Learning Objectives

• Understand the meaning of the term correlation.
• Analyze the relationship between two variables.
• Calculate the different measures of correlation.
• Examine the degree and direction of the relationships.

Chapter 6: Correlation

Introduction

• Understanding correlation and its significance in analyzing relationships between two variables.
• Examples:
  • Temperature vs. Ice-cream sales
  • Supply vs. Price of tomatoes

Techniques for Measuring Correlation

1. Scatter Diagrams
   • Visual representation of the relationship between two variables.
   • Helps in identifying the nature of the relationship (positive, negative, or no correlation).
2. Karl Pearson's Coefficient of Correlation
   • Measures the degree of linear relationship between two variables.
   • Value of r lies between -1 and 1:
     • r = 1: Perfect positive correlation
     • r = -1: Perfect negative correlation
     • r = 0: No correlation
   • Should be used when there is a linear relationship.
3. Spearman's Rank Correlation
   • Used when data cannot be precisely measured.
   • Measures the relationship between ranks assigned to variables.
   • Not affected by extreme values.

Types of Correlation

• Positive Correlation: Both variables move in the same direction.
  • Example: Increased income leads to increased consumption.
• Negative Correlation: Variables move in opposite directions.
  • Example: Decreased price of apples leads to increased demand.

Important Formulas

• Karl Pearson's Coefficient of Correlation (r):
  r = $\frac{N \sum{XY} - \sum{X} \sum{Y}}{\sqrt{(N \sum{X^2} - (\sum{X})^2)(N \sum{Y^2} - (\sum{Y})^2)}}$
• Spearman's Rank Correlation (r_s):
  r_s = $1 - \frac{6 \sum{D^2}}{n(n^2 - 1)}$
  • Where D is the difference in ranks and n is the number of observations.

Conclusion

• Correlation analysis provides insights into the relationship between variables but does not imply causation.
• Understanding the nature of correlation helps in making informed decisions based on data.