Pearson's Correlation Coefficient

Pearson's $r$ packages both direction and strength into one number:

• Sign (+/−): Positive $r$ → points slope upward. Negative $r$ → points slope downward.
• Magnitude (distance from zero): The closer $|r|$ is to 1, the stronger (tighter) the relationship.

Strength guidelines:

Pearson's r measures linear correlation strength and direction: r = +1 is perfect positive, r = −1 is perfect negative, r = 0 is no linear correlation. The sign gives direction; the magnitude (|r| close to 1) gives strength.

Pause, copy the r scale from −1 to +1, the sign rule (negative = negative direction; positive = positive direction), and the magnitude rule (closer to ±1 = stronger association; closer to 0 = weaker) into your book.

Pearson's r ranges from −1 (perfect negative linear) to +1 (perfect positive linear), with 0 meaning no linear association. The sign matches the direction you read from the scatterplot. For strength, HSC uses two thresholds: |r| ≥ 0.8 is strong, 0.5 ≤ |r| < 0.8 is moderate, and |r| < 0.5 is weak.

When interpreting $r$, always state: (1) the direction, (2) the strength, and (3) what it means in context of the two variables.

Examples:

• $r = 0.87$: strong positive linear correlation, as [x variable] increases, [y variable] tends to increase strongly.
• $r = -0.92$: strong negative linear correlation, as [x variable] increases, [y variable] tends to decrease strongly.
• $r = 0.41$: weak positive linear correlation, as [x variable] increases, [y variable] shows a slight tendency to increase, but the relationship is not consistent.
• $r = -0.05$: essentially no linear correlation, knowing [x variable] tells us almost nothing about [y variable].

Real example: For a study of age (x) and resting heart rate (y), $r = -0.68$ means "there is a moderate negative linear correlation between age and resting heart rate, as age increases, resting heart rate tends to decrease moderately."

Interpreting r in context: |r| ≥ 0.8 is strong, 0.5 ≤ |r| < 0.8 is moderate, |r| < 0.5 is weak. Always state the interpretation in terms of the actual variables, not just the number.

Pause, copy the three classification thresholds: |r| ≥ 0.8 strong, 0.5 ≤ |r| < 0.8 moderate, |r| < 0.5 weak, and the rule that every r description must use the actual variable names into your book.

The thresholds |r| ≥ 0.8 (strong), 0.5–0.8 (moderate), and < 0.5 (weak) let you classify any r value, but r has three important limitations: it only measures linear association (a perfect curve could give r near 0), it is sensitive to outliers (one extreme point can dramatically change r), and a strong r does not mean one variable causes the other.

Pearson's $r$ has important limitations that examiners test:

• $r$ only measures linear relationships. Two variables can have a perfect curved (non-linear) relationship with $r \approx 0$. Low $r$ does not mean no relationship, just no linear one.
• $r$ does not imply causation. A high $r$ tells you the variables are strongly associated, but it does not prove that one causes the other. (We will explore this in Lesson 4.)
• Outliers can distort $r$. A single outlier can pull $r$ toward 0 or toward ±1, making the relationship look weaker or stronger than it is for the main cluster of data.

Limitations of r: it only measures linear association (not curved relationships), is sensitive to outliers, and never proves causation, a high r value means association only, not that one variable causes the other.

Pause, copy the three limitations of Pearson's r: it only detects linear (not curved) relationships, it is sensitive to outliers, and a high r value does not establish that one variable causes the other into your book.

For each $r$ value below, state the direction, classify the strength, and write a sentence of interpretation. Assume x = advertising spend ($000s) and y = monthly sales ($000s).

1. $r = 0.88$
2. $r = -0.31$
3. $r = 0.05$
4. $r = -0.97$