The Pearson correlation coefficient (r) is a statistical measure that quantifies the strength and direction of the linear relationship between two continuous variables. It ranges from -1 to +1, where +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. It is the most widely used measure of correlation in statistics and research.

The value of r is interpreted by its magnitude and sign. The sign (+ or -) indicates direction: positive means as one variable increases, the other also increases; negative means as one increases, the other decreases. The magnitude indicates strength: |r| between 0.00–0.19 is very weak, 0.20–0.39 is weak, 0.40–0.59 is moderate, 0.60–0.79 is strong, and 0.80–1.00 is very strong. For example, r = 0.85 indicates a very strong positive correlation.

Pearson correlation measures linear relationships between continuous, normally distributed variables. Spearman correlation measures monotonic relationships and works with ordinal data or non-normal distributions. Spearman uses ranked data rather than raw values, making it more robust to outliers. Use Pearson when data is continuous and approximately normal; use Spearman when data is ordinal, non-normal, or contains outliers.

R-squared (r²) is the coefficient of determination, calculated by squaring the correlation coefficient. It represents the proportion of variance in one variable that can be explained by the other variable. For example, if r = 0.8, then r² = 0.64, meaning 64% of the variance in one variable is explained by the other. The remaining 36% is due to other factors or random variation.

No. Correlation does not imply causation — this is a fundamental principle in statistics. Two variables can be strongly correlated without one causing the other. They might both be influenced by a third variable (confounding factor), the correlation might be coincidental, or the causal direction might be reversed. Establishing causation requires controlled experiments, temporal precedence, and ruling out alternative explanations.

Pearson correlation assumes: (1) both variables are continuous and measured on interval or ratio scales, (2) the relationship is linear (not curvilinear), (3) both variables are approximately normally distributed, (4) there are no significant outliers that distort the relationship, (5) the data represents a random sample, and (6) homoscedasticity — the variance around the regression line is roughly constant. Violations of these assumptions may require Spearman correlation instead.

A correlation is statistically significant if the p-value is less than the chosen significance level (typically 0.05). Significance depends on both the magnitude of r and the sample size (n). Larger samples can detect smaller correlations as significant. This calculator computes the t-statistic and p-value using the formula t = r√((n-2)/(1-r²)) with n-2 degrees of freedom. A significant correlation means the observed relationship is unlikely to be due to chance.

While Pearson correlation can be calculated with as few as 3 data points, meaningful results typically require at least 20–30 paired observations. Smaller samples produce unstable correlation estimates that may not generalise. For reliable results in research, aim for at least 50+ observations. The larger the sample, the more precise and trustworthy your correlation estimate will be.