Positive and Negative Associations in Context
In “Describing Direction in a Scatterplot,” you learned to move from left to right across a scatterplot and notice whether the response tends to increase or decrease. Now practice applying that idea to familiar pairs of measurements. For example, as outdoor temperature rises, a household’s heating cost might tend to fall. That would be a negative association between temperature and heating cost.
The key is to describe how the two variables tend to move together. A positive association means that larger values of the explanatory variable tend to accompany larger values of the response variable. A negative association means that larger values of the explanatory variable tend to accompany smaller values of the response variable. These labels describe direction; they do not say whether a relationship is good or bad.
Keep the variable roles clear. As in “Explanatory and Response Variables in Scatterplots,” the explanatory variable goes on the horizontal axis, and the response variable goes on the vertical axis. Ask: As the explanatory variable increases, what tends to happen to the response? If the response tends to increase, the association is positive. If it tends to decrease, the association is negative.
The word “tends” matters. An association describes an overall pattern, not a rule that every individual observation must follow. One day with unusually high sales, or one month with an unexpectedly large heating bill, does not automatically reverse the direction. Look across the paired observations and identify the general tendency, as you practiced when describing scatterplots with DUFS.
A Simple Direction Check
When you have a scatterplot or a table of paired measurements, use this short check. First identify which variable is explanatory and which is the response. Then imagine moving from smaller to larger values of the explanatory variable. Finally, describe what the response tends to do. This keeps the classification tied to the variables instead of to a vague impression that the graph “goes up” or “goes down.”
Identify what is increasing and what is being observed in response. Include units and what each point represents when that information is provided.
Read across the scatterplot from left to right, or compare the paired values in order of the explanatory variable.
If the response generally rises, call the association positive. If it generally falls, call it negative. Do not require every point to follow the trend.
Name the two measurements and describe how they tend to change together. Avoid adding a cause or explanation that the data do not establish.
This classification is about the direction of the overall association. It is separate from strength, which describes how closely the points follow the pattern, and form, which describes whether the pattern is roughly straight or bends. Those features were introduced in “Judging Strength of an Association” and “Recognizing Linear and Nonlinear Form.” A relationship can be positive or negative whether it is strong or weak, and whether it is roughly linear or nonlinear.
Worked Example: Outdoor Temperature and Heating Cost
Suppose a fictional homeowner records the average outdoor temperature and monthly heating cost for the same home during eight months. Temperature is measured in degrees Celsius, and heating cost is measured in dollars. These invented observations are for practice, not a real study.
| Month | Average outdoor temperature (°C) | Heating cost (dollars) |
|---|---|---|
| A | -8 | 265 |
| B | -5 | 244 |
| C | -2 | 229 |
| D | 1 | 211 |
| E | 4 | 190 |
| F | 7 | 174 |
| G | 10 | 161 |
| H | 12 | 168 |
Identify the direction. The explanatory variable is average outdoor temperature, and the response variable is monthly heating cost. As temperature increases from the colder months to the warmer months, the heating cost generally decreases. The last month has a slightly higher cost than the month before it, but the overall tendency is downward.
Conclusion in context. For this home during the months recorded, outdoor temperature and monthly heating cost have a negative association: higher outdoor temperatures tend to go with lower heating costs.
The final observation is a useful reminder that a negative association does not require every increase in temperature to pair with a decrease in cost. A billing period, changes in household routines, or other conditions could affect one month. The direction describes the general pattern in the paired data, not a perfect step-by-step rule. The pattern alone also does not establish exactly why costs differ.
Positive Associations: The Response Tends to Rise
A positive association is not simply any graph that looks cheerful or any relationship that seems desirable. “Positive” has a specific statistical meaning: larger values of one variable tend to occur with larger values of the other. For example, if warmer days generally have higher cold-drink sales, temperature and sales show a positive association.
Some individual days might not follow that tendency. A rainy day could have lower sales than a slightly cooler day, for example. That does not necessarily change the classification if the overall pattern still shows higher sales at higher temperatures. As in “Spotting Outliers in Bivariate Data,” examine how a point fits the broader pattern rather than allowing one unusual observation to stand in for the whole plot.
Worked Example: Outdoor Temperature and Cold-Drink Sales
A fictional stand records the afternoon temperature and the number of cold drinks sold on eight days. Temperature is measured in degrees Celsius, and sales are counted in drinks. Each observation represents one day. The invented data include small departures from the overall pattern.
| Day | Afternoon temperature (°C) | Cold drinks sold |
|---|---|---|
| A | 17 | 31 |
| B | 19 | 36 |
| C | 20 | 34 |
| D | 22 | 43 |
| E | 23 | 45 |
| F | 25 | 42 |
| G | 27 | 55 |
| H | 29 | 61 |
Identify the direction. The explanatory variable is afternoon temperature, and the response is the number of cold drinks sold. In general, days with higher temperatures have more cold-drink sales. Sales dip from 19°C to 20°C and again from 23°C to 25°C, but these exceptions do not outweigh the overall upward tendency.
Conclusion in context. For the eight days recorded, afternoon temperature and the number of cold drinks sold have a positive association: higher temperatures tend to go with more drinks sold.
The conclusion describes an association in these observations. It does not establish that temperature caused the higher sales; for example, the days might differ in other ways. It also does not claim that every warmer day will have higher sales. “Tend to go with” communicates the overall direction without turning it into a guarantee.
Negative Associations: The Response Tends to Fall
A negative association occurs when larger values of the explanatory variable tend to pair with smaller values of the response. The word “negative” does not mean that one variable is bad or that the relationship is harmful. It identifies the direction in which the two measurements tend to move together.
For instance, a larger amount of weekly exercise might be associated with a lower resting pulse in a group of adults. A negative direction describes that pattern in the observations. Because the data are observational, it would be too strong to conclude from the association alone that exercise caused each person’s resting pulse to be lower.
Worked Example: Weekly Exercise and Resting Pulse
Imagine a fictional health class recording weekly exercise time and resting pulse for eight adults. Exercise time is measured in hours per week, and resting pulse is measured in beats per minute. Each pair of values belongs to one adult.
| Adult | Exercise time (hours per week) | Resting pulse (beats per minute) |
|---|---|---|
| A | 0.5 | 82 |
| B | 1.0 | 79 |
| C | 1.5 | 80 |
| D | 2.0 | 75 |
| E | 2.5 | 77 |
| F | 3.0 | 72 |
| G | 3.5 | 74 |
| H | 4.0 | 68 |
Identify the direction. The explanatory variable is weekly exercise time, and the response variable is resting pulse. As exercise time increases across these adults, resting pulse generally decreases. There are small increases between some neighboring observations, but the overall tendency is downward.
Conclusion in context. Among the eight adults observed, weekly exercise time and resting pulse have a negative association: adults reporting more hours of exercise per week tend to have lower resting pulse rates in beats per minute.
This describes the pattern in the observed adults; it does not show that increasing one individual’s exercise time will necessarily lower that person’s resting pulse. Other differences among adults could be related to both measurements. Keeping the conclusion descriptive is consistent with the caution about association and cause discussed in “Writing Descriptions in Context.”
Direction Is Not the Same as Meaning or Cause
The labels positive and negative describe how paired values tend to move, not whether the relationship is beneficial, harmful, or surprising. A positive association between temperature and cold-drink sales does not mean the association is “good.” A negative association between outdoor temperature and heating cost does not mean that either variable is undesirable. Use the words as statistical descriptions of direction.
Direction also does not tell you whether the association is strong or weak. If points are spread widely around an upward trend, the direction may still be positive even though the association is weak. If points lie close to a downward trend, the direction is negative and the association may be strong. As emphasized in “Describing a Scatterplot With DUFS,” a complete description can identify direction alongside strength and other features, but those are distinct parts of the description.
Finally, an association is not by itself a cause-and-effect conclusion. A scatterplot can show that two variables tend to change together, but the pattern alone does not rule out other variables or establish why the pattern occurs. State what the observations show and avoid supplying an explanation that the study does not support.
Common Mistakes and AP Exam Tips
- Calling a relationship positive because it is beneficial: “Positive” means that larger values of the explanatory variable tend to go with larger values of the response. It does not mean good, healthy, or desirable.
- Requiring every pair of observations to move in the same direction: Real data often include exceptions. Classify the overall tendency, and use “tends to” or “generally” rather than claiming that every observation follows it.
- Reversing the direction in words: Start with the explanatory variable and state what tends to happen to the response as it increases. For temperature and heating cost, warmer outdoor temperatures tend to go with lower heating costs—not the reverse.
- Using “the graph goes up” without naming the measurements: A full-credit contextual description identifies the variables and states how they are associated. For example, say “higher outdoor temperatures tend to go with lower monthly heating costs.”
- Confusing direction with strength or form: Positive and negative tell you the direction. They do not tell you how closely the points follow a pattern or whether the pattern is straight or curved.
- Claiming cause from an association: A negative or positive association alone does not show that changing one variable causes a change in the other. Describe the observed relationship without overstating what it proves.
For full credit, identify the two variables, name which one is the explanatory variable, and describe what tends to happen to the response as that variable increases. Finish by labeling the association positive or negative. If the observations do not show a clear upward or downward tendency, do not force either label; describe the pattern as having no clear direction instead.
Check Your Understanding
For each pair, decide whether the described overall tendency is positive or negative, and explain the direction in context.
- As the number of years a car has been used increases, its resale value in dollars tends to decrease. What is the direction of the association?
- As the number of pages read increases, the amount of time spent reading in minutes tends to increase. What is the direction of the association?
- For the heating-cost example, why does one month with a higher cost than the preceding month not necessarily change the overall classification?
- A scatterplot shows that larger values of a study-time variable tend to go with higher test scores, although several students do not follow the trend. How should the direction be described?
- In your own words, explain why a positive association does not necessarily mean that the relationship is beneficial or that one variable causes the other.