Correlation

AP Statistics· 12 min read

1. Defining the Pearson Correlation Coefficient★★☆☆☆⏱ 3 min

The Pearson r value is a standardized metric that removes the units of both variables, so changing the scale of x or y (for example converting inches to centimeters) will never alter the value of r.

📘 Definition

Pearson Correlation Coefficient

r

A unitless value ranging strictly between -1 and 1 that measures the strength and direction of the linear relationship between two quantitative variables

Example:

An r of 0.9 indicates a strong positive linear association, while an r of -0.7 indicates a moderate negative linear association

r = \frac{1}{n-1} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right)

📐 Worked Example

Calculate r for the following 3 data points: (1, 2), (2, 4), (3, 5)

1
First calculate the mean and standard deviation for x and y: $\overset{x}{ˉ} = 2$ , $s_{x} = 1$ , $\overset{y}{ˉ} = 11/3 \approx 3.67$ , $s_{y} \approx 1.53$
2
Convert each x and y value to z-scores: x z-scores = [-1, 0, 1], y z-scores = [-1.09, 0.22, 0.87]
3
Multiply paired z-scores, sum them, and divide by n-1: $r = \frac{( - 1 *- 1.09 ) + ( 0 * 0.22 ) + ( 1 * 0.87 )}{2} = 0.98$

✓ Quick check

Test your understanding of r properties

What is the maximum possible value of r?
- 1
- 100
- infinity
- 0
Reveal answer
1 —
r can never exceed 1 or be less than -1

2. Interpreting r Values from Scatterplots★★☆☆☆⏱ 3 min

r Range	Association Strength	Typical Scatterplot Pattern
0.7 to 1.0	Strong positive	Points tightly clustered around an upward sloping line
0.3 to 0.7	Moderate positive	Points loosely clustered around an upward sloping line
-0.3 to 0.3	Very weak or none	Points scattered randomly with no clear trend
-0.7 to -0.3	Moderate negative	Points loosely clustered around a downward sloping line
-1.0 to -0.7	Strong negative	Points tightly clustered around a downward sloping line

📐 Worked Example

Match 4 scatterplots to r values: -0.9, 0.2, 0.7, -0.4

1
Scatterplot 1 (tight downward line) = -0.9
2
Scatterplot 2 (random no trend) = 0.2
3
Scatterplot 3 (loose upward line) = 0.7
4
Scatterplot 4 (very loose downward line) = -0.4

Exam tip:

AP graders will deduct points if you forget to include the word 'linear' when describing what r measures

3. Factors That Distort Correlation Values★★★☆☆⏱ 3 min

📐 Worked Example

Show how adding a single outlier point (10, 10) changes r for the original 3-point dataset we calculated earlier (r=0.98)

1
Original 3 points: (1,2), (2,4), (3,5), r=0.98
2
Add outlier (10, 10), new n=4
3
Recalculate r: new r ≈ 0.97, a negligible change if the outlier follows the existing trend
4
If we add outlier (10, 0) instead, new r drops to ≈ -0.3, completely reversing the direction of the observed correlation

✓ Quick check

Identify the distortion factor

A researcher only tests students who scored 80% or higher on a math exam, and finds almost no correlation between study time and test score. What is the most likely cause?
- Restricted range
- Causation
- Perfect correlation
- Non-linear trend
Reveal answer
Restricted range —
Limiting your sample to a narrow subset of the full data range reduces observed correlation strength

4. Correlation vs Causation★★★☆☆⏱ 3 min

Even an r value of 0.99 does not prove that changes in x cause changes in y. The association could be driven by a third confounding variable, or be completely coincidental.

📐 Worked Example

The correlation between annual ice cream sales and annual drowning deaths in the US is r=0.9, a very strong positive correlation. Explain why this is not a causal relationship.

1
Ice cream sales do not cause drowning, and drowning deaths do not cause ice cream sales
2
The hidden confounding variable is average outdoor temperature: more people buy ice cream and swim when the weather is warm, driving both metrics up at the same time

5. Common Pitfalls

Wrong move:

Claiming a strong r value proves one variable causes the other

Why:

Correlation only measures linear association, and provides no evidence of causal direction

Correct move:

Explicitly state 'no causal conclusion can be drawn from observational data alone'

Wrong move:

Using r to describe a non-linear relationship

Why:

r only quantifies linear trends, so it will drastically understate the strength of curved associations

Correct move:

Always inspect the scatterplot first before calculating or interpreting r

Wrong move:

Reporting r as a percentage (e.g. saying r=0.8 means 80% correlated)

Why:

r is a standardized unitless value, not a proportion or percentage

Correct move:

Describe r as a measure of strength and direction of linear association, no percentage conversion

Wrong move:

Ignoring outliers that drastically shift r towards 1 or -1

Why:

A single extreme point can create a false strong correlation in otherwise uncorrelated data

Correct move:

Flag outliers and report r both with and without the outlier for full transparency

Wrong move:

Assuming r=0 means no relationship exists between variables

Why:

r=0 only means no linear relationship, a perfect curved relationship can have an r of exactly 0

Correct move:

Confirm a zero r value with a scatterplot to rule out non-linear patterns

6. Quick Reference Cheatsheet

Property	AP Exam Rule	Mistake to Avoid
Range of r	Always between -1 and 1 inclusive	Claiming r > 1 or r < -1
Interpretation Phrasing	State direction, strength, and linearity explicitly	Forgetting to mention 'linear' in your description
Causation Rule	No causal claim allowed for observational data	Saying 'X causes Y' in your FRQ response
Outlier Impact	Outliers can drastically shift r value	Treating r as a fixed value unaffected by individual points

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · Paper 1
MCQ on r interpretation
2022 · Paper 2 FRQ
r calculation for linear data
2021 · Paper 1
Correlation vs causation identification

What's Next

Mastering correlation is the critical foundation for moving to simple linear regression, the next core topic in this unit. You will learn how to use the correlation coefficient to calculate the slope and intercept of the least-squares regression line, interpret the coefficient of determination R² as the proportion of variation explained by the model, and assess residuals to check if a linear model is appropriate for your data. These skills are tested in nearly every AP Stats FRQ set focused on bivariate data, so solidifying your understanding of r now will prevent costly mistakes later when working with more complex regression calculations.