Lecture Notes on Correlation Coefficient - Pearson's R

Introduction

• Previous discussions involved interpreting graphs using descriptive terms like:
  • Strong positive association
  • Moderate negative association
  • Weak negative association
  • No association
• Transition from descriptive terms to numerical representation using correlation coefficient: Pearson's r

Pearson's r: Overview

• Pearson's r is a correlation coefficient that quantifies the degree of association between two variables in a scatter plot.
• Range: Between -1 and 1
  • 1: Perfect positive correlation
  • -1: Perfect negative correlation
  • 0: No association

Understanding Correlation

• Perfect Positive Correlation
  • Example: Cost of bananas based on weight purchased (linear relationship)
• Perfect Negative Correlation
  • Example: Distance from home decreases as you drive toward home at constant speed

Intervals of Correlation Coefficients

• No Association: r between -0.25 and 0.25
• Weak Positive: r between 0.25 and 0.5
• Weak Negative: r between -0.25 and -0.5
• Moderate Positive: r between 0.5 and 0.75
• Moderate Negative: r between -0.5 and -0.75
• Strong Positive: r between 0.75 and 1
• Strong Negative: r between -0.75 and -1

Describing Associations

• Given r = 0.6
  • Classified as moderate positive
  • Sketch: Dots moving upwards, slightly spread out

Estimating r from a Graph

• If the graph trends downwards and dots are close, r might be approximately -0.7

Linear vs Non-linear Associations

• Pearson's r is only meaningful for linear associations.
• Non-linear patterns in data yield meaningless r values.

Calculating Pearson's r

• Complex Calculation
  • Involves calculating average coordinates, differences, standard deviations, and standardized areas
• Steps:
  • Find average x and y coordinates
  • Measure distances from averages
  • Standardize using standard deviation
  • Multiply standardized values for each dot
  • Sum these products and divide to find r

Practical Use

• Calculation by hand is complex and rare
• Use calculators or software like Excel for efficient computation

Conclusion

• Understanding the intuition behind r calculation shows why certain quadrants of a scatter plot contribute positive or negative values
• Calculators and software tools simplify the determination of Pearson's r