Chapter 6: Scatterplots, Association and Correlation

Scatterplot

A graph showing the relationship between two quantitative variables. It helps reveal patterns, trends, relationships, and outliers.

Direction:

• Positive: As one variable increases, so does the other.

• Negative: As one variable increases, the other decreases.

Form

• Linear: Points follow a straight-line trend.

• Nonlinear: Curved or unusual patterns not suitable for linear methods.

Strength

• Strong relationships have points closely following a form.

• Weak relationships look like vague clouds with no clear trend.

Unusual Features

• Outliers: Points far from the trend.

• Clusters: Subgroups that stand apart from the main pattern.

Roles for Variables

• Explanatory (Predictor): Goes on the x-axis.

• Response: Goes on the y-axis.

• The role depends on how we think about the variables.

Correlation (r)

• Measures the strength and direction of a linear relationship between two quantitative variables.

• Always between -1 and +1.

• Positive r = positive association.

• Close to 0 = weak linear relationship.

Conditions for Correlation

1. Quantitative Variables: Only for numerical data.

2. Straight Enough: Must be approximately linear.

3. Outlier Condition: Outliers can distort correlation.

Properties of Correlation

• No units.

• Symmetric: Correlation of x with y = correlation of y with x.

• Unaffected by shifting/scaling.

• Sensitive to outliers.

Correlation ≠ Causation

• A strong correlation does not imply one variable causes the other.

• Lurking variables may influence both variables.

Straightening Scatterplots

If the pattern is curved but consistent, we may be able to transform the data to make it linear