Scatter Diagrams and Correlation

Univariate vs. Bivariate Data

Statistical analysis often distinguishes between univariate and bivariate data. Univariate data involves a single variable measured for each individual, typically used to describe characteristics. Bivariate data involves two variables measured for each individual, allowing for the study of relationships between them. The explanatory variable (predictor, independent variable) is usually plotted on the x-axis, while the response variable (dependent variable) is plotted on the y-axis.

• Univariate: Describes one variable for an individual.

• Bivariate: Explains the relationship between two variables for an individual.

Scatter Diagrams (Scatterplots)

A scatter diagram is a graphical tool used to display the relationship between two quantitative variables. Each point represents an individual, with coordinates (explanatory variable, response variable).

• Explanatory variable on x-axis

• Response variable on y-axis

• Each point: (x, y)

Describing Scatterplots

Scatterplots are described by their strength, direction, and type:

• Strength: Tightness of points around a trend (strong, moderate, weak)

• Direction: Positive (both variables increase), negative (one increases, other decreases), or no linear relation

• Type: Linear or non-linear

Correlation Coefficient (Pearson's r)

The linear correlation coefficient (r) measures the strength and direction of the linear relationship between two quantitative variables. It is unitless and ranges from -1 to +1.

• r = +1: Perfect positive linear relation

• r = -1: Perfect negative linear relation

• r ≈ 0: No linear relation

• Not resistant: Outliers can affect r

Formula:

Correlation vs. Causation

Correlation does not imply causation. An observed association may be due to a lurking variable, which affects both explanatory and response variables.

Least-Squares Regression

Regression Line and Residuals

When a linear relationship exists, the least-squares regression line is used to predict values. It minimizes the sum of squared residuals (errors).

• Residual: (difference between observed and predicted y)

• Positive residual: Observed y above predicted

• Negative residual: Observed y below predicted

Equation of the Least-Squares Regression Line

The regression line is given by:

• (slope)

• (y-intercept)

Interpreting Slope and Y-Intercept

• Slope: For each unit increase in x, y changes by the slope value, on average.

• Y-intercept: Predicted value of y when x = 0 (only meaningful if x = 0 is reasonable).

Scope of the Model

Predictions should only be made within the range of observed x-values. Extrapolation outside this range is unreliable.

Diagnostics on the Least-Squares Regression Line

Coefficient of Determination (R2)

The coefficient of determination () measures the proportion of variation in the response variable explained by the regression line.

• Closer to 1: better explanatory power

Residual Analysis

Residual plots help assess the adequacy of the linear model:

• Random scatter: Linear model appropriate

• Patterned residuals: Linear model not appropriate

• Constant error variance: Required for linear model

• Outliers: Identified via residual plots or boxplots

Influential Observations

An influential observation significantly affects the slope, y-intercept, or correlation coefficient. These are typically outliers in the explanatory variable.

Contingency Tables and Associations

Contingency Tables

A contingency table (two-way table) relates two qualitative variables, summarizing their joint frequencies.

Marginal and Conditional Distributions

• Marginal distribution: Frequency or relative frequency of either row or column variable.

• Conditional distribution: Relative frequency of each category of the response variable given a specific value of the explanatory variable.

Simpson's Paradox

Simpson's Paradox occurs when an association between two variables reverses or disappears upon introduction of a third variable. This highlights the importance of considering all relevant variables in analysis.

Summary Table: Types of Relationships in Scatterplots

Additional info: Academic context and examples were expanded for clarity and completeness. All included images directly reinforce the statistical concepts discussed.