Mathematics · 9th Grade · Statistical Reasoning and Data · Weeks 10-18

Scatter Plots and Correlation

Creating and interpreting scatter plots to visualize relationships between two quantitative variables.

• Common Core State Standards
• CCSS.Math.Content.HSS.ID.B.6
• CCSS.Math.Content.HSS.ID.C.7

About This Topic

Scatter plots give students their first graphical tool for exploring relationships between two quantitative variables. Each point represents an individual case with two measured attributes, and the overall pattern of points reveals whether and how the variables relate. In 9th grade, students describe correlation qualitatively (positive, negative, none) and connect the visual pattern to the correlation coefficient r, which is introduced or previewed depending on the specific course sequence.

The crucial conceptual move at this level is separating the pattern in the plot from what that pattern means causally. Strong correlation means the variables move together, but it does not explain why. Students naturally seek causal explanations, so this topic provides a valuable opportunity to build the skepticism about data claims that appears regularly in US news media.

Active learning is especially effective here because scatter plot interpretation requires calibrated judgment. Students looking at the same plot often describe the pattern differently, and structured peer discussion where they must agree on a description before moving forward builds the precise vocabulary and perceptual accuracy that later statistical work requires.

Key Questions

1. Analyze what the pattern of points on a scatter plot reveals about the relationship between variables.
2. Differentiate between positive, negative, and no correlation.
3. Explain why correlation does not imply causation.

Learning Objectives

• Create scatter plots to visually represent the relationship between two quantitative variables from a given dataset.
• Analyze the pattern of points on a scatter plot to describe the direction and strength of the relationship between variables.
• Differentiate between positive, negative, and no correlation based on the visual distribution of points on a scatter plot.
• Explain why a strong correlation between two variables does not necessarily imply a causal relationship, using a concrete example.

Before You Start

Coordinate Plane

Why: Students need to be able to plot and interpret points in a two-dimensional coordinate system to create scatter plots.

Quantitative Data Representation

Why: Students should have experience working with numerical data and understanding what different values represent.

Key Vocabulary

Scatter Plot	A graph that displays the relationship between two quantitative variables. Each point on the plot represents a pair of values for the two variables.
Correlation	A statistical measure that describes the extent to which two variables change together. It indicates the direction and strength of a linear relationship.
Positive Correlation	A relationship where as one variable increases, the other variable also tends to increase. Points on the scatter plot generally rise from left to right.
Negative Correlation	A relationship where as one variable increases, the other variable tends to decrease. Points on the scatter plot generally fall from left to right.
No Correlation	A relationship where there is no discernible pattern between the two variables. Points on the scatter plot appear randomly scattered.

Watch Out for These Misconceptions

Common Misconception

A correlation of 0 means the two variables have no relationship at all.

What to Teach Instead

A correlation of 0 means there is no linear relationship. Two variables can have a strong curved relationship and still produce an r value near 0. Showing a U-shaped scatter plot where points clearly follow a pattern but r is approximately 0 is the most effective correction for this persistent misunderstanding.

Common Misconception

The correlation coefficient r tells you how steep the line of best fit will be.

What to Teach Instead

r describes the strength and direction of the linear relationship, not its steepness. Two scatter plots can have the same r but very different slopes, depending on the scales of the variables. Comparing two such plots side by side in a partner activity makes this distinction concrete.

Common Misconception

Positive correlation means both variables always have large values at the same time.

What to Teach Instead

Positive correlation means that as one variable increases, the other tends to increase too, regardless of the absolute magnitude of either variable. Both variables could be small in value and still show strong positive correlation. Using scatter plots with small-scale data helps students focus on the direction of change rather than the size of the numbers.

Active Learning Ideas

Inquiry Circle: Create and Interpret a Real-Data Scatter Plot

Provide groups with a dataset of two quantitative variables such as hours of sleep versus average GPA or temperature versus energy use across US cities. Groups create the scatter plot, describe the form, direction, and strength of the association in complete sentences, and identify any points that appear to be outliers with a written justification.

35 min·Small Groups

Think-Pair-Share: Describe What You See

Show a scatter plot without labels or context. Students individually write a description of the pattern using specific vocabulary (positive or negative, strong or weak, linear or nonlinear), then compare descriptions with a partner and reconcile any differences in language or interpretation before sharing with the class.

15 min·Pairs

Gallery Walk: Matching Correlation to Scatter Plot

Post scatter plots at stations alongside cards showing r values such as -0.9, -0.4, 0.1, 0.7, and 0.95. Students rotate to each station and match the plot to the most appropriate r value, writing one sentence explaining the visual feature that most influenced their choice.

25 min·Small Groups

Real-World Connections

• Market researchers analyze scatter plots to see if there is a relationship between advertising spending and product sales for a new consumer good, helping to optimize marketing budgets.
• Environmental scientists use scatter plots to investigate the relationship between average daily temperature and the number of reported heat-related illnesses in a city, informing public health advisories.
• Economists examine scatter plots to explore potential links between a country's GDP per capita and its life expectancy, contributing to discussions about economic development and public health policy.

Assessment Ideas

Exit Ticket

Provide students with a small dataset (e.g., hours studied vs. test scores for 5 students). Ask them to: 1. Plot the data on a scatter plot. 2. Describe the apparent correlation (positive, negative, or none) and its strength. 3. Write one sentence explaining why this correlation does not prove that studying causes higher scores.

Discussion Prompt

Present students with a scatter plot showing a strong positive correlation between ice cream sales and drowning incidents. Ask: 'What is the relationship shown in this plot? Is it reasonable to conclude that eating ice cream causes drowning? What other factor might explain both of these trends?'

Quick Check

Show students three different scatter plots, each representing positive correlation, negative correlation, and no correlation. Ask students to label each plot with the correct type of correlation and briefly justify their choice based on the pattern of points.

Frequently Asked Questions

What does the direction of a scatter plot tell you?

A positive direction means that as the x-variable increases, the y-variable tends to increase. A negative direction means that as x increases, y tends to decrease. No clear direction means there is no consistent trend between the variables. Direction is the first characteristic to identify before assessing strength or attempting to fit a line to the data.

What is the correlation coefficient and what does it measure?

The correlation coefficient r is a number between -1 and 1 that measures the strength and direction of a linear relationship between two quantitative variables. Values near 1 or -1 indicate strong linear correlation. Values near 0 indicate little to no linear relationship. The sign of r matches the direction of the scatter plot: negative for downward trends and positive for upward trends.

How does active learning help students interpret scatter plots accurately?

Scatter plot interpretation requires calibrated judgment since the boundary between strong and moderate correlation is not a bright line. When students compare their assessments in pairs or small groups, they must use vocabulary precisely and resolve disagreements through reasoning. Collaborative analysis of real datasets from students' own lives makes these judgment calls feel meaningful rather than arbitrary exercises in classification.

Why does correlation not imply causation?

Two variables can move together because both are driven by a third factor (a confounding variable), because the relationship is coincidental, or because one truly causes the other. Correlation alone cannot distinguish among these possibilities. Establishing causation requires controlled experiments with random assignment, not observational data. Always ask what else could explain the pattern before drawing causal conclusions from a scatter plot.