Estimation of Correlation Coefficient

Overview

The correlation coefficient is a numerical measure that describes the strength and direction of a linear relationship between two variables. It is denoted by $r$ and ranges from $-1$ to $+1$:

• $r = +1$ : Perfect positive linear relationship.
• $r = -1$ : Perfect negative linear relationship.
• $r = 0$ : No linear relationship. The correlation coefficient is calculated using statistical software, a calculator, or by formula. For estimation purposes, the value of $r$ can also be inferred from a scatter plot.

Steps to Estimate Correlation Coefficient

1. Plot the Data: Create a scatter plot of the two variables to visually inspect their relationship.
2. Examine the Trend:

• If the points form an upward-sloping line, the correlation is positive.
• If the points form a downward-sloping line, the correlation is negative.

3. Determine the Strength:

• Strong correlation: Points are closely clustered around a straight line.
• Weak correlation: Points are more scattered.

4. Match the Pattern to Approximate Values:

• Points tightly aligned : $r \approx \pm 0.9$ or higher.
• Moderate spread : $r \approx \pm 0.5$
• Significant scatter : $r \approx \pm 0.2$ or lower.

Key Considerations

• Correlation does not imply causation.
• Outliers can significantly affect the correlation coefficient and should be considered when interpreting results.

Worked Examples

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Summary

• The correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables.
• Range:
  • $r = +1$ : Strong positive relationship.
  • $r = -1$ : Strong negative relationship.
  • $r = 0$ : No linear relationship.
• Use scatter plots to estimate r based on the pattern and strength of the relationship.
• Key points:
  • Correlation does not imply causation.
  • Outliers can distort the value of $r$