Rucete ✏ AP Statistics In a Nutshell
2. Exploring Two-Variable Data
This chapter introduces relationships between two categorical variables using two-way tables and graphical methods, and relationships between two quantitative variables using scatterplots, correlation, and least squares regression.
Relationships Between Two Categorical Variables
• Two-way tables summarize data for two categorical variables, showing frequencies or counts for each combination.
• Marginal frequencies are the row and column totals; relative frequencies convert counts to percentages.
• Conditional relative frequencies measure association between variables, based on row or column conditions.
Graphical Displays for Two Categorical Variables
• Side-by-side bar charts compare conditional distributions across groups.
• Segmented bar charts stack bars to show conditional relative frequencies within groups.
• Mosaic plots adjust bar widths to reflect group sizes and heights for conditional percentages.
• Association between variables is indicated when distributions differ across groups.
Simpson’s Paradox
• Simpson’s paradox occurs when a trend appears in different groups of data but disappears or reverses when groups are combined.
• Example: A surgeon’s survival rates appear worse overall but are better for each subgroup based on patient health condition.
• Always consider lurking variables that may influence observed associations.
Relationships Between Two Quantitative Variables
• Scatterplots graphically display relationships between two quantitative variables.
• Description of scatterplots includes DUFS: Direction (positive or negative), Unusual features (outliers, clusters), Form (linear or nonlinear), and Strength (weak, moderate, strong).
• Positive association: As one variable increases, the other tends to increase.
• Negative association: As one variable increases, the other tends to decrease.
Form, Direction, Strength, and Outliers
• Form: Linear or nonlinear pattern of points.
• Direction: Positive or negative association.
• Strength: How closely points follow a form (tight cluster vs. dispersed).
• Outliers: Points that fall far from the overall pattern; clusters are also noted as unusual features.
Correlation (r)
• Correlation r measures the direction and strength of a linear relationship between two quantitative variables.
• Properties of correlation:
• Always between −1 and +1.
• Positive r indicates positive association; negative r indicates negative association.
• Close to ±1: Strong linear relationship; near 0: Weak linear relationship.
• Switching x and y or changing units does not affect r.
• Correlation is strongly affected by outliers.
Least Squares Regression (LSR) Line
• A regression line models the relationship between two quantitative variables by minimizing the sum of squared residuals.
• Equation of the least squares regression line: ŷ = a + bx, where:
• ŷ is the predicted value of y,
• b is the slope (change in predicted y for a one-unit increase in x),
• a is the y-intercept (predicted y when x = 0).
Interpreting the Slope and Y-Intercept
• Slope (b): For each additional unit increase in x, the predicted y changes by b units.
• Intercept (a): Predicted value of y when x = 0 (may not always make sense in context).
Making Predictions
• Use the LSR line to predict y for a given x-value.
• Interpolation: Predicting within the range of observed x-values; generally reliable.
• Extrapolation: Predicting outside the range of observed x-values; risky and often unreliable.
Residuals and Residual Plots
• Residual: The difference between an observed y-value and the predicted y-value: residual = actual y − predicted ŷ.
• Positive residual: Actual y is above predicted ŷ.
• Negative residual: Actual y is below predicted ŷ.
Interpreting Residual Plots
• A residual plot displays residuals on the vertical axis and x-values on the horizontal axis.
• Good fit: Residual plot shows random scatter (no pattern).
• Bad fit: Curved patterns or fanning (changing spread) suggest that a linear model may not be appropriate.
Standard Deviation of the Residuals (s)
• Measures the typical distance between observed y-values and predicted ŷ-values.
• Smaller s means predictions are more accurate.
Coefficient of Determination (r²)
• r² represents the proportion of variability in y explained by the linear relationship with x.
• Example: r² = 0.85 means 85% of the variation in y is explained by x using the regression line.
• Higher r² indicates a stronger explanatory power of the model.
Transforming Data to Achieve Linearity
• If scatterplots or residual plots show nonlinearity, data transformations may help.
• Common transformations: logarithmic, square root, reciprocal.
• After transformation, re-examine scatterplots and residuals to check if linearity is improved.
Beware of Influential Points and Outliers
• Outliers have large residuals but may not change the regression line much.
• Influential points significantly affect the slope or position of the regression line.
• Points with extreme x-values are more likely to be influential.
Cautions About Correlation and Regression
• Correlation does not imply causation; a strong correlation between two variables does not mean that one causes the other.
• Lurking variables: Hidden variables that may influence both variables being studied, creating a spurious association.
• Correlation measures only linear relationships; it does not capture nonlinear associations.
• Correlation and regression are not resistant to outliers; a single unusual point can distort the results.
Association vs. Causation
• Observational studies can identify associations but cannot establish causation because they lack controlled conditions.
• Only well-designed experiments with random assignment allow valid cause-and-effect conclusions.
• Even strong, consistent associations across multiple studies do not prove causality without experimental evidence.
Summary of Regression Analysis
• Always check scatterplots and residual plots before relying on a regression model.
• Use r and r² to assess the strength and usefulness of the linear model.
• Interpret slope, y-intercept, residuals, and standard deviation of residuals (s) carefully and in context.
• Beware of extrapolation and influential points that could mislead predictions.
In a Nutshell
Exploring two-variable data involves examining relationships between categorical variables using two-way tables and graphical displays, and between quantitative variables using scatterplots, correlation, and least squares regression. Correlation measures strength and direction for linear relationships, but does not imply causation. Regression analysis models these relationships to make predictions, and residual analysis checks the model’s validity. Awareness of outliers, influential points, and the limits of correlation ensures proper interpretation and responsible statistical reasoning.