Exploring Relationships Between Two Quantitative Variables (AP Statistics Unit 2)

Two-variable (bivariate) data

Data consisting of paired observations on the same individuals; each individual contributes an ordered pair (x, y).

Paired observations

Two measurements taken on the same individual that form an ordered pair for bivariate analysis.

Scatterplot

A graph of paired quantitative data where each individual is represented by a point (x, y) showing the relationship between two quantitative variables.

Explanatory variable

The variable suspected to help explain, influence, or predict changes in another variable; usually plotted on the horizontal (x) axis.

Response variable

The variable you want to understand or predict; usually plotted on the vertical (y) axis.

Association

A tendency for two variables to vary together (without implying that one causes the other).

Direction (of a relationship)

Whether y tends to increase or decrease as x increases (positive, negative, or no clear association).

Positive association

As the explanatory variable x increases, the response variable y tends to increase.

Negative association

As the explanatory variable x increases, the response variable y tends to decrease.

No clear association

Changes in x do not show a consistent tendency in y.

Form (of a relationship)

The overall shape of the relationship in a scatterplot (commonly linear or nonlinear).

Linear form

A relationship in which points in a scatterplot cluster around a straight line.

Nonlinear form

A relationship in which points in a scatterplot cluster around a curve (e.g., bend, leveling off, U-shape).

Strength (of a relationship)

How tightly the points follow the overall form (strong if close to the form; weak if widely scattered).

Outlier (in a scatterplot)

A point that falls far from the rest of the data, meaning it is inconsistent with the overall pattern (not merely “large”).

Unusual x-value

A point with an x-coordinate far from the others; such points can have especially large impact on a fitted line in later regression work.

Cluster

A group of points separated from other groups in a scatterplot, often suggesting a hidden categorical variable.

Confounding

A situation where a third variable affects both variables of interest, potentially creating or masking an association.

Correlation (r)

A number that summarizes the strength and direction of a linear relationship between two quantitative variables.

Correlation coefficient (r) range

Correlation always satisfies −1 ≤ r ≤ 1.

Unitless (property of correlation)

Correlation has no units because it is based on standardized values, allowing comparisons across different measurement units.

Standardized value (z-score)

A value expressed as the number of standard deviations from the mean: z = (value − mean)/standard deviation.

Not resistant (property of correlation)

Correlation can change dramatically due to a single outlier because it is based on means and standard deviations.

Symmetry (of correlation)

The correlation between x and y is the same as the correlation between y and x; r does not depend on which variable is explanatory.

Transformation effect on correlation

Adding a constant to a variable or multiplying by a positive constant does not change r; multiplying by a negative constant leaves |r| the same but flips the sign.