Chapter 4: Scatterplots and Correlation - Detailed Notes

Chapter 4: Scatterplots and Correlation

Explanatory and Response Variables

  • Response Variable: Measures the outcome of a study.
  • Explanatory Variable: Influences or explains changes in the response variable.

Scatterplots

  • The most effective graph for displaying relationships between two quantitative variables.
  • Both variables are measured on the same individuals:
    • Horizontal Axis (x-axis): Explanatory variable (if applicable);
    • Vertical Axis (y-axis): Response variable.
  • Each point on the scatterplot represents an individual and its corresponding values for both variables.
Steps to Create a Scatterplot (Four-Step Process):
  1. STATE: Formulate the research question as a statement about the association between the two variables.
  2. PLAN: Create the scatterplot according to the guidelines stated above.
  3. SOLVE: Analyze the scatterplot for any observable relationships.
  4. CONCLUDE: Draw conclusions based on the analysis (details on this step are explored further in the chapter).
Example of a Scatterplot:
  • Comparing the percent of high school graduates taking the SAT to the state’s mean SAT Mathematics score.
    • Percent taking SAT: 23, 55, 80, etc.
    • Mean SAT Math Scores: 566, 551, 538, etc.

Interpreting Scatterplots

  • Analyze patterns and deviations in the scatterplot:
    • Overall Pattern: Direction, form, and strength of the relationship.
    • Outliers: Individual data points that deviate significantly from the general pattern.
Direction of Association:
  • Positive Association:
    • Above-average values of one variable accompany above-average values of the other, and vice versa for below-average values.
  • Negative Association:
    • Above-average values of one variable accompany below-average values of the other (and vice versa).

Adding Categorical Variables to Scatterplots

  • Differentiate categories using distinct colors or symbols.
  • Example: mean SAT Mathematics score against the percent of graduates taking the SAT, differentiating by regions (Midwest and Northeast).

Measuring Linear Association

  • Scatterplots reveal strength, direction, and form of relationships.
  • Correlation (r): Measures the direction and strength of the linear relationship between two quantitative variables.
    • Calculation involves means and standard deviations of both variables.
    • Correlation formula:
    • [ r = \frac{1}{n-1} \sum (xi - \bar{x})(yi - \bar{y}) ]

Facts about Correlation

  1. No distinction between explanatory and response variables in correlation.
  2. Unitless and invariant to changes in measurement units of the involved variables.
  3. Positive values indicate a positive association; negative values indicate a negative association.
  4. Correlation values range between -1 and 1.
Cautions about Correlation:
  • Both variables must be quantitative for calculation to be meaningful.
  • Correlation does not represent nonlinear relationships accurately.
  • Sensitive to outliers, which can heavily influence the correlation coefficient.
  • Not a complete summary of two-variable data concisely; therefore, further analysis may be necessary to understand the relationship fully.