Correlation Coefficient

Definition

A correlation coefficient is a measure of the strength and direction of the linear association between two variables, ranging from −1 (perfect negative) to +1 (perfect positive). A value of zero indicates no linear relationship. Pearson's correlation is the most common variant; Spearman's rank correlation is preferred for ordinal data or non-linear monotonic relationships.

Why It Matters

Correlation is often the first step in exploring relationships between variables before moving to more complex modelling. It helps identify promising predictor variables, detect multicollinearity, and assess convergent validity in psychometrics. However, correlation does not imply causation, and it only captures linear associations — two variables may have a strong non-linear relationship while showing near-zero correlation.

Example

A study examines the relationship between hours spent studying and exam scores. Pearson's r = 0.65 indicates a moderately strong positive linear association: students who study more tend to score higher. A scatterplot would show points trending upward from left to right. The squared correlation (r² = 0.42) tells us that approximately 42% of the variability in exam scores can be linearly explained by study hours.

Related Terms

• Regression Analysis
• P-Value
• Statistical Significance
• Effect Size (r itself is an effect-size measure)

Software Notes

• SPSS: Analyze > Correlate > Bivariate. Select Pearson and/or Spearman. Check "Flag significant correlations" to highlight p-values below your alpha level.
• R: cor(x, y, method = "pearson") or cor.test(x, y) for significance testing. For Spearman: cor(x, y, method = "spearman").
• Stata: correlate x y for Pearson correlations. spearman x y for rank correlation. pwcorr x y, sig displays significance stars.