Understanding your correlation result
The table below shows a commonly used descriptive scale for interpreting the strength of |r|. These bands are a convention, not a statistical test, and different fields sometimes use different thresholds.
| |r| range | Strength of linear association |
|---|---|
| 0.00 – 0.19 | Very weak |
| 0.20 – 0.39 | Weak |
| 0.40 – 0.59 | Moderate |
| 0.60 – 0.79 | Strong |
| 0.80 – 1.00 | Very strong |
- Pearson's r specifically measures linear association. Two variables can have a strong, clearly patterned nonlinear relationship (for example, a U-shaped curve) while still producing an r close to 0, because r cannot detect curvature.
- Correlation does not imply causation: a high correlation can arise from one variable causing the other, from a third confounding variable influencing both, from reverse causation, or from coincidence in the data.
- Because r is calculated from squared deviations, it is sensitive to outliers — a single extreme data point can substantially inflate or deflate the correlation coefficient.
- Pearson's r assumes the underlying relationship is approximately linear and that the data are measured on an interval or ratio scale; for monotonic but nonlinear relationships, a rank-based measure such as Spearman's correlation is often more appropriate.
What is the Pearson correlation coefficient?
The Pearson correlation coefficient (r) measures the strength and direction of the linear relationship between two paired numeric variables. It ranges from −1 to +1: values near +1 indicate a strong positive linear relationship (as x increases, y tends to increase), values near −1 indicate a strong negative linear relationship (as x increases, y tends to decrease), and values near 0 indicate little to no linear relationship.
This calculator also reports R², the coefficient of determination, which equals r² and represents the proportion of the variance in y that is statistically explained by a linear relationship with x. It further fits the least-squares regression line, the straight line that minimizes the sum of squared vertical distances between the line and the data points.
A strong correlation between two variables does not establish that one causes the other. Correlation is a purely statistical association; establishing causation requires additional evidence, such as a controlled experiment or a well-designed study that rules out confounding variables and reverse causation.
How to use this correlation calculator
- Enter the x values as a comma-separated list, for example 1, 2, 3, 4.
- Enter the corresponding y values as a comma-separated list, in the same order and with the same length as the x values, for example 2, 4, 6, 8.
- Provide at least 3 paired data points — the correlation coefficient is not meaningful with fewer.
- Read the correlation coefficient (r), R², the slope and intercept of the regression line, and the regression line equation itself.
The formula behind Pearson correlation and linear regression
Pearson's r is the sum of the products of each variable's deviation from its own mean, divided by the square root of the product of the summed squared deviations: r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √[Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²].
The least-squares regression line, ŷ = slope × x + intercept, uses slope = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² and intercept = ȳ − slope × x̄, chosen to minimize the sum of squared vertical distances between the line and each data point.
Worked example: x = 1, 2, 3, 4 and y = 2, 4, 6, 8 — a perfect straight line through the origin with slope 2. This gives r = 1 (perfect positive linear correlation), R² = 1, slope = 2, intercept = 0, and regression line ŷ = 2x.
Common mistakes
- Interpreting correlation as causation — a strong correlation between two variables does not establish that one causes the other; a third confounding variable, reverse causation, or coincidence can produce the same statistical pattern.
- Assuming Pearson's r captures any relationship between two variables, when it only measures linear association — a strong curved (nonlinear) relationship can produce an r near 0.
- Ignoring the influence of outliers, which can substantially inflate or deflate r because the calculation is based on squared deviations.
- Confusing r with the regression slope — r is a unitless measure of linear association strength and direction between −1 and +1, while the slope carries the units of y per unit of x and depends on the scale of the data.
- Extrapolating the regression line beyond the range of the observed x values, where the linear relationship estimated from the data may no longer hold.
Häufig gestellte Fragen
What does a correlation coefficient of 1 mean?
A Pearson correlation coefficient of r = 1 means the two variables have a perfect positive linear relationship: every data point falls exactly on a straight line with a positive slope, and as x increases, y increases proportionally.
Does correlation mean causation?
No. Correlation measures a statistical association between two variables, not a cause-and-effect relationship. A high correlation can result from one variable causing the other, from a third confounding factor affecting both, from reverse causation, or simply from coincidence — establishing causation requires additional evidence beyond correlation.
What is R² (the coefficient of determination)?
R² equals the square of the Pearson correlation coefficient (r²) and represents the proportion of the variance in the y variable that is statistically explained by its linear relationship with x. An R² of 0.81 (from r = 0.9) means 81% of the variance in y is associated with the linear relationship with x.
Can two variables be strongly related but have a low correlation coefficient?
Yes. Pearson's r only measures linear relationships. Two variables can be strongly, predictably related through a curved (nonlinear) pattern — such as a U-shape or exponential curve — and still produce an r close to 0, because r cannot capture that kind of relationship.
How is the regression line calculated?
The least-squares regression line minimizes the sum of squared vertical distances between the line and each data point. Its slope equals Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)², and its intercept equals the mean of y minus the slope times the mean of x.
How many data points do I need for a meaningful correlation?
This calculator requires at least 3 paired data points to compute a result, but correlation coefficients from very small samples are highly sensitive to individual data points and can be misleading; larger samples give more stable and reliable estimates of the true underlying relationship.
Quellenangaben
- Pearson K. Note on regression and inheritance in the case of two parents. Proceedings of the Royal Society of London 1895; 58: 240-242.
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods — correlation and linear regression. nist.gov.
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. W. H. Freeman (correlation, least-squares regression).
- Evans JD. Straightforward Statistics for the Behavioral Sciences. Brooks/Cole, 1996 (descriptive strength bands for |r|).
- Chatterjee S, Hadi AS. Regression Analysis by Example. Wiley (least-squares regression and its assumptions).