Week 2

  • Lecture 1: One variable at a time (bar charts, histograms, boxplots, mean, median, SD)

  • Lecture 2: Relationships between two variables (scatterplots, correlation, regression)

2⃣ Scatterplots

  • Show relationship between two quantitative variables

  • X-axis: explanatory (independent) variable

  • Y-axis: response (dependent) variable

  • Each point = one individual’s data

3⃣ Explanatory vs. Response Variables

  • Response: outcome measured

  • Explanatory: variable that influences or explains the response

4⃣ Interpreting Scatterplots

  • Form: linear, curved, clusters, no pattern

  • Direction: positive, negative, none

  • Strength: tightness of pattern

  • Outliers: points that break the pattern

5⃣ Categorical Variables in Scatterplots

  • Use color or shape to show categories

  • Simpson’s Paradox: trend reverses when data is grouped vs. aggregated

6⃣ Correlation (r)

  • Measures strength & direction of linear relationship

  • Range: −1 to +1

    • r > 0 → positive

    • r < 0 → negative

    • r = 0 → no linear relationship

  • Unitless, sensitive to outliers

  • Does not imply causation

7⃣ Regression Line (Least Squares)

  • Predicts y from x

  • Equation: ŷ = b₀ + b₁x

    • b₁ = slope

    • b₀ = y-intercept

  • Minimizes squared vertical distances (residuals)

8⃣ Coefficient of Determination (r²)

  • r² = % of variation in y explained by x

  • r² = 1 → perfect prediction

  • r² = 0 → no predictive power

  • Example: r² = 0.76 → 76% of y explained by x

9⃣ Residuals

  • Residual = observed y − predicted ŷ

  • Residual plot: checks for patterns

    • Random scatter → good fit

    • Curved or funnel shape → bad fit

🔟 Outliers vs. Influential Points

  • Outlier: unusual y-value

  • Influential: unusual x-value that shifts regression line

1⃣1⃣ Cautions with Regression & Correlation

  • Always plot data first

  • Don’t extrapolate beyond data range

  • Correlation ≠ causation

  • Averages inflate r and r²

  • Lurking variables can distort relationships

1⃣2⃣ Simpson’s Paradox

  • Trend in aggregated data reverses when data is split into groups

  • Example: Airline delays by city vs. overall

1⃣3⃣ Two-Way Tables (Categorical Data)

  • Rows: one variable

  • Columns: another variable

  • Marginal distribution: totals for one variable

  • Conditional distribution: distribution of one variable given a value of the other

  • Use row %, column %, or total % depending on context