cor() — Correlation

Shows how strongly two things move together, from −1 to +1.

cor(x, y)

Key idea

• Correlation, written r, is one number from -1 to +1 showing how strongly two things move together in a straight line. (e.g. -1 to +1)
• A value near +1 means a strong positive link: both rise together. (e.g. r = 0.9)
• A value near -1 means a strong negative link: one rises as the other falls. (e.g. r = -0.85)
• A value near 0 means little or no straight-line link. (e.g. r = 0.02)
• Correlation is not causation: a link does not prove one thing causes the other.

Definition

$$r=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x}{)}^2}\sqrt{\sum (y_i-\bar{y}{)}^2}}$$

$x_i,y_i$

each pair of values

$\bar{x},\bar{y}$

their means

$r$

the correlation, from -1 to +1

$$r=\frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum (x-\bar{x}{)}^2}\sqrt{\sum (y-\bar{y}{)}^2}}\approx 0.98$$

Where you'd use it

• You record each student’s study hours and exam score. cor might return 0.85, a strong positive number telling you that students who study more clearly tend to score higher.
• Tracking daily temperature against ice-cream sales, cor could come out near 0.9, confirming a strong link — though remember it shows they move together, not that heat directly causes the sales.
• For cars, comparing weight with fuel used per mile gives a positive cor around 0.7, so heavier cars tend to burn more fuel, while a value near 0 would mean almost no straight-line link.

Frequently asked questions

What does cor measure?

The strength of a linear relationship, from −1 to +1. 0 means no linear link (a curve can still exist).

Does correlation prove causation?

No — a hidden third variable can drive both.