# Correlation

> AP Statistics · AP 2024-2026 Statistics Syllabus
> Source: https://www.owlsprep.com/study/ap-statistics-u14-correlation/

This module covers Pearson correlation calculation, interpretation, factors that skew r values, and critical distinctions between correlation and causation, aligned to AP Stats scoring rubrics.

**Prerequisites:** [Scatterplot construction and interpretation](https://www.owlsprep.com/study/ap-statistics-u14-scatterplots/); [Basic descriptive statistics for quantitative variables](https://www.owlsprep.com/study/ap-statistics-u11-quantitative-descriptive-stats/)

## Learning objectives

- Calculate and interpret the Pearson correlation coefficient r for bivariate quantitative data
- Distinguish between correlation and causation to avoid invalid causal claims
- Identify how outliers, non-linear relationships, and restricted range impact the value of r
- Match scatterplot patterns to approximate r values between -1 and 1

## Defining the Pearson Correlation Coefficient

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.

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

*Notation:* r

*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: $\bar{x}=2$, $s_x=1$, $\bar{y}=11/3 \approx 3.67$, $s_y \approx 1.53$
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: $r = \frac{(-1*-1.09)+(0*0.22)+(1*0.87)}{2} = 0.98$

**Check your understanding**

Test your understanding of r properties

1. What is the maximum possible value of r?

   - 1
   - 100
   - infinity
   - 0

   *Why:* r can never exceed 1 or be less than -1

## Interpreting r Values from Scatterplots

**Exam command terms**

AP exam FRQs use specific command terms for correlation questions, with strict scoring requirements:

- **Describe the association** — You must explicitly state direction, strength, linearity, and any outliers to earn full credit

- **Interpret r** — You cannot reference causation, and must specify that it measures *linear* association

| 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

## Factors That Distort Correlation Values

> **Critical Pre-Check**
>
> r is extremely sensitive to outliers, restricted range, and non-linear trends, and can produce misleading values if you do not inspect the scatterplot first

**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

**Check your understanding**

Identify the distortion factor

1. 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

   *Why:* Limiting your sample to a narrow subset of the full data range reduces observed correlation strength

## Correlation vs Causation

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

> **tip**
>
> The only way to reliably establish causal relationships between two variables is to run a well-designed randomized controlled experiment, not to observe correlation in observational data

## Common pitfalls

- **Wrong:** Claiming a strong r value proves one variable causes the other
  - Why it fails: Correlation only measures linear association, and provides no evidence of causal direction
  - Correct: Explicitly state 'no causal conclusion can be drawn from observational data alone'
- **Wrong:** Using r to describe a non-linear relationship
  - Why it fails: r only quantifies linear trends, so it will drastically understate the strength of curved associations
  - Correct: Always inspect the scatterplot first before calculating or interpreting r
- **Wrong:** Reporting r as a percentage (e.g. saying r=0.8 means 80% correlated)
  - Why it fails: r is a standardized unitless value, not a proportion or percentage
  - Correct: Describe r as a measure of strength and direction of linear association, no percentage conversion
- **Wrong:** Ignoring outliers that drastically shift r towards 1 or -1
  - Why it fails: A single extreme point can create a false strong correlation in otherwise uncorrelated data
  - Correct: Flag outliers and report r both with and without the outlier for full transparency
- **Wrong:** Assuming r=0 means no relationship exists between variables
  - Why it fails: r=0 only means no linear relationship, a perfect curved relationship can have an r of exactly 0
  - Correct: Confirm a zero r value with a scatterplot to rule out non-linear patterns

## 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 |

## 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.

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