10.6Correlation Coefficient
Interpret the correlation coefficient r and evaluate the strength of a linear relationship. Understand r², the limits of correlation, and why correlation never implies causation.
Why This Matters
The correlation coefficient r gives you a precise, objective measure of how strongly two variables are related. It's used in every quantitative field — from psychology research to financial modeling — and is a core concept in AP Statistics.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does the correlation coefficient give us a precise, numerical measure of the strength and direction of a linear relationship?
Lesson Overview
The correlation coefficient, denoted r, is a number between −1 and +1 that measures the strength and direction of the linear relationship between two variables. The sign of r tells us the direction (positive or negative), and the absolute value |r| tells us the strength — the closer |r| is to 1, the stronger the relationship. A value of r = 0 indicates no linear relationship. The coefficient of determination, r², tells us the proportion of variation in y that is explained by the linear relationship with x. While r is a powerful tool, it only measures linear relationships and never implies causation.
Correlation Coefficient (r) — Interpretation Guide
r = −1
Perfect Negative
−1 < r ≤ −0.7
Strong Negative
−0.7 < r ≤ −0.3
Moderate Negative
−0.3 < r < +0.3
Weak / None
+0.3 ≤ r < +0.7
Moderate Positive
+0.7 ≤ r < +1
Strong Positive
Scatter Plots — How r Looks Visually
r = +1
Perfect Positive
r ≈ +0.9
Strong Positive
r ≈ +0.5
Moderate Positive
r ≈ 0
No Linear Correlation
r ≈ −0.9
Strong Negative
r = −1
Perfect Negative
Key Properties of the Correlation Coefficient
Range
−1 ≤ r ≤ +1. Values outside this range are impossible.
Sign
Positive r → positive correlation. Negative r → negative correlation.
Magnitude
Closer to ±1 → stronger. Closer to 0 → weaker.
Units
r has no units. It is a pure number.
Symmetry
Swapping x and y gives the same r.
Not causation
A high r does not mean x causes y.
r vs. r² (Coefficient of Determination)
| r | r² | Interpretation of r² | Strength |
|---|---|---|---|
| 1.00 | 1.00 | 100% of variation in y explained by x | Perfect |
| 0.90 | 0.81 | 81% of variation explained | Very Strong |
| 0.80 | 0.64 | 64% of variation explained | Strong |
| 0.70 | 0.49 | 49% of variation explained | Moderate–Strong |
| 0.50 | 0.25 | 25% of variation explained | Moderate |
| 0.30 | 0.09 | 9% of variation explained | Weak |
| 0.00 | 0.00 | 0% of variation explained | None |
Note: r² is always between 0 and 1. It tells us the proportion of variation in y that is explained by the linear relationship with x.
r² — How Much Variation Does the Model Explain?
The shaded portion shows the percentage of variation in y explained by the linear model. The unshaded portion is unexplained (due to other factors).
Correlation vs. Causation
Causation
X directly causes Y to change. Example: More rain → more plant growth.
Correlation (No Causation)
X and Y move together, but a lurking variable causes both. Example: Ice cream sales and drowning rates both rise in summer (lurking: hot weather).
Worked Examples
A study finds r = 0.92 between hours studied and test scores. Interpret this value.
Sign: positive → as hours studied increase, test scores tend to increase.
|r| = 0.92, which is close to 1 → strong linear relationship.
A dataset has r = −0.45. Interpret this value.
Sign: negative → as x increases, y tends to decrease.
|r| = 0.45 → moderate (between weak and strong).
Two datasets have r = 0.85 and r = −0.85. Which has the stronger linear relationship?
|0.85| = 0.85 and |−0.85| = 0.85.
Both have the same strength. The sign only indicates direction.
A linear model has r = 0.9. Calculate r² and interpret it.
r² = (0.9)² = 0.81
81% of the variation in y is explained by the linear relationship with x.
The remaining 19% is due to other factors.
A scatter plot of shoe size vs. math score has r = 0.05. What can you conclude?
|r| = 0.05 ≈ 0 → very weak (essentially no) linear relationship.
Shoe size is not a useful linear predictor of math score.
Note: r = 0.05 does not mean there is absolutely no relationship — just no linear one.
Guided Practice
Guided Practice Video: Correlation Coefficient
Review how to interpret the correlation coefficient r — its sign, strength, and relationship to the scatter plot — before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Interpret r = −0.78. State the direction, strength, and what it means in context for a dataset about age of a car (x) vs. resale value (y).
Hint: Sign → direction. |r| → strength (is 0.78 close to 0 or 1?).
Two datasets: Dataset A has r = 0.60; Dataset B has r = −0.80. Which has the stronger linear relationship?
Hint: Compare |r| values. The sign does not affect strength.
A linear model has r = 0.7. Calculate r² and interpret it.
Hint: r² = r × r. Then express as a percentage of variation explained.
A scatter plot of ice cream sales vs. drowning rates has r = 0.88. A student concludes that ice cream causes drowning. What is wrong with this conclusion?
Hint: What is the difference between correlation and causation? What might explain both variables?
A dataset has r = 0.02. A student says 'there is no relationship between the variables.' Is this fully accurate? Explain.
Hint: r = 0 means no linear relationship. Could there be a nonlinear relationship?
Key Vocabulary
Correlation Coefficient (r)
A number between −1 and +1 that measures the strength and direction of the linear relationship between two variables.
Coefficient of Determination (r²)
The square of r. Tells the proportion of variation in y explained by the linear relationship with x.
Perfect Correlation
r = +1 (perfect positive) or r = −1 (perfect negative). All data points lie exactly on a line.
No Linear Correlation
r ≈ 0. No linear pattern between the two variables (though a nonlinear pattern may exist).
Strength
How closely the data points cluster around the line of best fit. Measured by |r|.
Direction
Positive (r > 0) or negative (r < 0), indicating whether the variables increase or decrease together.
Interactive Practice — 5 Questions
Which r-value indicates the strongest linear relationship?
r = −0.75 indicates:
If r = 0.8, what is r²?
r = 0 means:
Which r-value is impossible?
Independent Practice
Independent Practice
Interpret r = 0.95. State the direction and strength of the linear relationship.
Which is stronger: r = 0.75 or r = −0.80? Explain your reasoning.
A model has r = 0.6. Calculate r² and interpret it. What percentage of variation in y is explained by x?
A scatter plot of height vs. weight has r = 0.72. Does this mean height causes weight? Explain using the concept of correlation vs. causation.
Rank these r-values from weakest to strongest linear relationship: 0.40, −0.85, 0.10, −0.60. Explain your ranking.
Common Mistakes
Thinking r = −0.9 is weaker than r = +0.6 because it's negative.
Strength is determined by |r|. r = −0.9 (|r| = 0.9) is stronger than r = +0.6 (|r| = 0.6).
Confusing r with r² — saying 'r = 0.8 means 80% of variation is explained'.
r² (not r) gives the proportion of variation explained. If r = 0.8, then r² = 0.64, meaning 64% of variation is explained.
Concluding causation from a strong r value — e.g., 'r = 0.95 proves X causes Y'.
r measures the strength of a linear association, not causation. A lurking variable may explain the relationship.
Using r to describe a non-linear relationship — e.g., applying r to a curved scatter plot.
r only measures the strength of a linear relationship. A low r doesn't mean no relationship — it may be non-linear.
Math Tips
r is always between −1 and +1. Values outside this range are impossible.
Strength is determined by |r| — the absolute value. r = −0.9 is stronger than r = +0.5.
r² (coefficient of determination) tells you the percentage of variation in y explained by x. r = 0.8 → r² = 0.64 → 64% of variation explained.
r = 0 means no linear relationship — but a strong nonlinear (curved) relationship could still exist.
Correlation ≠ causation. A high r-value does not mean one variable causes the other.