Pearson Product-Moment Correlation
Pearson Product-Moment Correlation Coefficient · Also known as: pearson r, product-moment correlation, bivariate correlation, Pearson Korelasyon Analizi
The Pearson product-moment correlation coefficient (r) is a parametric measure of the direction and strength of the linear association between two continuous variables. Introduced by Karl Pearson in 1895, it remains the most widely used bivariate correlation statistic in the social, health, and natural sciences. The coefficient ranges from −1 (perfect negative linear relationship) to +1 (perfect positive), with 0 indicating no linear association.
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When to use it
Use Pearson r when both variables are continuous and measured on an interval or ratio scale, the underlying relationship is expected to be linear (verify with a scatter plot), both variables are approximately normally distributed (check with Shapiro-Wilk for n < 50), the sample contains no influential outliers (check with a scatter plot or z-scores), and at least 30 observations are available. When normality or linearity does not hold, or when outliers are present and cannot be removed, switch to Spearman's rank correlation. For ranked or ordinal data, Kendall's tau is also appropriate.
Strengths & limitations
- Provides a single, scale-free coefficient that quantifies both the direction and the strength of a linear relationship.
- Directly interpretable: r² gives the proportion of shared variance, making practical significance easy to communicate.
- Extremely well established across all empirical disciplines, with clear effect-size benchmarks (Cohen, 1988).
- Forms the mathematical foundation for simple and multiple linear regression.
- Sensitive to outliers: a single extreme data point can substantially inflate or deflate r.
- Measures only linear association; a strong curvilinear relationship can produce r near zero.
- Requires both variables to be continuous and approximately normally distributed.
- A significant r does not imply causation between the two variables.
- Unreliable with fewer than 30 observations due to high sampling variability.
Frequently asked
What does an r of 0.60 mean in practice?
An r of 0.60 indicates a strong positive linear relationship. Its square, r² = 0.36, means that 36 % of the variance in one variable is statistically accounted for by the other. Cohen (1988) classifies |r| ≥ 0.50 as a large effect in the behavioral sciences, though context and domain norms always matter.
When should I use Spearman instead of Pearson?
Switch to Spearman's rho when either variable is ordinal, the relationship is monotone but not linear, the data contain influential outliers you cannot justify removing, or normality assumptions are clearly violated — particularly with small samples.
Does a significant Pearson r prove causation?
No. Correlation measures the strength and direction of association, not causation. A third variable may drive both, or the association may be coincidental. Causal inference requires experimental design or appropriate quasi-experimental methods.
How do I report Pearson r in a manuscript?
Report the coefficient, its 95% confidence interval, the sample size, and the two-tailed p-value — for example: r(98) = .54, 95% CI [.39, .66], p < .001. Also report r² as a measure of explained variance and classify the effect size using Cohen's (1988) benchmarks.
Sources
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates. DOI: 10.4324/9780203771587 ↗
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). SAGE. ISBN: 978-1446249185
How to cite this page
ScholarGate. (2026, June 1). Pearson Product-Moment Correlation Coefficient. ScholarGate. https://scholargate.app/en/statistics/pearson-correlation
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Kendall Tau CorrelationStatistics↔ compare
- Partial CorrelationStatistics↔ compare
- Simple Linear RegressionStatistics↔ compare
- Spearman CorrelationStatistics↔ compare