10.1Scatter Plots & Association
Create and interpret scatter plots and describe associations between two variables.
Why This Matters
Scatter plots reveal relationships between two numerical variables. They are used in science, education, business, and social research to recognize trends, compare data, and make informed predictions.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How can a scatter plot reveal the relationship between two numerical variables?
Lesson Overview
A scatter plot is a graph that displays the relationship between two numerical variables. Each data point is plotted as an ordered pair (x, y) on a coordinate grid. The explanatory variable (independent) is placed on the x-axis, and the response variable (dependent) is placed on the y-axis. After plotting the points, we look for a pattern — called an association — that describes how the two variables relate to each other. Associations can be described by their direction (positive, negative, or none), form (linear or nonlinear), and strength (strong or weak). We also look for clusters, gaps, and outliers that affect the overall pattern.
How to Describe an Association — 4-Step Method
| Step | Feature | Options | What to Look For |
|---|---|---|---|
| 1 | Direction | Positive, Negative, None | Does the pattern go up, down, or neither as x increases? |
| 2 | Form | Linear, Nonlinear | Do the points follow a straight line or a curve? |
| 3 | Strength | Strong, Weak | Are the points tightly clustered or widely spread? |
| 4 | Special Features | Clusters, Gaps, Outliers | Are there groups, empty regions, or unusual points? |
Positive Linear
As x↑, y↑ · tight cluster
Negative Linear
As x↑, y↓ · tight cluster
No Association
No trend · random scatter
Strong Association
Points hug the line closely
Weak Association
Points spread far from line
Nonlinear Association
Curved pattern · not a line
Worked Examples
A student records the number of hours studied and the test score for 6 students: (1, 55), (2, 62), (3, 70), (4, 75), (5, 85), (6, 92). Identify the explanatory and response variables, then describe the association.
Explanatory variable (x-axis): hours studied — this is the variable that may explain the score.
Response variable (y-axis): test score — this responds to the number of hours studied.
Plot the ordered pairs on a coordinate grid.
Direction: as hours increase, scores increase → positive association.
Form: the points follow an approximately straight-line pattern → linear.
Strength: the points are clustered fairly closely → strong.
A scatter plot shows the relationship between outdoor temperature (°F) and the number of hot chocolates sold at a café. As temperature increases, hot chocolate sales decrease. The points are spread moderately around a line. Describe the association.
Direction: as temperature increases, sales decrease → negative association.
Form: the points follow an approximately straight-line pattern → linear.
Strength: the points are moderately spread → moderate (or weak-to-moderate) association.
A scatter plot shows the relationship between a person's shoe size and their IQ score. The points appear randomly scattered with no visible trend. Describe the association.
Direction: no upward or downward trend is visible.
Form: no recognizable pattern.
Strength: not applicable — there is no association.
A scatter plot of age (years) vs. height (cm) for children ages 2–10 shows points that curve upward steeply at first, then level off. Describe the association.
Direction: as age increases, height increases → positive association.
Form: the points follow a curved pattern, not a straight line → nonlinear.
Strength: the points are close to the curve → strong.
A scatter plot shows the following data: most points follow a strong positive linear trend, but one point at (8, 20) is far below the rest of the pattern. Describe the scatter plot fully.
Overall pattern: strong, positive, linear association.
Identify the unusual point: (8, 20) is far below the trend — this is an outlier.
Note: the outlier does not represent the overall pattern.
Guided Practice
Guided Practice Video: Scatter Plots and Association
Review how to construct and interpret scatter plots — including identifying association direction, strength, and form — before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗A scatter plot shows the relationship between the number of absences and final exam grade. As absences increase, grades decrease. The points are tightly clustered around a line. Describe the association.
Hint: Identify direction (up or down?), form (straight or curved?), and strength (tight or spread?).
A researcher plots hours of TV watched per day (x) and GPA (y). Identify which variable is explanatory and which is the response variable. Explain your reasoning.
Hint: The explanatory variable is the one that may cause or explain changes in the other. Which variable might influence the other?
Plot the following data on a coordinate grid and describe the association: (1, 8), (2, 6), (3, 5), (4, 3), (5, 2), (6, 1).
Hint: Plot each ordered pair. Then look at the overall direction of the points from left to right.
A scatter plot shows data points that are spread widely and randomly across the graph with no visible pattern. What type of association does this represent?
Hint: If there is no upward or downward trend, what do we call this type of association?
A scatter plot shows a strong positive linear association, but one point at (10, 2) is far below all the others. What is this point called, and how should it be described?
Hint: A point that is far from the overall pattern has a specific name. Does it change the overall description of the association?
Key Vocabulary
Scatter Plot
A graph that shows the relationship between two numerical variables by plotting ordered pairs on a coordinate grid.
Explanatory Variable
The independent variable (x-axis). It is the variable that may explain or cause changes in the other variable.
Response Variable
The dependent variable (y-axis). It is the variable that responds to or is affected by the explanatory variable.
Association
A relationship or pattern between two variables shown in a scatter plot. Described by direction, form, and strength.
Positive Association
As the x-values increase, the y-values also increase. The data points trend upward from left to right.
Negative Association
As the x-values increase, the y-values decrease. The data points trend downward from left to right.
No Association
There is no visible pattern between the two variables. The data points appear scattered randomly.
Linear Association
The data points follow an approximately straight-line pattern.
Nonlinear Association
The data points follow a curved or other non-straight pattern.
Strong Association
The data points are clustered closely around the trend line or curve.
Weak Association
The data points are spread far from the trend line or curve.
Cluster
A group of data points that are close together, separated from the rest of the data.
Gap
A region in the scatter plot where there are no data points.
Outlier
A data point that is far from the overall pattern of the scatter plot.
Interactive Practice — 5 Questions
A scatter plot shows that as x increases, y also increases. The points are tightly clustered near a line. This is a:
Which scatter plot shows NO association?
A scatter plot shows temperature (x) vs. hot chocolate sales (y). As temperature increases, sales decrease. This is a:
Which variable is plotted on the x-axis of a scatter plot?
A scatter plot of shoe size vs. reading ability in adults shows no pattern. What can you conclude?
Independent Practice
Independent Practice
Create a scatter plot for the data: (1,3), (2,5), (3,4), (4,7), (5,8), (6,9). Describe the association (direction, form, strength).
A scatter plot shows hours of TV watched (x) vs. GPA (y). As TV hours increase, GPA decreases. Points are moderately clustered. Describe the association.
Identify the association for each: (a) shoe size vs. reading ability in adults; (b) temperature vs. hot chocolate sales; (c) study hours vs. test score.
A scatter plot shows no clear pattern between a student's birth month and their math grade. What type of association is this? Explain.
Error Analysis: A student says "The scatter plot shows a positive association, so more exercise causes higher test scores." Identify two errors in this statement.
Common Mistakes
Confusing the explanatory and response variables — plotting them on the wrong axes.
The explanatory variable (cause/independent) goes on the x-axis. The response variable (effect/dependent) goes on the y-axis.
Connecting the dots on a scatter plot — treating it like a line graph.
Scatter plots show individual data points. Do not connect them. Each point is independent.
Describing a scatter plot as 'random' when there is a visible trend.
Look carefully for a direction (positive or negative) and shape (linear or curved) before concluding there is no association.
Forgetting to describe form and strength — only saying 'positive' or 'negative'.
A complete description includes direction, form (linear/nonlinear), and strength (strong/weak).
Math Tips
Always label your axes with the variable name and units when creating a scatter plot.
The explanatory variable goes on the x-axis. Think: x explains, y responds.
Do not connect the dots. Scatter plots show individual data points — they are not line graphs.
Use all three features (direction, form, strength) when describing an association in context.
An outlier does not define the pattern. Describe the overall trend first, then note the outlier separately.
No association ≠ no relationship. There may be a nonlinear relationship even when there is no linear pattern.