Pearson Correlation

Definition

  • The Pearson product-moment correlation coefficient, commonly referred to as Pearson’s r, measures the degree of relationship between two variables, X and Y, using a single numerical value known as the correlation coefficient.

Key Concepts

  • Correlation: When two variables are mathematically associated, they are termed as correlated.

  • Correlation Coefficient (r):
      - This coefficient, denoted as r, ranges from -1 to +1.
      - A negative value of r indicates that as one variable (X) increases, the other variable (Y) decreases, suggesting an inverse relationship.
      - A positive value of r implies that as X increases, Y also increases, or vice versa; thus, there is a direct relationship.
      - An r value of 0 indicates no correlation, meaning there is a lack of linearity and no association between the variables.

Formula of Pearson's r

  • The formula for calculating Pearson's r is given by:

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

Where:

  • r = Pearson's r

  • x = x variable

  • y = y variable

  • n = number of samples in the population

Interpretation of Pearson's r

  • Legend:
      - Range | Interpretation
      - -------------- | ----------------
      - ±0.00 to ±0.10| No Correlation
      - ±0.11 to ±0.25| Negligible Correlation
      - ±0.26 to ±0.50| Moderate Correlation
      - ±0.51 to ±0.75| High Correlation
      - ±0.76 to ±0.99| Very High Correlation
      - ±1.00 | Perfect Correlation

Choosing the Right Statistical Treatment

  • The choice of statistical treatment depends on:
      - Independent Variable (IV) Level of Measurement
      - Dependent Variable (DV) Level of Measurement

  • Purpose of Treatment: Includes differences, associations, predictions, or impacts.

Statistical Treatments

  • Continuous IV
      - Continuous DV:
        - Treatment: Simple Regression
      - Categorical DV (2 groups):
        - Treatment: t-test
      - Categorical DV (3 or more groups):
        - Treatment: ANOVA
      - 2 or more Continuous IV
        - Continuous DV:
          - Treatment: Multiple Regression
      - Association/Prediction/Impact/Effect
        - Treatment: Pearson's r