AP Statistics Unit 2 Study Guide: Exploring Two-Variable Data (Tables, Scatterplots, Correlation, Regression, and Causation)

One-variable data

Data consisting of measurements on a single variable, analyzed by describing its distribution (center, spread, shape).

Two-variable data

Data with two variables measured on each individual, used to study whether and how the variables change together.

Association

A relationship between two variables where knowing the value of one provides information about the likely value of the other.

Categorical variable

A variable that places individuals into categories or groups (e.g., blood type, political party).

Quantitative variable

A variable that records numerical values for which arithmetic operations make sense (e.g., height, time).

Explanatory variable (x)

The variable used to explain or predict changes in another variable; often called the predictor.

Response variable (y)

The outcome variable you want to predict or understand; it responds to changes in the explanatory variable.

Paired data

Two-variable data where each individual contributes a matched pair of values (x, y); the pairing must be kept intact.

Two-way table

A table that displays counts for combinations of two categorical variables.

Contingency table

Another name for a two-way table of counts for two categorical variables.

Joint frequency

An interior cell count in a two-way table for a specific row-and-column category combination.

Marginal frequency

A row total or column total in a two-way table (a total for one variable, ignoring the other).

Table total (n)

The grand total of all counts in a two-way table; the overall sample size.

Marginal distribution

The distribution (often as proportions/percentages) of one categorical variable, ignoring the other variable.

Conditional relative frequency

A proportion computed within a given row or column (within a condition) to compare groups for association.

Segmented bar chart

A display for two categorical variables where each bar represents a group and segments show conditional relative frequencies (each bar totals 100%).

Mosaic plot

A display for two categorical variables that shows conditional proportions and can also reflect different group sizes via widths/areas.

Simpson’s paradox

A situation where an overall association changes direction or disappears when data are split into meaningful subgroups due to a lurking variable.

Bivariate quantitative data

A dataset containing two quantitative variables measured on the same individuals.

Scatterplot

A graph of paired quantitative data where each individual is plotted as a point (x, y) to visualize a relationship.

Direction (association)

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

Form (scatterplot)

The overall shape of a relationship in a scatterplot (e.g., linear, curved).

Strength (scatterplot)

How closely points cluster around the form (e.g., weak, moderate, strong).

Outlier

An unusual point that does not fit the overall pattern; it can strongly affect correlation and regression.

Correlation (r)

A numerical measure of the direction and strength of a linear relationship between two quantitative variables.

Correlation properties (linear, unitless, not resistant)

Key facts about r: it measures only linear association, has no units, ranges from -1 to 1, and can be greatly affected by outliers.

Standardized value (z-score)

A value expressed in standard deviation units: z = (value − mean) / standard deviation.

Coefficient of determination (r^2)

For linear regression, the proportion of variation in y accounted for by the regression of y on x (between 0 and 1).

Least-squares regression line

The line that minimizes the sum of squared vertical residuals: Σ(y − ŷ)^2.

Regression equation (ŷ = a + bx)

The equation of a regression line giving predicted response ŷ from explanatory value x, with intercept a and slope b.

Predicted value (ŷ)

The model’s predicted value of the response variable y for a given x.

Slope (b)

In ŷ = a + bx, the predicted change in y for a 1-unit increase in x (with units of y per unit of x).

Intercept (a)

In ŷ = a + bx, the predicted value of y when x = 0; may or may not be meaningful in context.

Extrapolation

Using a regression model to predict outside the observed range of x; risky because the pattern may not continue.

Residual

The difference between an observed and predicted value: residual = y − ŷ (the vertical distance from a point to the regression line).

Residual sign (positive vs negative)

Positive residual: model underestimated (point above line). Negative residual: model overestimated (point below line).

Residual plot

A graph of residuals versus x (or versus ŷ) used to check whether a linear model is appropriate.

Curvature (residual plot pattern)

A systematic curved pattern in a residual plot, suggesting the relationship is not linear and a different model may be needed.

Changing spread (fan shape)

A residual-plot pattern where variability increases or decreases with x, suggesting non-constant variability.

Standard deviation of residuals (s)

A typical prediction error size in y-units: s = sqrt( Σ(y − ŷ)^2 / (n − 2) ).

Regression outlier

A point with an unusually large residual (far above or below the regression line compared to other points).

Influential point

A point whose removal would noticeably change the regression line (slope/intercept) and possibly r and r^2.

Leverage

A measure of how far an x-value is from the mean of x; high-leverage points have strong potential to affect the regression line.

Transformation (to achieve linearity)

Changing variables (often using log/ln or power transforms) to make a curved relationship more nearly linear and improve a linear model.

Lurking variable

A third variable that affects both variables being studied and may explain or distort an apparent association.

Confounding

When the effects of two variables on a response cannot be separated (common in observational comparisons).

Observational study

A study where researchers observe and record data without assigning treatments; supports association but not causation.

Controlled experiment

A study where researchers impose treatments (often with random assignment); can support cause-and-effect conclusions if well designed.

Random assignment

Randomly assigning individuals to treatments; supports a cause-and-effect conclusion (for the participants).

Random sampling

Randomly selecting individuals from a population; supports generalizing results to the population.