Scatterplots and Correlation

By the end of this lesson, you will be able to construct and describe scatterplots, calculate and interpret the correlation coefficient r, and explain the critical difference between association and causation.

Scatterplots

A scatterplot displays the relationship between two quantitative variables. Each point represents one individual.

Convention: the explanatory variable (x) goes on the horizontal axis; the response variable (y) goes on the vertical axis. (Note: not all scatterplots have a clear explanatory/response distinction.)

Describing a scatterplot (DOFS):

• Direction: positive, negative, or no association
• Outliers: points that deviate from the pattern
• Form: linear or nonlinear
• Strength: how closely points follow the form

Positive association: as x increases, y tends to increase.

Negative association: as x increases, y tends to decrease.

The Correlation Coefficient r

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

r measures the strength and direction of the linear relationship between two quantitative variables.

Properties of r

• Always between −1 and 1: $-1 \leq r \leq 1$
• r = 1: perfect positive linear association
• r = −1: perfect negative linear association
• r = 0: no linear association (could still be a strong nonlinear relationship)
• r close to 0: weak linear association
• r close to ±1: strong linear association

Interpreting |r|:

• |r| ≥ 0.8: strong
• 0.5 ≤ |r| < 0.8: moderate
• |r| < 0.5: weak

Critical Properties

r is unit-free. Switching from inches to centimeters doesn't change r.

r is symmetric. The correlation of x with y equals the correlation of y with x.

r is NOT resistant to outliers. A single influential point can dramatically change r.