This tool computes the Pearson product-moment correlation coefficient (r) for two paired data series, measuring how strongly they move together in a straight line. It returns r on a scale from −1 to +1, the coefficient of determination r², and a plain-English strength reading. It is built for statistics coursework, research, and data analysis where you want to quantify the linear relationship between two variables without needing a statistics package.
How the Pearson formula works
Paste two series of numbers; values pair up by position. The tool computes the mean of each series, then applies the Pearson formula — the covariance of the two variables divided by the product of their standard deviations:
r = Σ[(xᵢ − x̄)(yᵢ − ȳ)] ÷ √[Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)²]
A result near +1 means a strong positive linear relationship (as x rises, y rises proportionally), near −1 a strong negative one (as x rises, y falls), and near 0 little or no linear relationship. Squaring r gives r², which represents the proportion of variance in one variable explained by a linear fit to the other.
Worked example
For x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5, the calculator returns r ≈ 0.775 and r² ≈ 0.60. That signals a moderately strong positive relationship: as x increases, y tends to increase, and about 60% of y’s variation is explained by the linear trend. A perfectly proportional pair such as y = 2x would return r = 1.00 exactly.
Interpreting the strength bands
Conventions vary by field, but a common guide:
| |r| range | Typical interpretation | |---|---| | 0.90 – 1.00 | Very strong | | 0.70 – 0.89 | Strong | | 0.50 – 0.69 | Moderate | | 0.30 – 0.49 | Weak | | 0.00 – 0.29 | Negligible or none |
These are heuristics, not thresholds. A “weak” r in a noisy field (such as psychology or economics) can be a meaningful finding, while a “strong” r in a tightly controlled physical experiment might be expected and unremarkable.
What Pearson r cannot tell you
Correlation is not causation. A high r only describes how closely two variables move together linearly — it cannot show that one causes the other. Two unrelated time series that both trend upward will correlate strongly by coincidence.
Pearson r assumes linearity. If the relationship between x and y is curved (e.g. y = x²), r can return near 0 even when the variables are perfectly related. For non-linear relationships, a rank-based measure such as Spearman’s rho is more appropriate.
Outliers distort r substantially. A single extreme value can drag the coefficient far from what the bulk of the data would show. If your series has obvious outliers, inspect them before treating the result as representative.
Paste your two series — every value pairs by position, and all calculation happens in your browser.