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

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  • Marginal distribution: the total for one categorical variable

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  • 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: