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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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Regression, Correlation, and Causation
At a Glance
Binder SectionProbability & Statistics
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
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.
2
A student scoring 86 has residual 86−82=4.
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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Warm-Up
Guided
Independent
Challenge
1
Quick Start
Warm-Up
A model predicts 48, but the observed value is 53. Find the residual.
2
We Do
Guided Practice
Slope 2.5 means predicted y rises 2.5 units per one x unit.
A negative residual means the observed value is below the prediction.
r near -1 indicates a strong negative linear association.
A curved residual pattern suggests a linear model may be inappropriate.
Predicting far outside the observed x-values is extrapolation.
3
You Do
Independent Practice
A model predicts 48, but the observed value is 53. Find the residual.
Create and solve a second example with different values.
4
Stretch Your Thinking
Challenge & Real-World Practice
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.
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- I can identify the important information.
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- I can justify and verify my conclusion.
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Question 1
What should happen before calculating?
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Question 2
Which response gives the strongest evidence?
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What is an effective accuracy check?
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Why are units or labels important?
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What should a student do after finding an error?
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