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Regression, Correlation, and Causation

Students fit and interpret models for bivariate data and distinguish association from causal evidence.

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Beginner 15–20 minutes Probability & Statistics
Probability & Statistics Page 8 of 8
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At a Glance

Regression, Correlation, and Causation
Binder SectionProbability & Statistics
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
Essential Question

When does a relationship between variables support prediction, and when can it support a causal claim?

Build Understanding

A regression model summarizes the relationship between variables. For a linear model, slope gives the predicted change in y for each one-unit increase in x, and the intercept must be interpreted in context.

Correlation measures the direction and strength of a linear association from -1 to 1. Residuals are observed minus predicted values; a random residual pattern supports a model. Neither correlation nor regression alone proves causation.

Visual Model
4Thousands 3Hundreds 2Tens 7Ones

4,327 = 4,000 + 300 + 20 + 7

Goals

Learning Targets

  • A line of best fit summarizes a trend.
  • Interpret slope and intercept in context.
  • Residual = observed minus predicted.
  • Correlation measures linear association.
  • Extrapolation beyond the data can be unreliable.
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Interactive Vocabulary

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Worked Examples

Example 1

A regression line for study time x and score y is y=4x+62.

1

At x=5, the predicted score is 82.

Example 2

Ice-cream sales and sunburns rise together

1

This correlation does not show that ice cream causes sunburn; warm weather affects both.

Ask and explain

Common Questions

Is a larger correlation always better? Magnitude indicates stronger linear association, while sign gives direction.
Does r = 0 mean no relationship? It means no linear relationship.
Can a regression line prove causation? No.
Notice and correct

Common Misconceptions

Saying correlation proves causation

Consider study design, random assignment, and lurking variables.

Treating a prediction as an exact result

Regression estimates a typical response and residuals show variation.

Using a model far outside the observed domain

Avoid unsupported extrapolation.

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Regression, Correlation, and Causation lesson and examples

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Quick Start

Warm-Up

Warm-up 1

Directions: Solve the problem. Show your thinking, then enter only your final answer.

A model predicts 48, but the observed value is 53. Find the residual.

Picture the dataUse the picture to organize what you know before you solve.
look for center, spread, and patterns

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We Do

Guided Practice

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You Do

Independent Practice

Independent 1

Directions: Solve independently. Show your work, check it, then enter only your final answer.

A model predicts 48, but the observed value is 53. Find the residual.

Picture the dataUse the picture to organize what you know before you solve.
look for center, spread, and patterns

Choose how you want to work or answer.

Independent 2

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Create and solve a second example with different values.

Picture your planUse the picture to organize what you know before you solve.
What do Iknow?Which rulefits?What must Ifind?

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4
Stretch Your Thinking

Challenge & Real-World Practice

🧩 Challenge

Find or create a context with strong correlation but no reasonable causal link. Identify a lurking variable, explain why the correlation can still support limited prediction, and state what study design would strengthen a causal claim.

IXL

Khan Academy

DeltaMath

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  • I can identify the important information.
  • I can use accurate notation, labels, and units.
  • I can justify and verify my conclusion.
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Question 1

What should happen before calculating?

2
Question 2

Which response gives the strongest evidence?

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Question 3

What is an effective accuracy check?

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Question 4

Why are units or labels important?

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Question 5

What should a student do after finding an error?

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