Pearson Product-Moment Correlation
Pearson Correlation Calculator
Measure the linear relationship between two continuous variables. Enter paired raw data values separated by commas.
Configure data
n = 0
n = 0
Descriptive Statistics
Scatterplot & Line of Best Fit
Effect Size (Shared Variance)
APA 7th edition reporting statement
Copy-paste template
Calculation Details
Interpretation Report
Step-by-step analysis & plain-language conclusions
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Hypotheses
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Assumptions check
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Computed statistical values
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Statistical decision
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Effect size & practical significance
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Conclusion & reporting
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All interpretations are generated from your data. Review before use in academic submissions.
Reference · Statistical Theory
How Pearson Correlation Works
Complete theoretical foundation, assumptions, effect size interpretation, and primary references for the Pearson Product-Moment Correlation.
What is Pearson r?
Developed by Karl Pearson (1895), the Pearson Product-Moment Correlation Coefficient ($r$) measures the strength and direction of the linear relationship between two continuous variables. The value of $r$ always ranges from -1.0 to +1.0. A positive value indicates that as one variable increases, the other also increases. A negative value indicates that as one variable increases, the other decreases. A value of 0 indicates no linear relationship.
r = SP_xy / √(SS_x × SS_y)
where:
SP_xy = Σ(x_i - x̄)(y_i - ȳ) [Sum of products]
SS_x = Σ(x_i - x̄)² [Sum of squares X]
SS_y = Σ(y_i - ȳ)² [Sum of squares Y]
t = r × √((n - 2) / (1 - r²)) [t-statistic for significance]
df = n - 2
Assumptions
- Continuous variables — both variables must be measured at the interval or ratio level. Use Spearman's rho for ordinal data.
- Paired observations — each participant must have a pair of values (one for X, one for Y).
- Linearity — the relationship between variables must be linear. Inspect a scatterplot to verify. If the relationship is curved, Pearson $r$ will underestimate the strength of the association.
- No significant outliers — Pearson's $r$ is highly sensitive to outliers, which can artificially inflate or deflate the correlation.
- Bivariate normality — both variables should be approximately normally distributed, especially for small sample sizes, to validly compute the $p$-value.
Effect size measures
- Correlation Coefficient (r) — Used directly as an effect size. Cohen (1988) suggested the following conventions in psychology/behavioral sciences: |r| = 0.10 (small), |r| = 0.30 (medium), |r| = 0.50 (large).
- Coefficient of Determination (r²) — Represents the proportion of variance in one variable that can be explained by the variance in the other variable. For example, if $r = 0.50$, then $r² = 0.25$, meaning 25% of the variance is shared between the two variables.
APA 7th edition reporting format
Template: "A Pearson product-moment correlation was computed to assess the relationship between [Variable X] and [Variable Y]. There was a [positive/negative], [weak/moderate/strong] correlation between the two variables, r([df]) = .xx, p = .xx."
Report the $r$ statistic to two decimal places without a leading zero (since it cannot exceed 1.0). Report exact p-values (e.g., p = .032) unless p < .001. State the degrees of freedom ($N - 2$) in parentheses.
Primary references
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
Pearson, K. (1895). Note on regression and inheritance in the case of two parents. Proceedings of the Royal Society of London, 58, 240-242.