Correlation Analysis Notes

Bivariate Data

  • Involves two variables, describing relationships using correlation analysis.

  • Relationships may indicate:

    • Degree of Association

    • Cause and Effect

    • Predictive Ability

    • Reliability of Test

Correlation Analysis

  • Statistical method to determine if a relationship between variables exists.

  • Measures the association or strength of the relationship between two variables (x and y).

Scatterplot

  • A graph showing points collected from bivariate data on a Cartesian plane.

  • Helps visualize the relationship between two variables.

Types of Linear Correlation

  • Positive Linear Correlation (r = 1): y increases predictably as x increases.

  • No Correlation (r = 0): No relationship between x and y.

  • Negative Linear Correlation (r = -1): y decreases predictably as x increases.

Degree of Association

  • Trend line indicates the direction of correlation.

  • Closeness of points to the trend line indicates the strength of the relationship:

    • Strong correlation (perfect positive or negative)

    • Moderate correlation

    • Weak or no correlation

Pearson Product-Moment Correlation (r)

  • Quantifies the linear relationship between two random variables, x and y.

  • Formula: r=n(∑xy)−(∑x)(∑y)[n∑x2−(∑x)2][n∑y2−(∑y)2]r = {n(\sum xy) - (\sum x)(\sum y) \over \sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}

Interpretation of Correlation Coefficient

  • Ranges from +1 to -1.

    • r = +1: Perfect positive correlation.

    • r = -1: Perfect negative correlation.

    • r = 0: No correlation.

  • Verbal interpretations:

    • 0.00: No correlation

    • ±0.01 to ±0.20: Slight Correlation

    • ±0.21 to ±0.40: Low Correlation

    • ±0.41 to ±0.70: Moderate Correlation

    • ±0.71 to ±0.80: High Correlation

    • ±0.81 to ±0.99: Very High Correlation

    • ±1.0: Perfect Correlation