Correlation Coefficient Notes

Correlation Coefficient

Pearson Correlation Coefficient (r)

• The calculation of correlation coefficient is symbolized as $r$.
• Also known as Pearson correlation coefficient for interval scale.
• Pearson was a student of Francis Galton, who introduced eugenics and was interested in the heights of parents and their children.
• $r$ tells the general trend of the relationship between two variables and provides an exact calculation of the association.
• A positive sign means the correlation is positive, and a negative sign means it's negative (related to the slope).
• $r$ is bounded by -1 and 1.
  • 1 or -1 indicates a perfect correlation, meaning all data points fall on a straight line.
  • 0 indicates no correlation.
• The magnitude of the correlation defines the strength; ignore the sign when determining strength.
  • For example, $-0.99$ is a stronger correlation than $0.75$.

Calculation of r

• Formula: $r = \frac{\sum(Z<em>X * Z</em>Y)}{N}$
  • Where $Z<em>X$ and $Z</em>Y$ are the z-scores for $X$ and $Y$ values, respectively.
  • $N$ is the sample size (not degrees of freedom).
• Steps:
  1. Convert all scores to z-scores.
  2. Calculate the cross-product of the z-scores for each person.
  3. Sum the cross-products.
  4. Divide by the number of people in the study ($N$).

Hypothesis Testing for Correlation Coefficient

• Null Hypothesis: There is no association between two variables ($r = 0$).
• Alternative Hypotheses:
  • Two-tailed: There is an association between two variables (direction not predicted).
  • One-tailed: Predicts a positive or negative association (directional hypothesis).
• Test Statistic: T statistics
  • Convert $r$ to a t-score using the formula: $t = r \sqrt{\frac{N-2}{1-r^2}}$
  • Degrees of freedom: $df = N - 2$

R-squared ($r^2$)

• The square of the correlation coefficient.
• Represents the proportion of total variance in one variable that can be explained by the other variable (proportion of variance accounted for).