Calculate correlation coefficient (Pearson r and Spearman ρ) between two datasets. Find the strength and direction of relationships with R², scatter plots, regression lines, and step-by-step calculations.
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Calculate correlation coefficients to measure the relationship between two variables. Compute Pearson r for linear relationships, Spearman ρ for monotonic relationships, and coefficient of determination (R²). Includes interactive scatter plots with regression lines and detailed statistical analysis.
The correlation coefficient is a statistical measure that describes the strength and direction of a linear relationship between two variables. It ranges from -1 to +1, where +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no linear relationship. Pearson's r measures linear correlation, while Spearman's ρ measures monotonic relationships and is more robust to outliers.
Pearson Correlation Formula
r = Σ(xᵢ - x̄)(yᵢ - ȳ) / √[Σ(xᵢ - x̄)² × Σ(yᵢ - ȳ)²]Measure relationships between variables in experiments, from drug dosage effects to environmental factors.
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Use Pearson (r) when your data is continuous, normally distributed, and you expect a linear relationship. Use Spearman (ρ) when data is ordinal, has outliers, is non-normal, or when you're testing for any monotonic relationship (not just linear). Spearman is also better for ranked data or when relationships are curved but consistently increasing or decreasing.
R² is the square of the correlation coefficient and represents the proportion of variance in one variable that can be explained by the other. For example, r = 0.8 gives R² = 0.64, meaning 64% of the variability in Y can be explained by its linear relationship with X. R² ranges from 0 to 1.
No, correlation does not imply causation. A strong correlation between X and Y could mean: X causes Y, Y causes X, both are caused by a third variable, or the relationship is coincidental. Establishing causation requires controlled experiments and careful study design.
Generally: |r| ≥ 0.7 is strong, 0.4 ≤ |r| < 0.7 is moderate, and |r| < 0.4 is weak. However, interpretation depends on context - in physics, r = 0.9 might be considered weak, while in psychology, r = 0.4 might be considered substantial.
When values are tied (equal), they receive the average of the ranks they would have occupied. For example, if values 8, 8, 8 would occupy ranks 3, 4, 5, each receives rank 4. This fractional ranking method is standard practice.