Exploring Bivariate Data: every key term you need

Exploring Bivariate Data

Examining relationship between two variables; key questions: Is there association? How strong? What's the pattern?

Scatterplot

Plot with one variable on x-axis, other on y-axis; shows association pattern (linear, curved, clustered, none).

Correlation Coefficient (r)

Measures strength/direction of LINEAR relationship: r=-1 (perfect negative), r=0 (no linear), r=+1 (perfect positive).

Correlation Properties

Unitless, ranges -1 to +1. Same r if x/y swapped. r ≠ causation. Sensitive to outliers. Only measures linear relationships.

Causation vs Correlation

Strong correlation ≠ causation. May have: reverse causation, confounding variables, lurking variables, coincidence.

Least Squares Regression Line

Line minimizing squared vertical distances (residuals) from points. Equation: ŷ = a + bx where b=r(sy/sx).

Regression Line Interpretation

Slope b: for each 1-unit increase in x, y increases by b units (on average). Intercept a: predicted y when x=0.

Predictions & Extrapolation

Use regression to predict y for given x (interpolation is OK). Extrapolation (beyond data range) unreliable.

Residuals

Actual y - Predicted ŷ. Residual plot (residuals vs x) should show random scatter (no pattern) if linear model appropriate.

Coefficient of Determination (R²)

R² = r²; proportion of y's variation explained by x. R²=0.85 means 85% of variation in y explained by x.

Outliers & Influential Points

Outlier: unusual y value. Influential: point far from others on x-axis, dramatically changes regression line.

Transformations

When linear model fails, transform data (log, square root) to achieve linearity; analyze transformed data.

Categorical Variables

Use indicator (dummy) variable (0/1) as predictor. Regression slope compares mean outcome between categories.

Simpson's Paradox

Trend reverses when data subdivided by group. Shows importance of examining relationships within subgroups.

Ecological Fallacy

Drawing individual-level conclusions from group-level data (e.g., states' data doesn't apply to individuals in states).

Spurious Correlation

Strong correlation exists due to confounding variable, not direct causation. Classic example: ice cream sales & drownings (temperature).

Two-Way Tables

Frequency table for two categorical variables. Examine marginal (row/column) distributions and conditional distributions.

Conditional Distribution

Distribution of one variable given specific value of another. Shows how relationship differs within subgroups.

Independence

Two categorical variables independent if conditional distributions same as marginal distribution (no association).

Chi-Square Association

Measures strength of association between categorical variables. Higher value = stronger association.

Mosaic Plots

Visual display of two-way table; tile size represents cell frequency; useful for categorical relationships.

Unit 2 Key Ideas

Correlation measures linear association; r close to ±1 = strong, r≈0 = weak. Regression predicts y from x but doesn't imply causation. Residual plots reveal if linear model appropriate.

More AP Statistics guides

• Designing Studies
• Probability
• Random Variables & Distributions
• Sampling Distributions
• Inference for Means