3.1: Correlation and Variate Relationships

Two-Variable (Bi-Variate) Relationships

• Explanatory variable: a variable that attempts to explain or influence observed outcomes
  • What is being used to make the prediction
  • Displayed on the x-axis
• Response variable: a variable that measures some outcome
  • What is being predicted
  • Displayed on the y-axis

Describing Scatterplots and Bi-Variate Data: FUDS

• Form: linear, curve, u-shape, etc.
• Unusual Points: outliers, influential points
  • Outlier: a point with a large residual (usually decreases the correlation)
  • Influential: a point which draws the line toward it (usually increases the correlation)
• Direction: positive or negative association (or neither)
  • Positive association—as one variable increases, so does the other
  • Negative association—as one variable increases, the other decreases
• Strength: how closely the points follow the form
  • Strong, weak, moderately strong/weak

Residuals

• Individual points with large residuals are outliers in the y direction because they lie far from the line that describes the overall pattern
• Individual points that are extreme in the x direction may not have large residuals, but can be very important; such points are influential if removing them would markedly change the results of the calculation

Correlation (r)

• Gives the direction and strength of a linear relationship
  • Does not imply causation
• Makes no distinction between explanatory and response variables
  • Can switch x’s and y’s and they would still be correlated
• Both variables must be quantitative
• Standardized and will not change if we change/convert units of measurement from x, y, or both
• r itself has no units
• Positive r = positive association
  • Negative r = negative association
• Correlation only measures strength and direction of linear relationships
• -1 ≤ x ≤ 1 always
• The closer r is to 1 or -1, the stronger the linear form
  • The closer r is to 0, the weaker the linear form and the more scattered the points are
• r does not tell the whole story

Displaying Data

Two-Way Tables

• Two-way table: a table that displays data for two categorical variables about the same group of individuals

  

• Marginal distribution: the total for one categorical variable

  

• The yellow box shows the marginal distribution for gender, and the purple box is the marginal distribution of opinions

• Conditional distribution: the distribution within just one value of one variable

  • Often uses language of the probability of A “given” B

Segmented Bar Graphs

• Also known as segmented bar charts

• Segmented bar graph: a chart that displays categorical data as a percentage of the whole

  • Similar to a pie chart

  

Mosaic Plots

• Mosaic plot: a segmented bar graph used to compare groups where the widths of the bars are proportional to the size of the groups

• Mosaic plots of the same data from the previous section: