Linear Correlation of Bivariate Data

Introduction

Linear correlation of bivariate data explores the relationship between two variables to determine how one may change in response to the other. This topic includes understanding Pearson’s product-moment correlation coefficient (\( r \)), scatter diagrams, regression lines, and the implications of correlation and causation in real-world contexts.

Key Concepts

Correlation and Pearson’s Product-Moment Correlation Coefficient (\( r \))

• Definition: Measures the strength and direction of a linear relationship between two variables. Values of \( r \) range from -1 to 1:
  • \( r = 1 \): Perfect positive correlation
  • \( r = -1 \): Perfect negative correlation
  • \( r = 0 \): No correlation
• Calculation:

  The formula for \( r \) is:

  \( r = \frac{\sum (x_i – \bar{x})(y_i – \bar{y})}{\sqrt{\sum (x_i – \bar{x})^2 \sum (y_i – \bar{y})^2}} \)

  where:

  • \( x_i, y_i \): Individual data points
  • \( \bar{x}, \bar{y} \): Means of \( x \) and \( y \)

  Hand calculations enhance conceptual understanding, while technology simplifies computation for larger data sets.

• Critical Values: Students should be familiar with critical values of \( r \) for significance testing when provided.
• Important Note: Pearson’s \( r \) is only meaningful for linear relationships.

Solved Example 1: Calculating \( r \)

Problem: Given the following data, calculate the correlation coefficient (\( r \)):

Solution:

1. 1. Calculate the means: \( \bar{x} = 3 \), \( \bar{y} = 4 \).
   2. Compute \( (x_i – \bar{x}) \), \( (y_i – \bar{y}) \), and their products:

1. Calculate the sums: \( \sum (x_i – \bar{x})(y_i – \bar{y}) = 6 \), \( \sum (x_i – \bar{x})^2 = 10 \), \( \sum (y_i – \bar{y})^2 = 6 \).
2. Substitute into the formula:

   \( r = \frac{6}{\sqrt{10 \cdot 6}} = \frac{6}{\sqrt{60}} \approx 0.77 \).

Interpretation: There is a strong positive correlation between \( x \) and \( y \).

Scatter Diagrams and Lines of Best Fit

• Scatter Diagrams: Graphical representation of bivariate data to visualize relationships.
• Lines of Best Fit:
  • Drawn by eye, passing through the mean point (\( \bar{x}, \bar{y} \)).
  • Helps in identifying trends and predicting values.

Solved Example 2: Line of Best Fit

Problem: Draw the line of best fit for the data:

Solution:

1. Calculate \( \bar{x} = 3 \), \( \bar{y} = 4 \).
2. Use technology or manual methods to determine the regression equation: \( y = 0.8x + 2.6 \).
3. Plot the points on a scatter diagram and draw the line passing through \( (3, 4) \).