Overview
This lecture introduces graphical and numerical methods for exploring relationships between variables, focusing on scatterplots and the use of Pearson's correlation coefficient to assess linear associations.
Visualizing Relationships Between Variables
- Graphs and summary statistics help analyze relationships between two or more variables.
- Scatterplots display data points for two numerical variables, with each axis representing one variable.
- Patterns in scatterplots can indicate the type and strength of a relationship.
Scatterplots and Relationship Types
- A positive relationship occurs when both variables increase together, seen as an upward trend in a scatterplot.
- A negative relationship occurs when one variable increases as the other decreases, shown as a downward trend.
- When data falls roughly along a straight line, the relationship is linear.
- Nonlinear relationships do not align with a straight line, even if a relationship exists.
- Unrelated variables show no discernible pattern in a scatterplot.
Scatterplot Matrices
- A scatterplot matrix shows multiple scatterplots to visualize relationships among several variable pairs at once.
- Different color lines on matrices can illustrate positive, negative, or no relationships.
Quantifying Relationships: Pearson's Correlation Coefficient
- Pearson's correlation coefficient (r) measures the strength and direction of a linear relationship between two variables.
- r values range from -1 (perfect negative linear relation) to +1 (perfect positive linear relation).
- An r value close to 0 indicates a weak or no linear relationship.
- Example values: r = -0.86 (strong negative), r = 0.70 (strong positive), r = 0.17 (weak positive), r = -0.25 (weak negative).
Key Terms & Definitions
- Scatterplot — a graph that displays values for two numerical variables using dots.
- Linear Relationship — a relationship where data points form a straight line in a scatterplot.
- Scatterplot Matrix — a grid of scatterplots for multiple pairs of variables.
- Pearson's correlation coefficient (r) — a statistic measuring direction and strength of a linear relationship, ranging from -1 to +1.
Action Items / Next Steps
- Review scatterplot examples and practice identifying relationship types.
- Memorize the definition and range of Pearson's correlation coefficient (r).