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Representing the Relationship Between Two Quantitative Variables

When we have two quantitative variables measured on the same individuals, we often want to understand if there’s a relationship between them and, if so, describe its nature. This involves both graphical and numerical summaries.

Scatterplots

A scatterplot is the most effective graphical tool for displaying the relationship between two quantitative variables. Each individual in the data is represented as a point on the graph.

• Explanatory Variable (Independent Variable): Plotted on the horizontal ( $ x $ ) axis. This variable is thought to explain or cause changes in the response variable.
• Response Variable (Dependent Variable): Plotted on the vertical ( $ y $ ) axis. This variable measures an outcome of a study.

Example: If we’re studying how the number of hours studied relates to exam scores, hours studied would likely be the explanatory variable, and exam scores the response variable.

Describing a Scatterplot (DOFS)

When analyzing a scatterplot, we look for four key characteristics:

1. Direction:

   • Positive Association: As the explanatory variable increases, the response variable tends to increase. The points trend upwards from left to right.
   • Negative Association: As the explanatory variable increases, the response variable tends to decrease. The points trend downwards from left to right.
   • No Association: There is no clear upward or downward trend. The points appear randomly scattered.

2. Outliers: Individual points that fall outside the overall pattern of the relationship. These points can significantly influence numerical summaries and models.

3. Form: The general shape of the relationship.

   • Linear: The points cluster around a straight line. This is the most common form we study in AP Statistics.
   • Non-linear (Curved): The points follow a curved pattern (e.g., parabolic, exponential).
   • No Clear Form: The points show no discernible pattern.

4. Strength: How closely the points follow the form.

   • Strong: The points are tightly clustered around the form.
   • Moderate: The points show a clear form but with some scatter.
   • Weak: The points are widely scattered, but a form might still be discernible.

Example of DOFS Description

“The scatterplot shows a strong, positive, linear relationship between hours studied and exam scores, with no obvious outliers.”

Choosing Appropriate Variables

It’s crucial to correctly identify the explanatory and response variables based on the context of the problem.

Correlation

While scatterplots provide a visual description, the correlation coefficient (denoted by $ r $ ) is a numerical measure that quantifies the strength and direction of a linear relationship between two quantitative variables.

• Range: $ r $ always falls between -1 and 1, inclusive ( $ -1 \le r \le 1 $ ).
• Direction:
  • $ r > 0 $ : Positive linear association.
  • $ r < 0 $ : Negative linear association.
  • $ r \approx 0 $ : Very weak or no linear association.
• Strength:
  • Values close to $ \pm 1 $ indicate a strong linear relationship.
  • Values close to $ 0 $ indicate a weak linear relationship.
  • $ r = \pm 1 $ means a perfect linear relationship (all points fall exactly on a straight line).

Formula for Correlation Coefficient: $$ r = \frac{1}{n-1} \sum \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right) $$ Where:

• $ n $ is the number of observations.
• $ x_i $ and $ y_i $ are individual data points.
• $ \bar{x} $ and $ \bar{y} $ are the means of $ x $ and $ y $ .
• $ s_x $ and $ s_y $ are the standard deviations of $ x $ and $ y $ .

Important Notes about Correlation:

• Correlation does not imply causation. A strong correlation between two variables does not mean one causes the other. There might be Lurking Variables involved.
• Correlation only measures linear relationships. A strong non-linear relationship might have a correlation close to 0.
• Correlation is not resistant to outliers. Outliers can significantly affect the value of $ r $ .
• Changing units of measurement does not change $ r $ . If you change from feet to meters, $ r $ remains the same.