Study Notes on Pearson Correlation Coefficient

Lecture Overview

• This is lecture five of module eight.

• Topic: Pearson Correlation Coefficient.

Pearson Correlation Coefficient

• Definition: A statistical measure that describes the strength and direction of a relationship between two variables.

• Notation:

  • Sample: represented by r

  • Population: represented by the Greek letter rho (ρ)

Range of Values

• Both r and ρ range from -1 to 1.

  • Positive correlation: When r > 0 (e.g., 0.2, 0.8).

  • Negative correlation: When r < 0 (e.g., -0.2, -0.8).

  • No correlation: When r = 0.

Strength and Direction of Correlation

• As the absolute value of r moves away from zero, the strength of the correlation increases:

  • Example: |0.9| and |-0.9| are stronger than |0.1| and |-0.1|.

• Interpretation of r:

  • Tells you both the strength and the direction of the correlation.

Correlation and Regression Line

• The equation of the line of best fit (regression line) is: $y = b x + a$ Where:

  • b = slope

  • a = y-intercept

• Slope Interpretation:

  • Positive slope: If the line slopes upwards, then r > 0.

  • Negative slope: If the line slopes downwards, then r < 0.

  • Flat line: If the slope is zero, then r = 0.

Relationship between r and Slope

• If r > 0, then the slope b > 0.

• If r < 0, then the slope b < 0.

• If r = 0, then the slope b = 0.

• Key Difference:

  • r is restricted to the range [-1, 1].

  • Slope (b) can take any value:

  • - ext{infinity} < b < + ext{infinity}

  • Examples:

  • b = 4391 or b = -1382467.

Calculating r

• Can be calculated if both predictor and criterion variables are continuous.

• It is possible to calculate r if the predictor is categorical, but it may not be useful.

Handling Categorical Predictor with Continuous Criterion

• If y (criterion) is continuous and x (predictor) is categorical:

  • Recommended to use a bar graph.

  • Use a statistical measure called Cohen's d.

Definitions

• Continuous Variable:

  • A variable that can take on an infinite number of values.

  • Examples:

    • Temperature

    • Height

    • Survey scores

    • Percentages

• Categorical Variable:

  • A variable that can only take certain distinct values.

  • Examples:

    • Gender identity

    • Race

    • Country of origin

    • Favorite sports

Future Lectures

• In lectures six and seven, the focus will shift to

  • Bar graphs

  • Cohen's d statistics