10.2Correlation
Identify positive, negative, strong, weak, and no correlation in scatter plots. Learn to describe the direction and strength of linear relationships between two variables.
Why This Matters
Understanding correlation helps you interpret data responsibly — including the critical distinction that correlation does not imply causation. This concept is foundational in AP Statistics, Psychology, Economics, and scientific research.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do we describe and measure the strength and direction of a linear association?
Lesson Overview
Correlation describes the relationship between two variables in a scatter plot. We describe correlation using two characteristics: direction (positive, negative, or none) and strength (strong or weak). A positive correlation means that as one variable increases, the other tends to increase. A negative correlation means that as one variable increases, the other tends to decrease. No correlation means there is no visible pattern. Strength refers to how closely the data points cluster around an imaginary line — the tighter the cluster, the stronger the correlation.
Types of Correlation — Visual Guide
Strong Positive
Weak Positive
Strong Negative
No Correlation
Correlation Strength Spectrum
Strong Neg.
−1 to −0.7
Weak Neg.
−0.7 to −0.3
None
−0.3 to +0.3
Weak Pos.
+0.3 to +0.7
Strong Pos.
+0.7 to +1.0
Describing Correlation — Direction × Strength
| As x increases… | Direction | Points close to line? | Strength |
|---|---|---|---|
| y tends to increase | Positive | Yes — tightly clustered | Strong |
| y tends to increase | Positive | No — widely scattered | Weak |
| y tends to decrease | Negative | Yes — tightly clustered | Strong |
| y tends to decrease | Negative | No — widely scattered | Weak |
| y shows no pattern | None | N/A — random cloud | None |
How to Describe Correlation — 4-Step Method
- Look at direction: Do the points go up-right (positive) or down-right (negative)? Or is there no pattern?
- Look at strength: Are the points tightly clustered near a line (strong) or widely scattered (weak)?
- Write a complete description: "There is a strong positive correlation between study hours and test scores."
- Correlation ≠ causation: Just because two variables are correlated does not mean one causes the other.
Worked Examples
A scatter plot shows hours of TV watched vs. GPA. As TV hours increase, GPA decreases. The points are tightly clustered. Describe the correlation.
Direction: as TV hours increase, GPA decreases → negative direction.
Strength: points are tightly clustered → strong.
A scatter plot shows shoe size vs. math test score. The points are randomly scattered with no visible pattern. Describe the correlation.
Direction: no pattern visible.
Strength: not applicable.
A scatter plot shows age of a car (years) vs. its resale value ($). As age increases, value decreases, but points are loosely scattered. Describe the correlation.
Direction: as age increases, value decreases → negative.
Strength: loosely scattered → weak.
A scatter plot shows temperature (°F) vs. ice cream sales ($). As temperature rises, sales rise, and points are very close to a line. Describe the correlation.
Direction: both increase together → positive.
Strength: very close to a line → strong.
Explain why correlation does not imply causation. Give an example.
Correlation means two variables tend to change together.
Causation means one variable directly causes the other to change.
Example: Ice cream sales and drowning rates are both high in summer (positive correlation), but ice cream does not cause drowning — both are caused by hot weather (a lurking variable).
Guided Practice
Guided Practice Video: Correlation
Review how to interpret the correlation coefficient (r), describe the strength and direction of a linear association, and distinguish correlation from causation before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗A scatter plot shows minutes of exercise vs. resting heart rate. As exercise increases, heart rate decreases. Points are tightly clustered. Describe the correlation.
Hint: Identify direction first (does y increase or decrease as x increases?), then strength (how tightly clustered?).
A scatter plot shows number of absences vs. final grade. As absences increase, grades decrease, but points are loosely scattered. Describe the correlation.
Hint: Direction: grades go down as absences go up. Strength: loosely scattered = weak.
A scatter plot shows a student's height vs. their favorite color (coded 1–5). The points show no pattern. Describe the correlation.
Hint: Is there a visible trend? If not, what type of correlation is this?
A scatter plot shows years of education vs. annual salary. As education increases, salary increases, and points are moderately clustered. Describe the correlation.
Hint: Both variables increase together. Are the points tight or loose?
Error Analysis: A student says 'There is a negative correlation between temperature and hot chocolate sales, so cold weather causes people to buy hot chocolate.' Identify the error.
Hint: Does correlation prove causation? What is the student confusing?
Key Vocabulary
Correlation
A statistical relationship between two variables. Described by direction (positive/negative/none) and strength (strong/weak).
Positive Correlation
As x increases, y tends to increase. Points slope upward from left to right.
Negative Correlation
As x increases, y tends to decrease. Points slope downward from left to right.
No Correlation
No visible pattern between x and y. Points appear randomly scattered.
Strong Correlation
Data points cluster tightly around a line. The relationship is clear and consistent.
Weak Correlation
Data points are loosely scattered around a line. The relationship is present but not consistent.
Linear Correlation
The relationship between two variables follows a roughly straight-line pattern.
Nonlinear Correlation
The relationship between two variables follows a curved pattern, not a straight line.
Interactive Practice — 5 Questions
A scatter plot shows that as x increases, y decreases, and the points are tightly clustered. This is:
Which describes a weak positive correlation?
A scatter plot of shoe size vs. IQ score shows randomly scattered points. This is:
Which statement about correlation is TRUE?
As temperature increases, heating costs decrease. The points are very close to a line. This is:
Independent Practice
Independent Practice
Describe the correlation for each scatter plot: (a) points tightly clustered going up-right; (b) points loosely scattered going down-left; (c) points randomly scattered with no trend.
A scatter plot shows hours of sleep (x) vs. reaction time (y). As sleep increases, reaction time decreases, and points are tightly clustered. Describe the direction, form, and strength.
Two scatter plots: Plot A has r = 0.85; Plot B has r = −0.90. Which has a stronger linear relationship? Which is positive and which is negative?
Error Analysis: A student says "There is a positive correlation between ice cream sales and drowning rates, so ice cream causes drowning." Identify the statistical error and explain the concept of a lurking variable.
A scatter plot shows age of a car (x) vs. resale value (y). Describe the expected direction and strength of the association. Explain your reasoning.
Common Mistakes
Confusing correlation with causation — assuming that because two variables are correlated, one causes the other.
Correlation shows a relationship, not a cause. A third variable (lurking variable) may explain both.
Saying a weak correlation means no relationship — e.g., r = 0.3 means nothing is happening.
A weak correlation still suggests a slight trend. 'No correlation' means r ≈ 0 with no visible pattern.
Confusing the direction of correlation with its strength — thinking negative correlation is weaker than positive.
Strength depends on how close |r| is to 1, not the sign. r = −0.9 is stronger than r = +0.4.
Describing a non-linear relationship as 'no correlation' because it doesn't fit a straight line.
A curved pattern (like a U-shape) can show a strong relationship that isn't linear. Correlation (r) measures only linear association.
Math Tips
Always describe correlation using both direction (positive/negative/none) AND strength (strong/weak).
Correlation ≠ causation. A lurking variable may explain why two unrelated things appear correlated.
Strength is about how tightly the points cluster around a line — not about the slope of the line.
A negative correlation is not "weaker" than a positive one. r = −0.9 is stronger than r = +0.4.
No correlation means no linear pattern. A curved relationship can still be strong — it is just nonlinear.