ap stats unit two

Explanatory Variable (x)

Variable that explains or predicts changes in the response variable.

Response Variable (y)

The variable that measures the outcome of a study.

Scatterplot

A graph showing the relationship between two quantitative variables.

Direction of Association

Indicates whether the relationship is positive, negative, or neither.

Form of Association

The shape of the data pattern (linear, curved, clusters, no pattern).

Strength of Association

How closely the data follow a form; described as strong, moderate, or weak.

Outlier (two-variable data)

A point that deviates from the overall pattern of the relationship.

Correlation (r)

A number between −1 and 1 that measures the strength and direction of a linear relationship.

Properties of Correlation

Correlation is unitless, unaffected by units or switching x and y, and describes only linear relationships.

Correlation Does Not Imply Causation

Even strong correlation does not mean one variable causes changes in the other.

Least-Squares Regression Line (LSRL)

The line that minimizes the sum of squared residuals.

LSRL Equation

ŷ = a + bx, where a is the y-intercept and b is the slope.

Slope Interpretation

For each 1-unit increase in x, predicted y changes by b units.

Y-Intercept Interpretation

The predicted value of y when x = 0 (only meaningful when x = 0 makes sense).

Residual

Observed value minus predicted value (y − ŷ).

Positive Residual

The model underestimates the actual value (point above the line).

Negative Residual

The model overestimates the actual value (point below the line).

Residual Plot

A plot of residuals vs. x used to assess if a linear model is appropriate.

Coefficient of Determination (r²)

The proportion of variation in y explained by the regression on x.

Interpretation of r²

About r² × 100% of the variation in y is explained by the linear model.

Influential Point

A point that greatly affects the slope or equation of the regression line.

Outlier (regression)

A point with a large residual.

High-Leverage Point

A point with an extreme x-value that can pull the regression line.

Extrapolation

Predicting outside the range of observed x-values; unreliable.

Interpolation

Predicting within the observed range of x-values; usually reliable.

Standard Deviation of Residuals (s)

The typical prediction error of the regression model.

Transformations (log, ln, etc.)

Used to straighten nonlinear relationships for linear modeling.

Association vs. Causation

Causation can only be established by controlled experiments, not observational data.