Lecture Notes on Correlations and Pearson Correlations

Introduction to Correlations

  • Chapter six focuses on correlations and how to compute them by hand.

  • A correlation examines the relationship between two variables, looking at how they relate to one another.

  • Example: Association between height and weight.

Review of Correlations

  • Definition: Correlation refers to the statistical relationship between two variables.

  • Research Question Example: "Do people who are taller tend to weigh more?"

  • Important to identify the type of relationship predicted, especially in larger research projects.

  • Ethical Consideration: Cannot manipulate variables like personality traits (e.g., extroversion vs. introversion) experimentally due to genetic factors.

Graphical Representation of Correlations

  • Graph Types Learned: Bar graphs, line graphs, and scatter plots.

  • Scatter Plot: A tool to visually represent the relationship between two variables.

  • Example: If observing height and weight, points may trend upward indicating a positive relationship.

Types of Relationships

  • Positive Relationship: As one variable increases, the other also increases.

    • Example: Increase in temperature leads to an increase in beer sales.

  • Negative Relationship: As one variable increases, the other decreases.

    • Example: Increase in temperature leads to a decrease in coffee sales.

  • Curvilinear Relationship: Relationship initially appears positive, plateaus, then becomes negative.

    • Example: GPA vs. studying habits.

    • Overstudying can cause performance to decline, despite initial positive effects.

Understanding Relationships via Graphs

  • Positive Relationship: Graph slopes up to the right; both variables move in the same direction.

  • Negative Relationship: Graph slopes down to the right; one variable rises as the other falls.

  • No or Weak Relationship: Points scattered with no noticeable slope (a flat line).

  • Curvilinear: U-shaped curve in the graph.

Pearson Correlation Coefficient

  • Noted as r. Ranges from -1 to +1.

    • -1: Perfect negative correlation

    • +1: Perfect positive correlation

    • 0: No correlation

  • The magnitude of r indicates strength; regardless of the sign, larger absolute values denote stronger relationships.

  • Slope and Line of Best Fit: Shows how closely the data points cluster around a line, reflecting the strength of the relationship.

Practical Examples

  • Example Data Points: Neuroticism vs. burnout plotted on a scatter plot.

  • Line of Best Fit: Represented by linear regression analysis, minimizing distances between data points and the line.

Review of Relationships

  • Four Types of Relationships:

    • Positive

    • Negative

    • No/Weak relationship

    • Curvilinear

  • Cutoff Scores for interpretation:

    • Strong: r from 0.7 to 1.0

    • Moderate: r from 0.3 to 0.69

    • Weak: r below 0.29

Misinterpretation of Correlation

  • Catphrase: "Correlation does not imply causation."

    • In correlational studies, independent variables are not manipulated, hence causative conclusions cannot be drawn.

  • Extraneous Variables: Third variables that can confound results. Example: Healthy eating might correlate with overall health, but exercise could be the confounding factor.

  • Funny Example: More home appliances correlating with contraception use due to income level being the confounding variable.

The Third Variable Problem

  • Identifies potential spurious effects where a confounding variable influences both variables of interest.

  • Partial Correlations measure associations, controlling for a third variable to evaluate the remaining relationship.

Importance of Range

  • Discusses the concept of restrictive range when variables are limited in their variability.

  • Example: An SAT score range of 1000-1150 is restrictive compared to a full range of 400-1600, affecting the representativeness of findings.

Steps to Compute Pearson Correlation by Hand

  1. Find Means and Standard Deviations for both variables (X and Y).

  2. Calculate Z-scores for each data point using the formula:

    • z = \frac{x - \mu}{\sigma}

  3. Multiply Z-scores for each corresponding X and Y obtaining a new row of products.

  4. Sum the products from the previous step.

  5. Divide by the number of scores (n) to find the Pearson r, where:

    • r = \frac{\Sigma(zx \cdot zy)}{n}

Completing the Correlation’s Interpretation

  • State findings clearly, e.g., "There is a strong negative correlation between stress and health; as stress increases, health decreases."

  • Report the Pearson r value in your conclusion.

Additional Practice Problems

  • Engage with provided practice problems in lab sessions to reinforce computational skills and understanding of correlations.

Conclusion

  • Recap of the importance of correlations, their interpretation, and practical computation for future applications in research.