Correlation Coefficient and Scatter Plots

Understanding the Correlation Coefficient (r)

• Definition: Measures the strength and direction of a linear relationship between two variables.
• Range:
  • Greater than or equal to -1
  • Less than or equal to +1
• Interpretation:
  • r = 1: Perfect positive linear correlation.
  • r > 0: As one variable increases, the other increases (positive slope).
  • r = 0: No correlation.
  • r = -1: Perfect negative linear correlation.
  • r < 0: As one variable increases, the other decreases (negative slope).

Analyzing Scatter Plots

Sketching Lines of Best Fit

• A line that best describes the behavior of the data.

Examples

1. Second and Fourth Scatter Plots:

   • Observation: Easy to sketch a line of best fit.
   • Correlation:
     • Points close to the line imply strong correlation.
     • Negative Slope: As one variable increases, the other decreases.
     • Estimate: Strong negative correlation; select r ≈ -0.9.

2. Another Scatter Plot:

   • Observation: Positive slope.
   • Correlation:
     • Points close to the line imply strong correlation.
     • Positive Slope: As one variable increases, the other also increases.
     • Estimate: Strong positive correlation; select r ≈ 0.9.

3. First Scatter Plot:

   • Observation: Positive slope.
   • Correlation:
     • Points not as close to the line.
     • Estimate: Positive correlation but weaker; select r ≈ 0.6.

4. Last Scatter Plot:

   • Observation: Negative slope.
   • Correlation:
     • Points further from the line.
     • Estimate: Negative correlation but weaker; select r ≈ -0.6.

Conclusion

• The correlation coefficient provides insights into the strength and direction of relationships in data, helping in making informed choices about linear relationships.