Correlation coefficient calculator
Two columns of paired data, one number that captures how they move together, plus the significance test that tells you whether to trust it.
A correlation coefficient measures how strongly two variables move together, on a scale from -1 (a perfect inverse relationship) through 0 (none) to +1 (a perfect positive one). Paste two paired columns and this calculator returns Pearson's r for a linear relationship or Spearman's rho for a rank-based one, each with its t statistic, degrees of freedom, exact p-value, and the coefficient of determination r-squared.
Pearson's r
0.976
a strong positive correlation (p = < .001).
How the coefficient is computed
Pearson's r is the covariance of the two variables divided by the product of their standard deviations, which is the same as standardising both variables and averaging their products:
r = Σ(x - x̄)(y - ȳ) / √[Σ(x - x̄)² Σ(y - ȳ)²]
Spearman's rho applies this same formula after replacing each value with its rank, with tied values sharing the average rank. The significance test converts r into a t statistic, t = r × √[(n - 2) / (1 - r²)], with n - 2 degrees of freedom, and the p-value is the two-tailed area of the t distribution. For Pearson, the calculator also reports the 95% confidence interval for r built on the Fisher z transformation, and r-squared gives the proportion of shared variance.
The guides behind this tool
Frequently asked questions
- What is the difference between Pearson and Spearman correlation?
Pearson's r measures the strength of a straight-line relationship between two interval variables and assumes the data are roughly normal. Spearman's rho measures whether the relationship is monotonic by correlating the ranks instead of the raw values, so it suits ordinal data, skewed distributions, or curved-but-consistent relationships. When the data are clean and linear the two are close; when there are outliers or non-linearity, Spearman is the more robust choice.
- What does the correlation coefficient r tell you?
The correlation coefficient r summarises both the direction and the strength of a linear relationship on a scale from -1 to 1. A value near 1 means the two variables rise together, a value near -1 means one falls as the other rises, and a value near 0 means little linear relationship. The sign gives the direction and the magnitude gives the strength, but r alone does not establish that one variable causes the other.
- Is a correlation of 0.5 strong?
By Cohen's widely used benchmarks, a correlation of about 0.1 is small, 0.3 is medium, and 0.5 is large, so 0.5 is generally considered a strong relationship in social-science research. The interpretation is field-dependent, though: in some experimental settings 0.5 is modest, while in messy survey data it is substantial. Squaring it gives r-squared of 0.25, meaning 25% of the variance in one variable is shared with the other.
- Does correlation imply causation?
No, a correlation only shows that two variables move together, not that one causes the other. A third variable may drive both, the causal direction may be reversed, or the association may be coincidental in the sample. Establishing causation requires a design that rules out these alternatives, such as a randomised experiment or a carefully controlled longitudinal model, not a correlation coefficient on its own.