reports that this set of points shows no clear pattern.
Distinguish irregularity from a changing pattern. The response values here fluctuate without a clear structure. If instead a plot showed a clear, organized pattern that first fell and then rose as watering time increased, it would show an association, even though it would not have one overall positive or negative direction. The correct description depends on the pattern, not merely on the absence of a steady rise or fall.
Worked Example: A Clear Pattern with No Single Direction
A fictional robotics club compares a robot’s programmed travel distance with the time needed to complete a course. Distance is the explanatory variable, in meters; completion time is the response variable, in seconds. The following measurements are invented for practice.
| Program | Travel distance (meters) | Completion time (seconds) |
|---|---|---|
| A | 1 | 14 |
| B | 2 | 9 |
| C | 3 | 6 |
| D | 4 | 5 |
| E | 5 | 7 |
| F | 6 | 11 |
| G | 7 | 17 |
Trace the pattern across the range. Completion time first decreases as travel distance increases, reaching 5 seconds at 4 meters. For the larger distances shown, completion time increases. So it would not be accurate to summarize the whole plot as simply positive or simply negative.
Do not call it no association. The values show a clear pattern: completion times are lower at middle distances and higher toward the ends of the displayed range. A suitable description is: “There is an association between programmed travel distance and completion time, but it has no single overall positive or negative direction across these observations.” The key distinction is that the pattern is visible, even though it changes direction.
Common Mistakes and AP Exam Tips
- Describing the axes instead of the association: “The \(x\)-values are between 10 and 35” does not describe direction. State how the response tends to change as the explanatory variable increases.
- Reversing the direction: Read the graph from left to right, not from top to bottom. A positive association rises as \(x\) increases; a negative association falls as \(x\) increases.
- Claiming that every point follows the trend: Direction is an overall tendency. Use “tends to increase” or “tends to decrease,” not “always increases” or “always decreases,” unless every displayed point actually supports that claim.
- Calling every non-monotone pattern no association: A plot with a clear pattern that changes direction still shows an association. “No apparent association” is for a plot with no clear pattern.
- Using causal language without support: Saying one variable “causes” the other goes beyond describing a scatterplot. Prefer “is associated with” unless the study design supports a causal conclusion.
- Leaving out context: “There is a negative association” may identify direction, but a stronger AP response names the variables: for example, “Higher outdoor temperatures tend to be associated with fewer hot drinks sold.”
For a concise, full-credit direction statement, identify the variables and describe the response’s tendency as the explanatory variable increases. If there is no clear pattern, say “no apparent association.” If the pattern is clear but changes direction, state that it is an association without one overall positive or negative direction.
Check Your Understanding
For each situation, focus on the direction or pattern and express your conclusion in context.
- A scatterplot compares weekly cycling time and distance traveled. As cycling time increases, the plotted distances generally increase. Write a direction statement that names both variables.
- A scatterplot compares outdoor temperature and the number of mittens sold at a fictional shop. The points generally fall as temperature increases. What is the direction, and how would you describe it in context?
- A scatterplot shows points scattered without a clear pattern as the explanatory-variable values increase. What phrase describes the plot, and what does that phrase not prove?
- A plot shows the response decreasing over the first part of the explanatory-variable range and then increasing over the rest, in a clear pattern. Should you call this no apparent association? Explain.
- Why is “as practice time increases, test scores always increase” usually less careful than “as practice time increases, test scores tend to increase”?