Correlation Coefficient: Reading Scatter Plots

What is the correlation coefficient?

The correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables. It ranges from $-1$ to $+1$.

Interpreting $r$

• $r = +1$: perfect positive correlation — as $x$ increases, $y$ increases at a constant rate. Points lie exactly on a line with positive slope.
• $r = -1$: perfect negative correlation — as $x$ increases, $y$ decreases at a constant rate.
• $r = 0$: no linear correlation — no straight-line pattern. (There might still be a curved relationship!)

Strength guide

• $|r| > 0.7$: strong linear relationship
• $0.3 < |r| < 0.7$: moderate linear relationship
• $|r| < 0.3$: weak or no linear relationship

Step-by-step: estimating $r$ from a scatter plot

Step 1 — Direction. Do points trend upward (positive $r$) or downward (negative $r$)?

Step 2 — Tightness. How close are the points to forming a line? Tight cluster → $|r|$ close to 1. Scattered cloud → $|r|$ close to 0.

Step 3 — Estimate. A tight upward trend might be $r \approx 0.9$. A loose downward trend might be $r \approx -0.5$.

The critical warning: correlation ≠ causation

Ice cream sales and drowning deaths are positively correlated. Does ice cream cause drowning? No! Both increase in summer (the hidden variable). High $r$ means two things move together — it does NOT mean one causes the other.

Try it in the visualization

Adjust the target $r$ and see how the scatter plot changes shape. A high $|r|$ gives a tight cluster around the regression line; $r \approx 0$ gives a formless cloud.