Scatterplots and Correlation

Scatterplots

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

• Explanatory variable ($x$): plotted on the horizontal axis
• Response variable ($y$): plotted on the vertical axis

Describing Scatterplots (DOFS)

1. Direction: Positive, negative, or no association
2. Outliers: Unusual points
3. Form: Linear, curved, clusters
4. Strength: Weak, moderate, strong

Correlation Coefficient ($r$)

The correlation $r$ measures the strength and direction of a linear relationship.

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

Properties of $r$

1. $-1 \leq r \leq 1$
2. $r > 0$: positive association
3. $r < 0$: negative association
4. $|r|$ close to 1: strong linear relationship
5. $|r|$ close to 0: weak or no linear relationship
6. $r$ has no units (dimensionless)
7. $r$ is not affected by changes in units (adding, multiplying)
8. $r$ is the same regardless of which variable is $x$ or $y$

Interpreting $r$

| $|r|$ | Strength | |-------|----------| | 0.8 – 1.0 | Strong | | 0.5 – 0.8 | Moderate | | 0.0 – 0.5 | Weak |

Cautions About Correlation

1. Correlation ≠ Causation: Association does not imply cause-and-effect
2. $r$ only measures linear relationships (a curved pattern may have $r \approx 0$)
3. $r$ is sensitive to outliers
4. $r$ should only be used for quantitative variables
5. Always look at the scatterplot — don't rely on $r$ alone

Influential Points

An influential point substantially changes the regression line or correlation when removed.

• Points with extreme $x$-values are often influential
• Outliers may or may not be influential

AP Tip: Always plot the data before calculating $r$. The correlation coefficient can be misleading without seeing the actual pattern (recall Anscombe's quartet).