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Correlation &
Causation

Learn to interpret patterns in data, describe correlations qualitatively, and understand why correlation does not imply causation.

Pre-Algebra

What is Correlation?

Correlation describes a relationship between two variables. When one variable changes, the other tends to change in a predictable pattern. We can observe correlation by looking at scatter plots: points that form a pattern suggest the variables are correlated.

Three Types of Correlation

Positive
As one variable increases, the other also increases. Points slope upward.
Negative
As one variable increases, the other decreases. Points slope downward.
None
No clear pattern. The variables do not appear related. Points are scattered randomly.
Positive Negative None

Why Does This Matter?

Every day, people use data to make decisions: doctors choosing treatments, businesses launching products, governments setting policies. If we confuse correlation with causation, we can make costly or even dangerous mistakes.

Health
A study finds people who eat chocolate score higher on memory tests. Does chocolate boost memory?
Business
A company runs ads and sales rise. Did the ads cause the increase, or was it a holiday?

What is Causation?

Causation means one event directly produces another event. There is a clear mechanism: changing variable \( A \) makes variable \( B \) change. Causation is much harder to prove than correlation.

Cause (A) Effect (B) directly causes

The Golden Rule

Correlation does NOT imply Causation

Just because two variables move together does not mean one causes the other. There could be a hidden third variable, pure coincidence, or a reversed cause-and-effect relationship.

Why Correlation Fails as Proof

Lurking Variable
A hidden third factor causes both variables to change, creating a false link.
Coincidence
Two unrelated things happen to move together by pure chance in the same time period.
Reverse Causation
We think A causes B, but actually B causes A. The direction is backwards.
Lurking Variable Variable A Variable B false correlation

Correlation vs Causation

Correlation Causation
Definition Two variables move together One variable directly causes the other
Proof needed Observe a pattern in data Controlled experiment or mechanism
Direction No direction implied Clear: A causes B
Example Ice cream sales and drownings rise together Studying more leads to higher test scores
Strength Weak to strong pattern Direct mechanism exists

How to Tell Them Apart

1
Look for a pattern. Do the two variables move together? If yes, you may have a correlation.
2
Ask "Could a lurking variable explain this?" Think of any third factor that might be driving both variables.
3
Check the direction. Does changing A actually change B, or could B be causing A?
4
Look for a controlled experiment. Only a properly designed experiment can prove causation.
5
State your conclusion carefully. Say "is associated with" for correlation, "causes" only when proven.

Example 1: Ice Cream & Drownings

Ice Cream Sales ($) Drownings trend

Example 2: Study Hours & Scores

Hours Studied Test Score (%) 0 25 50 75 100 1 2 3 4 5

Example 3: Spurious Correlation

Data shows a strong positive correlation between the number of movies Nicolas Cage appeared in per year and the number of people who drowned in swimming pools. This is a real data pattern!

Nicolas Cage Films Pool Drownings ? Pure coincidence!

Strength of Correlation

Correlations can be strong or weak. A strong correlation means points cluster tightly around the trend line. A weak correlation means points are spread out, with only a slight pattern.

Strong Positive Weak Positive

Reading Scatter Plots

1
Identify the variables. What does the \( x \)-axis represent? What about the \( y \)-axis?
2
Look for a direction. Do points go up (positive), down (negative), or show no trend?
3
Judge the strength. Are points tightly clustered (strong) or widely scattered (weak)?
4
Describe the correlation. Use words like "strong positive," "weak negative," or "no correlation."

Common Mistakes to Avoid

✕ Assuming Cause
Seeing a correlation and immediately concluding one variable causes the other without evidence.
✕ Ignoring Lurking Variables
Forgetting to consider a hidden third factor that might explain both variables.
✕ Confusing Direction
Getting cause and effect backwards: saying A causes B when it is actually B causing A.

Summary Cheat Sheet

Concept Key Idea Language to Use
Positive Correlation Both variables increase together "As X increases, Y tends to increase"
Negative Correlation One increases while the other decreases "As X increases, Y tends to decrease"
No Correlation No pattern between variables "X and Y show no relationship"
Causation One variable directly causes the other "X causes Y" (only with evidence)
Lurking Variable Hidden factor drives both variables "A third factor may explain both"

Question 1 of 5

A study finds that cities with more fire stations also have more fires. What is the best explanation?

Question 2 of 5

A scatter plot shows that as temperature increases, lemonade sales increase. The points cluster tightly around an upward trend line. How would you describe this?

Question 3 of 5

Which of the following is the best example of causation?

Question 4 of 5

Data shows that as the number of hours of sleep increases, the number of mistakes on a test decreases. What type of correlation is this?

Question 5 of 5

A news headline reads: "People who eat breakfast earn higher salaries." A classmate says, "So eating breakfast makes you richer!" What is wrong with this reasoning?