Exploring Bivariate Data: every key term you need (+ practice quiz)

More AP Statistics guides

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

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.

Correlation Coefficient r

r = (1/(n-1)) Σ z_x·z_y. Measures strength and direction of a LINEAR relationship between two quantitative variables; -1 ≤ r ≤ 1, unit-free.

Properties of r

Unchanged by swapping x and y, by changing units, or by adding constants. Strongly affected by outliers. r near 0 does not mean no relationship, only no linear one.

Least-Squares Regression Line

ŷ = a + bx, the line minimizing the sum of squared vertical residuals. Always passes through (x̄, ȳ).

Slope Formula

b = r · (s_y / s_x). Slope has units of y per unit of x, and its sign always matches the sign of r.

Intercept Formula

a = ȳ - b·x̄. Interpretation as predicted y when x = 0 is only meaningful if x = 0 is within the data range.

Residual

Observed y minus predicted y (y - ŷ). Positive residual: the line underpredicts. The residuals from LSRL always sum to zero.

Residual Plot

Residuals vs x (or vs ŷ). Random scatter with no pattern supports a linear model; a curve or funnel shape indicates the model is inadequate.

Coefficient of Determination r^2

Fraction of the variation in y explained by the linear relationship with x. Template: 'r^2 % of the variability in [y] is explained by the LSRL on [x].'

Standard Deviation of Residuals s

s = sqrt(Σ residual^2 / (n-2)). Typical size of a prediction error, in units of y.

Extrapolation

Using the LSRL to predict outside the range of observed x-values. Unreliable because the linear pattern may not continue.

Outlier vs Influential Point

Outlier: large residual. Influential point: removing it changes the slope or r substantially, usually because it is extreme in x. A point can be one, both, or neither.

Leverage

Points far from x̄ in the x-direction have high leverage; they pull the line toward themselves and often have small residuals despite being influential.

Lurking Variable in Regression

A variable not included in the analysis that drives both x and y, so strong correlation does not imply x causes y.

Transforming for Linearity

If y grows exponentially with x, log(y) vs x is linear. If y follows a power model y = a·x^p, log(y) vs log(x) is linear.

Exponential Model Back-Transform

If log(ŷ) = a + bx, then ŷ = 10^a · (10^b)^x; each unit increase in x multiplies predicted y by 10^b.

Computer Regression Output

Read 'Coef' column: Constant row = intercept, predictor row = slope. S = residual SD, R-Sq = r^2 as a percent. r = ±sqrt(R-Sq) with sign of the slope.

Interpreting Slope

Template: 'For each additional [unit of x], the predicted [y] increases/decreases by [b] [units of y].' Always say 'predicted' or 'on average.'

Association for Categorical Data

Two categorical variables are associated when conditional distributions of one variable differ across levels of the other. Compare row or column percentages, not counts.

Segmented Bar Graph

Displays conditional distributions side by side; bars of identical proportions indicate no association.

Simpson's Paradox

An association seen in every subgroup reverses when the groups are combined, because of an unequal distribution of a lurking variable across groups.

Direction, Form, Strength, Unusual

Describe scatterplots with DUFS: direction (positive/negative), unusual features (outliers, clusters), form (linear/curved), strength (weak/moderate/strong), always in context.

Regression to the Mean

For r < 1, predicted ŷ is fewer standard deviations from ȳ than x is from x̄. Extreme x-values are predicted to have less extreme y-values.